Copyright by Sarah Ann McKinnon 2010 The Dissertation Committee for Sarah Ann McKinnon certifies that this is the approved version of the following dissertation: Municipal-level estimates of child mortality for Brazil: A new approach using Bayesian Statistics Committee: _____________________________ Joseph E. Potter, Co-Supervisor _____________________________ Robert A. Hummer, Co-Supervisor _____________________________ Daniel A. Powers _____________________________ Thomas W. Pullum _____________________________ Carl P. Schmertmann _____________________________ Renato M. Assunção Municipal-level estimates of child mortality for Brazil: A new approach using Bayesian Statistics by Sarah Ann McKinnon, BA; MPH Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December 2010 Dedication I dedicate this dissertation to my sister, Jessica Berumen, who in the last three years has shown me the meaning of real strength and perseverance. I also dedicate this dissertation to the coolest person I know, Oliver Roderick. v Acknowledgements This dissertation would not have been possible without the guidance and assistance of a number of people. I would like to express my deepest gratitude to Joe Potter, who has been my mentor since the day I arrived at UT. Joe has not only provided me with invaluable help on this dissertation but also has shaped my life and career. I am truly fortunate to have been able to work with him. I would also like to give sincere thanks to Bob Hummer who tirelessly read and reread multiple versions of this dissertation, always giving great comments and suggestions for improvements. I want to acknowledge my other wonderful committee members, Tom Pullum, Dan Powers, Carl Schmertmann, and Renato Assunção, who were instrumental in helping me accomplish this work. In particular, I want to thank Carl Schmertmann and Renato Assunção for helping me unravel the mysteries of WinBUGS and Bayesian Statistics. I would be remiss if I did not acknowledge the administrative and computing services staff at the Population Research Center, especially Mary de la Garza who was a constant source of assistance and encouragement. Finally, I want to give thanks to my mother, Kathleen O‟Rourke, without whom neither I nor this dissertation would exist. vi Municipal-level estimates of child mortality for Brazil: A new approach using Bayesian Statistics Sarah Ann McKinnon, Ph.D. The University of Texas at Austin, 2010 Supervisors: Joseph E. Potter and Robert A. Hummer Current efforts to measure child mortality for municipalities in Brazil are hampered by the relative rarity of child deaths, which often results in unstable and unreliable estimates. As a result, it is not possible to accurately assess true levels of child mortality for many areas, hindering efforts towards constructing and implementing effective policy initiatives for the reduction of child mortality. However, with a spatial smoothing process based upon Bayesian Statistics it is possible to “borrow” information from neighboring areas in order to generate more stable and accurate estimates of mortality in smaller areas. The objective of this study is to use this spatial smoothing process to derive estimates of child mortality at the level of the municipality in Brazil. Using data from the 2000 Brazil Census, I derive both Bayesian and non-Bayesian estimates of mortality for each municipality. In comparing the smoothed and raw estimates of this parameter, I find that the Bayesian estimates yield a clearer spatial pattern of child mortality with smaller variances in less populated municipalities, thus, more accurately reflecting the true mortality situation of those municipalities. These estimates can then be used, ultimately, to lead to more effective policies and health initiatives in the fight for the reduction of child mortality in Brazil. vii Table of Contents Chapter 1. Introduction ................................................................................................... 1 Overview of Child Mortality ........................................................................................... 2 Child Mortality in Brazil ............................................................................................. 3 Child Mortality and Female Education in Brazil ........................................................ 8 Policies for the Reduction of Child Mortality ........................................................... 12 Study Aims ................................................................................................................ 13 Chapter 2. Data and Methods ....................................................................................... 19 Indirect Estimates of Child Mortality ........................................................................... 22 Spatial Smoothing Methods .......................................................................................... 40 Locally Weighted Averages Spatial Smoothing........................................................ 40 Nonparametric Regression Spatial Smoothing .......................................................... 41 Empirical Bayesian Spatial Smoothing ..................................................................... 42 Bayesian Spatial Smoothing ...................................................................................... 45 Constructing Bayesian Estimates of Child Mortality .................................................... 50 Diagnostic Tests ........................................................................................................ 53 Constructing Bayesian Estimates of Child Mortality using Female Education ............ 76 Diagnostic Tests ........................................................................................................ 77 Chapter 3. Results ......................................................................................................... 99 Crude Municipal-Level Estimates of Child Mortality. ................................................. 99 Municipal-Level Bayesian Estimates of Child Mortality ........................................... 106 Municipal-Level Bayesian Estimates of Child Mortality and Female Education ....... 114 Chapter 4. Discussion ................................................................................................. 124 The Case of Ceará ....................................................................................................... 127 viii References ....................................................................................................................... 134 Vita .................................................................................................................................. 141 ix List of Tables Table 2.1. Proposed relationship between women's age and probability of mortality for children prior to specific ages ........................................................................................... 25 Table 2.2. Multiplying factors for estimating the proportion of children born alive who die by age a, q(a), from the proportion dead among children ever born to women 15-20, 20-25, etc........................................................................................................................... 27 Table 2.3. Results of Sullivan's regression model for constructing multipliers to be used in creating an estimate of q(a) from the proportion dead among children ever born to women in five-year age groups ......................................................................................... 29 Table 2.4. Results of Coale and Trussell's regression model for constructing multipliers to be used in creating an estimate of the probability of mortality from the proportion dead among children ever born to women in five-year age groups .......................................... 31 Table 2.5. Proportion of children born to women aged 20-29 who would have died (d20-29) prior to age a based on the age distribution of children born to women aged 20 to 29 [c20- 29(a)] and the probability of death to children prior to age a [q(a)]. ................................. 34 Table 2.6. Example of the Markov property: The effect of rain or sun on day 1 on the probability of rain or sun on later days ............................................................................. 48 Table 2.7. Percentage difference between Monte Carlo error and the standard deviation from 50,000 to 100,000 iterations: Brazil, 2000 ............................................................... 67 Table 2.8. Percentage difference between Monte Carlo error and the standard deviation from 50,000 to 100,000 iterations: Brazil, 2000 ............................................................... 90 Table 3.1. Descriptive statistics for municipal-level crude estimates of child mortality: Brazil, 2000 ..................................................................................................................... 101 Table 3.2. Descriptive information on municipalities with insufficient data for creating crude estimates of child mortality: Brazil, 2000 ............................................................. 102 Table 3.3. Descriptive statistics for municipal-level Bayesian estimates of child mortality: Brazil, 2000 ..................................................................................................................... 110 Table 3.4 provides descriptive statistics of the Bayesian estimates for the country as a whole as well as for each region. Overall, there is very little change in these statistics from those in Table 3.3 (which presented results for Bayesian estimates of child mortality simulated without using the education covariate). The overall and regional mean municipal-level  )5(q values are all identical, although the median values for Brazil, the North, the Northeast, and the South did increase slightly. In terms of measures of x dispersion, there was no difference in the values for variance and standard deviation and very little change in the range of values. ........................................................................ 116 Table 3.4. Descriptive statistics for municipal-level Bayesian estimates of child mortality incorporating female education: Brazil, 2000 ................................................................. 117 Table 4.1. Descriptive statistics for estimates of child mortality using crude and Bayesian methods and UNDP child mortality rates: Brazil, 2000 ................................................. 126 xi List of Figures Figure 1.1. Regions of Brazil .............................................................................................. 5 Figure 1.2. Child mortality rates for Belo Campo, Bahia: 1998-2008. ............................ 15 Figure 1.3. Municipal-level child mortality rates*: Brazil, 2000. .................................... 16 Figure 1.4. Municipal-level child mortality rates* by the log of population size:............ 17 Figure 2.1. Municipalities of Brazil. ................................................................................. 22 Figure 2.2. Factors determining d25, the proportion dead among children born to women at exact age 25: c25(a), the age distribution of children ever born to women aged 25, and q(a), the proportion dead by age a under prevalent mortality risks. ................................. 24 Figure 2.3. Mean age of children )c( by mother‟s age group in Brazil, 1996. ................ 33 Figure 2.4. Quartiles of the number of women aged 20-29 in each municipality. ........... 37 Figure 2.5. Confidence intervals of the proportion of children dead to women aged 20-29 (d20-29): 5 most populated and 5 least populated municipalities. ...................................... 39 Figure 2.6. Spatial neighbors of Alta Floresta D‟Oeste, Rondônia, Brazil. ..................... 51 Figure 2.7. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. ............ 55 Figure 2.8. Lags and autocorrelation values for spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. ...................... 57 Figure 2.9. Lags and autocorrelation values for the standard deviation of β (ς.β): Brazil, 2000................................................................................................................................... 59 Figure 2.10. BGR statistics for the proportion of children who had died (pi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. .. 61 Figure 2.11. BGR statistics for spatially structured random effects (βi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. .. 62 Figure 2.12. BGR statistics for the standard deviation of β (ς.β) after 40,000 iterations: Brazil, 2000. ...................................................................................................................... 63 Figure 2.13. Posterior density distribution for the proportion of children who had died (pi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. ...................................................................................................................... 64 xii Figure 2.14. Posterior density distribution for spatially structured random effects (βi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000................................................................................................................................... 65 Figure 2.15. Posterior density distribution for the standard deviation of β (ς.β) after 40,000 iterations: Brazil, 2000. ......................................................................................... 66 Figure 2.16. History of values for 40,000-60,000 iterations (thinned by 20) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. .................................................................................................. 69 Figure 2.17. History of values for 40,000-60,000 iterations (thinned by 20) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20- 29: Brazil, 2000. ................................................................................................................ 72 Figure 2.18. History of values for 40,000-60,000 iterations (thinned by 20) of the standard deviation of β (ς.β): Brazil, 2000. ...................................................................... 75 Figure 2.19. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. ............ 79 Figure 2.20. Lags and autocorrelation values for spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. .................. 81 Figure 2.21. Lags and autocorrelation values for the education effect (e): Brazil, 2000. . 82 Figure 2.22. Lags and autocorrelation values for the standard deviation of β (ς.β): Brazil, 2000................................................................................................................................... 83 Figure 2.23. BGR statistics for the proportion of children who had died (pi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. .. 84 Figure 2.24. BGR statistics for spatially structured random effects (βi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. .. 85 Figure 2.25. BGR statistics for the education effect (e) after 50,000 iterations: Brazil, 2000................................................................................................................................... 86 Figure 2.26. BGR statistics for the standard deviation of β (ς.β) after 50,000 iterations: Brazil, 2000. ...................................................................................................................... 86 Figure 2.27. Posterior density distribution for the proportion of children who had died (pi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. ...................................................................................................................... 87 Figure 2.28. Posterior density distribution for spatially structured random effects (βi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000................................................................................................................................... 88 xiii Figure 2.29. Posterior density distribution for the education effect (e) after 50,000 iterations: Brazil, 2000. ..................................................................................................... 89 Figure 2.30. Posterior density distribution of the standard deviation of β (ς.β) after 50,000 iterations: Brazil, 2000. ..................................................................................................... 89 Figure 2.31. History of values for 50,000-60,000 iterations (thinned by 30) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. .................................................................................................. 92 Figure 2.32. History of values for 50,000-60,000 iterations (thinned by 30) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20- 29: Brazil, 2000. ................................................................................................................ 95 Figure 2.33. History of values for 50,000-60,000 iterations (thinned by 30) of education effect (e): Brazil, 2000. ..................................................................................................... 98 Figure 2.34. History of values for 50,000-60,000 iterations (thinned by 30) of the standard deviation of β (ς.β): Brazil, 2000. ...................................................................... 98 Figure 3.1. Municipal-level crude estimates of child mortality: Brazil, 2000. ............... 100 Figure 3.2. Municipal-level crude estimates of child mortality by the log of sample of women aged 20 to 29: Brazil, 2000. ............................................................................... 103 Figure 3.3. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Brazil, 2000. ................................................................... 104 Figure 3.4. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% smallest sample sizes, Brazil, 2000................................................................................................................................. 105 Figure 3.5. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% largest sample sizes, Brazil, 2000................................................................................................................................. 106 Figure 3.6. Municipal-level Bayesian estimates of child mortality: Brazil, 2000. ......... 107 Figure 3.7. Spatial random effects (βi) on municipal estimates of child mortality: Brazil, 2000................................................................................................................................. 108 Figure 3.8. Municipal-level Bayesian estimates of child mortality by the log of the sample of women aged 20 to 29: Brazil, 2000................................................................ 111 Figure 3.9. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Brazil, 2000. .................................................. 112 xiv Figure 3.10. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% smallest sample sizes, Brazil, 2000. .......................................................................................................... 113 Figure 3.11. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% largest sample sizes, Brazil, 2000. .......................................................................................................... 114 Figure 3.12. Municipal-level Bayesian estimates of child mortality incorporating female education: Brazil, 2000. .................................................................................................. 115 Figure 3.13. Spatial random effects (βi) on municipal estimates of child mortality incorporating female education: Brazil, 2000. ................................................................ 116 Figure 3.14. Municipal-level Bayesian estimates of child mortality by log of women aged 20 to 29 incorporating female education: Brazil, 2000. .................................................. 118 Figure 3.15. The municipalities of Ielma Marinho and Natal: Rio Grande do Norte, Brazil, 2000. .................................................................................................................... 119 Figure 3.16. Municipal-level measures of female education: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. .................................................................................................... 120 Figure 3.17. Municipal-level crude estimates of child mortality: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. ....................................................................................... 121 Figure 3.18. Municipal-level Bayesian estimates of child mortality: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. ....................................................................................... 122 Figure 3.19. Municipal-level Bayesian estimates of child mortality incorporating female education: Ielma Marinho, Rio Grande do Norte, Brazil, 2000...................................... 123 Figure 4.1. Municipal-level crude estimates of child mortality: Ceará, 2000. ............... 129 Figure 4.2. Municipal-level Bayesian estimates of child mortality: Ceará, 2000........... 131 Figure 4.3. Municipal-level Bayesian estimates of child mortality incorporating child mortality: Ceará, 2000. ................................................................................................... 132 1 Chapter 1. Introduction Although child mortality rates (deaths to children under the age of 5 per 1,000 live births) in Brazil have decreased substantially in the last several decades, rates remain elevated when compared to other countries in Latin America (UNICEF 2006; WHO 2006). Additionally, child mortality rates have been found to be much higher in certain geographical regions and within disadvantaged social groups. The Task Force on Child Health and Maternal Health, established by the United Nations to monitor and evaluate progress towards improvements in maternal and child health, has concluded that “deep inequities in health status and access to healthcare both between and, equally important, within countries” (Freedman et al. 2005 p. xi) are a major contributor to the difficulties experienced by countries such as Brazil trying to reduce child mortality across all groups. According to the task force, efforts to improve maternal and child health must shift from a “one-size-fits-all” approach to one that includes “the intimate spaces of families, households, and communities” (p.2). Included in their recommendations is that initiatives for assessing and addressing health status and access to care must have a much smaller geographical focus, stating that “until initiatives genuinely draw on context- specific knowledge and local capacity, health initiatives will not succeed at scale” (p.22). Current efforts to estimate child mortality levels in small geographical areas have been hampered by small population sizes and the relative rarity of child deaths, which has often resulted in unstable and noisy estimates. However, advancements in the field of Bayesian Statistics have made it possible to generate more stable estimates of mortality in areas of small population sizes by “borrowing” information from similar areas. The objective of this study is to use Bayesian methods to derive reasonably reliable and fully 2 replicable estimates of child mortality for 5,505 municipalities in Brazil. These estimates can then be used, ultimately, to lead to more effective policies and health initiatives in the fight for the reduction of child mortality in Brazil. Overview of Child Mortality The child mortality rate (deaths to children under the age of 5 per 1,000 live births) of an area has long been considered to be an important indicator of health and development. Biologically, children have much weaker immune systems than adults and, thus, are far more vulnerable to environmental or social deficiencies (Caldwell 1996). In addition, they are unable to care for themselves and are, thus, completely dependent on others. As a result, children are generally the group first and most strongly affected by standards of living. Likewise, advances in health or social conditions are often first observed in improvements in child mortality (Omran 1971). Studies on child mortality have amassed a huge list of possible determinants, including individual- and community- level factors such as maternal age, electricity, race, income, sanitation, water source, urban/rural residence, region of residence, household composition, occupation, female education, and access to health care (Caldwell 1979; Merrick 1985; Eberstein 1989; Casterline, Cooksey, and Ismail 1992; Majumder and Islam 1993; AlMazrou et al. 1997; Victora et al. 2003; Mogford 2004; Heaton and Forste, 2003; Wang 2003; Basu and Stephenson, 2005; Pradhan and Arokiasa 2006; Andoh et al. 2007). Worldwide, the most common direct causes of child mortality are pneumonia, diarrhea, malaria, measles and AIDS, all of which are preventable and many of which are treatable (UNICEF 2006). Child mortality rates vary considerably and are characterized by strong differentials. According to the World Health Organization, child mortality rates 3 in 2005 were highest in Sierra Leone, with an estimated 282 deaths per 1,000 live births and lowest in Iceland and Singapore, where there are only 3 deaths per 1,000 live births (WHO 2006). The level of mortality has been found to be highly correlated with the relative development of a country, with 25 times higher rates in the least developed countries compared to the most developed countries (153 deaths per 1,000 children vs. 6 deaths per 1,000 children) (UNICEF 2006). Additionally, within-country variation in child mortality has been well-documented, with rates often varying substantially across different regions and social groups (Mosley and Chen 1984; Moser, Leon, and Gwatkin 2005). In the last several decades, child mortality rates have declined substantially with worldwide estimates falling from 191 deaths per 1,000 live births in 1960 to 76 deaths per 1,000 live births in 2005 (UNICEF 2006). While every nation has experienced some level of decline, there is a great amount of disparity in the overall amount of decline. Whereas countries such as South Korea, Oman, and Portugal have had a greater than 95% decline in rates of child mortality, Rwanda, Zambia, Zimbabwe, and Liberia have yet to achieve a 20% decline in their rates. In addition, in many countries declines in child mortality have slowed considerably. Between 1990 and 2005, eight countries had no change in child mortality rates and, in an additional seventeen countries, child mortality rates actually increased (UNICEF 2006). Child Mortality in Brazil Consistent with worldwide patterns, child mortality in Brazil fell from 77 deaths per 1,000 live births in 1960 to 33 deaths per 1,000 live births in 2005 (UNICEF 2006). However, according to the World Health Organization (2006), child mortality rates in 4 Brazil are higher than those of the United States and Canada as well as a number of other Latin American countries including Argentina, Colombia, Ecuador, Mexico, Panama, and Venezuela. In fact, Brazil‟s child mortality rates are most similar to those of the Dominican Republic, Belize, Nicaragua, and Honduras, countries whose development indices are far below those of Brazil (World Bank 2007). Child mortality in Brazil is characterized by large regional differentials, which appear to be strongly associated with social and economic conditions. The country of Brazil is often separated into five different regions (Figure 1.1), each reflecting broad differences in geography, environment, population, and development (Denslow and Tyler 1984; Thomas, Strauss, and Henriques 1990; Sastry 1996; Hudson 1997; UNDP 2003; IBGE 2009a). The poorest region of the country, the Northeast, has very low levels of education and income, high unemployment, underdeveloped infrastructure services such as water, sanitation, and electricity, and a high concentration of black residents. In this region, child mortality rates have traditionally been extremely high (relative to the other regions). Estimates of child mortality rates from the 1940 Census indicate that there were approximately 310 deaths per 1,000 live births in the Northeast (Carvalho 1974). In contrast, in the South region of the country, which has comparatively higher income and education levels as well as a very low concentration of nonwhite residents, child mortality rates were estimated to be as low as 140 deaths per 1,000 live births in 1940. Even as overall mortality levels in Brazil declined, differences in child mortality between the Northeast and the South persisted. According to findings from the 1970 Census, child mortality rates were estimated at 280 deaths per 1,000 live births in the Northeast and 70 deaths per 1,000 live births in the South (Carvalho 1974). And, more 5 recently, infant mortality rates estimated from the 2000 Brazil Census continued to be more than twice as high in the Northeast as in the South (23.6 vs. 64.3 deaths per 1,000 live births) (Alves 2003). Figure 1.1. Regions of Brazil Source: 2000 Brazilian Census. Although differentials in child mortality are greatest between the Northeast and South regions, other areas demonstrate notable patterns as well. The North region, which also tends to be relatively poor and underdeveloped, had estimates of child mortality rates that exceeded the national averages in the 1940, 1950, and 2000 Census (Carvalho 1974; Alves 2003). The Southeast region, which has the largest regional population and a well- developed economy based largely on industrial production, had consistently below- 6 average rates of child mortality in 1940, 1950, and 1970. Likewise, while the 2000 infant mortality rate for all of Brazil was 32.2 deaths per 1,000 live births, the rate for the Southeast region was 27.5 deaths per 1,000 live births. Finally, the Center-West region, home to the capital of Brazil (Brasilia), tends to fall somewhere between the North/Northeast and South/Southeast divide. Living standards are below average, yet residents of the capital have some of the highest incomes of all cities in the country (Denslow and Tyler 1984; IBGE 2009a). Estimates of child mortality rates were relatively low in the Center-West region from 1940-1970 while the infant mortality rate for the region was very similar to the national rate in 2000 (31.0 deaths per 1,000 live births). However, for Brasilia, the infant mortality rate in 2000 was estimated to be 23.2 deaths per 1,000 live births, the second lowest rate in the entire country (Carvalho 1974; Alves 2003). In addition to sizeable regional differentials in overall child mortality, the five regions of Brazil have also experienced large variations in mortality declines as well. According to Brazilian Census estimates from 1940 and 1970, deaths to children under the age of five declined by almost 35% (Carvalho 1974). During that time, areas in the Northeast and Center-West regions experienced the smallest reductions in child mortality. In fact, in one area in the Northeast, child mortality declined by only 11%. In contrast, the greatest decline in mortality occurred in the São Paulo area of the Southeast region, with a 54% reduction in mortality followed by the North region with a 50% reduction. The South region also experienced substantial declines in child mortality, with all areas reporting at least a 44% decline in estimates of child mortality. Between 1991 and 2000, the largest declines in infant mortality occurred in the Southeast and the 7 Center-West (20% in both areas) while the smallest declines occurred in the Northeast and North (14 and 16%, respectively) (Alves 2003). Although regional differentials in levels and declines in child mortality Brazil are both large and persistent, there is also evidence of even greater differentials at smaller geographic levels. When Alves (2003) calculated 2000 infant mortality rates by state and the federal district of Brasilia, he found that Rio Grande de Sul in the South region, Brasilia in the Center-West region, and São Paulo in the Southeast region had the lowest rates (18.7, 23.2, and 23.5 deaths per 1,000 live births, respectively). The highest rates were found in the Northeast states of Alagoas, Pernambuco, and Sergipe (84.5, 74.1, and 68.5 deaths per 1,000 live births). While states in the Northeast all exhibited elevated mortality rates compared to those in the South and Southeast, there was a very large range, with the highest rate (84.5 deaths per 1,000 births) in the state of Alagoas and the lowest rate (53.9 deaths per 1,000 births) in Bahia, a 36% difference. Likewise, the most extreme infant mortality rates differed by 44% in the North, 31% in the Center-West, 23% in the Southeast, and 36% in the South. In addition, Alves found that infant mortality rates for smaller areas also varied tremendously. Specifically, he found that rates in municipalities in the South were as low as 8.5 deaths per 1,000 births while in the Northeast rates were as high as 110 deaths per 1,000 live births – a rate that is remarkably similar to those found in a number of very poor African countries (United Nations Development Programme 2003). Recent work by Castro and Simões (2010) has explored spatio-temporal patterns in inequality in micro-regions 1 in Brazil. According to their findings, inequality in infant mortality at the level of the micro-region is consistent throughout Brazil, with the least 1 The micro-region is composed of a group of municipalities. There are 557 micro-regions in Brazil. 8 advantaged areas demonstrating the highest levels of infant mortality. Although overall levels of infant mortality have fallen over time in Brazil, levels of inequality have remained stable. Child Mortality and Female Education in Brazil One of the most consistent determinants of child mortality both worldwide and within Brazil is female education. According to Caldwell (1979), education of women can influence child mortality in three ways. First of all, women with higher levels of education tend to be less fatalistic and, thus, are more likely to seek help for sick children. Secondly, mothers with more education are “more capable of manipulating the modern world” (p. 410) and will not only know where to obtain quality healthcare but also how to demand such care. Finally, Caldwell believes that female education represents a shift in family structure towards greater equality among the parents. In such a situation, the mothers are involved in familial decision-making and can make the care of children a priority. They also have the freedom to seek out healthcare needs for their children without being dependent on their husband‟s approval. Therefore, while education is often used as a proxy measure for socioeconomic status, in studies of child mortality, its meaning should be seen as representing much more. In Brazil, a number of studies have found that female education is consistently associated with child mortality rates. Using 1970 Brazil Census and 1976 household survey data (PNAD), Merrick (1985) reported strong differentials in child mortality by maternal education. Child mortality rates (by age of the mother) were between 3 and 11 times greater among mothers with no education compared to mothers with 10 or more years of education. In addition, Merrick found that education was strongly associated 9 with declines in child mortality between 1970 and 1976, with maternal education responsible for 32% of the decline and husband‟s education responsible for another 33%. Studies that included community-level variables have also reported a significant association between education and child mortality. For example, Goldani et al. (2002) studied differences in estimates of infant mortality for geographical areas in Porto Alegre, Brazil differentiated by quintiles of the percentage of mothers with less than six years of education. The authors found that, from 1995 to 1998, areas with the greatest proportions of low-educated mothers had significantly higher infant mortality rates compared to areas with the lowest proportions of low-educated mothers. Only in the final year of the study, 1999, did they fail to find a statistically significant difference between the areas. Alves (2003) also found that município-level measures of literacy were a significant determinant of the decline in infant mortality seen in the comparison of 1991 and 2000 Brazil Censuses, with a 10% increase in literacy rates resulting in a 3.6% decrease in infant mortality. Data for specific areas of the country also indicate a strong relationship between child mortality and female education. In a study using 1991 Brazil Census data for Rio de Janeiro, Iyer and Monteiro (2004) found that women in the lowest education quartile had a five times greater risk of reporting a death than women in the highest quartile. Likewise, Sastry (2004) analyzed 1970-1991 Census data for the state of São Paulo, Brazil and found that even as child mortality rates decline, maternal education continued to be strongly associated with child mortality. In regression models that included controls for household wealth, household water supply, household sanitation, age, and migrant status, child mortality rates were highest among the least educated women and lowest 10 among the most educated women for every census year. In 1970, the ratio of child mortality rates between illiterate women and women with more than a secondary education was 1.79. By 1980, the ratio increased somewhat to 1.95 and, by 1991, the ratio increased substantially to 4.35. In a study of socioeconomic determinants of child mortality in the South/ Southeast (excluding the states of Minas Gerais and Espírito Santo) and Northeast regions, Sawyer and Soares (1983) found that in 1970 child mortality rates among mothers with more than an elementary school education compared to mothers with no education were 52% lower in the urban South/Southeast, 59% lower in the rural South/Southeast, and 54% lower in the urban Northeast. In the rural Northeast, child mortality rates were only 10% lower and this finding was not statistically significant. Likewise, Thomas, Strauss, and Henriques (1990) used national household data from the Estudo Nacional da Despesa Familiar (ENDEF) collected between 1974 and 1975 to conduct child survival analyses for the South/Southeast (including Brasilia) and the Northeast region of Brazil and found that maternal education was a significant predictor in both regions, although its effect was somewhat stronger in the Northeast. In the urban Northeast, compared to child survival rates among illiterate mothers, survival rates were 7% higher if mothers were literate, 12% higher if they had completed elementary school, and 18% higher if they had at least completed secondary school. In the urban South/Southeast, the survival rate was 5% higher if mothers were literate, 7% higher if they had completed elementary school, and 11% higher if they had at least completed secondary school. In rural areas, child survival rates were 4% higher in both the Northeast and South/Southeast among literate vs. illiterate mothers while they were 10% 11 higher in the Northeast and 6% higher in the South/Southeast if they had completed elementary school. In a study using data from the 1986 Demographic and Health Survey, the 1980 Brazil Census, and 1988 meteorological records for Northeast and South/Southeast Brazil, Sastry (1996) found that, again, maternal education was significantly associated with child mortality in both the Northeast and South/Southeast regions of Brazil. In the Northeast, he found that mothers with 3 or more years of education had a 28% lower risk of child mortality compared to mothers with less than 3 years of education. In the South/Southeast region, he determined that each year of maternal education resulted in an 8% lower mortality risk. In addition, Sastry also found that maternal education influenced the effect of other determinants as well and that this influence varied between the two regions. For example, in the Northeast, he found that determinants of child mortality among women with low levels of education included sanitation, water supply, trash collection, and access to health care. In the same region, determinants for women with high levels of education included electricity and access to specialized health care. In the Southern regions, number of television stations and access to health care were the only variables associated with lower child mortality among women with less education, while sanitation and access to specialized health care were associated with lower child mortality among women with more education. Due to the consistent association between child mortality and female education, it is reasonable to assume that areas with high levels of female education should have lower levels of child mortality and vice-versa. Therefore, efforts to construct reliable measures 12 of child mortality for small geographical areas in Brazil can be more successful if they also incorporate information on the level of female education in the area. Policies for the Reduction of Child Mortality The Task Force on Child Health and Maternal Health, established by the United Nations to monitor and evaluate progress towards improvements in maternal and child health, has concluded that “deep inequities in health status and access to healthcare both between and, equally important, within countries” (Freedman et al. 2005 p. xi) are a major contributor to the difficulties experienced by countries such as Brazil in achieving reductions in child mortality. According to the task force, efforts to improve maternal and child health must shift from a “one-size-fits-all” approach to one that includes “the intimate spaces of families, households, and communities” (p.2). Included in their recommendations is that initiatives for assessing and addressing health status and access to care must have a much smaller geographical focus, stating that “until initiatives genuinely draw on context-specific knowledge and local capacity, health initiatives will not succeed at scale” (p.22). In Brazil, areas that have implemented policy initiatives that shift away from a “one-size-fits” all approach to one that focuses on the specific needs and issues of communities have been successful at lowering child mortality. For example, the state of Ceará located in the Northeast region of Brazil has typically had very high levels of child mortality, school drop-outs, and malnutrition (UNICEF 2001). However, beginning in 1987 this state began to implement a number of policies to address their particular circumstances. To address water shortages that were common in this semi-arid state, they built water tanks for many of the poorer families (Castro and Simões 2010). Additionally, 13 the state undertook a decentralization of social services for its population from the level of the state to the level of the municipality. As a part of this decentralization, each municipality in the state was assigned community health agents to address not only the health care needs of the people but also to disseminate information (Fuentes and Niimi 2002). These efforts led to a 32% decline in the infant mortality rate. As a result of these improvements, 34 of the 184 municipalities in the state received the UNICEF-Municipal Seal of Approval for municipal-level improvements in health and mortality (UNICEF 2001). Study Aims In order to successfully implement context-specific policy for the reduction in child mortality in Brazil, as well as evaluate the success of the policies, the first step is to develop reliable local-area estimates of child mortality. At this time, child mortality rates are published for each municipality in Brazil via a website entitled Portal ODM (www.portalodm.com.br) which was developed under the coordination of the United Nations Development Programme (UNDP). The website provides a link to each municipality and all available rates from 1998 to 2008 (see Figure 1.2 for an example of child mortality rates for the municipality Belo Campo, Bahia). While constructing municipal-level child mortality rates is an important first step to following the recommendations of the UN task force, there are several problems with these rates. First of all, they are calculated using vital statistics data which, in Brazil, according to a recent study of the Live Birth Information System in Brazil, information on births occurring between 2003 and 2005 was inadequate for 13% of the total population and as high as 28% for the North region (Szwarcwald 2008). 14 Secondly, attempting to measure relatively rare events such as child mortality for small geographical areas (which often have small populations) can result in highly unstable estimates (Clayton and Kaldor 1987). This is evident in Figure 2.1 in which the child mortality rates for the municipality Belo Campo range from a low of 11.8 child deaths per 1,000 live births in 1998 to a high of 44.6 child deaths just two years later. It is highly unlikely that there is any logical explanation for such a huge variation in child mortality rates but, instead, variation is more likely due to the fact that the number of live births is quite low (327 live births were registered in 2008 - IBGE 2004). And, with an average child mortality rate of 33 deaths per 1,000 live births, relatively few or more deaths in a given year can greatly impact the child mortality rate. 15 Figure 1.2. Child mortality rates for Belo Campo, Bahia: 1998-2008. Source: Portal ODM 2009 Figure 1.3 maps child mortality rates for each municipality in Brazil for the year 2000 using child mortality rates from the Portal ODM website. Of the 5,505 municipalities presented, 738 (13.5%) are missing a child mortality rate for this year. Additionally, although it is well established that child mortality is higher in the northern regions of Brazil and lower in the southern regions (Carvalho 1974; Alves 2003), this is not evident in the map. Overall, the pattern is very sporadic and there is a great amount of variation in municipal-level child mortality rates in all regions of the country, indicating that the child mortality rates provided by the Portal ODM website may be inaccurate. 16 Figure 1.3. Municipal-level child mortality rates*: Brazil, 2000. Source: Portal ODM 2009 *Deaths to children under the age of 5 per 1,000 live births To see how variation in child mortality rates relates to sample size, Figure 1.4 plots the 2000 child mortality rates by the log 2 of the population size of each municipality. Variation in child mortality rates, as expected, is far greater in the municipalities with the smallest population sizes and decreases as population size increases, indicating unstable and, potentially, inaccurate rates for those areas. 2 Due to the large differences in population sizes in municipalities in Brazil and the large concentration of municipalities with less than 500,000 residents, it would be difficult to discern the true range in values for the less populated areas without taking the log of the population size. 17 Figure 1.4. Municipal-level child mortality rates* by the log of population size: Brazil, 2000. Source: UNDP 2009 *Deaths to children under the age of 5 per 1,000 live births Thus, while it has been recognized that the construction of municipal-level measures of child mortality is important for understanding and addressing inequities in child mortality in Brazil, these measures are hampered by both the source of data and the small population sizes of municipalities. However, using an alternate data source and advances in the field of Bayesian Statistics, it is possible to address these issues and generate more stable estimates for municipalities in Brazil. The aim of this study is to use Brazil Census data and the relatively new Bayesian methods to “borrow” information from other similar areas in order to derive more reliable estimates of child mortality for the 5,505 municipalities in Brazil. Using these improved estimates, I am able to demonstrate that levels of child mortality in the municipalities in the state of Ceará are 18 much more accurate using the Bayesian approach and reflect improvements that have occurred as a result of specific policy initiatives tailored to those specific communities. 19 Chapter 2. Data and Methods There are three data sources that can be used to create (or estimate) child mortality rates for Brazil. The first, vital statistics, is often the preferred data source for calculating child mortality rates because they include information on all births and deaths occurring in an area. However, in many developing countries, vital statistics cannot be used for this purpose because the quality of the data is often unreliable due to the fact that many births and deaths go unregistered. As illustrated above, a large proportion of births in Brazil are lacking adequate information, with even higher proportions of missing data in the poorer regions of the country (Szwarcwald 2008). Although improvements in vital statistics collection are occurring in Brazil, the data continue to have some major flaws and cannot be considered reliable for the calculation of municipal-level estimates of child mortality. Recently, a number of developing countries with weak vital statistics systems have begun using large-scale, nationally-representative surveys which include detailed reproductive histories as a source of data for calculating child mortality rates. One such survey, the Demographic and Health Survey (DHS), has been conducted in Brazil in 1986, 1991, and 1996 and has been used in a number of studies of child mortality (e.g., Sastry and Burgard 2005; Victora and Barros 2001; Wagstaff 2000; Sastry 1997; Curtis, Diamond, and MacDonald 1993). However, the main problem with the DHS data is that they are not representative of geographical levels below that of the nation or region. Specifically, in 1996 DHS data was only collected in 286 municipalities, a mere 5% of all municipalities in the country (analysis of 1996 DHS data). Thus, while it is an excellent 20 source for creating child mortality rates for the nation as a whole, it is not possible to use this data to construct estimates for each of the 5,505 municipalities in Brazil. When vital registration systems are weak and nationally representative surveys do not contain sufficient data, a third dataset, the Census, can be used to effectively measure child mortality. Even in lesser developed countries, Census coverage rates tend to be very high (only 0.8% of units were missed in the 2000 Brazilian census - Silva 2005). Additionally, compared to survey data, Census data have two benefits. First of all, data collection is carried out in every municipality, allowing for the creation of municipality- level measures, and, secondly, the sample size is far greater. For example, almost 13,000 women 3 were interviewed for the 1996 DHS compared to more than 5 million for the 2000 Brazil Census. Thus, the primary data used for this study are 2000 Brazil Census data. In Brazil, 2000 Census data are collected using two types of questionnaires. The first, the short form, includes basic demographic questions such as the sex, age, relationship to head of household and literacy while the long form includes all questions on the short form as well as questions on household conditions and amenities, income and occupation status, literacy and education, race, religion, marital status, migration, and parity (Silva 2005). The long form questionnaire is administered to a 20% sample of households in municipalities with less than 15,000 residents and a 10% sample of households in municipalities with 15,000 or more residents. For both the long and short form questionnaires, there are no missing values on any variables for the 2000 Census. 3 In the DHS only women between 15 and 49 were interviewed; thus, the number given for the Brazil Census was also limited to women in the same age group. 21 Through a variety of methods, imputation procedures are used to fill in values for missing responses in addition to replacing inconsistent or erroneous values (Silva 2005). For the purposes of this study, I use microdata derived from the long-form questionnaire. The main unit of analysis of this study is the municipality (n=5,505) which is identified by a geographic identifier included with the Census data (Figure 2.1). The construction of all initial variables is done using SAS software, Version 9.2 of the SAS System for Windows (SAS Institute Inc 2008). GeoDa (Anselin, Syabri and Kho 2006) is used to create neighborhood matrices and WinBUGS (Bayesian inference Using Gibbs Sampling - Lunn et al. 2000) is used to perform Bayesian spatial smoothing. All maps are created using ArcMap Version 9.3 (Esri Inc 2008) and all figures are created using Stata 10.0 (StataCorp 2007). Municipal-level measures of female education are constructed by summing the total number of years of schooling for every woman aged 25 and above in each municipality. The sums are then divided by the sum of women aged 25 and above to calculate the mean number of years of schooling for women aged 25 and above in each municipality. While the long form questionnaire does ask each woman how many children they had and the number of children still alive, it does not include sufficiently detailed information on their reproductive histories to allow for the direct calculation of child mortality rates. Yet, it is possible to use the Census information that is available to create indirect estimates of child mortality. 22 Figure 2.1. Municipalities of Brazil. Source: 2000 Brazilian Census. Indirect Estimates of Child Mortality Indirect estimates of child mortality, first introduced by William Brass in 1968 (Brass and Coale 1968), have a long history in the field of demography and are often used to identify and track mortality trends for countries with incomplete registration systems. Rather than relying on incomplete or erroneous vital statistics records, the Brass method uses information that can be easily obtained to calculate the proportion of 23 children who have died to women in certain age groups which, in turn, can be converted into the cumulative probability of death prior to a certain age. According to Brass, two factors determine the proportion of children who have died to women at age x [dx]: the probability of death to children prior to age a [q(a)] and the age distribution of the children born to the women if none of the children had died [cx(a)]:  αx 0 x (a)q(a)dacxd (1) where the age distribution is determined by the proportion of women bearing a child at age x [f(x))]:     αx 0 x f(x)dx a)f(x (a)c (2) Assuming that fertility and mortality trends are nearly constant, Brass showed that dx would be approximately equal to q(a) with the actual age a dependent on the shape of mortality, shape of fertility, and the age distribution of the children to the women. For example, in Figure 2.2 Brass compared q(a) values (derived from life tables from countries with high levels of mortality) with values of cx(a) and dx obtained from the women aged 25 from the African populations he studied and demonstrated, that for this group of women, the proportion of children who had died was equal to the probability of death to children before the age of 2.5. 24 Figure 2.2. Factors determining d25, the proportion dead among children born to women at exact age 25: c25(a), the age distribution of children ever born to women aged 25, and q(a), the proportion dead by age a under prevalent mortality risks. Source: Brass and Coale 1968. By shifting the ages of the women and, thus, the age distribution of the children born, the relationship between dx and q(a) also shifts so that the proportion of children who have died is equal to the probability of death of children prior to a different age. Again, assuming that fertility and mortality patterns are constant, younger women will have more recently begun their childbearing experiences and, thus, the age distribution of their children will shift to the left resulting in a smaller proportion of children who are dead and a younger age at which dx and q(a) are equal. Conversely, for older women, the length of time since the onset of childbearing will be greater, the age distribution of the children will be shifted to the right, the proportion of children who had died would be higher, and the age at which dx and q(a) are equal will be greater. Thus, by simply varying the ages of the women, Brass was able to estimate the probability of mortality for 1 2 3 4 5 6 7 8 9 10 age a Proportion dead at age a -.28 -.20 -.10 -.02 Proportion at age .15- .10- .05- .01- c25(a) q(a) d25 25 children “from the lower limit of childbearing to the highest age for which data about children ever born (living and dead) are available” (Brass and Coale 1968: 107). In the end, Brass concluded that the proportion of children who have died to women in five year age groups beginning with age 15 and ending with age 65 could be used to estimate the probability of death prior to age 1 through age 35 (Table 2.1). Table 2.1. Proposed relationship between women's age and probability of mortality for children prior to specific ages Age of women x Age of children for whom cumulative mortality is best defined 15-20 q(1) 20-25 q(2) 25-30 q(3) 30-35 q(5) 35-40 q(10) 40-45 q(15) 45-50 q(20) 50-55 q(25) 55-60 q(30) 60-65 q(35) Source: Brass and Coale 1968 While Table 2.1 provides a very neat and straightforward summary of the relationship between dx and q(a), Brass admitted that the relationship only holds true “when the fertility and mortality schedules are „standard‟ ones with age patterns roughly like those found in African populations” (Brass and Coale 1968: 107). Brass believed that differences between groups, especially in terms of fertility schedules, could impact the age distributions of children and, thus, the proportion of children who have died and the exact age at which dx and q(a) will be equal. For example, women who begin childbearing at very young ages will have much older children when compared with women of the same age who have delayed childbearing. As a result, the first group of 26 women will have a higher proportion of children who have died and the value for dx will be equal to the value of q(a) at an older age a. To account for the effect of differences in fertility timing across different groups, Brass proposed the use of a multiplier when using dx to estimate q(a): q(a) = (multiplier)dx (3) Assuming that the age of fertility onset would have the greatest impact on the relationship between q(a) and dx, Brass simulated different fertility conditions by substituting new values for s (the age of childbearing onset) in his formula for the proportion of women at age x who gave birth: f(x) = k(x-s)(s+33-x) 2 (4) 4 Brass then substituted the new values of f(x) into the equation for dx and compared the new value of dx with the corresponding value for q(a) based upon Table 1 and, thus, the value of the multiplier is simply the number needed to make dx equal to q(a) again. For each age group Brass created eight possible multipliers to adjust for different fertility behaviors (Table 2.2). The choice of multiplier is dependent upon the fertility conditions in the population of interest which is defined using the ratio of the parity among women aged 15-20 and women aged 20-25         2 1 P P which Brass found to be a fairly accurate estimate of age of childbearing onset. 4 Note that k is a constant that refers to the total number of children born at the end of childbearing and, thus, has no impact on the multipliers. 27 Table 2.2. Multiplying factors for estimating the proportion of children born alive who die by age a, q(a), from the proportion dead among children ever born to women 15-20, 20-25, etc. 2 1 P P Values 0.387 0.330 0.268 0.205 0.143 0.09 0.045 0.014 Mother’s Age Group q(a) Multiplying Factors 15-20 q(1) 0.859 0.890 0.928 0.977 1.041 1.129 1.254 1.425 20-25 q(2) 0.938 0.959 0.983 1.010 1.043 1.082 1.129 1.188 25-30 q(3) 0.948 0.962 0.978 0.994 1.012 1.033 1.055 1.081 30-35 q(5) 0.961 0.975 0.988 1.002 1.016 1.031 1.046 1.063 35-40 q(10) 0.966 0.982 0.996 1.011 1.026 1.040 1.054 1.069 40-45 q(15) 0.938 0.955 0.971 0.988 1.004 1.021 1.037 1.052 45-50 q(20) 0.937 0.953 0.969 0.986 1.003 1.021 1.039 1.057 50-55 q(25) 0.949 0.966 0.983 1.001 1.019 1.036 1.054 1.072 55-60 q(30) 0.951 0.968 0.985 1.002 1.020 1.039 1.058 1.076 60-65 q(35) 0.949 0.965 0.982 0.999 1.016 1.034 1.052 1.070 Source: Brass and Coale 1968 Although Brass believed that the relationship between dx and q(a) was most influenced by fertility characteristics, he did test his method under different mortality conditions. Using the four different mortality patterns represented by the Coale-Demeny model life tables (Coale and Demeny 1983), 5 Brass created comparable sets of multipliers and compared them against his original multipliers. In the end, he concluded that the multipliers would not vary when the probability of mortality at each age follows the same pattern as that of his standard life table. However, for the model life tables that did not demonstrate a similar pattern (e.g. the South model), Brass only found substantial 5 In 1966, Coale and Demeny created model life tables using patterns in the probability of mortality from 192 life tables. Based upon the fact that life tables with similar mortality patterns also tended to be geographically close, Coale and Demeny created four regional model life tables. The North model life table is characterized by low mortality at less than one year of age and above 50 years of age. Life tables used in the construction of the South model tended to have low mortality between 40 and 60 years of age but high mortality below 5 and above 65 years of age. The mortality pattern for East model life table is high infant mortality and high mortality above 50 years of age. Finally, the West model has fairly constant mortality for all age groups. 28 differences in the multipliers created for the youngest age groups. Brass concluded that “in the presence of uncertainty about the shape of the mortality schedule precise estimates cannot be expected… (but) imprecision resulting from the age pattern of mortality does not appear important” (Brass and Coale 1968 p. 113). Accordingly, Brass did not propose any modifications to the multipliers based upon mortality conditions. In 1972, Jeremiah Sullivan published an article proposing a modified method for obtaining multipliers. In contrast to the method used by Brass which relied primarily on algebraic formulas and simulations, Sullivan derived his multipliers using empirical data and regression methods. By combining fertility data from 46 different countries (often for multiple time periods) with 40 mortality schedules from the Coale-Demeny model life tables, he was able to generate 650 different observations that included fertility information as well as values of both q(a) and dx. Sullivan then tested regression models using different fertility parameters to predict values of multipliers or the ratio of the probability of dying prior to certain ages to the proportion of children who died         x d q(a) . He found that the ratio of average parity for women aged 20-25 and women aged 25-30         3 2 P P was the best predictor of the values of the multipliers resulting in the following regression equation:          3 2 x P P BA d q(a) (5) To account for difference in mortality patterns, Sullivan separated countries by their respective regional life table and conducted four separate regression equations. Sullivan‟s method results in two regression coefficients (A and B) of which there are 12 different 29 values (Table 2.3). Selection of appropriate coefficients is dependent on the mortality pattern of the group of interest and the age of the mothers for whom dx is calculated. Then, to calculate the value of the multiplier value to convert dx into an estimate of q(a), one simply fills in the regression equation using the values of A and B and the value for 3 2 P P for the particular study group. Table 2.3. Results of Sullivan's regression model for constructing multipliers to be used in creating an estimate of q(a) from the proportion dead among children ever born to women in five-year age groups Regression Equations Mortality Pattern Regression coefficients A B West 1.30 -0.54 North 1.30 -0.63 East 1.26 -0.44 South 1.33 -0.61 West 1.17 -0.40 North 1.17 -0.50 East 1.14 -0.33 South 1.20 -0.44 West 1.13 -0.33 North 1.15 -0.42 East 1.11 -0.26 South 1.14 -0.32 Source: Sullivan 1972.           3 2 2520 )2( P P BA d q           3 2 3025 )3( P P BA d q           3 2 3530 )5( P P BA d q 30 Shortly after Sullivan presented his method, Coale and Trussell (1974) published a subsequent study that used the same basic technique but included a much greater amount of data. As a result of a newly finished project at the Princeton Office of Population Research, Coale and Trussell were able to generate more than 1,500 fertility schedules, as compared to the 65 generated by Sullivan. In their words, the new fertility schedules represented “essentially the full range of age structures of fertility likely to be found in large human populations” (Coale and Trussell 1974: 185). Using the newly-generated data, Coale and Trussell determined that both the ratio of average parity for women aged 15-19 and women aged 20-25         2 1 P P and the ratio of average parity for women aged 20-25 and women aged 25-30         3 2 P P were the best predictors of the multipliers needed to convert dx into an estimate of q(a) and, thus, presented the following regression equation:                   3 2 2 1 x P P C P P BAk (6) In contrast to Sullivan‟s conclusions, this modified approach results in three, rather than two, regression coefficients (Table 2.4). Additionally, to create multipliers one must supply both 2 1 P P as well as 3 2 P P from the population of interest. Coale and Trussell also incorporated difference in mortality characteristics by calculating separate regression analyses according to the four mortality patterns exhibited in the Coale-Demeny regional model life tables. 31 Table 2.4. Results of Coale and Trussell's regression model for constructing multipliers to be used in creating an estimate of the probability of mortality from the proportion dead among children ever born to women in five-year age groups Mortality Pattern Age group of women A B C North 15-19 1.1119 -2.9287 0.8507 20-24 1.2390 -0.6865 -0.2745 25-29 1.1884 0.0421 -0.5156 30-34 1.2046 0.3037 -0.5656 35-39 1.2586 0.4236 -0.5898 40-44 1.2240 0.4222 -0.5456 45-49 1.1772 0.3486 -0.4624 South 15-19 1.0819 -3.0005 0.8689 20-24 1.2846 -0.6181 -0.3024 25-29 1.2223 0.0851 -0.4704 30-34 1.1905 0.2631 -0.4487 35-39 1.1911 0.3152 -0.4291 40-44 1.1564 0.3017 -0.3958 45-49 1.1307 0.2596 -0.3538 East 15-19 1.1461 -2.2536 0.6259 20-24 1.2231 -0.4301 -0.2245 25-29 1.1593 0.0581 -0.3479 30-34 1.1404 0.1991 -0.3487 35-39 1.1540 0.2511 -0.3506 40-44 1.1336 0.2556 -0.3428 45-49 1.1201 0.2362 -0.3268 West 15-19 1.1415 -2.7070 0.7663 20-24 1.2563 -0.5381 -0.2637 25-29 1.1851 0.0633 -0.4177 30-34 1.1720 0.2341 -0.4272 35-39 1.1865 0.3080 -0.4452 40-44 1.1746 0.3314 -0.4537 45-49 1.1639 0.3190 -0.4435 Regression equation: Source: Coale and Trussell 1974 In this study, I construct indirect estimates of child mortality rates [  )5(q ] for each municipality in Brazil in the year 2000. Although traditionally women between the ages                   3 2 2 1 x P P C P P BAk 32 of 30 and 34 are used to estimate the probability of death among children prior to age five, I opt to use younger women, between the ages of 20 and 29. Because the intent of this study is to examine mortality levels in the year 2000, it is preferable to use younger women as their experience is more heavily influenced by current levels of child mortality. To illustrate this idea, I examine the mean age of children (living and deceased) for women of different age groups )c( using DHS data from 1996 (Figure 2.3). According to the results of this analysis, women aged 30 to 34 gave birth, on average, 7.7 years before the survey. As most child deaths are concentrated in the first year of life, the mortality conditions of these women‟s children are unlikely to reflect current conditions. In contrast, the mean age of children among women between 20 and 24 years of age was 2.4 years, and the mean age of children among women aged 25-29 was 4.9 years 6 . 6 Although the youngest mean age of children (and, thus, the most recent birth histories) belonged to women less than 20 years of age, it has been shown that mortality estimates among this group are often inflated. This is primarily due to the fact that women who have children at younger ages represent a unique population whose risk factors (e.g. socioeconomic status) often differ from the population as a whole (Preston, Heuveline, and Guillot 2001). 33 Figure 2.3. Mean age of children )c( by mother‟s age group in Brazil, 1996. Source: DHS 1996 In order to use women between the ages of 20 and 29, I needed to make some modifications to the method detailed above. First, I determine how the proportion of children who have died to women between the ages of 20 and 29 relates to the probability of death to children prior to a specific age. To accomplish this, I first find the q(a) values for each single year of age using the 2000 Brazil Life Table (Table 2.5). I then find the age distribution of children (living and deceased) for women ages 20 to 29 [c20-29(a)] from 1996 DHS data (Table 2.5). By multiplying the two values together, I calculate the proportion of children born to women aged 20 to 29 who would have died prior to each age according to the age distribution of these women and the Brazilian Life Table probabilities of death prior to each age. Finally, by summing all the proportions of 34 children who would have died for each age group, I obtain the total proportion of children who would have died among women ages 20 to 29 (Σd20-29) if they had the age distribution of children born to women in this age group and the mortality risks of the 2000 Life Table. Table 2.5. Proportion of children born to women aged 20-29 who would have died (d20-29) prior to age a based on the age distribution of children born to women aged 20 to 29 [c20- 29(a)] and the probability of death to children prior to age a [q(a)]. a q(a) c20-29(a) d20-29 0 0.0260 0.1059 0.0028 1 0.0315 0.1154 0.0036 2 0.0338 0.1071 0.0036 3 0.0351 0.1136 0.0040 4 0.0362 0.1030 0.0037 5 0.0369 0.1002 0.0037 6 0.0375 0.0834 0.0031 7 0.0379 0.0787 0.0030 8 0.0382 0.0619 0.0024 9 0.0385 0.0552 0.0021 10 0.0387 0.0335 0.0013 11 0.0390 0.0229 0.0009 12 0.0394 0.0093 0.0004 13 0.0399 0.0059 0.0002 14 0.0405 0.0024 0.0001 15 0.0413 0.0014 0.0001 16 0.0423 0.0002 0.0000 17 0.0435 0.0000 0.0000 18 0.0448 0.0000 0.0000 Σc20-29(a) = 1.00 Σd20-29 = 0.0350 Sources: DHS 1996; IBGE 2009b 35 Finally, I then compare the value for Σd20-29, 0.0350, with the original q(a) values found in the Brazil Life Table and find that the value of Σd20-29 is most similar to the value of q(3), 0.0351. Therefore, the proportion of deaths to children for women aged 20- 29 is most comparable to the probability of death to children prior to the age of 3. I repeated this process separately for each of the five regions of the country and found the relationship between q(a), c20-29(a), and d20-29 was consistent for all areas. However, to be consistent with the most commonly used definition of child mortality (deaths to children under the age of five), I want to convert this value so that it can be used as an estimate of deaths to children under the age of five. Again, I return to the Brazil Life Table, this time for multiple years, and find that there is a consistent relationship between q(3) and q(5) which is defined with the following regression equation: q(5) ~ -0.0018 + 1.1017q(3) (7) To validate this measure, I compared the value of q(5) for the country as a whole I obtained using this process with published rates for the year 2000. The value I obtain was 0.037 which is identical to the value published by the US Census Bureau (U.S. Census Bureau, International Data Base 2010) and somewhat higher than the value of 0.033 published by the World Bank (Inter-agency Group for Child Mortality Estimation 2010). Thus, by applying this equation to the proportion of children who have died to women between the ages of 20 and 29,  )3(q , for each municipality I am able to obtain municipal-level indirect estimates of child mortality,  )5(q , for the year 2000. 36 While the Census includes an enormous amount of data, it does not overcome the limitations of small geographical areas such as municipalities. In Brazil, the population sizes of municipalities vary considerably which, in turn, means that the number of women available in each municipality for the construction of estimates of child mortality rates also varies. In 2000, approximately one-quarter of all Brazilian municipalities had less than 5,000 residents, while almost one-half were home to less than 10,000 people. After limiting Brazil Census microdata to women between the ages of 20 and 29, one- quarter of the municipalities had less than 80 women and one-half had less than 150 (Figure 2.4). 37 Figure 2.4. Quartiles of the number of women aged 20-29 in each municipality. Source: 2000 Brazilian Census. With an average child mortality rate of 33 deaths per 1,000 births in Brazil, it is likely that in many of the municipalities it will be difficult to create accurate estimates of child mortality because just a few more or less child deaths can greatly impact the estimates in less populated areas. This idea is illustrated in Figure 2.5, which displays the confidence intervals surrounding the proportion of children who have died to women between 20 and 29 years of age (d20-29) for the 5 municipalities with the largest population sizes and the 5 municipalities with the smallest population sizes. The 38 confidence intervals for the most populated municipalities are quite small; however, the intervals for the five least populated municipalities are, in comparison, extremely large, reflecting very unstable data. 39 Figure 2.5. Confidence intervals of the proportion of children dead to women aged 20-29 (d20-29): 5 most populated and 5 least populated municipalities. Source: 2000 Brazilian Census. 40 Spatial Smoothing Methods When dealing with unstable estimates due to small population sizes, spatial smoothing methods can be employed to improve the quality of estimates. It is well established that geographically close areas often share a number of similarities, and exhibits spatial autocorrelation (Clayton, Bernardinelli, and Montomoli 1993). Thus, by “borrowing” information from nearby (and likely quite similar) areas it is possible to strengthen and improve estimates for areas with small population sizes (Kafadar 1994). There are a number of different spatial smoothing methods that have been developed for a variety of uses. Below I review some of the more common spatial smoothing methods used in the fields of public health and demography. Locally Weighted Averages Spatial Smoothing The locally weighted averages approach to creating spatially smoothed estimates consists of creating an average estimate for each area using the estimate of that area as well as estimates in all nearby areas. In the example provided by Waller and Gotway (2004), the smoothed rate of area i ( i r ~ ) is calculated by summing the rates of all areas within region j divided by the number of areas within region j:      N 1j ij N 1j jij i w rw r ~ (8) where wij = 1 if areas fall within region j and wij = 0 for those who do not. There are a number of different ways to specify a region (or neighborhood). One such way, disk smoothing, consists of defining a neighborhood as all areas that fall within a certain 41 distance from the center of each area (known as the centroid). Alternatively, an adjacency approach defines neighbors as only those who are physically connected to each other. In a study of low birth weight (LBW) infants by zip codes in New York State, Talbot et al. (2000) proposed using a variable window size approach wherein “ the spatial filter is defined in terms of constant or near constant population size rather than constant geographic size” (p. 2400). Using this approach, the authors created initial neighborhoods that included all zip codes within 0.75 miles of each centroid. They then expanded the neighborhoods to the next closest zip codes until they reached the target number of births. Whereas they found that non-spatially smoothed estimates of LBW rates were highly varied with very little spatial pattern, they found rates constructed using locally weighted averages were far more stable and demonstrated clear spatial patterns. Nonparametric Regression Spatial Smoothing In nonparametric regression spatial smoothing, rather than simply calculating an average, estimates for each area are obtained by fitting a statistical model and calculating estimates for each area that maximizes the log-likelihood. This is shown in Equation 9: ))  N 1j jij ξ|log(f(yw (9) One of the most commonly used nonparametric regression spatially smoothing techniques, first introduced by Cleveland in 1979 and later modified by Cleveland and Devlin in 1988, loess (or locally weighted polynomial regression), uses independent variables from nearby areas in order to create locally weighted regression analyses for estimating values in each area. Nonparametric regression spatial smoothing uses a distance decay function (Haining 2003) wherein wij is defined by the kernel function 42           b ss kernw ji ij (10) where si and sj identifies the distance between the centroid of municipality i and municipality j. Thus, municipalities of closer proximity have more of an effect with the effect decreasing as the distance increases. The value of b determines the number of areas to be included with a larger value allowing for more areas and, as a result, more spatial smoothing. Empirical Bayesian Spatial Smoothing Spatial smoothing using Empirical Bayes (EB) and full Bayes (discussed in the next section) is based upon a theorem developed by an 18 th Century mathematician and Protestant minister named Thomas Bayes who believed that knowledge of previous events can be used to help determine the probability of an event occurring in the future (Bayes 1763). Today, almost 250 years later, the field of Bayesian Statistics is still based on this idea. Rather than relying solely on data, Bayesian methods combine data with additional information in order to create stronger and more stable measures. This additional information, known as priors, is often obtained from some previous (or prior) information already known about the topic. When using Bayesian statistics for spatial smoothing, priors are not derived from previous information but instead, based on the assumption of spatial autocorrelation, are derived from data in other areas and define the area-specific effect on the estimates. In the end, Bayesian spatial smoothing estimates represent a compromise between the data in one area and the data in all other (or neighboring areas) (Bernardinelli and Montomoli 1992). 43 In a full Bayesian approach, priors are expressed in the form of distributions with specific shapes, means, and variances. Combining observed data with a prior distribution results in a posterior distribution in which each parameter value is now represented by a distribution of values determined by both the data and the prior distribution, θ (Lynch 2007): data)|f(θ )(dataf θ)f(θ)|f(data (11) However, constructing a posterior distribution using this approach can be an extremely difficult process that involves complex integrations and is virtually impossible when working with complicated models. To avoid this, Empirical Bayes uses the observed data to set the most likely parameter values, known as hyperparameters, for the prior distribution to determine a shrinkage factor, i.e. how much the local estimate shrinks towards the mean estimate using this approach (Bernardinelli and Montomoli 1992). The hyperparameter values are determined by minimizing the Mean Squared Error (MSE) differences between estimates in individual areas and aggregate estimates (Marshall 1991). Most examples of Empirical Bayes spatial smoothing for disease (or mortality) mapping, begin with observed counts (Yi) which are defined as random variables with a Poisson distribution and a conditional mean of niξi where ni refers to the person-years at risk in area i and ξi refers to the risk of death or disease for a person in area i (Clayton and Kaldor 1987; Marshall 1991). Based on maximum likelihood, the best predictor of ξi is  i x i i n Y (12) 44 The next step in Bayesian analysis would be to define a prior distribution for the area- specific effects with mean (mi) and variance (Ai). In Empirical Bayesian analyses, however, the prior mean and variance are estimated from the data itself using the global weighted mean and variance. Using ni as the weight, Marshall (1991) defined the global weighted sample mean ( m~ ) as:    1i i i i n nx m~ i (13) and the global weighted sample variance (s 2 ) as:     i i i 2 ii2 n )m~x(n s (14) The weighted mean and variance are then used to calculate the Bayes shrinkage factor for each municipality  i C ~ i 2 2 n/m~n/m~s n/m~s   (15) The shrinkage factor is then be used to calculate EB estimates of the probability of death among children born to women aged 20-29 for each municipality:  i ~  )m~x(C ~ m~ ii  (16) While the above example provides an EB approach using global means and variances, this may be problematic because it can ignore potentially important spatial variation in outcomes. Marshall (1991) recommended defining neighborhoods and using neighborhood means and variances to estimate prior parameters in order to create local 45 EB estimates, which are more strongly affected by areas of closer proximity and, presumably, with more similar characteristics. Assunção et al. (2005) implemented this approach to estimate age-specific fertility schedules for 3,800 municipalities in Brazil. Rather than shrinking local estimates towards a national average, they identified, for each municipality, its neighborhood as the nearest k municipalities needed to obtain a total of 21,000 women and at least one birth in each age group. They then calculated EB estimates for each municipality by shrinking the estimates towards the neighborhood means. Bayesian Spatial Smoothing A full Bayesian Spatial Smoothing approach differs from Empirical Bayes in that the parameters of the prior distribution are not estimated from aggregate data but, instead, are considered to be random variables with their own distributions, resulting in a hierarchical model. More specifically, the first level of the model is defined by the observed data itself. In the second level of the model, the prior distribution defines spatial dependence between nearby areas through its hyperparameters (Besag, York, and Mollié 1991; Bernardinelli and Montomoli 1992). In the third the level of the model, hyperprior distributions for the hyperparameters are defined. The hyperprior distributions provide information on prior belief about how similar neighboring areas should be. The end result of the full Bayesian approach is a posterior distribution and, not, as in the examples above, a single point value. The posterior distribution is characterized by a compromise between the estimates in each area and the estimates in neighboring areas. Additionally, because the estimates in the neighboring areas are also influenced by the estimates in their neighboring areas, the estimates in each area are indirectly influenced 46 by those areas as well (Potter et al. 2010). An important aspect of Bayesian spatial smoothing is that this compromise depends largely on the data within each area. If the sample size is sufficiently large and, thus, the estimates are stable, the effect of the prior distribution on the data is relatively small. However, in areas in which the data is more unstable, the prior distribution (and, thus, the influence of neighboring areas) will be greater. As stated previously, constructing a posterior distribution is an extremely difficult process. However, in the 1990s, a sampling method known as the Markov Chain Monte Carlo (MCMC) was developed that made it possible to simulate a posterior distribution (Gilks, Richardson, and Spiegelhalter 1996). The Markov chain is a series of random states wherein each future state is only dependent on the current state and is independent of any past states, known as the Markov property. To illustrate the Markov property, suppose that the probability of rain on any day depends only upon whether or not it rained the day before and there is a 20% chance of sun following a rainy day (p(rain1,sun2) = 0.2) and an 80% chance of rain on two consecutive days (p(rain1,rain2) = 0.8). Conversely, if it were sunny the previous day, the probability of rain the next day is 40% (p(sun1,rain2) = 0.4) and the probability of sun is 60% (p(sun1,sun2) = 0.6). To calculate the probability of rain on the third day, given that it rained on the first day, it is necessary to calculate the probability of rain on the second day given that it was raining on the first day (0.8) multiplied by the probability of rain on the third day given that it was rainy on the second day (0.8) as well as the probability of sun on the second day given that it was rainy on the first day (0.2) multiplied by the probability of rain on 47 the third day given that it was sunny on the second day (0.4), resulting in a 0.72 probability of rain on the third day: p(rain1,rain2)p(rain2,rain3) + p(rain1,sun2)p(sun2,rain3) = p(rain1,rain3) (17) 0.8(0.8) + 0.2(0.4) = 0.72 However, if it was sunny on the first day, the probability of rain on the third day would not be 0.72. Instead, the probability of rain on the third day, given that it was sunny on the first day, would be the probability of sun on the second day given that it was sunny on the first day (0.6) multiplied by the probability of rain on the third day given that it was sunny on the second day (0.4) and the probability rain on the second day given that it was sunny on the first day (0.4) multiplied by the probability of rain on the third day given that it was rainy on the second day (0.8), resulting in a 0.56 probability of rain on the third day: p(sun1,sun2)p(sun2,rain3) + p(sun1,rain2)p(rain2,rain3) = p(sun1,rain3) (18) 0.60(0.40) + 0.4(0.8) = 0.56 Thus, on the third day the probability of rain does depend on the weather on the first day (the initial state) and does not satisfy the Markov property. Following the same logic, the probability of rain on the fourth day, given that it was rainy on the first day, would decline to 0.69 while the probability of rain, given that it was sunny on the first day, would increase to 0.62 (Table 2.6). With each successive day, the two probabilities of rain would become more similar and by the eighth day, the probability of rain is equal (0.67). At this point, the Markov property states that the probability of rain or sun is no longer affected by whether it was rainy or sunny on the 48 first day. After the eighth day, the probability of rain remains constant, resulting in a Markov chain with a stable probability distribution. Table 2.6. Example of the Markov property: The effect of rain or sun on day 1 on the probability of rain or sun on later days Rain on day 1 Sun on day 1 Probability of Rain Probability of Sun Probability of Rain Probability of Sun Day 2 0.80 0.20 0.60 0.40 Day 3 0.72 0.28 0.56 0.44 Day 4 0.69 0.31 0.62 0.38 Day 5 0.68 0.32 0.65 0.35 Day 6 0.67 0.33 0.66 0.34 Day 7 0.67 0.33 0.66 0.34 Day 8 0.67 0.33 0.67 0.33 Day 9 0.67 0.33 0.67 0.33 Day 10 0.67 0.33 0.67 0.33 When using MCMC to simulate a posterior distribution, a Markov chain consists of a set of states in which each state contains a value for the parameter of interest. The chain begins in a starting state (defined by an initial probability distribution) and then moves successively to additional states. After a number of steps, the Markov chain should eventually stabilize so that the value of the parameter in each successive state is determined only by the current state and a probability distribution defined by the combined effect of the data and prior distribution. Once the posterior distribution has been effectively simulated, it is then possible to sample values from the Markov chain 49 which accurately represent the values from the posterior distribution (Besag, York, and Mollié 1991). There are several different MCMC methods that can be used to simulate samples of the posterior distribution. One of the most common sampling methods, Gibbs sampling, has been used in Bayesian Statistics for over two decades. Gibbs sampling consists of assigning starting values for all parameters. In the first iteration, the first parameter is assigned an “updated” value obtained by randomly sampling the conditional probability distribution given the values of all other parameters and the observed data/prior distribution (Besag, York, and Mollié 1991; Bernardinelli and Montomoli 1992). Next, the second parameter is also assigned a new value sampled from the probability distribution given the new value of the first parameter, the starting values of the other parameters, and the data. This process continues until all parameters have been assigned new values resulting in the completion of one Markov chain. Then, in the next iteration, the first parameter is given a new value dependent, again, on the probability distribution, the data, and the new values assigned to all the other parameters in the first iteration. This process continues until eventually the chain converges so that the values of all parameters are determined by the combined effect of the observed data and the probability distribution, the prior distribution. Compared with the other spatial smoothing methods presented above, a full Bayesian approach to spatial smoothing is the only method that not only can incorporate information from neighboring areas but also can specify a level of certainty about the area-specific effects. For example, Empirical Bayes consists of estimating single values for the hyperparameters from the observed data and then using those estimates to find a 50 single spatially smoothed estimate for each area. There is no way to specify any degree of certainty or uncertainty about the estimated values for the prior mean and variance (Besag, York, and Mollié 1991). In contrast, in a full Bayesian approach in which the area-specific effect for Bayesian spatial smoothing are specified by a prior distribution, uncertainty about the prior distribution can be incorporated into the model. Additionally, the end results of a full Bayesian approach is a distribution of estimates, allowing for the calculation of credibility intervals which can be used to assess confidence in the Bayesian estimates (Bernardinelli and Montomoli 1992). Constructing Bayesian Estimates of Child Mortality In this study, I use an adjacency matrix to identify neighboring areas. Neighbors are defined as municipalities that are physically connected to one another. As an example, Figure 2.6 shows that the municipality Alta Floresta D‟Oeste has 7 neighbors. Overall, there are a total of 32,836 distinct adjacent pairs of municipalities in Brazil with an average of 6 neighbors per municipality. The smallest number of neighbors is 1 and the largest number of neighbors is 23. 51 Figure 2.6. Spatial neighbors of Alta Floresta D‟Oeste, Rondônia, Brazil. Source: 2000 Brazilian Census To use prior distributions obtained from neighboring areas, I employ a hierarchical Bayesian model in which the first level of the model consists of the level of mortality in a region in which Y (the number of child deaths reported by women aged 20 to 29 in each municipality i) is modeled using a binomial distribution 7 : Yi ~ binomial(pi ,ni) (19) 7 A binomial distribution is the preferred distribution when dealing with counts in small populations (Arató, Dryden, and Taylor 2006). 52 where pi is the probability of a child dying among women aged 20 to 29 in each municipality i and ni is the number of children born to women aged 20 to 29 in each municipality i. The probability of a child dying (pi) is modeled using a generalized linear model: logit(pi) = α + βi (20) where α is an unstructured random effect representing the global mean of the log-relative risks for all areas and βi is a spatially structured random effect representing the area- specific effects or the deviation from the global mean (Bernardinelli and Montomoli 1992). The second level of the hierarchical Bayesian model is the prior distributions for the random effects. The unstructured random effect or intercept term, α, is assigned an improper uniform prior distribution (Thomas et al. 2007). α ~ dflat() (21) The spatially structured random effect is assigned a conditional autoregressive (CAR) distribution: βi ~ CAR(τ) which identifies dependence between neighboring areas by making wi,i+1 = 1 if municipalities i and i+1 are neighbors and wi,i+1 = 0 if they are not.                i 1i 2 1ii1ii, ββw 2 τ expτ|βf (22) Additionally, the CAR model also includes the hyperparameter τ, the precision of the variance, which denotes geographical variability or how similar neighboring areas should be (Bernardinelli and Montomoli 1992). Due to uncertainty in the degree of similarity in 53 neighboring areas, in the third level of the hierarchical model, τ is assigned its own distribution, a hyperprior distribution, with a very weak gamma distribution: τ ~ γ(0.5, 0.0005) (23) To determine the standard deviation of β, τ is converted into ς.β:  /1. (24) Diagnostic Tests WinBUGS provides several diagnostic tests to ensure that the simulated posterior distribution is an accurate representation of the true posterior distribution. I selected a sample of 9 municipalities of varying sample sizes of women aged 20 to 29: 3 municipalities with less than 50 women, 3 municipalities with 50-200 women, and 3 municipalities with more than 200 women. For each of the municipalities, I recorded diagnostic information for the proportion of children who have died (pi) and the spatially structured random effects (βi). Additionally, I also recorded information on the standard deviation of β (ς.β) for the country as a whole. One issue that arises with MCMC sampling methods is autocorrelation between successive iterations of a chain. Because each sequence of a Markov chain is dependent upon the value of the preceding sequence, parameters in successive iterations often have similar values. Autocorrelation causes an artificial decrease in variances, making it appear as if convergence has been met before it actually has occurred (Lawson 2009). It is relatively easy to address issues of autocorrelation in Bayesian analyses by thinning the samples or selecting every kth iteration. As a result, successive parameter values will not be stored and issues of correlation should be improved. Additionally, by thinning, there is 54 less overall data that is stored which can be an important issue when dealing with a large number of parameters (Lynch 2007: 147). To identify issues of autocorrelation between successive chains, WinBUGS provides correlation values for each parameter at different lag times, in both graphical and tabular form. Values of 1 and -1 indicate complete correlation between chains while zero indicates no correlation. In Figures 2.7-2.9 I examine correlation values for the main parameters of interest. With no lag, it is clear that there is a high level of dependency between successive iterations as, in each of the graphs, the initial level of the bar is very close to 1. However, after a lag of only a single iteration, the values decrease substantially for most municipalities, although there is still some indication of autocorrelation in others. Figure 2.9, which looks at correlation for ς.β demonstrates the highest levels of dependency between chain sequences. After selecting every other iteration, the autocorrelation value is still very close to one at 0.86 and, after selecting every 10 th chain, it is still somewhat elevated at 0.26. However, with a lag of 20 the correlation declines substantially (0.050) and, thus, it appears that thinning the data and sampling every 20 th iteration should result in independent samples for all parameters. 55 Figure 2.7. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.217 10 0.027 20 0.015 30 -0.005 40 0.016 49 -0.001 p[784] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.189 10 0.002 20 -0.001 30 0.012 40 0.024 49 0.013 p[2457] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.130 10 -0.004 20 -0.018 30 -0.011 40 0.003 49 -0.011 Municipalities with 50-200 women aged 20-29 p[452] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.106 10 0.005 20 -0.007 30 0.019 40 0.000 49 0.000 p[578] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.186 10 -0.007 20 0.006 30 -0.013 40 -0.002 49 0.024 p[386] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.040 10 0.010 20 -0.004 30 -0.010 40 -0.004 49 -0.012 56 Figure 2.7 cont. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with more than 200 women aged 20-29 p[26] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.215 10 0.011 20 0.000 30 -0.009 40 0.002 49 0.006 p[186] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.047 10 -0.011 20 -0.001 30 0.013 40 -0.006 49 -0.016 p[161] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.036 10 0.012 20 -0.009 30 -0.011 40 0.000 49 -0.012 Source: 2000 Brazilian Census 57 Figure 2.8. Lags and autocorrelation values for spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.226 10 0.024 20 0.013 30 0.000 40 0.011 49 0.000 b[784] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.204 10 0.004 20 0.008 30 0.013 40 0.025 49 0.015 b[2457] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.133 10 -0.007 20 -0.020 30 -0.012 40 0.004 49 -0.008 Municipalities with 50-200 women aged 20-29 b[452] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.103 10 0.007 20 -0.002 30 0.020 40 -0.002 49 0.004 b[578] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.187 10 -0.008 20 0.009 30 -0.015 40 -0.005 49 0.024 b[386] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.042 10 0.012 20 -0.005 30 -0.011 40 -0.003 49 -0.012 58 Figure 2.8 cont. Lags and autocorrelation values for spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with more than 200 women aged 20-29 b[26] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.220 10 0.012 20 0.003 30 -0.012 40 0.003 49 0.007 b[186] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.049 10 -0.012 20 0.001 30 0.012 40 -0.006 49 -0.016 b[161] lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.041 10 0.012 20 -0.008 30 -0.012 40 0.000 49 -0.014 Source: 2000 Brazilian Census 59 Figure 2.9. Lags and autocorrelation values for the standard deviation of β (ς.β): Brazil, 2000. sigm a.b lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.859 10 0.262 20 0.050 30 0.007 40 0.015 49 0.039 Source: 2000 Brazilian Census After testing for (and eliminating) autocorrelation, it is important to ensure that the Markov chain has successfully converged to the posterior distribution. To examine convergence, WinBUGS provides information on the Brooks-Gelman-Rubin (BGR) statistic (Brooks and Gelman 1998). This is a multi-chain method in which several chains with different starting values are run simultaneously. Once convergence is reached, the initial value should no longer affect the current value and, thus, when the chains become very similar, it is likely that convergence has been reached. The BGR test statistic compares the between-chain variance (B) and the within-chain variance (P). The within- and between-chain variances are defined as the following: 2 . p 1i i i )γγ( 1p n B      (25) .  p 1i 2 i τ p 1 W (26) where n refers to the sample size, p refers to the number of chains, j i γ is the sample value, and 2 i τ is the variance of the i the chain. The BGR test statistic, R, is calculated using the following formula: ς.β 60 W B n 1 n 1n R    (27) Convergence can be assumed when the value of R for all parameters is close to 1.0. Additionally, the within- and between-chain variances should become stable and equal (or very close to equal) if convergence is met. To formally test for convergence, I compared the BGR statistics for two chains with different starting values which I ran for 40,000 iterations (thinned by 20 based upon the findings above). Based upon Figures 2.10-2.12 convergence has definitely been reached by 40,000 iterations for all parameters. Not only in every case is R very close to 1 but also the between- and within-chain variability are very stable and essentially equal. 61 Figure 2.10. BGR statistics for the proportion of children who had died (pi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 p[784] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 p[2457] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 1.5 Municipalities with 50-200 women aged 20-29 p[452] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 1.5 p[578] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 p[386] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 1.5 Municipalities with more than 200 women aged 20-29 p[26] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 p[186] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 p[161] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 Source: 2000 Brazilian Census R Between-Chain Variability Within-Chain Variability 62 Figure 2.11. BGR statistics for spatially structured random effects (βi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[784] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[2457] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 Municipalities with 50-200 women aged 20-29 b[452] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[578] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[386] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 Municipalities with more than 200 women aged 20-29 b[26] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[186] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 b[161] chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 Source: 2000 Brazilian Census R Between-Chain Variability Within-Chain Variability 63 Figure 2.12. BGR statistics for the standard deviation of β (ς.β) after 40,000 iterations: Brazil, 2000. sigm a.b chains 1:2 start-i teration 4001 100000 200000 300000 0.0 0.5 1.0 Source: 2000 Brazilian Census Another way to test for convergence using WinBUGS is to examine the posterior density distribution for the parameter values. When convergence has been reached, the distribution of the posterior samples will be bell-shaped and smooth although they do not need to be symmetric (Zhang 2008). In Figures 2.13-2.15, all of the density plots meet these criteria, indicating, again, that convergence has been met. R Between-Chain Variability Within-Chain Variability ς.β chains 1:2 64 Figure 2.13. Posterior density distribution for the proportion of children who had died (pi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chain 1 sam ple: 40000 0.0 0.05 0.1 0.15 0.0 10.0 20.0 30.0 40.0 p[784] chain 1 sample: 40000 0.0 0.025 0.05 0.075 0.0 20.0 40.0 60.0 p[2457] chain 1 sam ple: 40000 0.0 0.05 0.1 0.0 10.0 20.0 30.0 40.0 Municipalities with 50-200 women aged 20-29 p[452] chain 1 sample: 40000 0.0 0.05 0.1 0.0 10.0 20.0 30.0 40.0 p[578] chain 1 sample: 40000 0.0 0.025 0.05 0.075 0.0 20.0 40.0 60.0 p[386] chain 1 sample: 40000 0.02 0.04 0.06 0.08 0.1 0.0 20.0 40.0 60.0 Municipalities with more than 200 women aged 20-29 p[26] chain 1 sample: 40000 0.0 0.01 0.02 0.03 0.0 50.0 100.0 150.0 p[186] chain 1 sample: 40000 0.01 0.02 0.03 0.0 50.0 100.0 150.0 p[161] chain 1 sample: 40000 0.01 0.015 0.02 0.025 0.0 100.0 200.0 300.0 Source: 2000 Brazilian Census 65 Figure 2.14. Posterior density distribution for spatially structured random effects (βi) after 40,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chain 1 sam ple: 40000 -4.0 -2.0 0.0 0.0 0.25 0.5 0.75 1.0 b[784] chain 1 sample: 40000 -2.0 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 b[2457] chain 1 sam ple: 40000 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 Municipalities with 50-200 women aged 20-29 b[452] chain 1 sample: 40000 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 b[578] chain 1 sample: 40000 -2.0 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 b[386] chain 1 sample: 40000 -0.5 0.0 0.5 1.0 0.0 1.0 2.0 3.0 Municipalities with more than 200 women aged 20-29 b[26] chain 1 sample: 40000 -2.0 -1.5 -1.0 -0.5 0.0 0.0 0.5 1.0 1.5 2.0 b[186] chain 1 sample: 40000 -1.5 -1.0 -0.5 0.0 0.0 1.0 2.0 3.0 b[161] chain 1 sample: 40000 -1.0 -0.8 -0.6 -0.4 0.0 2.0 4.0 6.0 Source: 2000 Brazilian Census 66 Figure 2.15. Posterior density distribution for the standard deviation of β (ς.β) after 40,000 iterations: Brazil, 2000. sigm a.b chain 1 sam ple: 40000 0.5 0.6 0.7 0.8 0.9 0.0 10.0 20.0 30.0 Source: 2000 Brazilian Census To determine the number of iterations that should be run after convergence is reached, I use two additional statistics provided by WinBUGS: the standard deviation of the value of the chains and a Monte Carlo error (calculated by subtracting the mean of the sampled values from the total posterior mean). The general rule is that the number of iterations is sufficient when the percentage difference between the standard deviation and the Monte Carlo error is greater than 5% (Shaddick 2008). In Table 8, it is clear that an additional 40,000 iterations after convergence is more than acceptable because the percentage difference between the standard deviation far exceeds 5% for every parameter of interest. ς.β 67 Table 2.7. Percentage difference between Monte Carlo error and the standard deviation from 50,000 to 100,000 iterations: Brazil, 2000 Standard Deviation Monte Carlo error Percentage Difference p[1899] 0.0150 0.0001 99.3 p[784] 0.0088 0.0001 99.3 p[2457] 0.0140 0.0001 99.3 p[452] 0.0131 0.0001 99.3 p[578] 0.0081 0.0001 99.2 p[386] 0.0085 0.0001 99.3 p[26] 0.003 0.0000 99.2 p[186] 0.003 0.0000 99.4 p[161] 0.002 0.0000 99.3 p[1899] 0.480 0.0033 99.3 p[784] 0.341 0.0025 99.3 p[2457] 0.288 0.0019 99.3 p[452] 0.296 0.0020 99.3 p[578] 0.280 0.0023 99.2 p[386] 0.156 0.0011 99.3 p[26] 0.224 0.0017 99.2 p[186] 0.140 0.0009 99.4 p[161] 0.080 0.0006 99.3 ς.β 0.017 0.0001 99.2 Source: 2000 Brazilian Census Based upon these results and the above autocorrelation and convergence tests, I retain every 20 th iteration and run a total of 80,000 iterations, discarding the first 40,000 iterations as burn-in chains. Figures 2.16-2.18 provide a history of parameter values for the last 50,000 iterations. Although I have already determined that convergence has been met, these figures also indicate convergence as WinBUGS is consistently sampling within a small distribution of values across all chains. After completing all iterations, 68 WinBUGS provides sample statistics of these parameter values including the mean, median, and standard deviation. Using the mean value of pi as an estimate of q(3) I create municipal-level Bayesian estimates of child mortality [  )5(q ] using the formula:  )5(q ~ -0.0018 + 1.1017  )( 3q (28) To explore the spatial effect, I also map the mean values of each Bi. Additionally,  )5(q values as well as the 95% credibility intervals for the  )5(q values are available on-line at http://www.utexas.edu/cola/centers/prc/faculty/klh52265?tab=41. . 69 Figure 2.16. History of values for 40,000-60,000 iterations (thinned by 20) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.05 0.1 0.15 0.2 p[784] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.05 0.1 0.15 p[2457] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.05 0.1 0.15 70 Figure 2.16cont. History of values for 40,000-60,000 iterations (thinned by 20) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[452] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.05 0.1 0.15 p[578] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.02 0.04 0.06 0.08 0.1 p[386] chain 1 i teration 800000 900000 1000000 1100000 0.02 0.04 0.06 0.08 0.1 71 Figure 2.16cont. History of values for 40,000-40,000 iterations (thinned by 20) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with 50-200 women aged 20-29 p[26] chain 1 i teration 800000 900000 1000000 1100000 0.0 0.01 0.02 0.03 0.04 p[186] chain 1 i teration 800000 900000 1000000 1100000 0.01 0.02 0.03 0.04 p[161] chain 1 i teration 800000 900000 1000000 1100000 0.01 0.015 0.02 0.025 0.03 Source: 2000 Brazilian Census 72 Figure 2.17. History of values for 40,000-60,000 iterations (thinned by 20) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20- 29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chain 1 i teration 800000 900000 1000000 1100000 -3.0 -2.0 -1.0 0.0 1.0 2.0 b[784] chain 1 i teration 800000 900000 1000000 1100000 -2.0 -1.0 0.0 1.0 2.0 b[2457] chain 1 i teration 800000 900000 1000000 1100000 -1.0 0.0 1.0 2.0 73 Figure 2.17cont. History of values for 40,000-60,000 iterations (thinned by 20) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[452] chain 1 i teration 800000 900000 1000000 1100000 -1.0 0.0 1.0 2.0 b[578] chain 1 i teration 800000 900000 1000000 1100000 -2.0 -1.0 0.0 1.0 b[386] chain 1 i teration 800000 900000 1000000 1100000 -0.5 0.0 0.5 1.0 1.5 74 Figure 2.17cont. History of values for 40,000-60,000 iterations (thinned by 20) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with 50-200 women aged 20-29 b[26] chain 1 i teration 800000 900000 1000000 1100000 -2.0 -1.5 -1.0 -0.5 0.0 b[186] chain 1 i teration 800000 900000 1000000 1100000 -1.5 -1.0 -0.5 0.0 0.5 b[161] chain 1 i teration 800000 900000 1000000 1100000 -1.0 -0.8 -0.6 -0.4 -0.2 Source: 2000 Brazilian Census 75 Figure 2.18. History of values for 40,000-60,000 iterations (thinned by 20) of the standard deviation of β (ς.β): Brazil, 2000. sigm a.b chain 1 i teration 800000 900000 1000000 1100000 0.55 0.6 0.65 0.7 Source: 2000 Brazilian Census ς.β 76 Constructing Bayesian Estimates of Child Mortality using Female Education In Brazil, child mortality is highly associated with female education, both at the level of the individual (e.g. Merrick 1985; Sastry 2004) and the community (e.g. Goldani et al. 2002; Alves 2003). Therefore, the level of female education in a municipality should be a strong predictor of the level of child mortality. In the next part of this study, I incorporate information on the education level of each municipality in order to construct Bayesian estimates of child mortality. Using a hierarchical Bayesian model, I again begin with Y, the number of child deaths reported by women aged 20 to 29 in each municipality i, modeled using a binomial distribution with pi indicating the probability of a child dying among women aged 20 to 29 in each municipality i and ni the number of children born to women aged 20 to 29 in each municipality i.: Yi ~ binomial(pi ,ni) (29) In the next step, pi is modeled using a generalized linear model which includes the unstructured random effect (α) and the spatially structured random effects (βi) as in the previous model as well as a parameter for mean female education (e) logit(pi) = α + e(Edufi- Eduf) + βi (30) By including education in the model, I am effectively using both spatial and education neighbors to predict pi for each municipality i. Because I have very little previous knowledge on how much education should impact the estimates for each area, I define a very weak prior distribution for e by assigning a small value for the precision, thus, allowing the data to be the main determinant of the estimates: e~ N(0.0, 0.001) (31) 77 The unstructured random effect or intercept term, α, is assigned an improper uniform prior distribution α ~ dflat() (32) and to denote spatial dependence, as before, the prior distribution for the spatially structured random effect is specified using a conditional autoregressive (CAR) model: βi ~ CAR(τ) (33) defined as the following:                i 1i 2 1ii1ii, ββw 2 τ expτ|βf (34) which identifies dependence between neighboring areas and includes the hyperparameter τ, the precision of the variance, to denote how similar neighboring areas should be Again, with little knowledge on how spatially correlated neighboring areas are, I define a weak hyperprior distribution for the hyperparameter τ: τ ~ γ(0.5, 0.0005) (35) To determine the standard deviation of β, τ is converted into ϛ.β:  /1. (36) Diagnostic Tests To ensure that the simulated posterior distribution is an accurate representation of the true posterior distribution, I again perform a series of diagnostic tests. Using the same sample of 9 municipalities of varying sample sizes of women aged 20 to 29, I examine autocorrelation between successive iterations of chains (Figures 2.19-2.22) for each of the parameters: the proportion of children who have died (pi), the spatially structured random effects (βi), the education effect (e), and the standard deviation of β (ς.β). Both 78 the education effect and the precision of the hyperparameter demonstrate the highest dependency between successive iterations. However, by selecting every 30 th chain, the autocorrelation value for both parameter falls below one, indicating that sampling every 30 th iteration should result in independent iterations for all parameters. 79 Figure 2.19. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.230 10 0.017 20 -0.004 30 -0.006 40 0.009 49 0.003 p[784] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.214 10 0.033 20 0.004 30 0.007 40 0.007 49 0.021 p[2457] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.124 10 0.008 20 -0.007 30 0.004 40 -0.012 49 0.001 Municipalities with 50-200 women aged 20-29 p[452] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.110 10 0.000 20 0.001 30 0.004 40 -0.016 49 0.010 p[578] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.187 10 0.004 20 0.023 30 0.007 40 0.006 49 0.007 p[386] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.040 10 0.019 20 -0.008 30 0.010 40 -0.002 49 0.004 80 Figure 2.19 cont. Lags and autocorrelation values for the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with more than 200 women aged 20-29 p[26] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.246 10 0.000 20 -0.009 30 -0.013 40 -0.014 49 0.003 p[186] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.067 10 0.006 20 -0.005 30 -0.002 40 -0.013 49 -0.003 p[161] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.005 10 -0.009 20 0.002 30 0.001 40 0.004 49 -0.005 Source: 2000 Brazilian Census 81 Figure 2.20. Lags and autocorrelation values for spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.249 10 0.023 20 -0.009 30 -0.001 40 0.013 49 0.003 b[784] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.224 10 0.034 20 0.003 30 0.005 40 0.007 49 0.019 b[2457] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.125 10 0.009 20 -0.010 30 0.000 40 -0.012 49 0.002 Municipalities with 50-200 women aged 20-29 b[452] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.117 10 -0.004 20 0.001 30 0.005 40 -0.012 49 0.005 b[578] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.193 10 0.005 20 0.018 30 0.011 40 0.009 49 0.005 b[386] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.051 10 0.025 20 -0.002 30 0.012 40 -0.003 49 0.005 82 Figure 2.20 cont. Lags and autocorrelation values for spatially structured random effects (bi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with more than 200 women aged 20-29 b[26] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.254 10 0.002 20 -0.008 30 -0.008 40 -0.013 49 0.003 b[186] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.073 10 0.010 20 -0.004 30 0.002 40 -0.011 49 -0.003 b[161] chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.079 10 0.025 20 0.011 30 0.002 40 0.000 49 -0.004 Source: 2000 Brazilian Census Figure 2.21. Lags and autocorrelation values for the education effect (e): Brazil, 2000. e chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.927 10 0.502 20 0.224 30 0.075 40 0.021 49 0.001 Source: 2000 Brazilian Census 83 Figure 2.22. Lags and autocorrelation values for the standard deviation of β (ς.β): Brazil, 2000. sigm a.b chain 1 lag 0 20 40 -1.0 -0.5 0.0 0.5 1.0 Lag Value 1 0.878 10 0.302 20 0.125 30 0.063 40 0.015 49 -0.024 Source: 2000 Brazilian Census To determine when the Markov chain reaches convergence, I run two chains with very different initial values and examine the Brooks-Gelman-Rubin (BGR) statistic (Brooks and Gelman 1998). Figures 2.23-2.26 demonstrate that convergence has been reached after 50,000 iterations because R is very close to 1 and the between- and within- chain variability are stable and almost equal. Additionally, I also test for convergence by examining the posterior density for the parameter values. In Figures 2.27-2.30, all of the density plots are bell-shaped and smooth, providing further evidence that after 50,000 iterations, convergence has been met. ς.β 84 Figure 2.23. BGR statistics for the proportion of children who had died (pi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[784] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[2457] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Municipalities with 50-200 women aged 20-29 p[452] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[578] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[386] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Municipalities with more than 200 women aged 20-29 p[26] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[186] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 p[161] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Source: 2000 Brazilian Census R Between-Chain Variability Within-Chain Variability 85 Figure 2.24. BGR statistics for spatially structured random effects (βi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[784] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[2457] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Municipalities with 50-200 women aged 20-29 b[452] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[578] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[386] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Municipalities with more than 200 women aged 20-29 b[26] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[186] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 b[161] chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Source: 2000 Brazilian Census R Between-Chain Variability Within-Chain Variability 86 Figure 2.25. BGR statistics for the education effect (e) after 50,000 iterations: Brazil, 2000. e chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Source: 2000 Brazilian Census Figure 2.26. BGR statistics for the standard deviation of β (ς.β) after 50,000 iterations: Brazil, 2000. sigm a.b chains 1:2 start-i teration 7501 200000 400000 600000 0.0 0.5 1.0 Source: 2000 Brazilian Census R Between-Chain Variability Within-Chain Variability R Between-Chain Variability Within-Chain Variability ς.β chains 1:2 87 Figure 2.27. Posterior density distribution for the proportion of children who had died (pi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chain 1 sam ple: 50000 0.0 0.05 0.1 0.15 0.0 10.0 20.0 30.0 40.0 p[784] chain 1 sample: 50000 0.0 0.025 0.05 0.075 0.0 20.0 40.0 60.0 p[2457] chain 1 sam ple: 50000 0.0 0.05 0.1 0.0 10.0 20.0 30.0 40.0 Municipalities with 50-200 women aged 20-29 p[452] chain 1 sample: 50000 0.0 0.05 0.1 0.0 10.0 20.0 30.0 40.0 p[578] chain 1 sample: 50000 0.0 0.02 0.04 0.06 0.08 0.0 20.0 40.0 60.0 p[386] chain 1 sample: 50000 0.02 0.04 0.06 0.08 0.0 20.0 40.0 60.0 Municipalities with more than 200 women aged 20-29 p[26] chain 1 sample: 50000 0.0 0.01 0.02 0.03 0.0 50.0 100.0 150.0 p[186] chain 1 sample: 50000 0.01 0.02 0.03 0.0 50.0 100.0 150.0 p[161] chain 1 sample: 50000 0.01 0.015 0.02 0.025 0.0 100.0 200.0 300.0 Source: 2000 Brazilian Census 88 Figure 2.28. Posterior density distribution for spatially structured random effects (βi) after 50,000 iterations in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chain 1 sam ple: 50000 -2.0 -1.0 0.0 1.0 0.0 0.25 0.5 0.75 1.0 b[784] chain 1 sample: 50000 -2.0 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 b[2457] chain 1 sam ple: 50000 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 Municipalities with 50-200 women aged 20-29 b[452] chain 1 sample: 50000 -2.0 -1.0 0.0 1.0 0.0 0.5 1.0 1.5 b[578] chain 1 sample: 50000 -2.0 -1.0 0.0 0.0 0.5 1.0 1.5 2.0 b[386] chain 1 sample: 50000 -0.5 0.0 0.5 1.0 0.0 1.0 2.0 3.0 Municipalities with more than 200 women aged 20-29 b[26] chain 1 sample: 50000 -2.0 -1.5 -1.0 -0.5 0.0 0.0 0.5 1.0 1.5 2.0 b[186] chain 1 sample: 50000 -1.0 -0.5 0.0 0.0 1.0 2.0 3.0 b[161] chain 1 sample: 50000 -1.0 -0.75 -0.5 -0.25 0.0 0.0 2.0 4.0 6.0 Source: 2000 Brazilian Census 89 Figure 2.29. Posterior density distribution for the education effect (e) after 50,000 iterations: Brazil, 2000. e chain 1 sample: 50000 -0.2 -0.15 -0.1 -0.05 0.0 20.0 40.0 60.0 Source: 2000 Brazilian Census Figure 2.30. Posterior density distribution of the standard deviation of β (ς.β) after 50,000 iterations: Brazil, 2000. sigm a.b chain 1 sam ple: 50000 0.5 0.6 0.7 0.8 0.0 10.0 20.0 30.0 Source: 2000 Brazilian Census After convergence is reached, I run an additional 10,000 iterations of the Markov chain. According to Table 2.8, this is a sufficient number of iterations because the percentage difference between the standard deviation and the Monte Carlo error is far greater than 5% (Shaddick 2008). ς.β 90 Table 2.8. Percentage difference between Monte Carlo error and the standard deviation from 50,000 to 100,000 iterations: Brazil, 2000 Standard Deviation Monte Carlo error Percentage Difference p[1899] 0.0020 0.0000 99.1 p[784] 0.0084 0.0001 99.0 p[2457] 0.0141 0.0001 99.0 p[452] 0.0128 0.0001 99.1 p[578] 0.0081 0.0001 99.1 p[386] 0.0082 0.0001 99.0 p[26] 0.0032 0.0000 99.0 p[186] 0.0027 0.0000 98.9 p[161] 0.0015 0.0000 99.0 p[1899] 0.4533 0.0042 99.1 p[784] 0.3209 0.0029 99.1 p[2457] 0.2769 0.0027 99.0 p[452] 0.2866 0.0027 99.1 p[578] 0.2674 0.0025 99.1 p[386] 0.1565 0.0015 99.0 p[26] 0.2188 0.0019 99.1 p[186] 0.1375 0.0012 99.1 p[161] 0.0813 0.0009 98.9 e 0.0072 0.0001 98.9 ς.β 0.0169 0.0002 98.9 Source: 2000 Brazilian Census Based upon the diagnostic tests, I run a total of 60,000 iterations saving every 30 th iteration and discarding the first 50,000 iterations as burn-in chains. A history of the parameter values for the last 10,000 iterations is provided in Figures 2.31-2.18. After completing all iterations, I use the sample mean of the posterior distribution to map 91 municipal-level Bayesian estimates of child mortality [  )5(q ]. Additionally, I also map the spatial effect (Bi) as well as the education effect. Credibility intervals are provided to determine the significance of the association between education and child mortality at the level of the municipality. 92 Figure 2.31. History of values for 50,000-60,000 iterations (thinned by 30) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[1899] chain 1 i teration 1500000 1600000 1700000 0.0 0.05 0.1 0.15 0.2 p[784] chain 1 i teration 1500000 1600000 1700000 0.0 0.02 0.04 0.06 0.08 0.1 p[2457] chain 1 i teration 1500000 1600000 1700000 0.0 0.05 0.1 0.15 93 Figure 2.31cont. History of values for 50,000-60,000 iterations (thinned by 30) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 p[452] chain 1 i teration 1500000 1600000 1700000 0.0 0.05 0.1 0.15 p[578] chain 1 i teration 1500000 1600000 1700000 0.0 0.02 0.04 0.06 0.08 p[386] chain 1 i teration 1500000 1600000 1700000 0.02 0.04 0.06 0.08 0.1 94 Figure 2.31cont. History of values for 50,000-60,000 iterations (thinned by 30) of the proportion of children who had died (pi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with 50-200 women aged 20-29 p[26] chain 1 i teration 1500000 1600000 1700000 0.0 0.01 0.02 0.03 0.04 p[186] chain 1 i teration 1500000 1600000 1700000 0.01 0.02 0.03 0.04 p[161] chain 1 i teration 1500000 1600000 1700000 0.01 0.015 0.02 0.025 Source: 2000 Brazilian Census 95 Figure 2.32. History of values for 50,000-60,000 iterations (thinned by 30) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20- 29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[1899] chain 1 i teration 1500000 1600000 1700000 -2.0 -1.0 0.0 1.0 2.0 b[784] chain 1 i teration 1500000 1600000 1700000 -2.0 -1.0 0.0 1.0 b[2457] chain 1 i teration 1500000 1600000 1700000 -1.0 0.0 1.0 2.0 96 Figure 2.32cont. History of values for 50,000-60,000 iterations (thinned by 30) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with less than 50 women aged 20-29 b[452] chain 1 i teration 1500000 1600000 1700000 -1.0 0.0 1.0 2.0 b[578] chain 1 i teration 1500000 1600000 1700000 -1.5 -1.0 -0.5 0.0 0.5 1.0 b[386] chain 1 i teration 1500000 1600000 1700000 -0.5 0.0 0.5 1.0 1.5 97 Figure 2.32cont. History of values for 50,000-60,000 iterations (thinned by 30) of spatially structured random effects (βi) in select municipalities by sample size of women aged 20-29: Brazil, 2000. Municipalities with 50-200 women aged 20-29 b[26] chain 1 i teration 1500000 1600000 1700000 -2.0 -1.5 -1.0 -0.5 0.0 0.5 b[186] chain 1 i teration 1500000 1600000 1700000 -1.0 -0.5 0.0 0.5 b[161] chain 1 i teration 1500000 1600000 1700000 -0.8 -0.6 -0.4 -0.2 -5.55112E-17 Source: 2000 Brazilian Census 98 Figure 2.33. History of values for 50,000-60,000 iterations (thinned by 30) of education effect (e): Brazil, 2000. e chain 1 i teration 1500000 1600000 1700000 -0.1 -0.08 -0.06 -0.04 -0.02 Source: 2000 Brazilian Census Figure 2.34. History of values for 50,000-60,000 iterations (thinned by 30) of the standard deviation of β (ς.β): Brazil, 2000. sigm a.b chain 1 i teration 1500000 1600000 1700000 0.5 0.55 0.6 0.65 0.7 Source: 2000 Brazilian Census ς.β 99 Chapter 3. Results Crude Municipal-Level Estimates of Child Mortality. Crude (unsmoothed Brass) estimates of child mortality are mapped for each municipality in Figure 3.1. Of the 5,505 municipalities, 28% of the municipalities have  )5(q values of less than 0.025, 34% have values between 0.025 and 0.049, 24% have values between 0.050 and 0.099, and 4% have values of 0.100 and higher. Additionally, 10% of municipalities have insufficient data to create estimates of child mortality. In terms of spatial patterns, municipalities with higher estimates of child mortality appear to be somewhat more concentrated in the northern part of the country while municipalities with lower estimates are concentrated more in the southern areas. However, the spatial pattern is relatively weak because there is great variation in  )5(q values for municipalities in all regions of the country. 100 Figure 3.1. Municipal-level crude estimates of child mortality: Brazil, 2000. Source: 2000 Brazilian Census Further descriptive information about the crude estimates of child mortality is presented in Table 3.1. The mean  )5(q value for all municipalities is 0.043 with the Northeast demonstrating the highest value (0.062) and the South demonstrating the lowest (0.031). The variation in  )5(q values for all municipalities is quite large with a 101 range of 0.001 to 0.274, a variance of 0.00086, and a standard deviation of 0.029. Within the different regions, variation is highest for the North and lowest for the Southeast although the difference in standard deviations between the two regions is only 0.008. Table 3.1. Descriptive statistics for municipal-level crude estimates of child mortality: Brazil, 2000 Regions Brazil (n=4,971) North (n=432) Northeast (n=1,746) Southeast (n=1,485) South (n=905) Center West (n=403) Mean 0.042 0.062 0.032 0.031 0.034 0.043 Median 0.037 0.057 0.027 0.024 0.029 0.036 Minimum 0.001 0.004 0.001 0.002 0.002 0.001 Maximum 0.179 0.255 0.258 0.274 0.167 0.274 Variance 0.00057 0.00088 0.00049 0.00066 0.00050 0.00086 Standard Deviation 0.024 0.030 0.022 0.026 0.022 0.029 Missing 17 40 180 254 43 534 Source: 2000 Brazilian Census Additionally, it is evident in Table 3.1 that a sizeable number of municipalities in each region are missing estimates of child mortality. In total, 534 municipalities had no reported child deaths among women between the ages of 20 and 29, making it impossible to calculate  )5(q values for these areas. Although the proportion of municipalities with missing  )5(q values was highest in the South and Southeast, a lack of child deaths is highly associated with the population size of the municipalities. As seen in Table 3.2 the mean number of women between the ages of 20 and 29 in municipalities with no child deaths is only 62 compared with 343 for all other municipalities. Likewise, the number of 102 children born to women between the ages of 20 and 29 is also substantially lower in municipalities not reporting child deaths (65 vs. 395). Table 3.2. Descriptive information on municipalities with insufficient data for creating crude estimates of child mortality: Brazil, 2000 No child deaths (n=534) ≥ 1 child death (n=4,971) Number of women 20-29 Mean Minimum Maximum 62 8 259 343 11 101,214 Number of children born to women 20-29 Mean Minimum Maximum 65 6 289 395 10 79,690 Source: 2000 Brazilian Census As stated previously, one of the difficulties in constructing estimates of child mortality for areas with small population sizes is that the estimates tend to be unstable with a few more or a few less deaths greatly impacting estimates. To see how variation in  )5(q values relates to sample size, Figure 3.2 plots  )5(q values by the log of the number of women aged 20 to 29 sampled in each municipality. In this figure, there is much greater variability in  )5(q values in areas with smaller sample sizes with the variability decreasing as sample size increases. 103 Figure 3.2. Municipal-level crude estimates of child mortality by the log of sample of women aged 20 to 29: Brazil, 2000. Due to the strong association between child mortality and female education in Brazil, one would expect that, if municipal-level estimates of child mortality are accurate, areas with lower levels of education should exhibit higher  )5(q values while areas with higher levels of education should exhibit lower values. Figure 3.3, which plots municipal- level estimates of child mortality by the mean number of years of schooling of women aged 25 and above, does show some evidence of this relationship: child mortality levels tend to decrease as female education increases. In fact, the correlation coefficient of -0.44 is statistically significant. However, it is important to note that there is a high level of variability in estimates of child mortality at all levels of education. 104 Figure 3.3. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Brazil, 2000. Source: 2000 Brazilian Census When limiting the data to municipalities with the 10% smallest sample sizes (Figure 3.4), the relationship between child mortality and education is virtually nonexistent. Not only are the  )5(q values widely dispersed across all levels of education but the correlation coefficient (-0.08) is extremely weak and not statistically significant. In contrast, in Figure 3.5, which includes only municipalities with the largest sample sizes, the relationship between child mortality and education is strong and consistent. That is, there is clear visual evidence that higher child mortality in areas with lower levels of female education and vice-versa. Additionally, the correlation coefficient is strong (- 0.62) and statistically significant. These findings indicate, once again, that estimates of child mortality in areas with small populations may be very unreliable. 105 Figure 3.4. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% smallest sample sizes, Brazil, 2000. Source: 2000 Brazilian Census 106 Figure 3.5. Municipal-level crude estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% largest sample sizes, Brazil, 2000. Source: 2000 Brazilian Census Municipal-Level Bayesian Estimates of Child Mortality To attempt to address issues related to sample size and unstable (or missing) estimates of child mortality, I construct Bayesian estimates for each municipality, smoothing on data in neighboring areas. Figure 3.6 presents these estimates with the following breakdown: 26% of municipalities have  )5(q values of less than 0.025, 48% have values between 0.025 and 0.049, 24% have values between 0.050 and 0.099, and 1% have values of 0.100 and above. In comparison with Figure 3.1, there is much greater evidence of a spatial pattern in levels of child mortality using a Bayesian approach. Overall, there is far less variation in municipalities that are geographically close and clear and consistent evidence of elevated mortality levels in municipalities in the northern regions of the country and lower levels in municipalities in the southern regions. 107 Figure 3.6. Municipal-level Bayesian estimates of child mortality: Brazil, 2000. Source: 2000 Brazilian Census Figure 3.7 shows the spatial random effect (βi) or the deviation from the national mean for Bayesian estimates of child mortality for each municipality in Brazil. The lighter colors indicate that the spatial random effect results in a decrease in estimates of child mortality compared to the national average while the darker colors indicate an increase. This map indicates that by incorporating spatial priors into the estimates of child 108 mortality results in an increase in the  )5(q values for municipalities in the North and Northeast regions and a decrease in values for municipalities in the South, Southeast, and Center West regions. Figure 3.7. Spatial random effects (βi) on municipal estimates of child mortality: Brazil, 2000. Source: 2000 Brazilian Census 109 Table 3.3 presents descriptive statistics for municipal-level Bayesian estimates of child mortality. The mean  )5(q value for all municipalities is 0.040, a slight decrease from the mean crude value of 0.043 (Table 3.1). Likewise, the mean  )5(q values for each region are all slightly lower using a Bayesian approach. However, the overall pattern remains the same: the Northeast continues to have the highest mean value (0.061) and the South has the lowest (0.025). Most importantly, though, is that using a Bayesian approach results in substantial declines in variation in  )5(q values, indicating that estimates of child mortality are far more stable using this approach. For the country as a whole, the range in  )5(q values decreased from 0.001 to 0.274 for crude estimates to 0.013 to 0.202 for Bayesian estimates. Additionally, the variance decreased 55% from 0.0009 to 0.0004 and the standard deviation decreased 31% from 0.029 to 0.020. One of the principal advantages of using a Bayesian approach to estimate child mortality is that all municipalities now have estimates of child mortality. By “borrowing” data from neighboring areas, it is possible to create estimates of child mortality for areas in which it is not possible using only the indirect estimation approach. 110 Table 3.3. Descriptive statistics for municipal-level Bayesian estimates of child mortality: Brazil, 2000 Regions Brazil (n=4,971) North (n=449) Northeast (n=1,786) Southeast (n=1,665) South (n=1,159) Center West (n=446) Mean 0.041 0.061 0.029 0.025 0.031 0.040 Median 0.039 0.058 0.027 0.024 0.030 0.033 Minimum 0.019 0.023 0.014 0.013 0.017 0.013 Maximum 0.100 0.202 0.106 0.078 0.059 0.202 Variance 0.00012 0.00033 0.00008 0.00004 0.00005 0.00039 Standard Deviation 0.011 0.018 0.009 0.007 0.007 0.020 Missing 0 0 0 0 0 0 Source: 2000 Brazilian Census Figure 3.8 plots Bayesian estimates of child mortality by the sample size of women aged 20 to 29 in each municipality while Figure 3.11 plots the estimates by the log of the sample size. There is still considerable variation in estimates of child mortality in Brazil with greater variation among the areas with smaller numbers of women, although the level of variation is smaller when using a Bayesian approach. Compared with the crude estimates (Figure 3.2), the Bayesian estimates are more clustered around the mean. In addition, the variance is 66% smaller and the standard deviation is 31% smaller for the Bayesian methods compared with the crude estimates. 111 Figure 3.8. Municipal-level Bayesian estimates of child mortality by the log of the sample of women aged 20 to 29: Brazil, 2000. Source: 2000 Brazilian Census To assess the reliability of the Bayesian estimates of child mortality, Figures 3.9- 3.11 show Bayesian  )5(q values by the mean number of years of schooling of women aged 25 and above. Using only crude estimates of child mortality (Figures 3.3-3.5), the relationship between education and mortality tends to be fairly weak (especially in municipalities with small population sizes). However, after using a Bayesian approach to estimate child mortality, the relationship between these two variables improves substantially. Visually, it is much clearer in Figure 3.19 than Figure 3.3 that municipalities with lower female education have higher mortality levels and municipalities with higher female education have lower mortality levels. In addition, the correlation coefficient is much larger using Bayesian versus crude estimates of child mortality (-0.63 vs. -0.44). 112 Figure 3.9. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Brazil, 2000. Source: 2000 Brazilian Census However, the greatest improvement in  )5(q values occurs within municipalities with small samples of women. In Figure 3.10 the relationship between female education and child mortality is clear, both visually and via the statistically significant correlation coefficient. In fact, the correlation coefficient (-0.62) for the least populated municipalities does not differ greatly from that seen in the previous graph for all municipalities. In contrast, in Figure 3.4 which plotted  )5(q values by mean female education, there was no significant association and the correlation coefficient value was - 0.08. These findings indicate that using a Bayesian approach results in improvement in the reliability in estimates of child mortality. 113 Figure 3.10. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% smallest sample sizes, Brazil, 2000. Source: 2000 Brazilian Census Finally, Figure 3.11 plots  )5(q values by mean female education for the municipalities with the 10% largest samples of women aged 20 to 29. Overall, there is little difference between crude and Bayesian estimates when examining areas with larger sample sizes (correlation coefficients of -0.62 and -0.66, respectively), a result of the fact that the CAR model specifies that priors have a very limited effect when data are abundant and stable. 114 Figure 3.11. Municipal-level Bayesian estimates of child mortality by mean years of schooling of women aged 25 and above: Municipalities with the 10% largest sample sizes, Brazil, 2000. Source: 2000 Brazilian Census Municipal-Level Bayesian Estimates of Child Mortality and Female Education To further strengthen estimates of child mortality at the level of the municipality in Brazil, I next construct Bayesian estimates using information from neighboring areas and a covariate for female education (Figure 3.12). The breakdown of estimates of child mortality do not differ greatly from those seen in Figure 3.6: 23% of municipalities have  )5(q values of less than 0.025, 51% have values between 0.025 and 0.049, 25% have values between 0.050 and 0.099, and 1% have values of 0.100 and above. Again, there is a very clear spatial pattern in estimates of child mortality with the Northeast demonstrating the highest rates and the South/Southeast demonstrating the lowest. 115 Figure 3.12. Municipal-level Bayesian estimates of child mortality incorporating female education: Brazil, 2000. Source: 2000 Brazilian Census The spatial random effect (βi) is mapped for each municipality in Figure 3.13. By incorporating spatial autocorrelation, again, the  )5(q values increase in the Northern regions of the country and decrease in the Center West and Southern regions. 116 Figure 3.13. Spatial random effects (βi) on municipal estimates of child mortality incorporating female education: Brazil, 2000. Source: 2000 Brazilian Census Table 3.4 provides descriptive statistics of the Bayesian estimates for the country as a whole as well as for each region. Overall, there is very little change in these statistics from those in Table 3.3 (which presented results for Bayesian estimates of child mortality simulated without using the education covariate). The overall and regional mean 117 municipal-level  )5(q values are all identical, although the median values for Brazil, the North, the Northeast, and the South did increase slightly. In terms of measures of dispersion, there was no difference in the values for variance and standard deviation and very little change in the range of values. Table 3.4. Descriptive statistics for municipal-level Bayesian estimates of child mortality incorporating female education: Brazil, 2000 Regions Brazil (n=5,505) North (n=449) Northeast (n=1,786) Southeast (n=1,665) South (n=1,159) Center West (n=446) Mean 0.041 0.061 0.029 0.025 0.031 0.040 Median 0.040 0.059 0.027 0.025 0.030 0.034 Minimum 0.021 0.025 0.014 0.013 0.018 0.013 Maximum 0.960 0.203 0.103 0.078 0.058 0.203 Variance 0.00012 0.00032 0.00008 0.0004 0.0005 0.00039 Standard Deviation 0.011 0.018 0.009 0.007 0.007 0.020 Missing 0 0 0 0 0 0 Source: 2000 Brazilian Census Figure 3.14 presents the municipal-level  )5(q values by the log of the sample size of women aged 20-29. Again, the results are remarkably similar to those presented in Figures 3.10 and 3.11, with a 66% smaller variance and a 31% smaller standard deviation in  )5(q values using a Bayesian approach to estimate child mortality at the level of the municipality. 118 Figure 3.14. Municipal-level Bayesian estimates of child mortality by log of women aged 20 to 29 incorporating female education: Brazil, 2000. Source: 2000 Brazilian Census While the inclusion of education in the Bayesian model has very little impact on overall summary measures of child mortality estimates, it does have an effect on estimates for areas which differ from their neighbors in terms of female education. To illustrate this, I present findings for the municipality of Ielmo Marinho in the state of Rio Grande do Norte (Figure 3.15). Ielmo Marinho is located less than 50 kilometers from the capital of the state, Natal, with only one municipality between; yet, it is quite distinct from the capital city as well as other nearby municipalities. 119 Figure 3.15. The municipalities of Ielma Marinho and Natal: Rio Grande do Norte, Brazil, 2000. Source: 2000 Brazilian Census In terms of female education, Natal has one of the highest levels with 7.13 years, whereas Ielmo Marinho has one of the lowest with only 2.62 years (Figure 3.16). Additionally, of Ielma Marinho‟s eight contiguous neighbors, three have female education levels greater than 4 years and four have levels greater than 3 years. Only one municipality (Santa Maria) has a female education level that is similar to that of Ielma Marinho, below 3.00 years. Ielma Marinho Natal 120 Figure 3.16. Municipal-level measures of female education: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. Source: 2000 Brazilian Census Figure 3.17 displays the crude estimates of child mortality for each municipality in Rio Grande do Norte. Ielma Marinho has a  )5(q value of 0.048 which is lower than the many of the values seen in the neighboring municipalities. However, in the 2000 Census only 156 women between the ages of 20 and 29 in the municipality answered the long- form questionnaire, providing information on a total of only 232 children and 11 deaths. Thus, it is likely that the crude estimate of child mortality for this area is quite unstable ___ years = 2.63 121 and unreliable. And, in fact, the Bayesian  )5(q value for Ielma Marinho is higher (0.058) and more similar to the values in the neighboring areas (Figure 3.18). Figure 3.17. Municipal-level crude estimates of child mortality: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. Source: 2000 Brazilian Census ^ q(5) = 0.048 122 Figure 3.18. Municipal-level Bayesian estimates of child mortality: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. Source: 2000 Brazilian Census However, the spatially smoothed Bayesian estimates of child mortality for this area is constructed using only the Bayesian estimates of the other nearby areas and do not take into account the fact that Ielma Marinho appears to be different from these other areas, as exhibited by its lower levels of female education. This limitation is remedied in the final model which includes a covariate for education and, thus, borrows information from both spatial and education neighbors. Although Figure 3.19 appears to be almost ^ q(5) = 0.058 123 identical to Figure 3.18, the  )5(q value for Ielma Marinho increases to 0.062 due to the lower levels of education among women residing in the municipality. Figure 3.19. Municipal-level Bayesian estimates of child mortality incorporating female education: Ielma Marinho, Rio Grande do Norte, Brazil, 2000. Source: 2000 Brazilian Census ^ q(5) = 0.062 124 Chapter 4. Discussion Although the level of child mortality in Brazil has improved substantially in the last several decades, there are still many children throughout Brazil that are at a heightened risk of death. Those working towards further reductions of child mortality in countries such as Brazil now advocate that to effectively combat child mortality it is essential that efforts be focused more on the local levels as opposed to the level of the nation or state (Freedman et al. 2005). Consequently, the construction of reliable estimates of child mortality for small geographical areas is necessary for these efforts to move forward. It is often difficult to construct accurate estimates of child mortality for small geographical areas because population sizes also tend to be relatively small, resulting in unstable estimates. The purpose of this study was to present a method that would overcome this issue and allow for the creation of reliable and accurate estimates of child mortality at the level of the municipality in Brazil. Based upon the success of Potter et al. (2010) in mapping the fertility transition in Brazil using a Bayesian approach, I use a hierarchical Bayesian model to construct spatially smoothed estimates of child mortality. The first level of the model uses mortality data from a sample of women aged 20 to 29 from the 2000 Brazil Census. I next model the probability of a child dying among the women using a binomial model with a spatially structured random effect. The prior distribution for this random effect is constructed using a Conditional Autoregressive Model which incorporates spatial dependence among neighboring areas and allows for its impact to be greater for municipalities with weaker (more unstable) data. Hyperpriors for the prior parameters are intentionally vague so as to let the data direct the findings. 125 Due to the strong association between child mortality and female education (Caldwell 1979; Sawyer and Soares 1983; Merrick 1985; Thomas, Strauss, and Henriques 1990; Sastry 1996; Sastry 2001; Goldani et al. 2002; Alves 2003; Iyer and Monteir 2004), I next construct a hierarchical Bayesian model which incorporates municipal-level female education. Again, the first level of the model begins with the mortality data on women aged 20 to 29. I then model the probability of a child dying among the women using a binomial model with a covariate for female education and a spatially structured random effect. For this second approach, both the CAR model and its hyperprior distributions are identical to the one without education. For both approaches, I simulate a posterior distribution for the proportion of children who have died to women aged 20 to 29 using a sampling method known as the Markov Chain Monte Carlo (MCMC) and the Gibbs sampler. I then use the mean value obtained from the simulated distribution and construct estimates of child mortality for each municipality using the Brass indirect estimation approach (Brass and Coale 1968). Table 4.1 provides a summary of the different measures of child mortality presented in this study 8 . While the mean and median  )5(q values for the crude and Bayesian estimates are quite similar, the mean and median UNDP child mortality rates are quite a bit lower. This is likely due, in large part, to the fact that areas of lower socioeconomic status and higher child mortality have less adequate registrations of births and deaths (Szwarcwald 2008). In terms of confidence in the estimates (i.e., the stability of the estimates), the Bayesian estimates are much more stable compared with UNDP rates in terms of the range in values and more stable compared to the crude 8 For purposes of comparability, the UNDP child mortality rates are divided by 1,000. 126 estimates in terms of the range, variance, and standard deviation. Although stability of estimates is important, reliability is equally important and this is where the true advantage of the Bayesian estimates becomes apparent. While the correlation between education and Bayesian  )5(q values is -0.63, the correlation is much lower for crude  )5(q values (-0.44) and UNDP rates (-0.26) 9 . And, finally, it is very important to note that there are 534 municipalities with missing crude estimates of child mortality and 738 municipalities with missing UNDP child mortality rates, but there are no municipalities with missing Bayesian estimates of child mortality. Table 4.1. Descriptive statistics for estimates of child mortality using crude and Bayesian methods and UNDP child mortality rates: Brazil, 2000 UNDP child mortality rate a Crude  )5(q values Bayesian  )5(q values Bayesian  )5(q values b All Municipalities Mean 0.030 0.043 0.040 0.040 Median 0.026 0.036 0.033 0.034 Minimum 0.002 0.001 0.013 0.013 Maximum 0.313 0.274 0.202 0.203 Variance 0.0004 0.0009 0.0004 0.0004 Standard Deviation 0.021 0.029 0.020 0.020 Correlation with education -0.26** -0.44** -0.63** N/A Missing 738 534 0 0 Sources: Brazil Census 2000, Portal ODM 2009 a To make measures comparable – the child mortality rate is converted into deaths per 1 live birth: (Deaths to children under the age of 5 per 1,000 live births)/1,000 b With a covariate for mean female education * p<0.05 ** p<0.01 9 Due to the fact that the final Bayesian model incorporates female education when constructing estimates of child mortality, it would be erroneous to compare the correlation between education and child mortality for these estimates. 127 While the Bayesian estimates appear to be far more stable and accurate, incorporating female education into the hierarchical Bayesian model has very little impact on the overall estimates. However, the real benefit of this adaptation can be felt in individual municipalities whose level of education differs from that of nearby municipalities. In such cases, by incorporating education into the construction of estimates of child mortality, these differences are recognized and more reliable estimates can be obtained. The Case of Ceará The effectiveness of utilizing a Bayesian approach to estimate child mortality can be demonstrated by examining a region with mortality rates which differ from the larger region of the country. In this case, I chose to evaluate the state of Ceará, located in the Northeast region of the country. Ceará has traditionally had very high child mortality, but due to the implementation of policy initiatives directed specifically at its population needs, the state achieved a 32% reduction in child mortality between 1987 and 1998 (Fuentes and Niimi 2002). Additionally, a number of its municipalities have been recognized by UNICEF for substantial improvements in their health situation (UNICEF 2001). Yet, when attempting to measure the level of child mortality in the municipalities throughout the state (Figure 4.1), there is a wide range of estimates with no clear spatial pattern. Figure 4.1 also provides results on which municipalities were awarded the UNICEF Seal of Approval for improvements in their local areas, indicated by a bold black border. It is striking that there appears to be very little relationship between being awarded this honor and child mortality with these municipalities exhibiting both the highest and lowest levels of child mortality. This is likely due in part to the fact that many 128 of the municipalities have small populations resulting in unstable estimates (more than half of all municipalities had less than 200 women aged 20 to 29 sampled for the 2000 Brazil Census). 129 Figure 4.1. Municipal-level crude estimates of child mortality: Ceará, 2000. Source: 2000 Brazilian Census 130 In contrast, Figures 4.2 and Figure 4.3, which present Bayesian estimates of child mortality for Ceará have much clearer spatial patterns. There are overall fewer extreme values (very high or low levels of child mortality) and there is clear indication that levels of child mortality are lowest in the Northeastern region of the state. While the municipalities that were awarded the UNICEF Seal of Approval do not always exhibit the lowest estimates of child mortality, there is much more consistency across them and no extreme values. These findings, again, demonstrate the benefit of utilizing a Bayesian approach. 131 Figure 4.2. Municipal-level Bayesian estimates of child mortality: Ceará, 2000. Source: 2000 Brazilian Census 132 Figure 4.3. Municipal-level Bayesian estimates of child mortality incorporating child mortality: Ceará, 2000. Source: 2000 Brazilian Census 133 As illustrated in the above example, the Bayesian approach is much more accurate in identifying levels of mortality that are reflective of the true mortality situation of a community as well as outcomes of community actions. It is my hope and goal that these estimates (and this method) can be employed by those working towards improvements in child mortality to effectively understand how child mortality varies at the local level, eventually resulting in strategies that will lead towards continued improvements in child mortality rates throughout all areas of Brazil. However, Brazil is not a unique situation and there are many countries worldwide that struggle with measuring and addressing child mortality. It is my recommendation, then, that this method be employed in developing countries throughout the world in order to gain a greater understanding of the mortality conditions of their children and the best approaches for working towards improvements. 134 References AlMazrou, Y. Y., M. U. Khan, and K.M.S. Aziz. 1997. “Determinants of under-five mortality in Saudi Arabia.” Saudi Medical Journal. 18(1): 31-36. Alves, D. 2003. “Factors Explaining Infant Mortality in Brazil: 1991-2000.” Paper presented at the Fifty-fifth International Atlantic Economic Conference, Vienna, Austria. Andoh, S. Y., M. Umezaki, K. Nakamura, M. Kizuki, and T. Takano. 2007. “Association of household demographic variables with child mortality in Cote d'Ivoire.” Journal of Biosocial Science. 39(2):257-265. Anselin, L., I. Syabri and Y. Kho 2006. GeoDa: An Introduction to Spatial Data Analysis. Geographical Analysis 38 (1), 5-22. Arató, N. M., I. L. Dryden, and C. C. Taylor. 2006. “Hierarchical Bayesian modeling of spatial age-dependent mortality.” Computational Statistics and Data Analysis. 51:1347-1363. Assunção, R. M., C. P. Schmertamann, J. E. Potter, and S. M. Cavenaghi. 2005. “Empirical Bayes Estimation of Demographic Schedules for Small Areas.” Demography 42(3): 537-558. Basu, A. M. and R. Stephenson. 2005. “Low levels of maternal education and the proximate determinants of childhood mortality: a little learning is not a dangerous thing.” Social Science & Medicine. 60(9):2011-2023. Bayes, T. 1763. “An Essay towards solving a Problem in the Doctrine of Chances.” Philosophical Transactions of the Royal Society of London. 53:370–418. Bernadinelli, L. and C. Montomoli. 1992. “Empirical Bayes Versus Fully Bayesian Analysis of Geographical Variation in Disease Risk.” Statistics in Medicine 11:983–1007. Besag, J., J. York, and A. Mollié. 1991. “Bayesian Image Restoration with Two Applications in Spatial Statistics.” Ann. Inst. Statis. Math. 43(1):1-59. Brass, W and A. J. Coale. 1968. “Methods of Analysis and Estimation.” Pp. 88-139 in The Demography of Tropical Africa, edited by W. Brass, A. J. Coale, P. Demeny, D. F. Heisel, F. Lorimer, A. Romaniuk, and E. T. Van de Walle. Princeton, NJ: Princeton University Press. Brooks, S. P. and A. Gelman. 1998. “General Methods for Monitoring Convergence of Iterative Simulations.” Journal of Computational and Graphical Statistics. 7(4): 434-455. 135 Caldwell, J. C. 1979. “Education as a factor in mortality decline: An examination of Nigerian data.” Population Studies. 33(3):395-413. Caldwell, P. 1996. “Child survival: physical vulnerability and resilience in adversity in the European past and the contemporary Third World.” Social Science and Medicine. 43:609-619. Carvalho, J. A.. 1974. “Regional Trends in Fertility and Mortality in Brazil.” Population Studies. 28:401-21. Casterline, J. B., E. C. Cooksey, and A. F. Ismail. 1992. “Infant and Child-Mortality in Rural Egypt.” Journal of Biosocial Science. 24(2):245-260. Castro, M. C. and C. Simões. 2010. Spatio-Temporal Trends of Infant Mortality in Brazil. Paper presented at the annual meeting of the Population Association of America, Dallas, TX, April 15–17. Clayton, D. G., L. Bernardinelli, and C. Montomoli. 1993. :Spatial Correlation in Ecological Analysis.” International Journal of Epidemiology.22(6): 1193-1202. Clayton, D. G. and J. Kaldor. 1987. “Empirical Bayes Estimates of Age-Standardized Relative Risks for Use in Disease Mapping.” Biometrics. 43: 671-681. Cleveland, W. S. 1979. "Robust Locally Weighted Regression and Smoothing Scatterplots". Journal of the American Statistical Association. 74 (368): 829–836. Cleveland, W. S. and S. J. Devlin. 1988. "Locally-Weighted Regression: An Approach to Regression Analysis by Local Fitting". Journal of the American Statistical Association. 83 (403): 596–610. Coale, A. J. and P. Demeny. 1983. Regional Model Life Tables and Stable Populations. 2nd ed. New York, NY: Academic Press. Coale, A. J. and T. James Trussell. 1974. “Model Fertility Schedules: Variations in The Age Structure of Childbearing in Human Populations.” Population Index. 40(2): 185-258. Curtis, S. L., I. Diamond, and J. W. MacDonald. 1993. “Birth interval and family effects on Postneonatal Mortality in Brazil.” Demography. 30(1):33-43. Demographic and Health Survey (DHS). 1996. Brazil: Standard DHS, 1996. Available on-line at http://www.measuredhs.com/aboutsurveys/search/metadata.cfm?surv_id=85&ctry _id=49&SrvyTp=country. Denslow, D. and W. Tyler. 1984. “Perspectives on poverty and income inequality in Brazil.” World Development. 12(10):1019-1028. 136 Eberstein, I. W. 1989. “Demographic Research on Infant Mortality.” Sociological Forum. 4(3): 409-22. Esri Inc. 2008. Esri ArcMap 9.3. Redlands, Ca: Esri Inc. Freedman, L. P., R. J. Waldman, H. de Pinho, M. E. Wirth, A. M. R. Chowddhury, and A. Rosenfield. 2005. Who's got the power? Transforming health systems for women and children: Achieving the Millennium Development Goals. Sterling, VA: Earthscan. Fuentes, P. and R, Niimi. 2002. “Motivating municipal action for children: the Municipal Seal of Approval in Ceará, Brazil.” Environment & Urbanization. 14(2):123-133. Gilks, W. R., S. Richardson, and D. J. Spiegelhalter. 1996. Markov Chain Monte Carlo in Practice. New York, NY: Chapman and Hall. Goldani M. Z., R. Benatti, A. A. M. da Silva, H. Bettiol, J. C. W. Correa, and M. Tietzmann, M. A. Barbieri. 2002. “Narrowing inequalities in infant mortality in Southern Brazil.” Revista de Saude Publica. 36(4):478-83 Haining, R. P. 2003. “Exploratory spatial data analysis: numerical methods.” Pp. 226-270 in Spatial Data Analysis: Theory and Practice. New York, NY: Cambridge University Press. Heaton, T. B. and R. Forste. 2003. “Rural/urban differences in child growth and survival in Bolivia.” Rural Sociology. 68(3):410-433. Hudson Rex A., ed. 1997. Brazil: A Country Study. Washington, DC: Library of Congress. Available on-line at http://countrystudies.us/brazil/24.htm. Instituto Brasileiro de Geografia e Estatística (IBGE). 2009a. Síntese de Indicadpres Sociais: Uma Análise das Conições de Vida da Populção Brasileira, 2009. Available online at http://www.ibge.gov.br/home/estatistica/populacao/condicaodevida/indicadoresmi nimos/sinteseindicsociais2009/indic_sociais2009.pdf. Instituto Brasileiro de Geografia e Estatística (IBGE). 2009b. Revisão das Tábuas Completas de Mortalidade. Available online at http://www.ibge.gov.br/home/estatistica/populacao/ tabuadevida/2002/default_revisao.shtm. Instituto Brasileiro de Geografia e Estatística (IBGE). 2004. Pesquisa de Informações Básicas Municipais. Available online at http://www.ibge.gov.br/home/estatistica/ economia/perfilmunic/default.shtm Inter-agency Group for Child Mortality Estimation. 2010. Mortality rate, under-5 (per 1,000): Brazil. Available online at http://data.worldbank.org/indicator/SH.DYN.MORT. 137 Iyer, S. and M. F. G. Monteiro. 2004. “The Risk of Child and Adolescent Mortality among Vulnerable Populations in Rio de Janeiro, Brazil.” Journal of Biosocial Science. 36:523-46. Kafadar, K. 1994. “Choosing among two-dimensional smoothers in practice.” Computational Statistics & Data Analysis. 18: 419-439. Lawson, A. B. 2009. Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology. New York, NY: CRC Press. Lunn, D. J., A. Thomas, N. Best, and D. Spiegelhalter. 2000. “WinBUGS -- a Bayesian modelling framework: concepts, structure, and extensibility.” Statistics and Computing, 10:325--337. Lynch, S. M. 2007. Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. New York, NY: Springer. Majumder, A. K. and S. M. Shafiqul Islam. 1993. “Socioeconomic and environmental determinants of child survival in Bangladesh.” Journal of Biosocial Science. 25(3):311-8. Marshall, R. J. 1991. “Mapping Disease and Mortality Rates Using Empirical Bayes Estimators.” Applied Statistics. 40:238-94. Merrick, T. W. 1985. “The Effect of Piped Water on Early Childhood Mortality in Urban Brazil, 1970 to 1976.” Demography. 22:1-24. Mogford, L. 2004. “Structural determinants of child mortality in sub-Saharan Africa: A cross-national study of economic and social influences from 1970 to 1997.” Social Biology. 51(3-4):94-120 Moser, K. A., D. A. Leon, and D. R. Gwatkin. 2005. “How does progress towards the child mortality millennium development goal affect inequalities between the poorest and least poor? Analysis of Demographic and Health Survey data.” BMJ. 331:1180-1182. Mosley, H. and L. Chen. 1984. “An Analytical Framework for the Study of Child Survival in Developing Countries.” Population and Development Review. 10(supplement):25-48. Omran, A. R. 1971. "The Epidemiologic Transition: A Theory of the Epidemiology of Population Change." Milbank Memorial Fund Quarterly. 49(4):509-538. Portal ODM. 2009. Portal ODM – Acompanhamento Municipal dos Objetivos de Desenvolvimento do Milênio. Available on-line at http://www.portalodm.com.br/. 138 Potter, J. E., C. P. Schmertmann, R. M. Assunção, and S. M. Cavenaghi. 2010. “Mapping the Timing, Pace and Scale of the Fertility Transition in Brazil.” Population Development Review. 36(2): 283–307. Pradhan, J. and P. Arokiasa. 2006. “High infant and child mortality rates in Orissa: An assessment of major reasons.” Population Space and Place. 12(3):187-200. Preston, S. H., P. Heuveline, and M. Guillot. 2001. Demography: Measuring and Modeling Population Processes. Malden, MA: Blackwell Publishing. SAS Institute Inc. 2008. SAS: Version 9.2. Cary, NC: SAS Institute Inc. Sastry, N. 1996. “Community characteristics, individual and household attributes, and child survival in Brazil.” Demography. 33(2):211-229. Sastry, N. 1997. “What explains rural-urban differentials in child mortality in Brazil?” Social Science and Medicine. 44(7):989-1002. Sastry, N. 2004. “Trends in socioeconomic inequalities in mortality in developing countries: The case of child survival in São Paulo, Brazil.” Demography. 41(3):443-464. Sastry, N. and S. Burgard. 2005. “The Prevalence of Diarrheal Disease Among Brazilian Children: Trends and Differentials from 1986 to 1996.” Social Science & Medicine. 60(5):923–935. Sawyer, D.O. and E.S. Soares. 1983. “Child mortality in different contexts in Brazil: variation in the effects of socio-economic variables.” Pp. 145-160 in Infant and Child Mortality in the Third World. Paris: CICRED. Shaddick, G. 2008. Hints on using WinBUGS. Available online at http://www.stat.ubc.ca /~gavin/WinBUGSdocs/WinBUGS%20hints.pdf. Silva, P. L.2005. “Reporting and compensating for non-sampling errors for surveys in Brazil: current practice and future challenges.” Pp. 231-248 in Household Sample Surveys in Developing and Transition Countries. New York: United Nations. StataCorp. Stata Statistical Software: Release 10. College Station, TX: StataCorp LP; 2007. Sullivan, J. M. 1972. “Models for the Estimation of the Probability of Dying between Birth and Exact Ages of Early Childhood.” Population Studies. 26(1): 79-97. Szwarcwald, C. L. 2008. “Strategies for improving the monitoring of vital events in Brazil.” International Journal of Epidemiology. 37:738-744. 139 Talbot, T. O., M. Kulldorff, S. P. Forand, and V. B. Haley. 2000. “Evaluation of aptial filters to create smoothed maps of health data.” Statistics in Medicine. 19: 2399- 2408. Thomas, A., N. Best, D. Lunn, R. Arnold, and D. Spiegelhalter. 2007. GeoBUGS User Manual, Version 1.3. Available on-line at http://mathstat.helsinki.fi/openbugs/Manuals/ GeoBUGS/Manual.html. Thomas, D., J. Strauss, and M-H. Henriques. 1990. “Child survival, height for age and household characteristics in Brazil.” Journal of Development Economics. 33: 197- 234. United Children‟s Fund (UNICEF). 2001. “Child Friendly Cities." Available on-line at http://www.childfriendlycities.org/. United Children‟s Fund (UNICEF). 2006. “Progress for Children: A Child Survival Report Card." Available on-line at http://www.childinfo.org/mortality_ufmrcountrydata.php. United Nations Development Programme (UNDP). 2003. Human Development Report 2003. New York, NY: Oxford University Press, 2003. U.S. Census Bureau, International Data Base. 2010. International Data Base: Brazil. Available on-line at http://www.census.gov/ipc/www/idb/country.php. Victora, C. G. and F. C. Barros. 2001. “Infant mortality due to perinatal causes in Brazil: trends, regional patterns and possible interventions.” Sao Paulo Medical Journal. 119(1):33-42. Victora, C. G., A. Wagstaff, J. Armstrong Schellenberg, D. Gwatkin, M. Claeson, and J- P. Habicht. 2003. “Applying an equity lens to child health and mortality: more of the same is not enough.” The Lancet. 362:233-241. Wagstaff, A. 2000. “Socioeconomic inequalities in child mortality: comparisons across nine developing countries.” Bulletin of the World Health Organization. 78(1):19- 29. Waller, L.A. and C. A. Gotway. 2004. “Visualizing Spatial Data.” Pp. 68-117 in Applied Spatial Statistics for Public Health Data. Hoboken, N.J: John Wiley & Sons. Wang, L. 2003. “Determinants of child mortality in LDCs. Empirical findings from demographic and health surveys.” Health Policy. 65: 277-299. World Health Organization (WHO). 2006. “World Health Statistics 2005." Available on- line at http://www.who.int/healthinfo/statistics/mortchildmortality/en/index.html. World Bank. 2007. “World Development Indicators 2007.” Available on-line at http://go.worldbank.org/3JU2HA60D0. 140 Zhang, L. 2008. Bayesian Data Analysis Using %WinBUGS. SAS Conference Proceedings: PharmaSUG, Atlanta, Georgia. 141 Vita Sarah Ann McKinnon was born in Northampton, Massachusetts on April 15, 1975 to Kathleen O‟Rourke and Donald McKinnon. In 1997, she received her received her Bachelor of Arts degree in Anthropology from the University of Texas at El Paso and, in 2000, she received a Masters of Public Health degree from the University of Texas Houston Health Science Center. In August 2003, she entered the Graduate School at the University of Texas at Austin. Permanent address: 900 Aurora Circle, Austin, Texas 78757 This dissertation was typed by the author.