EOOOEMETha
The Library of The University of Texas
A FACTOR ANALYSIS OF SOME MEASURES OF SILENT READING ABILITY
T 9 AN ORIGINAL MANUSCRIP* THIS IS AM wtthOUT IT MAY NOT US '.IiHUUx THE AUTHOR ’ S PEHi’il SSI ON
Approved;
iTHIS IS AN ORIGINAL MANUSCRIPT IT MAY NOT IC CCPIID WITHOUT THE AUTHOiCS PERMISSION
A FACTOR ANALYSIS OF SOME MEASURES OF SILENT READING ABILITY
THESIS
Presented to the Faculty of the Graduate School of The University of Texas in Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Jesse Edward Franklin, B. A., M. A. (Floresville, Texas) Austin, Texas June, 1935
Library University of Texas Austin, Texas
To My Wife
FRANCES SAMPLE FRANKLIN
whose assistance in scoring the tests and in the statistical work has contributed; but, above all, whose constant encouragement and sacrifice has made this research possible, and to
MY FIVE DAUGHTERS
who are my inspiration and joy.
PREFACE
In presenting this study, the writer gratefully acknowledges his indebtedness to Dr. C. T. Gray for suggestion of the problem, for assistance in preparation of the manuscript, and for constant encouragement in the progress of the research. For suggestions as to the selection of tests and for constructive criticism of the manuscript, I am indebted to Dr. H. T. Manuel.
Gratitude is expressed to Dr. B. F. Holland for photographing the eye-movements and for other help with the laboratory work, and to Dr. L. A . Jeffress and Dr. Leigh Peck for constructive criticisms of the manuscript.
Acknowledgement is also made to Dr. I. I. Nelson, Principal, and to Misses Mary Cook and Annie Lee Durham, teachers of reading, in the University Junior High School, for their cooperation, and to the boys and girls who served as subjects.
To all the members of the Supervisory Committee, I wish to express my gratitude for suggestions and encouragement.
J. E. FRANKLIN
Austin, Texas June, 1935
TABLE OF CONTENTS
Chapter Page
I Introduction and Statement of the Problem 1
II The Theory of Factors. ..... 7
111 Subjects, Materials, and Testing Program 32
IV Statistical Procedures ..... 53
V Preliminary Study of the Test Results 60
VI Thurstone’s Simplified Mdthod of Multiple Factor Analysis 73
VII Results of Factor Analyses. . . 100
VIII Summary ..... 129
8ib1i0graphy.......... 132
LIST OF TABLES
Table Page
I Description of Test Results 61
II Intercorrelations of the Twenty- four Tests and Measures .... 72
111 Calculation of First Factor Weightings, Analysis II ... . 97
IV Calculation of First Factor Residuals, Analysis 11. ... . 98
V Calculation of Second Factor Weightings, Analysis II ... . £9
VI Factor Weightings, Analysis I. . . . 101
VII Factor Weightings, Analysis II . . . 105
VIII Factor Weightings, Analysis 111. . . 107
IX Factor Weightings, Analysis IV . . . 109
X Factor Weightings, Analysis V. . . . 11l
XI Factor Weightings, Analysis VI . . . 113
XII Results of the Six Analyses 128
XIII Weightings of Factor ”A H 120
XIV Comparative Weightings, Factor ”B” . 123
XV Comparative ’Weightings, Factor ”0° . 125
LIST OF FIGURES
Figure Page
I Distribution of the Scores on Tests which Were Combined in Producing the Four Composite Measures 56
II Geometrical Representation of the Correlation between Two Tests 80
CHAPTER I INTRODUCTION AND STATEMENT OF THE PROBLEM
Library University of Texas Austin, Texas
One of the more recently developed statistical techniques which should prove useful in arriving at a better understanding of many psychological problems is the multiple factor analysis. While the theory of factors has occupied the attention of a number of psychologists for more than a quarter of a century, and while their work has resulted in the advancement of our knowledge of mental abilities, it is only recently that developments in the multiple factor technique have begun to give promise of a much wider application of the concepts of factor analysis in a way which should prove most helpful in analyzing a variety of psychological and educational problems.
One of the problems which has engaged the attention of investigators for many years, and which has apparently delayed the development of more adequate concepts for mental organization, has been the seeming impossibility of isolating mental abilities for study. Since we can-
not isolate mental functions, our procedure has been to measure abilities as they operate as a part of the total mental organization in a way that allows us to see, at least in part, their relation to total mental organization. The correlation technique and other statistical devices have been invented as helpful tools in studying such relationships.
The factor analysis technique seems to supplement all other devices in this field of study and apparently yields valuable information concerning relationships.
The primary datum of factor analysis is a table of intercorrelations between test scores, or other variables. The factor analysis yields a simpler pattern of relationships, which is more easily comprehended than are the complexities offered by the table of intercorrelations. A factor analysis provides the answer to two important questions: (1) how many factors must be postulated in order to account for the observed correlations between the tests, or measures; and (2) how much of each factor is represented by each of the tests or measures. Thus, if we can account for the variations measured by a group of tests by postulating one, two, three, or four factors, or abilities operating to cause the variations, and can measure the amount of variation in each measure caused
by each of the factors postulated, then we can think more clearly concerning the nature and organization of the processes involved.
It would appear that a technique for thus analyzing relationship should be utilized in searching for a better understanding of all our present psychological and educational problems. It will, doubtless, be applied in the near future to many types of measurements where relationships are vital problems.
One of the most important fields in which it has not as yet been applied is the individual school subject. Since a study of this type should prove helpful in gaining a better understanding of the psychology of a particular subject, we are here proposing such procedure.
The school subject which would seem to deserve primary attention among such studies is that of reading. Reading is not only the most important school subject, in that the ability to read is fundamental to all other school subjects; but it is the subject which occupies more time in the school program and is utilized more universally after leaving school than any other one subject.
Reading also lends itself more readily to such a study than does any other school subject. Much experimental work has been done in the field of reading and a TxU large variety of tests and measures of reading ability have been devised. The setting seems ideal for such an investigation in the field of reading ability. Hence we propose to study reading ability by means of the factor analysis technique.
A consideration of the original proposal of our problem as ”A Factor Analysis of Reading Ability,” immediately suggested the necessity of delimitations. -Reading ability includes two major divisions, in each of which the elements of speed, purpose, and other general characteristics are far from the same. These divisions are oral and silent reading. Since measures of the two types of reading would hardly be comparable, it was necessary to study either oral or silent reading. Silent reading is the type most frequently employed by most persons, and instruction in the schools is largely for that type of reading. In consideration of these facts and realizing that most of our standardized reading tests are teste of silent reading, our first delimitation was the addition of the adjective ”silent” in our proposed subject. Hence no consideration is given in the study to oral reading as such. No attempt was made, however, to note or to control the amount of lip-movement or of sub-vocal oral reading engaged in by the subjects
in the test situations. We use the term ’’silent reading 1 * with the usual connotations.
The need of further delimitation is imposed by the technique to be employed. That is to say, the factor analysis technique is a statistical tool to be used in analyzing a table of intercorrelations into specific, group, and general factors which are characteristic of the measures represented in that particular table of intercorrelations. Hence our analysis is not of silent reading ability; but rather an analysis of certain measures of the ability.
While standardized reading tests and other recognized procedured for the measurement of reading are generally conceded to yield valid measures of reading ability, it should be borne in mind that the factors found by an analysis technique are statistical devices derived for the purpose of analyzing the table of intercorrelations into smaller number of common elements, rather than measures of any psychological or physiological characteristics of the reading process itself.
¥le might speculate concerning the nature of reading ability and the characteristics which cause one measure to show a high positive weighting of a certain factor while another test has a negative weighting of the same factor. It might also be interesting to postulate names and characteristics for the various factors which are found in a factor analysis of the different measures of reading. The factors, however, are not realities; but are statistical elements out of which can be formed a more simplified pattern of thinking about the complexities which are represented by the table of intercorrelations. It is possible, however, that such considerations of the more simplified patterns found by the analysis of the measures of reading may lead to a realignment of the principal factors in terms of which we think of reading ability.
Hence our present statement of the problem is ”A Factor Analysis of Some Measures of Silent Reading Ability.”
The techniques employed in the study are not original. Any original contribution in the present study is only in the utilization of the factor analysis technique in the study of a field to which it has not as yet been applied.
CHAPTER II THE THEORY OF FACTORS
The theory of factors was introduced by
in 1904, and most of the literature on the factor problem has revolved about his concept of the single common factor, "g”, and his two*factor method. During the thirty years which have elapsed since the theory of factors was introduced, valuable contributions have been made to the theory, and new statistical devices have been invented for the determination and measurement of specific, group, and general factors. This chapter presents a brief review of the major contributions in the field of factor analysis.
The first quantitative experimental evidence for the existence of a general ability underlying intellectual o activity was presented by Spearman in a published in-
vestigation of the relation between sensory discrimination and intellectual abilities. As a result of his finding a high positive correlation between the measures of "general sensitivity” and the measures of ’’general intelligence,” he drew the conclusions which have formed the basis of his theory of factors. The conclusions were: (a) “Whatever branches of intellectual activity are at all dissimilar, then their correlations with one another appear wholly due to their being all variously saturated with some common fundamental function (or group of functions)" and (b) "All branches of intellect-
ual activity have in common one fundamental function (or group of functions), whereas the remaining or specific elements of the activity seem in every case to be wholly 4 different from that in all the others.” While Spearman
reserved his discussion as to the physical nature of this fundamental function until a more objective study of its relation could be made, he maintained that, "as an imp- ortant practical consequence of this universal Unity of the Intellectual Function, the various actual forms of mental activity constitute a stable interconnected ’Hierarchy according to their different degrees of intellective saturation,”
In a subsequent investigation, Spearman and Krueger
verified the findings of Spearman’s former experiment. The concept of ’’broad” or group factors which were common to some of the abilities measured, but not common to all, also grew out of this study.
7 In 1912, Spearman and Hart considered the results
obtained and the theories of mental organization held by several other psychologists. They pointed out the in- effectiveness of determining the presence of a hierarchy in a table of correlations by the method suggested in Spearman’s first paper—that is, by the method of mere subjective estimation. By hierarchy in a table of correlation is meant that the variables can be so arranged that “the values of the coefficients are highest in one corner, and thence gradually diminish in both vertical and horizontal directions, the diminution of every column or row being in the same porportion.“
Because of the ineffectiveness of this method of determining the presence of a hierarchy in a table of coefficients of correlation, Spearman and Hart introduced the intercolumnar correlation coefficient as a criterion for hierarchy. This criterion consisted simply in singling out from a table of correlations, various pairs of columns or coefficients and finding the correlation between the two series of correlation coefficients constituting each pair. In order to prove the presence of a hierarchy, and hence the assumption of “ft General Common Factor, the correlation between columns should Q be positive and very high*” TxU
In this same paper, Spearman and Hart presented the tetrad equation, which later became the fundamental criterion of hierarchy, but it was not used until much later, ’’because they did not possess an expression for its probably error. This equation, which has come to
play so important a part in Spearman’s two-factor theory fend in all factor analysis based on tetrad differences, 11 was deduced by Burt from an equation previously used by Kreuger and Spearman.above;
The tetrad equation is
Tor - r bp
when a, b, p and q represent any four variables or tests. By the use of partial correlation formulae, Spearman and Hart furnished a simple proof of the equation and of the fact that when the equation holds good the correlation between the pairs of columns in a table of correlation coefficients becomes equal to 12 unity. While the converse proposition of the above
General Factor B g” is here interpreted in psychological terms as H some common fund of intellectual energy,” and that ”eyery intellectual act appears to involve, both the specific activity of a particular system of cortical neurones, and also the general energy of the whole equation (that is, if (1) is true, general factors "g" and specific factors "s” alone determine the observed variables) might be considered to be assumed in this article, it was not definitely stated and proved until later. It is interesting to note in passing that the
cortex.
A concise statement of the intercolumnar correlation criterion for hierarchy and its application to test data compiled by Simpson, was published by Spearman in 1914. ~ This new criterion was used to show the presence
of a hierarchy, which Simpson, through inspection, had failed to observe.
The first effective challenge of Professor Spearman’s position—that if a hierarchy can be formed in a table of intercorrelations the existence of a'General 16 said to be proved—was made by Thomson in 1916. By
means of correlations derived from a system of dice, Thomson showed that an excellent hierarchy could be
381846
made with specific and group factors only, without the presence of a general factor.
17 Spearman objected that Thomson’s dice arrangements
y e / were arbitrary and extremely improbable of chance, Thomson agreed to Spearman’s objection, but maintained that the production of a hierarchy in a situation knowm to contain no general factor, brought into serious question the use of hierarchy as a proof of the existence of the general factor. Thomson’s contention is that other statistical devices are needed to determine whether correlation is caused by a general factor or by overlapping factors.
In later papers, Thomson lB pl§duced further evidence,
by the use of dice, that hierarchies could be produced without the presence of a general factor. In view of the conclusion that the statistical devices did not exclude other explanations of mental organization, Thomson proposed the "sampling theory" of ability as an explanation of observed hierarchies: "The mind, in carrying out any activity such as a mental test, has two levels at which it can operate. The elements of activity at the lower level are entirely specific, but those at the higher level are such that they may come into play in different activities. Any activity is a sample of these elements. The elements are assumed to be additions like dice, and each to act on the ’all or none’ principle, not being in fact further 20 divisible." By this theory a form of behaviour is
determined by a sample or pattern of neurones, and correlation is due to overlapping samples of neurones. Instead of two forms of factors, "g" and "s", there may be any number of combinations varying from the "s" to the "g".
Thomson’s work elicited quite a wealth of controversial matter, but its main value seems to have been that it led the adherents of the two-factor theory to assume no longer that the converse of the original proposition was true and a proof of the converse was produced. It had previously been assumed that a hierarchy existed if ”g” and ”s” alone determined the intercorrelations, but no proof had been offered as to the converse. Thompson’s data questioned the validity of 21 2? the assumption. Garnett end Spearmanproved the converse
theorem for normally distributed variables. They established the proposition that if the tetrad equation is satisfied, only ”g” and ”s” are present and there are no overlapping group factors. Spearman later produced a
proof making the converse theorem applicable to any distribution.
24 Garnett" next studied the relations existing
between the factors "g" and "s" and gave a clear explanation of Thomson* s dice data. He showed that both types of factors were consistent with the facts of equiproportion or hierarchy and that by analysis the "g" and "s" could be obscured to such an extent that factors such as found by Thomson would appear. OR Spearman later pointed out several ways, including that
of dice, by which the "g" could be transformed and its presence concealed.
Garnett also pointed out that there are unlimited numbers of ways in which the variables may be separated into "g" and "s" aid that to obtain a unique "g" and "s", there must be at least four variables.
In a study of group factors, introduced
the “cosine law” and the fundamental system of equations for the theory of direction cosines in n-dimensional space. This geometric interpretation of correlation, whereby any normal variable can be expressed as a vector by coordinates mutually at right angles in space, has been utilized more recently by Thurstone in developing the technique of multiple factor analysis which will be discussed later.
As was noted earlier in the review, the tetrad equation was used only as an approximate criterion of hierarchy, due to the fact that no formula was available to measure its probable error. The deflation of 27 such a formula, in 1924, by Spearman and Holzinger,and
the large number of formulae which have since been derived have made for accuracy in the use of the tetrad
equation, and have established the tetrad difference as a valuable tool in rigorous factor analysis.
The tetrad difference, n F H , is stated "by the equation
• - r^ ~ o (2)
Its probable error was first stated as
“* 3 *f +4 j + V (i~r^ Z (i~^) Z J
This long formula is for the probable error of a single tetrad. A table of 14 variables would necessitate the calculation of 3,003 such probable errors. In order to provide a more usuable method of measuring the probable 28 error, Spearman and Holzinger proposed a new approximate
formula which deals simultaneously with all the tetrad differences derived from the whole set of correlatives. The formula for the probable error of the distribution of tetrad difference is
(4)
where Px r being the average of all the cor- 52 - the average squared deviation from f , and ft ~ the numbers of individuals entering into the correlations.
pq Spearman states that ’the square of this probable
error can easily be shown to equal the mean of the squared probable errors of all the tetrad differences taken separately?
While there is an abundance of literature on controversial matters as to the application of the factor theory and the psychological interpretation of ”g H and “s”, the advocates of the theory have made no significant change in their position or method of proof of the divisibility of intellectual activities into two factors.
Spearman’s attitude toward the more recent procedure of dividing variables into a number of factors, rather than his two, is that, although such division is possible, it is not as meaningful as his simple division into H g” and ”s”. He says, "These two factors, ’g‘
and ’s’, have shown themselves to be the divisions about which the most important statements and discoveries can be made. M
31 Again, Spearman states that "all such large
multiplications of factors in an ability does not arise from renouncing its primary and fundamental bisection into the original two factors, universal I g’ and nonuniversal ’s’; it only comes from submitting the nonuniversal factor to further and secondary divisions. Moreover, this second sub-division, unlike the primary bisection, is unstable. All these sub-divisions of an ability depend on what other abilities we choose to put into one and the same set; they therefore come and
go at will. Whereas the primary bisection into universal and non-universal factors remains inviolate. It is not dependent on any chance composition of a particular set of abilities, but instead marks the most fundamental feature in ability as a whole."
One of the more significant recent contributions in the field of factor analysis is to be found in "Cross-32 roads in the Mind of In this volume Kelley
utilizes the tetrad difference equation in the development of a number of formulae expressing relationship between correlations. Kelley extends Spearman’s use of the tetrad differences and uses the summations of tetrad differences as indicators of bonds between the variables.
The major steps in Kelley’s factor analysis technique is outlined as follows: "(a) determine tentative standard deviation values for the farious factors in each variable. This is done by finding the mean value of all the tetrads involving each pair of variables, and tentatively assigning factor values so that the bonds, other than that given by a general factor, running
throughout all the variables are largest for those pairs of variables giving the largest mean tetrads. (b) Refine these preliminary values so as to reduce the mean square error of estimate of the intercorrelation coefficient s.
It is interesting 'to note the psychological interpretation placed upon the factors by Kelley. He does not interpret his findings as supporting Spearman’s concept of tt g”. In fact he feels that there are at least two, and perhaps three, traits combined in the concept H g”. He says, ’’First there is a factor making for correlation between the variables due to maturity, race, sex differences, and differences of antecedent nurture. The second factor which the writer finds clearly indicated in his own data, and which is undoubtedly measuring is the verbal factor. Finally there may be a third factor—not variability in maturity, in sex, in race, in nurture; and not verbalpresent in Spearman’s The writer believes that if While Kelley has made valuable contributions in the field of factor analysis, his method is involved and cumbersome, involving as it does the calculation of numerous tetrad differences.
such a residual factor remains, after an allowance for 24 the things mentioned, it is very small.”
The contribution which will probably mean more in making for the general application of factor analysis to the more important psychological problems, in which it will be helpful to analyze the complete phenomena into the smaller number of essentials that are operative to produce them, is the theory of multiple factors introduced by Thurstone, in 1931 and since elaborated
and simplified.
Thurstone’s is a more generally applicable method of factor analysis, which has no restrictions as regards group factors and which does not restrict the number of general factors that are operative in producing the intercorrelations. Thurstone turns from the use of tetrad differences—so longemployed by Spearman and ■ others to determine equiproportion and hence show that the intellectual activities could be divided into two factors—-and generalizes the factor analysis technique to make it applicable to problems in which the two-factor pattern is too simple and for which several independent factors must be postulated. The tetrad difference method is shown to be a special case of the multiple factor method.
Thurstone’s technique for the analysis of a table of intercorrelations into multiple factors is trigonometric in its derivation. It makes use of the ’’cosine law” and the fundamental system of equations for the theory of directional cosines in n-dimensional space suggested by Garnett in a previous reference.By representing the tests as vectors with their intercorrelations representing the angular separations of the vectors within a sphere, and rotating the axes under certain conditions, he ingeniously derives the technique for analyzing the intercorrelations into the factors necessary to account for the intercorrelations.
37 Thurstone I "center of gravity" method of multiple
factor analysis, which has been utilized in the analysis of the intercorrelations in the present study will be described in some detail in a subsequent chapter. The steps in the derivation of the equations and an outline of the method of procedure in the solution of a factor analysis problem will be given there.
Since we have not attempted, in this brief review of the development of the factor theory, to include the uses which have been made of the factor analysis technique, a brief statement of some of its applications may be in order here, looking especially to its application to school subjects.
Spearman’s original work, which led to the development of the theory of factors, grew out of an experiment 38 in a school situation. Inspired by Galton’s work,
he started experimenting in a little village school to determine whether, as Galton had indicated, the abilities commonly taken to be ’’intellectual 9 had any correlation with each other or with sensory discrimination. Finding a certain correlation between the abilities measured, he he wanted to know ”how much” they correlated,
and as a means of determining “how much,” he evolved a theory of “correlation coefficients.” Noticing a systematic inter-relation between the magnitude of the coefficients of correlation he suggested the possibility of their hierarchial arrangements. Search for a criterion of hierarchy led to the development of the tetrad difference equation which has become the fundamental equation of the two factor theory.
Spearman’s work with factor analysis has been largely in the field of developing technique and equations, he has been especially interested in its application to mental organization or “intelligence.” Professor Spearman and his students have generally investigated mental traits singly and in relationship to the ”g” factor. By such investigations they have described several mental factors such as “cognition,” “preservation,” and “oscillation,” as well as the general factor “g”. Their investigations, however, have never been carried into the field of school subjects as such.
Kelley, centered his study of mental organization chiefly upon data obtained from situations more akin to regular school subjects. His analysis of measures of reading speed and reading power, arithmetic speed and power, and memory, indicated the existence of factors which he called (1) general factor (being dependent upon heterogeneity of race, sex, maturity, and antecedent nurture), (2) verbal factor, (3) number factor, (4) memory factor, (5) spatial factor, and (6) speed factor. Further application of Kelley’s technique to school subjects has not appeared.
The Thurstone technique of factor analysis lends itself readily to use with a variety of types of data. The much simpler calculations involved, and the ease with which a table of intercorrelations can be analyzed by its use, would indicate that the Thurstone technique will be applied in the solution of a variety of problems.
The first published application of the technique 40 was Thurstone’s study of vocational interests. In this
study he analyzed the intercorrelations of eighteen variables obtained by the Strong Interest Blank, and
concluded that the intercorrelations could be accounted for by the postulation of four factors; namely, (1) interest in science, (2) interest in language, (3) interest in people, and (4) interest in business.
A similar study of very recent origin is on the 41 vocational interests of boys, using data also obtained
by use of the Strong Interest Blank. These investigators found three clear-cut interest factors and some consistency in the indication of a fourth factor. They were able to identify at least three interests of boys which were identical with interests found in men. They could also measure the difference in the weightings or importance of the three factors in men and in boys, and thus indicate the general pattern of the interests of both boys and men.
Two studies employing the factor analysis technique in the study of personality have recently been made. One of these is the study made by the sdjust- ment questionnaire. He utilized the Thurstone Technique as a means of indicating the possible situations in which tetrad analysis might prove the presence of factors. As indicated elsewhere in this study, Perry found factoi patterns by use of the Thurstone Technique which were comparable to those found by the exceedingly laborious method of tetrad analysis.
The other study of personality by the factor analysis technique is a study by Flanagan 43 which is not yet available from the publishers.
The only published application of the factor analysis technique to school as such, is the general application made by Line, Rogers, and Kaplan. The
primary interest of this study was the evaluation of the factor analysis techniques as to their adequacy for studying relationships in school subjects. The invests gators were particularly concerned with the harmony
between the approaches of Spearman and Thurstone and the applicability of the two techniques to school problems.
They used thirteen variables, including five measures of mental traits and the teachers’ final grades in eight school subjects. Eighty pupils from the fifth grade served as subjects. Their analysis by use of the Thurstone Technique revealed the presence, and amount, of the same four factors which Spearman had previously determined by an involved procedure. They also found a general factor, common to all school subjects, but varying widely in weightings, or in the amount of it contained in each subject.
These findings would seem to indicate to us that the study of each of the school subjects by a similar technique should yield information concerning the nature of the mental processes demanded by the various school subjects, and the analysis of the individual school subjects would be indicated as the next step in the application of this technique.
I Spearman, C., “General Intelligence, Objectively Determined and Measured,” American Journal of Psychology, 1904, vol. 15, pp. 201-293.
2 Spearman, C., op. cit.
3 Spearman, C., op. cit.
Spearman, o*, op. cit.
s Spearman, 0., op. cit., p. 284.
°Krueger, F., and Spearman, C., "Die Correlation Zwischen verschiedenen geistigen Leistunfefahigkeiten," Zeitschrift fur Psychologic, 1906, vol. 44, pp. 50-114.
7 Hart, 8., and Spearman, C., ’’General Ability, Its Existence and Nature," British Journal of Psychology, 1912, vol. 5, pp. 51-84.
8 Hart, 8., and Spearman, C., op, cit., p. 58.
9 Hart, 8., and Spearman, C., op. cit,, p. 59.
w Dodd, S. 0., “The Theory of Factors," Psychological Review, 1928, vol. 35, p. 213.
Burt, Cyril, “Experimental Teste of General Intelligence,” British Journal of Psychology, 1909, vol. 3, p. 159. ~
12 '‘Hart, 8., and Spearman, C., op. cit., p. 80.
1 Spearman, C., "Manifold Sub-Theories of ’The Two Factors’," Psychological Review, 1920, vol. 27, pp. 159-172.
14 Hart, 8., and Spearman, C., op. cit., p. 79.
C., "The Theory of Two Factors,” Psychological Review, 1914, vol. 21, pp. 101-115.
G. H., 11 A Hierarchy Without a General Factor, 11 British Journal of Philosophy, 1916, vol. 8, pp. 271-281. ~
lz Spearman, 0., “Some Comments on Mr. Thomson’s Paper,” British Journal of Psychology, 1916, vol. 8, pp. 282-284.
18 Thomson, G. H., "The Hierarchy of Abilities,” British Journal of Psychology, 1919, vol. 9, pp. 337-344. Thomson, G. H., "The proof or Disproof of the Existence of General Ability," British Jouinal of Psychology, 1919, vol. 9, pp. 321—336.
20 Thompson, 0. H., op. cit., p. 344.
J. C. M., ”0n Certain Independent Factors in Mental Measurements,” Proceedings of the Royal Society of London, 1919, 96a, pp. 91—111. C., ”Manifold Sub-Theories of the Two Factors,” Psychological Review, 1920, vol. 27, pp. 173-190.
23 Spearman, C., ’’Correlations Between Arrays in a Table of Correlations,” Proceedings of the Royal Society of London, 1922, vol. 101 A, pp. 94-1007
24 Garnett, J. C. M., "The Single General Factor in Dissimilar Mental Measurements," British Journal of Psychology. 1920, vol. 10, pp. 245-556.
C., "Recent Contributions to the Theory of Two Factors," British Journal of Psychology, 1922, vol. 13, pp. 26-30.
Garnett, J. 0. M., “General Ability, Cleverness, and Purpose,” British Journal of Psychology. 1919, vol. 9, pp. 345-^66.
27 "Spearman, 0., and Holzinger, K., ”The Sampling Error in the Theory of Two Factors,” British Journal of Psychology. 1924, vol. 15, pp.
pQ Spearman, C., and Holzinger, K., ’’Notes on the Sampling Error of Tetrad Differences,” British Journal of psychology, 1925, vol. 16, pp. 86-88.
O., The Abilities of Man, 1927, Appendix p. XI.
3 sSpearman, 0., "What the Theory of Factors is Not," Journal of Educational Psychology, 1931, vol. 22, pp.*ll2-117.
Spearman, C., "The Factor Theory and Its Troubles III,” Journal of Educational Psychology, 1933, vol. 24, p 7
32 Kelley, Thurman L., Crossroads in the kind ¥f Man, 1928.
33 Kelley, Thurman L., op. cit. p. 80.
34 Kelley, Thurman L., op. cit., pp. 17-18.
•xr "Thurstone, L. L., ’’Multiple Factor Analysis,” Psychological Review, 1931, vol. 38, pp. 406-427.
o6 Thurstone, L. L., The Theory of Multiple Factors, 1933, p. 9. ~ — —
37 Thurstone, L. L., A Simplifled Multiple Factor Method, 1933. * ~ —
38 Murchison, C., History of Psychology in Aut obi o graphy. 1930 pp. 299-333."
3Q Kelley, Thurman L., Crossroads in the Mind of Man, 1928.
40 Thurstone, L. L., ”A Multiple Factor Study of Vocational Interests/ 1 Personnel Journal, 1931, vol. 10, pp. 198-205.
41 Carter, H. D., Pyles, M. K.., and Bretnall, E. P., W A Comparative Study of Factors in Vocational Interest Scores of High School Boys, n Journal of Educational Psychology, 1935, vol. 26, pp.
42v Perry, R. C., £ Group Factor Analysis of the Adjustment questionnaire, 1935.
43 ' Flanagan, J. 0., Factor Analysis in the Study of Personality. 1935. ’ “’ —
44 Line, W., Rogers, K. H., and Kaplan, £., “Factor- Analysis Techniques Applied to Public-School Problems,” Journal of Educational Psychology, 1934, vol. 25, pp. 58-65.
CHAPTER III SUBJECTS, MATERIALS, AND TESTING PROGRAM
The present study makes use of the factor analysis technique in studying the intercorrelations of twenty - four measurements, consisting largely of reading test scores. A description of the subjects, tests, and method employed in obtaining the measurements is included in this chapter.
A. The Subjects.
The subjects used in this study include 210 boys and girls of the class entering the sixth grade of the University Junior High School, Austin, Texas, in September, 1934. At the time the testing program was begun there were 218 pupils enrolled in grade 6a. An attempt was made to utilize the entire population of the grade; but sickness and absences for other reasons daring the period of testing reduced our population to 210 pupils for whom scores on the battery of tests and measures were complete.
A population of 210 was considered sufficiently large to insure reliable results in the study. Cor-
relations approximating or exceeding 0.50 were expected, and coefficients of correlations of such size would be much above the generally accepted criterion of reliability, which requires the coefficient of correlation to be at least four times its probable error. It was also felt that the time required to score the tests and make the calculations necessary to find the correlation coefficients for a larger group would be impractical.
The use of the entire population of the grade was desired in order to eliminate the effect of any factors of selection, other than those which had operated in the selection of the population of the entrants to this particular school and grade.
A word concerning the University Junior High School may be in order. This school is one of two Junior high schools for white children in the Austin public school system. While The University of Texas cooperates with the Austin Public Schools in the operation of this school to the extent of providing the building and having a part in the selection of its principal, the school is otherwise controlled and operated as any other school in the system. There are approximately 900 pupils enrolled in the three grades. Sth, 7th, and Sth, included in the school. The
territory served by the school is a geographical division including the north half of the city, while the other jonior high school enrolls the 6th, 7th, and Bth grade pupils from the other half of the city.
Those pupils completing the fifth grade work in four ward schools, and a small portion of these from two other ward schools, are eligible for admission into the sixth grade of the University Junior High School. One of these ward schools is located in the University neighborhood and probably has a student body well above the average in ability. Another is in a middle class residence section, and the other two are in the sections of the city inhabited largely by people engaged in manual labor. The percentage of pupils enrolled from schools other than these four is very small. The territory served by the school is occupied by white, English-speaking families, as is indicated by the fact that every pupil included in this study is from an English-speaking home, except one girl, in whose home Spanish is spoken.
The question might well be raised as to why the study was confined to the pupils of one school grade. It was proposed during the conception of the study that two groups at different grade levels be considered. Comparable treatment of two groups, one taken rather early in the stages of development of the reading habits and another after the habits had become more mature would undoubtedly be interesting and profitable. It was considered wise, however, to confine this study to a single group in order that the size of the population might increase the reliability of the coefficients of correlations found. It is hoped that a similar study may be made at another grade level for comparative purposes.
The question naturally follows as to why the sixth grade of the University Junior High School was chosen for this study. At least three reasons entered into the decision. First, the sixth grade in the Austin public school system is the first year of the junior high school, and in the group we have those who have recently come, in the main, from four different schools. If different types of teaching, or peculiar emphasis on certain phases of reading instruction are calculated to influence certain test scores of reading ability, this influence of one teacher or method in a particular school would be somewhat less noticeable in the entire group.
Another reason for choosing the sixth grade was that the reading habits are generally considered to
mature rather rapidly through the first four grades and that beyond this the rate of growth is less pro-45-46 nounced. Hence it was felt that in the measurements
o f reading ability at the sixth grade level, we would be getting at the more nearly matured patterns of reading habits, and be better able to analyze the reading performance where the habit was more stable.
Still another reason for selecting this particular grade and school was its availability, the University Junior High School being located on the University canpus. It also furnished a population approximating the size desired and it provided ample laboratory space, well equipped for carrying on the experimental work. The fact that the group contained practically no bilingual pupils was also a matter of consideration.
B. Materials.
Since the use of all available measures of silent reading ability was impossible in such a study, we
attempted to select tests and measures of silent reading ability which would together be somewhat representative of the entire field of measurements of the ability in question. A consideration of silent reading as such, and of the various materials and methods used in its measurement, seemed to indicate the following phases which should be included in our measurements: (1) rate of reading, (2) comprehension of materials read, (3) size of meaningful vocabulary, (4) the mechanics of reading, and, for interpretative reasons, (5) intelligence, and (6) teachers* marks in reading.
Our next problem was the selection of tests or techniques which should be used in obtaining the measurements of these phases of silent reading ability. The usual considerations, such as reliability, validity, ease of administration, scoring, and cost entered into the selection of the tests, along with a consideration of their peculiar fitness to measure a particular phase of reading ability or to measure the reading ability in a particular way.
An examination of the teste of reading rate indicated at least two types of rate teste. In one type the rate score is more or less independent of comprehension, while in another type the rate score takes
account of whether the material read is comprehended. It is not our purpose to discuss the merits of the two type® of tests. In order that any influence of these different methods of measuring rate might share in the analysis, a test of each type was included.
As a test of the first type, the“ Monroe Standard-47 ized Silent Reading Tgst“ was used. The rate score
on this test is in terms of the total number of words read per minute. The total number of words read during the four minutes allowed for the test is indicated by a number opposite the line on which the subject is reading when time is called. One fourth of this number gives the number of words read per minute. In order to make this score comparable with similar scores on Test I for lower grades, Monroe suggests the addition of twenty-nine to the actual score thus secured. It will thus be noted that in deriving this rate score no consideration is taken as to whether the subject comprehended the material which he read.
As a test of the second type, the “Chapman-Cook 40 Speed of Reading Test” was used. This test includes
thirty paragraphs of thirty words each. The paragraphs are of approximately equal difficulty. A paragraph is to be read and one word, H which spoils the meaning” is to be crossed out. The rate score is computed from the number of paragraphs comprehended, as indicated by tt the proper word being crossed out,” during the two and onehalf minutes allowed for the test. Thus a subject is given no credit on his rate score for reading the paragraph unless he comprehends it well enough to cross out the proper word.
The next phase of sildnt reading ability, in the measurement of which we were concerned, was comprehension. The measurement of comprehension is approached in reading tests from at least three points of view: (1) as comprehension measured in terms of H rate, rt (2) as comprehension measured in terms of "power," and (3) on a more analytical basis such as comprehension of general significance, noting details, predicting outcomes, and the like. One test of each general type was included in our battery.
_ _ 4.9 The surgesB test was employed in obtaining a
measure of comprehension in terms of rate. The scale consists of twenty (20) paragraphs of approximately sixty (60) words each. Accompanying each paragraph is a picture. The paragraph contains instructions for the subject to do something to the picture accompanying it. Each of the twenty paragraphs, with the operation accompanying it, is scaled to be approximately of equal difficulty. The score is the number of paragraphs correctly marked in the five minuted allowed for the test, and thus the test is a measure of comprehension in terms of the speed with which the subject comprehends and reads the paragraph and performs the accompanying operations.
50 The New Stanford Reading Test, paragraph meaning
section, was used to obtain a measure of reading comprehension as “power.” This test includes thirty-eight paragraphs of unequal length and graded difficulty. The first paragraphs are easy and the range of difficulty is scaled upward to the end of the test. Each paragraph contains one or more blanks to be filled in after the subject has read the paragraph. There are eighty blanks in the total test. The score is computed from the number of blanks correctly filled in during the twenty-five minutes allowed for the test. The time allowed is ample to permit careful reading of the entire test. The subjects, however, find certain paragraphs beyond their comprehension so that the number of blanks correctly filled gives an indication of their comprehension in terms of "power. 11
51 The Nelson Silent Reading Test, paragraph meaning
section was used in obtaining a measure of “power of comprehension” from the analytical point of approach. This tests consists of twenty-five paragraphs, each followed by three questions concerning the content of the paragraph. The questions are Of the four-response form. The three questions following each paragraph are calculated to measure three phases of comprehension: namely (1) ability to understand the general significance of a paragraph, (2) ability to note details, and (3) ability to predict the probable outcome. Scores are obtained separately for total correct responses to each of the three types of questions and a total comprehension score is obtained from their summation.
The paragraphs are arranged in ascending order of difficulty and the time allowed, twenty minutes, is such that most of the pupils will do about as many exercises as they are d)le to do.
Measures of the relative size of meaningful vocabularies were obtained from the word meaning section of the New Stanford Reading Test and the vocabulary section of the Nelson Reading Test, both of which were referred to above.
The word meaning section of the New Stanford Test contains eighty words used in sentences, each requiring the choice of one of five words to make the sentence true. The time, ten minutes, is long enough for most of the subjects to reach a point in the words, arranged as they are in ascending order of difficulty, where they no longer have significant meanings for them. The score is computed from the total number of words correctly chosen.
The vocabulary section of the Nelson Reading Test consists of one hundred sentences, with the choice of one word from a list of five being necessary to make the sentence complete. The words to be chosen are arranged in ascending order of difficulty and the time allowed, ten minutes, is ample for most of the students to reach the limit of their understanding of the words included in the list. The score is the number of words correctly chosen.
The photographic eye-movement technique was employed to obtain symptomatic measures of reading ability, for a detailed discussion of the development of this technique and a general description of the eye-movement camera, the reader is referred to Gray’s descript ion. 52
By use of the eye-movement technique we are able to get a record for each subject of three primary characteristics of eye-movements: namely (1) the number of fixations made in reading each line, (2) the average duration of the fixations, and (3) the number of regressive movements, or movements in a backward direction. From these records the following scores are derived for each subject; (1) the average number of fixations per line, (2) the average duration of the fixations, (3) the average perceptional time per line as represented by the product of the number of fixations per line and their average duration, and (4) the total number of regressive movements made in the seven lines.
The use of the eye-movement records in the measurement of reading ability is justified on the grounds that such records furnish an index to the general nature of the subject’s reading process and indicate symptoms of the stage of maturity of his reading habits. The three primary characteristics of eye-movements listed above 53 have been shown to be symptoms, respectively, of the
three fundamental elements of reading; namely, (1) the span of recognition utilized in the reading of printed material, (2) the rate of recognition regardless of the size of the recognition unit, and (3) the regularity, or rhythmical progress of the perceptions along the printed lines.
With a hope of facilitating interpretation, two mental tests were included in mur battery. One was the type usually employed, using verbal material. The other was a test of the non-verbal type, which would eliminate the influence of reading.
54 The Henmon-Nelson Test of Mental Ability was used
as a test of the verbal sort. The test consists of ninety items arranged in order of increasing difficulty. The answers are of the multiple choice type from five possible responses. The time allowed, thirty minutes, is ample for the sixth grade pupils to answer all items within their ability. The score is the number of correct responses.
55 The Fintner Non-Language Test was employed as a
test of the non-verbal type. This test is administered by demonstrations on the blackboard, and no reading is necessary for the understanding of the directions or for doing the exercises of the test. The test blank consists of the following six exercises; (1) Movement Imitation, i. e., reproducing the movements of a pointer afier it has moved from dot to dot in different ways on the blackboard, (2) Easy Learning, i. e., a very simple digit symbol test containing three elements, (3) Hard Learning, i. e., a more difficult digitsymbol test containing nine elements, (4) Drawing Completion, 1. e., drawing in the missing parts of pictures, (5) Reverse Drawings, i. e., reproducing geometrical forms as they would be when turned upside-down, and (6) Picture Reconstruction, 1. e., indicating by number the positions of the parts of pictures so as to make complete The score of each section is calculated by multiplying the t4tal number correct by a given weighting. The summation of these weighted scores was used as the total score in this study.
The other measure used in the study was the semester grade in reading received by the subjects at the end of the first semester of the school year 1934-35. Letter grades given by the teachers were transmuted into number grades on an arbitrary basis of ”A plus” as 98, ”A” as 95, ”A minus” as 92, ”B plus” as 88, etc.
C. Testing Program.
The testing program was carried on at the University Junior High School during January, 1935, and the eyemovement pictures were taken during February, 1935.
Before starting the testing program the experimenter visited the school and explained the purpose and proposed procedure to the principal and to the two teachers of reading. The interest and cooperation of the principal and teachers was most helpful throughout the program. The reading classes were visited and the general routine , size of class and possible disturbances, were noted. An effort was also made to become acquainted with as many of the pupils as possible before the testing started, in order to secure better rapport between the experimenter and pupils during the testing.
The tests were all given to the ®ix (6) regular class groups during their reading class periods, in their regular recitation rooms. Each group took the tests in the same order, and all took the same test either on the same day or one day later. That is to say, three reading classes took the New Stanford Reading Test during their regular reading class period on January 3. On January 4, the same test was given to the other three reading classes. On January 7, the Burgess, Monroe and Chapman-Cook tests were given to the first three classes and on the following day to the other classes. The Henmon-Nelson Test was given on January 10 and 11; the Nelson Reading Test January 14 and 15; and the Pintner Non-Language Tests on January 17 and 18.
This regular succession of the tests was proposed to equalise within each test any practice effect which might arise from the taking of tests. Thus, if the experience of taking the New Stanford Reading Test tends to produce a higher score on the Nelson Reading Test taken subsequently, then this effect should be somewhat equalized among the subjects because of their having taken the tests in the same order and the same number of days apart.
All tests were administered by the experimenter. The directions for the administration of each test were carefully followed and the time limits set up for each test were strictly observed. The class periods were fifty minutes in length and ample time was allowed for complete preparation on the part of the pupils before tests were distributed.
The splendid morale of the school was manifest in the fine interest of the pupils in the test, and the|r effort to do their best throughout all the tests.
The eye-movement records were secured by means of the eye-movement camera owned by the Department of Educational Psychology of The University of Texas. The equipment was moved to the University Junior High School and set up in a room equipped for its use. Two selections were prepared to be read by the pupils. The first, which served as a practice selection, was paragraph 6 from the Nelson Silent Reading Test, Form A. It consists of eight lines of material well within the range of comprehension of the average reader of the group. The test selection contained nine lines of easy material adapted from a reading selection found 56 in a fourth grade reader." The passage was apparently
one of the easier selections in the fourth reader and seemed, by comparison, less difficult than the average selection found in a numbers of fourth grade readers examined.
Our purpose in selecting a passage well within the range of comprehension of our poorer readers was that our eye-movement records might reveal the characteristic reading habits of the subjects, rather than the movements found where some difficulty is encountered in the reading material. By reducing comprehension as ’’power” to a minimum, our measure of the span and rate of recognition of the printed material, and of rhythmic progress along the lines, should be more characteristic of the subject’s reading habits.
The subjects were brought to the eye-movement camera in groups of twelve or thirteen and the procedure of securing the pictures was demonstrated to them. After being told in detail what each subject was to do and being shown how the record would appear, the group was allowed to wait in an adjoining room while the subjects were photographed one at a time.
When the practice selection had been completed the subject was asked two or three questions, in order to make sure that the passage had been carefully read. The test selection was then presented and the eyemovements were recorded on a moving film. Questions concerning this selection were also asked, and the experimenter was satisfied that the selection had been read with at least a fair degree of comprehension in each instance.
D. Summary Statement.
The basic conditions of the experiment have been described in this chapter. We have selected a population consisting of 210 subjects from the same school grade. This population is considered sufficiently large to insure reliable data.
The possible influence of bilingualism upon the results of reading tests has been reduced to a minimum. Selective factors, other than those operating in the selection of the grade population for one half of a city, have been eliminated by including the entire population of a grade in the study.
The subjects were chosen from a grade level where the growth in the reading habits is considered to be rather stable, and where the reading habits are rather mature in their development.
The battery of tests used included five well known and widely used standardized tests which are representative of the measures of the various phases of reading ability, and two distinctly different types of tests measuring intelligence. Teachers 1 semester grades in reading were also included. Records were obtained by use of the eye-movement technique. This technique of measuring in reading has been in use for a number of years and the contributions to the understanding of
reading ability which have been made possible through its use are considered most fundamental. The battery of tests, or measures, is thus considered adequate to produce data worthy of careful study in the interest of the psychology of reading.
Efforts were made to control carefully conditions in the testing program. The administration of all tests by the same person equalized the personal factor in the administration of the tests, and should make for more comparable results. The administration of the tests in regular classroom situations and during regular class periods made for uniformity of conditions and thus resulted in normal responses on the part of the subjects. Conditions were also controlled as to the order of succession of the tests and as to the interval between testing periods. Interest in the testing program was maintained throughout.
45 Gray, 0. T., B Types of Reading Ability Exhibited Through Tests and Laboratory Experiments,” Supplementary Education Monographs, 1917, vol. 1, No. 5. Buswell, G. T., "Fundamental Reading Habits,” Supplementary Education Monographs, 1922, No. 21.
47 Monroe, Walter S., Monroe ! s Silent Reading Test (Revised), Test 11, Form 2.
J. C., and Cook, Chapman-Co ok Speed of Reading: Test, Form A.
49 Burgess, M. A., A Scale for Leasuring Ability In Silent Reading, Picture Supplement 1.
5 0Kelley, T. L., Ruch, £. M., and L. M., The New Stanford Reading Test. Form X.
51 Nelson, M. J., The Nelson Silent Heading Test, Form A.
52 Gray, C. T., Def hencies in Heading Ability. 1922, Chapter XI. pp . 173-206: * ~
53 Buswell, 0. T., “Fundamental Reaching Habits, “ Sueplementary Educational Monographs, 1922, No. 21, p. 11.
V. A. 0., and Nelson, M. J., The Henmon- Nelson Test of Mental Ability, Form A.
55 Pintner, R., Non-Language Mental Tests,
56 'Johnson, Constance, “Care of Cut Flowers, H in Far and Near, A Fourth Reader, p. 231.
CHAPTER IV STATISTICAL PROCEDURES
A. Scoring the Tests and Measures.
All tests were scored by the experimenter and later rescored by his wife, Frances Sample Franklin. The scores were then compared, and any paper whose scores did not correspond was carefully rechecked by both graders.
The photographic eye-movement records were projected and charted by a University student. Random samples of the records were re-charted by the experimenter and identical scores found, indicating that the records had been accurately charted.
B. Tabulation of the Data.
The data from the test blanks, eye-movement charts, and teachers’ record books were tabulated on data sheets,and were checked twice by two persons. They were read first from the data sheets and checked on the original sources and later were verified by reading
from the original sources and checking-on the data sheets
C. Method of Combining Test Scores.
The data of this study include: scores on twentyfour measures or combinations of measures. The scores derived from any one of the measures, or combinations of such scores, constitutes what we term, herein, a variable. Four of the variables were obtained by combining the scores of two or more measures. The four composite scores thus obtained are: (1) variable 16, paragraph meaning, which includes scores on the New Stanford Paragraph Meaning section, variable 1, and the Nelson Paragraph Meaning section, variable 14; (2) variable 17, word meaning, which includes some of the New Stanford Word Meaning section, variable 2, and The Nelson Word Meaning Section, variable 10; (3) variable 18, rate., which includes the scores on the Chapman-Cook Speed of Reading Test, variable 4, and the Monroe Rate scores, variable 5; and (4) variable 19, comprehension, which includes the scores on the New Stanford Paragraph Meaning section, variable 1, the Monroe Comprehension section, variable 6, the Burgess Scale, variable 8, and the Nelson Paragraph Meaning section, variable 14.
These composite scores were obtained by a method outlined by Woodworth.The plan consists in finding
the difference between the individual’s score on a test and the average score (i. e., X-Average X), dividing this plus or minus difference (**/ by the sigma of the district k ) bution and calling the I > the "reduced score,” or sigma score. Sigma scores found in this way for the same individual on several tests may be combined by averaging them, assuming that the distributions of the test scores are approximately normal, or that the distributions are similar. Comparison of the distributions of the scores combined is shown in Figure I.
D. Method of Calculating the Coefficients of Correlation
Zero order coefficients of correlation were calculated from the
r = xy - N(Jr (5)
The elements of the formula, however, were not obtained by the usual scatter-diagram procedure. As was noted in M C H above, several of the test scores were converted to sigma scores in order to combine them into composite scores. Since this was to be done for a number of the tests, it was decided to convert all raw scores to sigma scores, and these sigma scores were used in calculating coefficients of correlation, as 59 60 suggested by Woodworth and Garrett.
The sigma score of an individual on a test H X” is his f ’x H (i. e., his deviation from the average, X minus Av. x ) divided by . Likewise his sigma score on test ”Y H is his M y” (i. e., his deviation from the average of test divided by{7y. Hence the product o f the sigma score on X any Y . The summation of these products for the entire population divided by N thus satisfies equation (5) which becomes
p = Ziah N (6)
where a and b are sigma scores. In other words, the coefficient of correlation is the average product of the sigma scores of the individuals.
All sigma scores were carried to four decimal places and their products used in computing the correlations were rounded off to four places. Four place coefficients of correlation were used throughout the study.
E. Multiple Factor Analysis.
61 The Thurstone ’’Center of Gravity” technique was
used in the factor analysis of our data. As the steps in the solution of a. multiple factor problem are rather involved, and as the technique is relatively new, it will be described at some length in a subsequent chapter.
The technique was applied to several groupings, each including a different number of variables.— 6 procedure of making several analyses by the same technique was considered preferable to applying other methods of analysis, such as Spearman’s and Kelley’s, to our data,
along with the Thurstone technique. Perry, in a study
of the adjustment questionnaire, applied these different techniques to his data and found comparable factor patterns by use of the different techniques. The present writer also applied the Thurstone technique to Kelley’s
data and found factor patterns comparable to those found by Kelley. Hence it was felt that more material of interpretative value could probably be secured through use of the more easily applied technique on several patterns of the variables than by using several techniques on fewer combinations of the variables.
"Woodworth, R. S., "Combining the Results of Several Tests,” Psychological Review, 1912, Vol. 19, pp. 97-123.
Henry C., Statistics in Psychology and Education, 1926, p. 168.
59 Woodworth, R. S., op. cit., pp. 104-106. 60 Garrett, Henry C., op. cit., p. 285.
cl Thurstone, L. L., A Simplified Multiple Factor Method. 1933.
Sperry, R. C., A Group Factor Analysis of the Adjustment Questionnaire, 1934. “
63 Kelley, Thurman L., Crossroads in the Mind of Man, 1938, p. 100. “~ ~~
FIGURE I
CHAPTER V PRELIMINARY STUDY OF THE TEST RESULTS
While the table of intercorrelations constitutes the primary data for a factor analysis, a consideration of the distribution of the scores and a comparison of the results with published norms for the various tests used, will probably prove of interest to most readers. Table I includes the grade norms for the tests, the mean score of the experimental group, the standard deviation of the distributions, and the range of scores obtained.
An examination of Table I indicates that the mean scores of the group tested are above the norms for the mid-year of the sixth grade. While some of the difference in mean scores can be accounted for by the fact that the norms are derived from scores of pupils in twelveyear school systems, as well as from systems including eleven years, as do the Texas schools, yet the difference is great enough in most instances to indicate that the test performance of the group is superior.
While Table I does not reveal the normality of the
distribution of the scores of each test, an examination of Figure I shows the approximate normality of certain of the distributions. By an inspection of the original data, the writer was assured of the approximate normality cf all the distributions.
The method most frequently employed in the study of test results is an inspectional analysis of the intercorrelations obtained. This method has been utilized in a number of studies of reading measures, both for the purpose of determining the comparative values of tests and for giving a better understanding of the nature of reading ability. For this reason it seems desirable to examine the intercorrelations of the measures used in this study.
Table 11, which appears at the end of this chapter, indicates the intercorrelations of the results of twenty two reading measurements and of two measurements of mental ability.
Attention is called to the fact that the probable error of the coefficients of correlation are not included in the table. The omission of this measure is justified from two points of view. First, the size of the probable error does not enter into the determination of the factors or of their weightings by the technique
employed. Hence their calculation as a part of the primary data of this study is unnecessary. Second, it is clear that all coefficients of correlation included in the table are reliable (that is, we can be sure that some real correlation exists) so long as the coefficient is as large as .20. This can be ascertained by applying the equation for the probable Error of a coefficient of correlation, or by referring to a table of Probable Errors for populations of various sizes. Application of the above method shows that all of our correlations, except fifteen, are reliable when the generally accepted criterion of reliability is used. Those results whose reliability might be questioned include fourteen correlations with the Pintner-Non-Language Test, and the correlation between the teachers’ reading grades and the number of regressive eye-movements.
An examination of the table of intercorrelations reveals several matters of interest, hence a few of the relationships will be briefly discussed.
First, the unusually high positive correlations which appear throughout the portion of the table representing reading tests are interesting. If the intercorrelations of the measurements obtained from the five reading tests are considered together their
mean correlation is .77. The high correlation between these tests, which measure reading ability in a variety of ways, seems to justify the concept that reading is a rather general ability which might be measured by widely different instruments. Such close relationship of the measures can probably be explained by postulating some general factor or group of factors whose mediating influence is common to the entire group of measures. A factor analysis should reveal such factors if they are operative in the measures.
The very high and almost identical correlations between the New Stanford total score, its Paragraph Meaning section and Word Meaning section, are indicative of the careful construction of the test. A similar close relationship between the total score and the constituent parts is noted in the Nelson Silent Reading Test.
The high and uniform correlations found between both the Nelson Silent Reading Test and the New Stanford Reading Test and all the other reading measures would indicate that these two tests are very representative measures of reading ability as measured by tests.
Second, an interesting relationship is noted in the fact that the measures of word meaning correlate
higher than do the measures of paragraph meaning with the reading rate tests. This may be interpreted as meaning that rate, as measured by rhe tests of reading rate, is perhaps determined more by the rate of recognition of words as such, than by the rate at which the subject is able to comprehend the larger thoughtunits. In order to explain this it seems necessary to postulate the mediating effect of some factor or factors which influence the measurements both of reading rate and of word meaning. Such relationship is not apparent, and the existence of a factor common to the measures of these two abilities does not appear to have been revealed. The factor analysis technique may be of value in determining the presence of such a factor.
In this connection the relatively low correlation between the Burgess scale, which measures comprehension in terms of rate, and the other measures of reading rate is also of interest. This low relationship may be accounted for by the fact that approximately one-fourth of the total time devoted to the test is used in drawing, rather than in actual reading. c4
Third, the relatively low correlation of teachers* reading grades with all the reading tests used seems to indicate that teachers’ marks are based not entirely upon reading ability as measured by tests. The influence of effort made, interest maintained, improvement attained, and other such items are possibly included in teachers’ grades, along with the estimate of actual ability in reading. It is interesting to note, however, that teachers’ marks correlated higher with the more representative measures of reading ability, such as the Nelson Reading Test and the New Stanford Reading Test (r«+.7l).
Fourth, a cursory glance at the correlations of the eye-movement records with the other measures of reading, reveals relatively low correlations with all the measures employed; the average intercorrelation being only .44. While this indicates a rather low validity for the eye-movement records as measures of general reading ability, it should be botne in mind that the validity of the measures for such purpose has not been claimed. The eye-movement records have been employed as symptomatic measures of the three fundamental elements constituting the reading activity; namely, (1) the span of recognition utilized, (2) the rate of recognition, regardless of the size of the recognition unit, and (3) the regularity or rhythmical progress of the perceptions along the printed line. Considering these claims for the measures, rather than evaluating them in the light of their correlations with reading tests, their validity would undoubtedly be much greater.
The uniformity with which each of the eye-movement measures correlates with the other variables is an indication of their reliability. The average correlation of the number of fixations with the other variables is .41. The- close correspondence of the correlation of each variable to this figure it of interest. The average correlations of the other three eye-movement measures with all the variables are: (1) duration of fixation, .46; (2) perception time, .51; and (3) the number of regressive movements, .31. The small deviation from these averages in each column indicates that the measures 65 are rather reliable. Eurich has shown the following
reliability coefficients for the eye-movement records: number of fixations .91; number of regressions .91; and average duration of fixations .86.
The reader should bear in mind when considering the relationships shown between the eye-movement records and other variables, that in each instance the correlations have been found after the eye-movement scores were reversed in order. That is to say, the smaller number of fixations per line have been considered the better scores, and hence given the higher positive values in terms of sigma scores. Similarly, the fewer regressive movements, the shorter perception time, and the shorter duration of fixations have been considered as the higher scores. Hence, where we might expect to find a negative correlation between the actual number of fixations per line and the speed of reading, it here appears as a positive correlation. The intercorrelations of the variables are unchanged except for sign.
Fifth, the correlations of the two mental tests with the other measures employed is worthy of notice. The consistency with which the Pintner Non-Language Test is seen to yield low positive correlations with 66 the other measures is in line with Pintner’s explan- nation that the low correlations found between the nonverbal intelligence tests and verbal tests as due to the fact that they are testing different aspects of intelligence.
The high correlations between the Henmon-Nelson Test of Mental Ability and the reading tests is characteristic of the relationship usually found between verbal tests of mental ability and reading tests. The correlations may be accounted for by suggesting that mental tests of the verbal sort measure the same ability as is measured by reading tests. There is the possibility, however, that the obtained correlations are due to the mediating influence of some factor or factors which are operative in producing variations of the scores on each of the tests.
By way of summary it may be said that the test results, described in terms of central tendencies and deviations, indicate that the subjects who provided the data for this study are normally distributed when described in terms of the abilities measured. The reading ability of the group is more than a year in advance of their school grade, when described in terms of published norms.
The table of intercorrelations presents a wealth of information conderning the interrelationships between the various measures used. The complexity of the interrelations, however, makes for difficulty in the formation of generalizations concerning the measures.
The high correlation found between all the tests of reading ability indicates that reading ability is rather general in nature and is measurable in a Variety of ways. The close relationship between the Nelson Reading Test and all other reading measures used shows that this test is more representative of reading ability as a general concept than any of the others used. The New Stanford Test appears only slightly less representative of reading ability.
Teachers’ grades in reading are not shown to be a valid criterion of reading ability as measured by tests.
The eye-movement measures do not appear to de valid measures of reading ability as a general concept. The uniformity, however, with which they correlate with the different measures of reading ability indicates a relationship between them and some other phases of the ability.
The study of the test results in terms of their description and by an inspectional analysis of their intercorrelations has produced a variety of information concerning the group tested and the tests used. It has also raised certain questions concerning the relationships indicated by the measures. The need of further analysis of the results is, however, clearly shown. We shall subsequently study these test results by means of the factor analysis technique.
A., ”An Experimental and Statistical Study of Reading and Reading Tests, M Journal of Educational Psychology, 1921, vol. 12, J. 387.
Eurich, Alvin C., "Additional Data on the Reliability and Validity of Photographic Eye-Movement Records," Journal of Educational Psychology, 1933, vol. 23, pp. SSO-SSZ, ~
66 Pintner, R., "Results Obtained With the Non- Language Group Test.” Journal of Educational Vu’-h-h’ / hum । M jinMt iM«r ii ।ii । ji, ii mmmm* Psychology, 1924, vol. io, p.
Variables Mo rm Mean O’of Ristri- but5 on Range , 1. M. 3, paragraph .78. 37.99 13.38 46-115 2. N. 8. word 78. 83.42 12.47 46-113 3. IT, S. Total 78. 85.96 12.29 52-112 4. Chapman-Cook 12.8 13.26 4.82 2-29 5, Monroe Rate 164. 161.66 40.73 77-266 6. Monroe comprehens ion 11.1 12.52 3.11 5-20 7• pintner Language MO ne 412.31 51.66 261-573 8. Burgess i 9. 10.35 » 3.25 ‘ i < 4-19 9. nenmon-Me Ison Mental 61. 66.72 11.71 26-88 10. pelson '"ord 41. 46.24 9.23 20-69 11. Melson gen. Signif. 12.5 14.08 3.37 6-24 12. Melson Mote Retails 12.5 14.27 3.46 4-24 13. Melson predict Outcomes 12.5 13.88 3.72 5.23 14. Melson Paragraph 38. 42.94 10.58 19-71 15. Melson Total 79. 89.15 18.32 42-139 16.-19.Composite paragraph Mone 0.0 Combined si gma scores 20. Reading grade Mo ne 85.83 7.16 65-95 21. Mo. of Fixations * T one 9.63 2.49 4-19 22, Murat ion of Fixations Mone 7.66 1.35 3.9-11.8 23. perceptual Time Mone 74.40 27.66 25-187 24, Regressive Movements Mone 12.44 7.41 0-47
TABLE I DESCRIPTION OF TEST RESULTS
Variable Numbei ' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 New Stanford, Paragraph 1 .7976 .9555 .7025 .6082 .6420 .3746 .6325 .7692 .7668 .7426 .7152 .7885 .7206 .7924 .8598 <7968 .6030 .7667 .6898 .3998 .4336 .5105 .2743 New Stanford, Word 2 .7976 .9516 .6917 .6845 .7753 .2332 .6663 .7696 .8248 .7305 .7023 .7158 .7358 .8261 .7247 .9022 .6799 .7287 .6664 .4315 .5119 .5426 .2749 New Stanford, Total 3 .9555 .9516 .6415 .6204 .7501 .2591 .6634 .7910 .8302 .7515 .7303 .7460 .7646 .8400 .7974 .8544 .6177 .7413 .7125 .4425 .5013 .5436 .2886 Chapman-Cook, Speed 4 .7025 .6917 .6415 .7399 .7531 .1437 .7687 .5797 .7472 .7957 .7778 .7599 .7550 .8183 .7137 .7112 •8357 .7356 .6907 .4151 .4869 .5090 .3703 Monro e Rate Mnn a 5 .6082 .6845 .6204 .7399 .8482 .1254 .7070 .5539 .6699 .7455 .7126 .7210 .7449 .7557 .6436 .6653 .8317 .7023 .5236 .4146 .4160 .5024 .3095 Lvi\J LLL V M Comprehension 6 .6420 .7753 .7501 .7531 .8482 .1664 .7481 .6786 .7840 .7810 .7973 .7826 .7942 .8212 .7469 .8007 .7689 .8070 .6420 .4122 .4330 .4781 .2429 Pintner Non-Lang. Mental 7 .3746 .2332 .2591 .1437 .1254 .1664 .1532 .4091 .2413 .1435 .2169 .1740 .1831 .2164 .2479 .1877 .1393 .2147 .2394 .0689 .0744 .0704 .0607 Burgess Scale 8 .6325 .6663 .6634 .7687 .7070 .7481 .1532 -6333 .6783 .7539 .7619 .7464 .7799 .7828 .6714 .6793 .7007 .7639 .5474 .4089 .4900 .5210 .3165 Henmon-Nelson Mental 9 .7692 .7696 .7910 .5797 .5539 .6786 .4091 .6333 .7400 .6471 .6956 .6677 .6932 .7551 .7089 .7609 .5684 .6804 .6602 .3254 .4063 .4125 .2049 Nelson, Vocabulary- 10 .7668 .8248 .8302 .7472 .6699 .7840 .2413 .6783 .7400 .7859 .7614 .7595 .7819 .9428 .7670 .8847 .6772 .7444 .6743 .3844 .5117 .4947 .2446 Nelson, General Significance 11 .7426 .7305 .7515 .7957 .7455 .7810 .1435 .7539 .6471 .7859 .9152 .9127 .9662 .9389 .8413 .7527 .7655 .7994 .6837 .3934 .5368 .5070 .2950 Nelson, Note Detoils 12 .7152 .7023 .7303 .7778 .7126 .7973 .2169 .7619 .6956 .7614 .9152 .8992 .9617 .9191 .7991 .7403 .7226 .7840 .6432 .3811 .5124 .4824 .3042 Nelson, Predict Outcome 13 .7885 .7158 .7460 .7599 .7210 .7826 .1740 .7464 .6677 .7595 .9127 .8992 .9650 .9611 .8327 .7328 .7236 .7886 .6704 .4180 .5114 .5316 .3301 Nelson, Paragraph 14 .7206 .7358 .7646 .7550 .7449 .7942 .1831 .7799 .6932 .7819 .9662 .9617 .9650 .9482 .8364 .7523 .7431 .7971 .6957 .4092 .5360 .5212 .3187 Nelson, Total 15 .7924 .8261 .8400 .8183 .7557 .8212 .2164 .7828 .7551 .9428 .9389 .9191 .9611 .9482 .8430 .8586 .7389 .8111 .7109 .4074 .5523 .5183 .3090 Composite, Paragraph 16 .8598 .7247 .7974 .7137 .6436 .7469 .2479 .6714 .7089 .7670 .8413 .7991 .8327 .8364 .8430 .7478 .6457 .7679 .6622 .3817 .4565 .4665 .2755 Composite, Word 17 .7968 .9022 .8544 .7112 .6653 .8007 .1877 .6793 .7609 .8847 .7527 .7403 .7328 .7523 .8586 .7478 .7021 .7326 .6372 .3866 .5215 .5024 .2615 Composite, Rate 18 .6030 .6799 .6177 .8357 .8317 .7689 .1393 .7007 .5684 .6772 .7655 .7226 .7236 .7431 .7389 .6457 .7021 .7077 .5271 .3585 .5159 .4589 .2608 Composite, Comprehension 19 .7667 .7287 .7413 .7356 .7023 .8070 .2147 .7639 .6804 .7444 .7994 .7840 .7886 .7971 .8111 .7679 .7326 .7077 .5954 .3662 .4754 .4651 .2809 Teachers ’ Reading Grade 20 .6898 .6664 .7125 .6907 .5236 .6420 .2394 .5474 .6602 .6743 .6837 .6432 .6704 .6957 .7109 .6622 .6372 .5271 .5954 .2561 .4654 .4067 .1438 Number of Fixations 21 •3998 .4315 .4425 .4151 -4146 .4122 .0689 .4089 .3254 .3844 .3934 .3811 .4180 .4092 .4074 .3817 .3866 .3585 .3662 .2561 .3077 .8131 .7606 Duration of Fixations 22 .4336 .5119 .5013 .4869 .4160 .4330 .0744 .4900 .4063 .5117 .5368 .5124 .5114 .5360 .5523 .4565 .5215 .5159 .4754 .4654 .3077 4946 .2925 Perceptual Time 23 .5105 .5426 .5436 .5090 .5024 .4781 .0704 .5210 .4125 .4947 .5070 .4824 .5316 .5212 .5183 .4665 .5024 .4589 .4651 .4067 .8131 .6946 .6923 No. Regressive Movements 24 .2743 .2749 .2886 .3703 .3095 .2429 .0607 .3165 .2049 .2446 .2950 .3042 .3301 .3187 .3090 .2755 .2615 .2608 .2809 .1438 .7606 .2925 .6923
TABLE II INTERCORRELATIONS OF THE TWENTY-FOUR TESTS AND MEASURES INCLUDED IN THE STUDY (All correlations are positive hence signs are not indicated)
CHAPTER VI THURSTONS’S SIMPLIFIED METHOD OF MULTIPLE FACTOR ANALYSIS
The method of factor analysis employed in this 67 study was first described by Thurstone in 1931, and the present modified method published in 1933. Because the
technique has been developed so recently and is, therefore, perhaps not familiar to all, it seems advisable to include here, as was indicated in a previous chapter, a description of the method with the derivation of its formulae and an outline of the calculations involved in the analysis of a table of intercorrelations by use of the technique.
The multiple factor analysis may be briefly described as a statistical technique devised for the purpose of analyzing a table of intercorrelations into a smaller number of common elements. The factor problem,
therefore, consists in determining the number of general, independent, and uncorrelated factors that are operative in producing a given table of intercorrelations for a number of variables. When the number of factors which must be postulated in order to account for the observed correlations is determined, a weighting of each of the factors for each of the variables is also calculated, thus indicating how much each factor is represented by each of the tests or variables.
A. Description and Statistical Basis of the Method.
The material for this description cf Thurstone’s ”Center of Gravity Method” of multiple factor analysis is taken from the two previous references to Thurstone’s 69 work and from his larger monograph on this subject.
Those who are interested in a more detailed description of formulae or theory will find these references helpful.
This description makes no claim of originality either in formulating the theory, in deriving the formulae, or in the method of applying the theory* We have simply
described each of the phases of the technique as developed by Thurstone. All equations have been taken directly from his works mentioned above without further reference being made, and in some instances, our description necessarily utilizes his phraseology.
The method assumes that the factors which operate to produce correlation between tests, combine in a linear manner to account for the total performance in a test. This is the assumption of the usual regression equation. This pattern may be expressed for two general factors as follows:
(8)
in which
- Standard Score in test ”A”. ~ Standard Score in ability No. 1. - Standard Score in ability No. 2. a f and coefficients, or weightings.
In its general application, the relationship expressed by such an equation should include a term to account for each general factor which might exist. It should also indlude a term to account for the specific factor peculiar to the test, and yet another term to account for
the chance error of measurement. It will be noted, therefore, that the present equation represents a case in which two general factors alone are assumed to account for the total variation in scores of test H A n .
The two abilities are assumed to be uncorrelated and the performance, ,on the test is here expressed as a function of the scores in the two general abilities. The coefficients or weightings, a, and , are essential properties of the test, in that these coefficients indicate the extent to which test »A” calls for each of the abilities, 1 and 2. The values of and X_ are attributes of the individual subject who makes the score
Since S is by definition a standard score, then the variation of the performance of a group of subjects, N, in test ”A H may be written as follows:
' /K (9)
also by
~ r ‘ x ' 6/(7 = (10)
I x ~ \ Km £ <7, <4 x, xJ 7 (11)
~ Q ' ~ (12)
But since X and X are standard scores by definition,
# (13)
Since the two abilities have been assumed to be uncorrelated and hence entirely independent, the cross products disappear so that
*/ x x „ Cz = = or /r //Z (14)
Hence, by equation (9), (12), and (14), we have
/ (15)
in which a ( and factor loadings of test ”A tt . This relationship can similarly, be shown to exist between the factor weightings of any test, and can be extended to any number of abilities or factors.
Thus, as in (8), we can express the score on test M B h by the equation
i-
In which case (15) would become
4 4? = / (17)
This same pattern may also be extended to apply for three factors as follows:
+
In which esse (15) would become
4 -i
or the pattern may be extended further to apply to any number, M N”, of factors, as
i~L 3 X 3
In which case (19) becomes
hh.fh/b, -+ b / j ai)
This simple relation is a useful element in the factor theory, for it shows a necessary limitation in the magnitude of the factor weightings that enter into the equation. This relationship means that the sum of the squares of the factor loadings in a test must equal unity. In practice, this perfect unity is seldom sought. The major portion of the variation on tests is usually caused by a relatively small number of group factors, and in most cases the smaller group factors, and the factor specific to a single test are insigni-
ficant. While the technique here described is applicable to any number of variables and any number of factors, in most instances four or five factors will adequately explain the intercorrelation.
B. Geometrical Interpretations.
Equation (15) and equation (17) determine the same circle with the center at origin. The geometrical representation of the two test performances can be made with interesting results. In Figure II the two coordinate axes may represent the two abilities and X respectively. Then, since a ( and the coefficients for test H A”, we may represent this test as 8. vector, oa, and test ”B” may be represented by vector ob. Or we may think of the tests as represented by points in the periphery of a circle. Representation of the tests in a three-factor system would require that the tests be indicated as points on the surface of a threedimensional sphere. This same reasoning may be extended by analogy to a system of M r H factors and HrMdimensional space.
The correlation between the two tests “A” and may also be interpreted geometrically. Since and are both standard scores, their respective standard
deviations are both unity and the correlation is reduced to
Sy 'ab /!/
By equations (8) and (16)
. (q, i / A l > (23)
— <7/A X, X. y ( 24 .
Y J Z 4/ Z y (25)
Since the abilities are uncorrelated by definition, the cross products vanish and by (13)
/I Z = <7 Z 4 v- A (26)
This equation (26) shows that the correlation is the cosine of the angle between the two vectors in Figure 11. A table of intercorrelations of M N” variables, or tests, could be geometrically presented in the same manner. In case the »N M tests, or variables, have only two factors operating to cause their variation, they could be represented as points on the preiphery of a circle; but in case where three factors are operative, we note that equation (19) is the equation of a sphere
and b , b , and b are the three coordinates of a point on the surface of the sphere. By similar reasoning we know that the space order of the geometric representation is equal to the number of postulated operative factors. Thus, if five factors are operative, we should have five-dimensional space, or if "r" factors are operative we should have “r’’-dimensional space.
In these cases each test or variable may be represented as a point on the surface of a sphere. Hence, if there are three factors operative in producing the intercorrelations, it should be possible to locate each test or variable as a point on the surface of a sphere. Since each coefficient of correlation is the cosine of the central angle, when three factors are operative, each test, or variable, would be represented by a point on the surface of the sphere, and the points would be so located that the cosines of each central angle is equal to the intercorrelation between the respective pair of tests.
The factor analysis problem, then, is to determine the coordinates a z , , and a 3 for test A; the coordinates b ; , , and b 3 for test B; and so on for each test in the series. It should be possible, then, to calculate the coefficients of correlation by sub-
stituting the coordinates in equation (26) extended for the proper number of variables. Within reasonable limits, these calculated coefficients should agree with the observed correlation.
Having all these tests located as points on the surface of a sphere, the coordinates of any of these points cannot be determined without first deciding where the coordinate axes are to be drawn. The location of these axes is arbitrary, since the correlations give us only the angular separation of the tests, end no information whatever as to where the coordinates axes should be located, other than that by definition the factors are uncorrelated and hence the axes must be orthogonal. Thus, we see that there is no basis for locating any sort of absolute coordinate axes, and hence no absolute factors or absolute mental abilities. The situation, however, is analogous to most other measures, since they are usually relative to some reference point; for example, elevation is relative to sea level and time is relative to certain reference points.
The method of locating the axes and thus determining the coordinates or factor weightings is the special contribution of this center of gravity method. The
method starts with equation (26), extended to the number of factors postulated. Writing the equation for three factors and applying it to the first column of a table of correlation coefficients, called herein a M correlational matrix,” we have
-f- O 3 H c ’ a <z 3 G
y. b Q (28)
in which M K n represents each of the tests in turn.
A similar summation may be written for each column of the correlation matrix. Then we have
?.>< v (29) <U<_ <3 -+ i~/l/ 3 2^ 3
~ v (30)
Now, if a , , and a r are constructed to be coordinates of a point ”a” in space of H r“ dimensions, and if the H r” coordinates for each other test have a similar interpretation, then the coordinates of the center cf gravity of all the H n n points on the sphere will be
;
If the tests are clustered in any direction of the r-dimensional space, the center of gravity will be located in a direction from the original toward such a cluster. If the first is passed through the center of gravity of the "n" points, the new coordinates of the center of gravity will be
( a ' °)
and hence
' ' (31)
so that by (30)
J (32$
and hence
(35)
Applying the relationship which obtains in equation (31) to equations of the type of (29), we have the equations fcr calculating the first factor loading of each test, such as
(34) ' __ /
The first factor residuals are then computed according to the form
r aa = 7 t a Q - = — — /_ (35) /"e V■— 4/ -4/ J?? /V/< — <7
and summing
But substituting values of "a", from (34) and of from (33) we have
/ (36)
and similarly the algebraic sum of each column of the matrix of first factor residuals must be zero. This furnished an excellent check on the arithmetical check work of computing the factor loadings.
■Proceeding to find the second factor, it is noted that the center of gravity of the first factor residuals is zero, and hence a different method of locating the coordinate axes must be used.
The number of pointe representing tests on the surface of the sphere is the same after the first factor has been removed, but the points are now distributed in (r-1)-dimenstonal space since one axis is no longer operative.
If an observer could be stationed at the origin, sd that he could see in (r-l)-dimensional space, he would discover a clustering of the points, in case a second factor is conspiciously present. In order to make the clustering more noticeable and also tb maximize the amount of the residuals that are accounted for by each successive factor, each point is thought of as represented hot only by itself, but also by its reflection through the origin, which is called the image. Thus if one point represents a certain test, then its image would represent the same test with reversed sign for the standard scores. If all the points and their images . are thought of as visible, then the second factor would reveal itself in the form of a cluster of points. The second axis should be located in the direction of such a cluster.
Since the center of gravity of the ”n” residual points is at the origin, it is clear that the center of gravity of the ”n” points and their ”n M images will also be at the origin. In order to determine the direction of the cluster, if one exists, we select the test with the highest sum of residual intercorrelations irrespective of sign,
This procedure is based on the fact that the coefficients and their residuals ere interpreted as cosines. Since the test nearest the center of the cluster will have smaller central angles with the other tests, and hence the cosines will be nearer,unity, the will be largest.
When the center of the cluster is Thus located, all tests are projected into the hemisphere in which the cluster centers. Thus if test ”B n correlates negatively with the test at the center of the cluster, we consider not test “B”, but its image test H ~B”, which will correlate positively with the central test. By changing the signs of all tests that correlate negatively with the central test, the center of gravity is now moved in the direction of the cluster and a second factor can be extracted by the same procedure as was used for the first factor. This same procedure can be repeated for the extraction of as many factors as may be necessary to account for rhe correlation, or until the residuals are so small that they can be ignored.
0. Outline of Calculations.
The steps in the calculation of a problem by the factor analysis technique described above will now be presented. The example used in demonstrating the steps is the analysis which is referred to elsewhere in this study as analysis 11. It involves the following nine variables: 1. New Stanford Reading Paragraph Meaning, 2. New Stanford Word Meaning, 3. Chapman-Cook Speed, 4. Monroe Speed, 5. Monroe Comprehension, 6. Burgess Comprehension, 7. Henmon—Nelson Mental, 8. N e ieon Word Meaning, and 9. Nelson Paragraph Meaning.
A table of correlations, hereafter referred to as the correlational matrix, is prepared as shown in Table 111, which appears at the end of this chapter. The diagonal cells, which should contain the self-correlations, or coefficients of reliability, ate left blank. The sign of each coefficient is recorded in the upper part of the narrow cell to the left of the coefficient.
Each diagonal cell of the matrix is now to be filled with the estimated communality coefficient of the particular test. By communality is meant the variance of the rest after both the error factor, and the factor
specific to the test, are eliminated. The coefficient of communality will always be less than the coefficient of reliability if there is any specific factor in the test. It will also be larger than the largest coefficient of correlation, disregarding sign. Hence in selecting an estimated value for the coefficient of communality of a test with reference to the other tests, the largest coefficient in each column, disregarding sign, is selected and recorded with a plus sign in the diagonal cell of that column. For example, the largest coefficient, disregarding sign, in the first column is .7976, which is recorded in the diagonal cell. The largest coefficient in the third column is .7687, hence it is recorded in the diagonal cell.
Since this first matrix contains the original coefficients, rather than residuals, and hence relationship of equation (35) is not applicable; need not be obtained. Neither is it necessary to since the first axis is to be passed through the center of gravity of the N points and no determination of the center of the cluster is needed.
Each column of coefficients is next added and recorded in the row . The horizontal sum of entries is , which is recorded beyond the right end of the column. In this example ~ 59.1888.
i 8 computed and recorded. In this example its value is .0168960883. The next step is to compute and record it. The value in this example is .1300.
The first factor loadings, or weightings, are calculated from relations as expressed in equations similar to (34), in the following manner:
Weighting of Test 1 or A^a z “ (-A. 1300) = A 8368 Weighting of Test 2 or B-b, ~ H. 1300) (4-6.7704) ABBO2
These values are recorded in the next to the bottom row marked th, indicating the factor weighting with its temporary sign. Since, in this matrix, all the tests are positively correlated, these factor loadings are repeated with the same positive signs in row K, as the factor loadings, or as the weighting of each test with this first factor.
After the influence of the first i actor has been removed, the residuals of each correlation is next computed. In equations similar to (35), we have the relationship
which means that the residual correlation between two tests is determined as the difference between the coefficients and the product of the first factor weightings of the two tests. As in Table IV, the residual corre-
lation between variable 1 and variable 2 of our example becomes
+. 7969 -(+.8368)(+.8802)= 4 .0610
The residuals for the other cells are computed in the same manner. These computations may conveniently be made with a slide rule or calculating machine, using a constant multiplicand for each column. The writer found the latter method preferable and so devised sheets to facilitate the calculations. The factor weightings are recorded with the temporary in the first column. Each coefficient of correlation, with its signs, is recorded in the upper part of the cell which corresponds to its position in the lower half of the correlational matrix. The products of the factor weightings are then recorded in the appropriate cells and the subtractions made. The residuals are then transferred to the new data sheet, Table V. It will be noted that the diagonal cells in the matrix contain an adjusted entry as will be explained later.
The algebraic sum of each column is now recorded, in the row marked . If the calculations have been correctly made, these sums should approach zero, as indicated in equation (36), and as will be seen in the example. If one of these sums deviates markedly from zero, an arithmetical error has been made.
Following this check of the work the diagonal entries are erased, but the signs are left standing. The highest residual coefficient of the respective columns, disregarding sign, is then recorded in the diagonal cell, and the sum of each column, disregarding sign, is found and recorded in the row .
The bolumn with the largest number in row/^^is then noted. In this example it is column seven. This indicates that test seven has more in common with the other tests, after the influence of the first factor has been removed, than does any other one test.
In order to determine the weighting of each of the tests with the second factor which is to be extracted, and in order to move the center of gravity toward the center of the cluster, the image of these tests which do not now correlate positively with the tests nearest the center of the cluster is now considered. To do this all of the signs in column seven must be made positive in the manner described in the paragraphs which follow.
It will be noted that the first entry in column seven is positive. This means that Test I correlates posit ivelyw it h test seven and hence it does not need be projected. Similarly with Test 2; the original sign (in the upper half of the sign cell) of the third entry is negative, meaning that test three now correlates negatively with test seven. In order, therefore, to obtain its coordinates with Test I, Test 3 must be projected through the origin and its image, rather than the test itself be considered.
This projection is made possible by reversing all signs in row three, including the sign of the variable itself. The reversed signs are placed in the lower half of the sign cell. Similar reasoning leads to the result of all signs in rows four, five, six, and nine. Tests 3, 4,5, 6, and 9 will not be considered, but their image Tests, -3, -4, -5, -6, and -9 will enter into the calculations.
The signs in the corresponding columns are now reversed. Since the signs in row three were reversed, all signs, including the sign of the variable itself, must now be reversed in column three. This reversed sign is recorded in the main cell just above the numerical entry. In cases where the sign in the sign cell has already been reversed, the last recorded sign, which is the one in the lower half of the sign cell, is now reversed. Similar reversals are made in columns four, five, six, and nine.
In each column whose signs were not reversed, that is columns one, two, seven, and eight, there is no sign in the main cell. In each of these columns, the last recorded sign for that entry is copied.
The second factor weightings are now obtained by the same procedure as was employed in obtaining the weighting of the first factor. That the algebraic sum of each column, is determined and summed to , which in this example is 3.8941. The next step is to which is .256798747, and extract the square l root, .5068, which, when multiplied by the summation of each test found in row, gives the factor weighting with its temporary sign,ZtK.
These factor loadings are then placed in the bottom row, K, and their real sign determined. For example. Test 3 was seen to correlate negat ivelywith Test 7, so it was projected through the origin, and thus the second factor weighting of Test -3 was found to be "A .1853. If the second factor has a positive influence upon the variation of Test -3, then it would have an equal, but negative or inhibiting, influence on Test 3 itself. Hence we must reverse the sign in order to obtain the factor weighting of Test 3. Similarly we must reverse the sign of each variable whose sign in the ’‘variable*’
column at the top of Table Vis negative. If the sign of the variable is positive in the variable column, the sign of the factor weighting is unchanged, and it K, becomes K. with with its sign unchanged. By similar procedure we determine the residual correlation after the second factor has been extracted and determine rhe weightings of the third factor. Similarly for the fourth, fifth, or as many other factors as appear significant.
6? Thurstone, L. L., ‘‘Multiple Factor Analysis,” Psychological Review, 1931, vol. 38, pp. 406-427. 68 Thurstone, L. L., A Simplified Multiple Factor Method and an Outline of tEe ~
69 Thurstone, L. L., The Theory of Multiple Factors, 1933.
Figure II
4 y 4 2 '-f' 3 * 4 4 6 4 7 8 -f 7 * 1 -I- 7/76 t 7976 4 7025 6082 + 6420 4 6325 4- 7692 4 7668 4 7206 + 2 7776 4 8ZW 4 n ' 6945 4 7753 4 6663 4 7696 4 #24* 4 7358 ' 3 4 70Z5 7 &U7 4 7687 7399 4 4 7687 4 5777 4 4 ' 4 -t 6o8z 4- 4 7377 * 84 82 4 8482 4 7070 4 5531 4 6699 4 744-1 ' 5 -1 4 7753 4 753! h 8482 4 8482 4 4 678b 4 784o 4 7942 * 6 4 -4 66 G 3 4 7687 h 7070 4 7481 7799 4 6333 4 6785 4 7799 4 7 4 76 9z 4 ■7696 4 5791' * 4 6786 4 6333 4 ■7696 7 7400 4 67 3Z + 8 -/ 7668 4 8148 4 74-71 6699 4 7840 4 6783 -f 74-oa 4 8248 4 78/9 * ? 4 7Zob 4 7358 -4 755° 744-9 4 7942 4 7799 4 6932 7 ■78/9 7 7942 £ 4 4 6-7704 4 65065 6.4047 / 687/7 4 63940 4 m 7 98 -K -4 8368 4 S8oz t 8458 1 8326 —1" 8933 8312 7 8043 7 8863 4 8840 / — (0X83 K 4 836g 4 4 8458' *8326 7 8133 8312 4 8043 7 8863 4 2840 \ - Tzr J3oz
TABLE III Calculation of First Factor Weightings. Analysis II.
836 8 37176 8502 + 7? 76 00,10 + 8249 + 174% osoo 8458 + 7OZ5 i-lM 0053 36,9/7 05 Z8 ■ 111 + 1G87 '0533 +■ 83X6 07,13 -f-6 8+0 +1111 0+84 31317 +7J4Z 0357 + +G13Z 150-0 4- + 8133 +64zo 37475 IO&5 + 1753 + 01/0 3 84-8Z 37738 'TolT- 4- 0507 4 W +43ZS +4'155 0030 3 G 41,3 3-13/G 0553 3 7577 "0557 37070 0G1ZL 0141 3- + i 0&S~6 +7777 3b18l_ ozw 8043 +7G1Z 3-073 0 "oigz. +■1696 37071 05/7 3~ ±3777 / 35537 1/58 +G78G 37/85 05/7 ±(p333 'D35Z 57b?i> IZZ1 + 8863 + 760$ 314/7 02SI 3 382+2 D+41 574-7Z 37410 ~ooz5- 3GG71 t137j_ 0b80 3-7840 317/7 0071 +G785 0584 41400 31/71. 027/ + 38248 31855 0373 3 + 8840 +izog 3-1317 OHI + 1358 O4-Z3 37550 +m 0073 31441 0087 3 41142 37817 0 045 4 3771? 313+8 "045/ + G152 0178 378/7 + 1835 oo/b 3 7742 + 78/5 0/27 -oo n "ooz* "00/6 ~00/8 -00/7 -00 54 ~OO/7 -0023
TABLE IV Calculation of First Factor Residuals. Analysis II.
-I- 4/ 4 4 4 -3 -4 4 — 4 -G 4 4 4 7 4 -4 4 y / 4 4 , 4 oG/o ~ 4 Oo5 $> fifth - 4 o G 4 4- o4G2 5 0 25 / / 4 -/Z ¥ . OG/o 4 aCH •4 O b If -f — 4 O/70 - 4 o & 5i -f 4 oG7 7 4 4 C>447 - 4 4 4 oof 3 4- 4 os zy 4 — —• 3 oW4 4 4 o 357 5 00 25 4 4 ■og57 4 4 fOC>G 4 4 4 -7 00 73 4 -4 4 4 4 4 4 OS 5/ T \ -4 7/3 3 4 4 7a f7/ 4 o744 4 4 4 ac to 4 4 0084 4 4 4055 4~ 4- of/o -f OO2b 4 4 7 a 44 4 4 /o 50 4- 4 oo5g 4 4 om 4 4 0077 4 4 00 4b 4 4 OG3O 4 4 OGO 3 4 -f 6(,S7 4 4 oi44 4 4 oo5G 3~ 4 OG3/ 4 4 4 4 oft/ 4 4 1 i 47 4 4 4 4 oC'7 - 4 fooG> — 4 —- 4 — 4 o 35~2 4 4 7/55 4 4 OZ77 — 4 0/ 75 f 4 4 oAbl 4 4 o447 4 OO 2rf — -4 - 4 00 7/ 4 ObhY 4 4 oz7l 4 4 OGfO 4 00 >G -4 4 Olli 4 4 4 4 4 4 ooi/f ¥ 4 oo4S 4 4 o45! 4 077f 4 4 00 /G 4 4 0451 £o - oo 1 J - oo Uf 0 OO 4 - *o If - oo f? - oo/G Oo D 4 O0I? - 00 UiZS 47a3 •Loot ■ w 3030 3417 4 4 4 w 4 Cootf 4 4 4 4 3030 4 1117 747. it\ 4 4- 2243 4 4 ■3o/fi ■1434 4 ■2133 4 4 153G 4 oUZ /_ 'Xr — ■ ■ UC', y?47 4 4 2273 3<>¥3 — 4434 - ■w 4 4 7540 P772 -
TABLE V Calculation of Second Factor Weightings. Analysis II.
CHAPTER VII RESULTS OF FACTOR ANALYSES
Presentation of Results
The factor analysis technique has been applied to six groupings of the measures obtained. The purpose of analyzing the variables in these different patterns is to determine whether the principal factors found in one grouping of the variables appear again when other combinations of the variables are analyzed. Such consistency seems essential to the reliability of the technique.
The first analysis envolves eleven variables, ineluding the four composite scores, the New Stanford total score, the Nelson total score, the Burgess score, the score on the Henmon-Nelson Mental Test, and the sepres on the three analytical measures of paragraphs meanings from the Nelson Test. The selection of this group was based on the writer's supposition that these were representative of the abilities being studied.
The results of the analysis are shown in Table VI. The first column of this table gives the names of the
tests, or variables, used in the analysis. The next four columns show the leadings of each variable, and the last column represents rhe percentage of the total variation of each test which is accounted for by the four factors determined in this analysis.
The factor weightings may be considered as the coefficients of correlation between the tests and the factors. Thus the weighting of the New Stanford Test
with the first factor, .8691, indicates a positive correlation between the New Stanford and the ability represented by the first factor. In bhe column containing the second factor there is a weighting of -F,2175 for the General Significance section of the Nelson Test. This indicates a negative correlation between this portion of the test and the ability represented by the second factor. In -uhe column containing the third factor values, we find that the New Stanford Test has a weighting of which indicates no correlation between this test and the ability represented by the third factor. The factor weightings have a maximum range of from —I to 41.
The factor weightings may be interpreted in terms of the degree of saturation of the test with the ability represented by the factor. Thus a positive weighting of a test with a given factor means that the scores on the test tend to vary in proportion to the extent to which the ability associated with this factor is utilized in the test performance. A negative weighting is interpreted as meaning that the scores on the test tend to decrease with any increase in the amount of this ability manifest in the test performance, and a zero weighting
indicates no relationship between the variation of test scores and the amount of the ability utilized by the subjects in the test performance.
The last column in each table shows the percentage of the variation of each test accounted for I>> the four factors. If the variation of the test scores is entirely accounted for by the four factors, this summation of the squares of the factor weightings of each test should be unity. The difference between the amount accounted for and unity represents the extent to which the test must be described b}r additional influences, which msy be represented as the effect of small group factors, of factors specific to the particular tests, and of chance errors in measurement.
The relatively small weighting for the fourth factor indicates that no further analysis need be made. The first factor has large weightings in all the tests, the second has only a few large weightings, and the third and fourth have no large weightings, which indicates the decreasing importance, or functioning, of the successive factors. The largest weighting of the fourth factor accounts for only five per cent of the total test variation, which indicates that very little of the remaining variation of the tests could be accounted for by the
additional factors which are common to this group of variables. The large percentage of variation accounted for by the four variables is a further indication of the adequacy of the four factors to describe the tests.
The first feature of interest in Table VI is the high positive correlation of each of the tests with the first factor. This first factor describes at least sixty per cent of the variation of each of the tests included in this analysis, and it accounts for ninety-four per cent of the variation of the Nelson Test scores.
The second factor is seen to be positive in the New Stanford Test, the Henmon-Nelson Test and the measures of word meaning; whereas it is negative, in the measures of reading rate and in the paragraph meaning sections of the Nelson Test.
The third factor appears with negative signs in the composite measure of rate and of word meaning; whereas it is positively correlated with the Nelson Test. The fourth factor appears to have little in common with any test other than the composite measure of word meaning.
A second analysis included the nine variables as indicated in Table VII.
The reason for selecting this group of tests was to determine whether the same high weightings of the first factor would again appear. Two measures used in Analysis I were also included in Analysis II, for the purpose of noting their comparative weightings when analyzed with two different groups of reading measures. The two tests included in both analyses are the Burgess scale and the Menmon-Nelson Mental Test.
The results, as presented in Table VII, reveal the presence of a first factor which is practically equal to the first factor shown in Analysis I. The two tests included in both groups also show practically identical weightings in the two analyses.
The second factor is shown to vary with the scores on the mental measures and with scores on the New Stanford Test, while it has a negative correlation with the tests of reading rate and comprehension as measured in terms of rate. The third factor indicates the Monroe Test and the Chapman-Cook Test as measures of reading rate, and the fourth factor may serve as a means of differentiating between mental ability and reading rate. A fifth factor was determined for this group, but its insignificance is seen in the fact that its highest weighting only accounts for one per cent of the variation on the Nelson Meaning Test.
The third analysis included the nineteen measures c obtained by means of the reading test. The results of Analysis 111 are presented in Table VIII.
A comparison of the first factor weightings with those obtained for the same variables in Analysis I and II gives some indication of the reliability of the technique. The weightings of the variables are relatively unchanged by the three groupings. The greatest change
in weighting was .0362, and the average variation of the weightings for the entire group was only .0064. The degree of saturation of the tests with this factor is again noted.
The second factor is also easily identified as the second factor revealed in the other two analyses. Its high positive are found ir the two mental tests and the New Stanford Test, whereas its negative weightings are noted in the three measures of rate and in the Nelson Test. The differentiation of the New Stanford Test and the Nelson Test, as shown by this factor, could hardly be revealed by any other technique. The demarcation which this factor makes between reading rate and intelligence test scores is unusual.
Hie third factor presents an interesting characteristic of the technique employed. A comparison of the weightings of each test with this factor reveals loadings comparable in size, but different in signs. This difference in signs may be explained by a, geometrical representation of the tests. The weightings may be conceived as being calculated from the coordinates of the test with the factor in one instance, and from the coordinates of its image in the otdier. Such an interchange of signs is produced when the addition of certain other measures causes the center of gravity of the distribution of the tests in space to be shifted beyond the origin. The factor common to the group of tests may be unchanged by the addition of the other tests, but the signs may be reversed.
When this allowance has been made for the reversal of the signs, the third factor is seen to be the same in the three analyses. This third factor is equally operative in word meaning tests and measures of reading rate, and is effective as an inhibiting force in the comprehension sections of the Nelson Test. This factor, which differentiates between comprehension on the one hand, and the combination of reading rate and word meaning on the other, presents an interesting field for speculation.
The fourth analysis includes only five variables. There were two reasons for this section; first, Thurstone’s
statement that the method is perhaps most suitable for a large number of variables such as thirty, fifty, or one hundred, together with the fact that he had used the technique with nine variables, suggested applying the method to this small group; and second, the consistently high weightings of these five variables with the second factor led to their selection. The New Stanford Test and the intelligence tests share in the positive weightings, whereas the konroe Test, rate score, is Representative of the negative functioning of this factor.
The results of Analysis IV are presented in Table IX. The consistency of the first factor weightings is noticeable, but it is not apparentfrom this analysis,
why the weightings of the Non-Language Mental Test should be higher in this combination of Variables than in any other. The second factor is of special interest, since
a further study of its weightings constituted one of the reasons for this analysis. A comparison of the loadings found here with those of the same variables in the other analyses, causes one to question whether this second
factor is the same as any other factor yet revealed. The high positive weighting of the Non-Language Test and the high weighting of opposite signs for the konroe Rate Test, is interesting. The weightings of the other three measures are much smaller than those shown in previous analyses. The additional fact that the third and fourth factors are also difficult to identify may indicate that this technique is not applicable to this small number of and hence justify the elimination of this analysis from further consideration.
In the fifth analysis, the variables used were chosen to test the reliability of this factor analysis method with special reference to the high weightings of the fl>rst factor revealed. Two reading tests and one reading rate test were combined with two intelligencetests and four measures obtained from the eye-movement records. The intercorreletions had previously revealed that the eye-movement records and the non-language intelligence test were not closely related to the tests of reading ability, and hence could, if the weightings are functions of the groupings of the variables rather than of the variables themselves, conceivably disturb the factor weightings.
The results of Analysis V are presented in Tgble X. While the loadings of the reading tests with the first factor have been slightly reduced in this combination, they are relatively high. The non-language test weighting is not changed in this combination, but the Henmon-Melson Mental Test is represented by a relatively low weighting.
The other factors found are not identifiable as factors previously revealed in our analyses. The high weighting found in this factor should, however, prove helpful in determining the relationship between the eye-movement measures and the results of reading tests.
The entire group of twenty-four measures obtained in this study were included in the sixth analysis, the results of which are presented in Table XI. The fact that the weightings of the two principal factors revealed in Analysis 111 are relatively unaffected by the addition of the five other measures is an indication of the reliabilty of the factor analysis technique.
The weightings of the eye-movement measures with the first factor show the extent touhich these symptomatic measures of the reading habits are influenced by the operation of this factor. "While these weightings indicate the presence of this common element in the eye-movement measures and reading tests, the second factor revealed in this analysis is seen to be peculiarly characteristic of the eye-movement measures in that a large weighting is indicated for each of the variables obtained from the eye-movement records, whereas small negative weightings are indicated for each of the reading tests.
It is interesting to note that after this second factor, which is peculiar to the eye-movement records, has been extracted the next factor determined is easily identified as the second factor found in Analysis 111,
and the fourth factor may be identified as the third factor in Analysis 111.
A fifth factor was also determined, and the weightings indicate that this factor has probably not been determined in any of the other analyses. Its chief point of interest is in the differentiation which it makes between the number of regressive movements and the composite rate score. A similar contrast is seen in the relation of the composite reading rate and of the Conroe Rate scores to this fifth factor.
Interpretation of Results
As has been explained earlier in this chapter, the size of the weightings of a test with a given factor indicate the repat ionship between the test and that factor. This relation may, in cases involving intellectual activities, be interpreted by considering the factor as representing some ability, which is possessed or utilized by different individuals in varying amounts. The utilization of a large amount of the ability accompanies good test performance and the absence of the ability is associated with low scores. In such an interpretation, the relationship between the ability and the test scores is an essential property of the test itself. The other felement which is operative in determining the influence of this factor on the individual test score
is the amount of the ability utilized by the subject in the test performance.
With this background, it is of interest to examine the result of the analyses.
Description of the tests in terms of factors.
In order to facilitate the description of a test in terms of its factor weightings in the different analyses, Table XII, arranged with the factor weightings of each variable in a single row, is placed at the end of this chapter.
By following the first row, the paragraph section of the New Stanford Test is described as being correlated with the first factor, as indicated by an average weighting of 4*8665. By squaring the average weighting , we find that seventy-five per cent of the variations of the scores on the paragraph section of the New Stanford Test are accounted for by the degree to which tnis first factor is utilized by the subjects. In like manner, Analysts 11, 111, and VI describe the test as correlating with the second factor, to the extent of an average coefficient of 4.3430, which would account for twelve per cent of the variation of the test scores. The third and fourth factors together describe only three
per cent of the variation. Thus the four factors determined in the analyses are seen to describe ninety per cent of the variation in the scores on this test. The remaining variation could be described in terms of less significant group factors and factors specific to this test.
Shile such detailed study of each test is not germane to the present work, it is interesting to note that three factors determined in the analyses are sufficient to describe a large amount of the variation on each test. While the three factors describe only thirty per cent of the variation in the Pintner Non- Language Test, it is important to note that they more adequately describe the reeding tests, accounting for at least seventy per cent of the variation in each test, and as much as ninety-five per cent of the variation in one of the reading tests. The average amount of the variation of the reading test scores that is described by the three factors is eighty-five per cent.
While the three factors describe more that ninety per cent of the variation of the New Stanford Test and the Nelson Test, they describe only seventy per cent of the variation of the Burgess Scale and less that eighty per cent of the variation in measures of reading rate. Such differences indicate that the various reading tests measure, in the main, a common ability, but that they measure it to different degrees. The further differentation between the measures of reading rate, word knowledge, paragraph meaning, and comprehension in terms of rate or power, is due to the operation of additional factors as contributing forces in certain of the tests and inhibiting forces in the others.
Description of the Factors
Before describing the factors which have been determined in these analyses, it should be made clear that the matter of assigning names to the factors or of describing them in terms of our general concepts of reading ability is entirely extraneous to the statistical analysis. The technique employed has yielded, for each test, factor weightings in terms of the coordinates of the test with the factors. There is no guarantee that the weightings will be so arranged that the factors can be readily described or named. Difficulty in recognizing a factor may be due to the axes being so located that they cut between the axes that are implied in one’s general understanding of the tests involved ar of the abilities presumably required by the test.
Recognizing these limitations placed upon any definite naming of the factors which have “been f ound, it is possible to deduce, in a general way, the nature of the obtained factors by an inspection of the weightings for each factor. Thus if the weightings indicate high correlation between a factor and certain tests, and an opposite relation between the same factor and other tests, it is possible to describe the factors in,terms of these relations.
An examination of the factors obtained in the present study seems to indicate three clearly distinguishable factors which are operative in the tests used’ and a fourth factor, which is characteristic of the eyemovement measures, is noted. The other factors determined either exerted a relatively small amount of influence in the tests, or else were difficult to identify in the several analyses.
The first factor derived, which shall be referred td as factor ”A”, is significant. It is operative as a contributing factor in each of the measures used, but is indicated as operative in accounting for as low as ten per cent of the variation on the Pintner Uon-Language Mental Test and as high as ninety-three per cent of the variation in tne total score on the Nelson Reading Test.
Table XIII presents the weightings of factor ”A“ in five of the analyses. Analysis IV is omitted from consideration because of the seeming inconsistency of the factors in this analysis in which only five variables were used. The la6t column indicates the pef-cent of the score variation that is accounted for by the average weighting of the test with this factor. An examination ef the percentages which are indicated for the seventeen variables representing reading tests, shows that factor ”A" is operative in each of the reading tests used, but operative in varying degrees of effectiveness. While the factor accounts for only sixty-two per cent of the variation in the Burgess Test it accounts for ninety per cent of the Nelson Test Variation. This factor alon<e accounts for seventy-seven per cent of the performance in the seventeen variables that were obtained from reading test scores. It is noticeable that the factor is less operative in tests of reading rate than in tests of comprehension and word meaning. Its presence is indicated in the symptomatic measures of reading habits end in the non-language test of mental ability.
Factor ”A” is thus seen to be common to all the measures used, and hence may be considered to represent a general ability. Whether this factor cannot be interpreted , as in Spearman’s ”g”, in terms of mental energy; or, as is Kelley’s general factor, in terms of heterogeneity of race, sex, and maturity; is not here determined. While it would seem reasonable to identify this ”A“ factor with the general factor indicated by both Spearman and Kelley as operative in all intellectual activities, its identity with such a factor is not here established. The analyses reveal, however, in the verbal measures of reading ability which were used in this study, the presence of a factor operating to determine more than three-fourths of the variation in test scores.
Turning to the next factor, an examination of the weightings determined in each of the analyses reveals the fact that the second factor in Analysis VI is clearly not identifiable as the second factor in other analyses. The third factor of this analysis, however, shows weightings comparable to those of the second factor in the other analyses. This factor, which shall be referred to as factor ”B”, includes this third factor of Analysis VI and the second factor of the other analyses.
Ine weightings of this ”B fl factor are seen to vary from—-. 3043 to 3934, indicating that the ability represented is not operative as a contributing force
in all of the tests. The factor may then be described in terms of the tests with which it is positively correlated and in terms of those in which it is found with opposite sign. In order to distinguish readily the situations in which the factor differentiates the tests, Table XIV is prepared, presenting the contrasted weightings of factor ”B”.
The factor is indicated as accompanying larger scores on the two tests of mental ability, on the New Stanford Test, and on the tests of word meaning. It is also associated with a small number of fixations per line. Contrasted with these measures, are the measures of rt?ading rate and of comprehension measured in terms of rate, the paragraph sections of the Nelson Test and the duration of eye-movement fixations.
The weightings of the two widely different mental tests with this factor are interesting. Their weightings with the first factor were .25 and .80, but their weightings with the ”B h factor are practically identical, indicating that while the tests are very different, there is an element common to both in practically the same degree.
The consistency with which the analyses differentiate between reading rate on the one hand and a group of measures, including the New Stanford Test, the mental
tests and tests of word meaning on the other, is an indication of the reliability of the technique employed. Whatever this ”B” factor is, it must be characterized in terms of its effectiveness in differentiating between these tests.
An examination of the remaining weightings reveals a degree of consistency in the operation of a third factor. This factor, which shall be referred to as factor <? C“, is indicated in Analysis VI, fourth factor, the third factor in Analysis 111, and the image of the third factor in Analysis I and 11. In speaking of the image of a factdr, reference is made to cases in which the center of gravity of the tests in space is on the side of the origin opposite the tests, and, hence, the sign determined for their factor weightings must be reversed in order to make them comparable to the other factors. An examination of the extreme positive and negative weightings of this factor is facilitated by their presentation in Table XV.
An interesting characteristic of this factor is seen in the fact that word meaning, which was differentiated from rate by the H Bf factor, is closely identified by the ”C M factor with rate as measured by the composite rate score and the Monroe Rate Test. The differentation between the Monroe Rate Test and the Chapman-Cook Test by this factor is also interesting, as it is the fact
that the Nelson Test is practically the only test with which the ”C# factor is consistently correlated with a negative sign. Whatever this factor may be, it is associated with an increase in word knowledge and in the rate of reading, as contrasted with the comprehension scores on the Nelson Reading Test. Further character- of the factor would not seem to be justified by the present results.
The lack of consistency shown by the weightings of the other factors made their identification in the various analyses questionable. One other factor, howevey, may be described. In Analysis VI, the first factor weightings were practically identical with the corresponding weightings of the other analyses. The second factor that was extracted, however, yielded insignificant negative weightings for all the tests, and large positive weightings for each of the four eye-movement measures. Whatever this factor is, it seems to have little influence in reading tests, and yet is the predominant factor in the eye-movement measures.
The measured results of these factor analyses may be summarized as follows: in the first place, a high degree of consistency is found in the factor weightings of each variable in the different analyses, indicating the reliability of the technique. Again, the analyses have yielded three factors which are capable of describing eighty-five per cent of the performance on the reading tests used. And finally, the three factors may be characterized as a general factor,“A”, and a second factor, *B B , which is closely related to the mental tests, to the New Stanford Test, and to the measures of word meaning; and bears the opposite relationship to reading rate and to comprehension measured in terms of rate; while the third factor, f, C H , may be characterized'as accompanying high scbres of reading rate and of word meaning, and as being negatively correlated with scores on the comprehension sections of the Nelson Test. This simple pattern, indicating the operation of three factors as determining eighty-five per cent of the variation in the formation of concepts of reading ability.
70 Thurstone, L. L., A Simplified Multiple Factor Method, 1933, p. 1.
Variables Factors Variation Accounted for 1 2 3 New Stanford-Totai <.8681 M.3349 f.0099 .88 Burgess Scale 4.8218 —.1373 ".1981 4 1330 .75 Henmon-Nelson Mental 4.7969 4.3048 -.0132 <1105 .74 Nelson Gen, Significance -.2175 41334 .96 Nelson Note Details 4.1327 <0203 .90 Nelson Predict Outcomes -.1933 -.0269 .94 Nelson Total — 0563 zf.1485 .99 Paragraph Meaning ,8796 -£•1510 •A 0476 .81 Word Meaning t .8715 4-. 2514 -.1008 ",2320 .89 Reading Rate ■t .7926 <0368 .71 Comprehension -f- .8682 -.0489 -.0625 <1240 .78
TABLE VI Factor Weightings, Analysis I
Variables Factors Variation Accounted 1 2 4 5 N.S. Paragraph f.836S <3885 <1535 *.0004 .80 N. S, Word A 3808 <3393 -.1343 <0503 <0074 ♦ 84 Chapman-Cook Speed <4.8458 SIS 58 <1744 -1.1628 -.0068 .81 Monroe Speed <8336 ".3043 -.1459 <0504 *.0053 • 81 Monroe Comprehensio <8933 ".1934 -.2722 A 0313 -.0095 .91 Burgess P. S. I. <8313 21*3 1.1732 —.1225 - .0336 .77 Henmon-Nelson Menta <8043 A 3093 -.0029 -.2687 *.0134 .83 Nelson Word <8863 £1536 ".0191 f ,0£®l — .1063 • 83 Nelson Paragraph 1.8840 1 -• <0677 -.0208 .81
TABLE VII Factor Weightings, Analysis II
variables Factors variation Accounted far 1 2 3 4 yew Stanford, paragraph 8730 4 .3934 —.1144 .93 Mew Stanford, word 4*8794 2572 4.2823 4.1095 .93 yew Stanford, Total 4*8899 4 .3 749 4.0870 4.2332 .99 Chapman-Cook Speed 4.8428 — .2312 4.0963 - .0657 .77 Monroe Rate 4.8067 - .2653 4 .2024 — .1396 .78 Monroe comprehension 4.8795 — .1264 4 .1770 —.0773 .85 pintner yon-Language 4.2645 4 .3133 — .1798 — .2988 .29 Burgess 4.8156 - .1536 - .0086 — .0502 .69 He nmo n-re Ison Men tai 4.8045 4 .3217 - .0747 - .1754 .78 Melson word 4.8942 4 .0866 4 .1844 - .0616 .83 Melson General significance 4*9195 -T.2202 — .1403 4- .1524 .90 yelson vote retails 4 . 9094 — .1971 — .2076 4- .0102 .90 Melson Predict Outcomes 4.9137 — .1800 —.2017 4 .1657 .93 Melson paragraph 4.9291 -—,2190 -.2244 4 .1516 .98 Melson Total 4.9691 — .1200 — .0176 4.1216 .95 Composite paragraph 4.8770 4 .0845 — .1710 4 .0745 .83 Composite word 4.8839 4 .1524 4.2516 "A.0780 .87 Composite Rate 4.8117 .2553 + .1980 -.1717 .78 Comprehension 4*8664 — .0253 -- .0388 4- .0265 .76
Table VIII Factor Weightings, Analysis III
Variables Factors Variation Accounted for 1 2 3 4 New Stanford Paragraph y.0228 y.1109 .81 New Stanford Word -A 87 39 -.2420 y».O219 .87 Pintner Non-Lang, cental . 4249 f .0907 .f.0070 .35 Henmon-Nelson Mental /.8709 .83 Monroe Reading Rate 7“ — — /•7072 .3984 f-.1398 -.0340 .67
TABLE IX Factor Weightings, Analysis IV
Variables Factors Variation Accounted for 1 2 3 4 New Stanford-Total A 8282 — 3315 .0394 .79 Monroe Hate /.7313 — 0625 .68 Pintner Non-Lang. A 2739 -.2055 .22 Henmon-Nelson Men. <.7520 -.4299 y-. 2722 .89 Nelson Reading Total A 8 402 -.0315 .1938 .88 No. of Fixations / .0341 .1183 .85 Duration of Fixation 6372 -.5461 /.0421 .71 Perceptual Time /.8185 4.4102 -.3221 ,96 Regressive movements P " +.5812 .73
TABLE X Factor Weightings, Analysis V
Variables Factors Variation Accounted for 1 2 3 4 5 K. S. Paragraph 4**3854 -41274 —.3471 -T.0893 4. 0935 .90 K. 5. ”’ord 4.8772 - .0781 -.2729 4.0546 — .1929 .89 K. S. Total 4.9394 —.0630 —.3917 — .0194 .94 Chapman-cook Speed 4.8485 -.0521 4.2375 -4.1124 —.0786 .78 Monroe Rate 4.8015 -.0551 4. 1544 4.4621 4 .2563 .94 Monro e pomprehe ns ion 4.8656 - .1671 -/..1334 4.2797 —.0683 .89 pintner Kon-Language 4*2524 —.0949 —.2939 —.0843 0222 .17 gurgess 4.8155 — .0227 4 .1816 4 .1264 y.0166 .72 penmon-relson Mental 4.7913 -.1869 -.3 052 —.0565 —.0466 .74 pelson vord 4.8835 —.1388 — .1345 -.0565 — .1070 .83 Kelson Gen. Signif. 4.9107 -.1319 -/.2310 — .1395 4-. 1473 .95 Kelson ”Ote Details 4.8973 — .1304 4.2198 —.1237 4.0941 .90 Kelson Outcomes 4.9032 —.0947 4.1925 —.1433 4.1768 .92 Kelson paragraph 4.9224 — .1184 4.2283 — .1762 4.1974 .98 Kelson Total 4*3573 — .1625 4.0692 — .1192 4.1171 .97 composite paragraph 4*8854 _ .1502 — .0716 —.0943 .80 Composite 48748 -.1144 — .1784 4.1046 — .0641 .81 Composite Rate 4*8033 —.0814 4.2554 4 .0166 — .5336 .98 comprehens ion 4.8532 — .1367 4.0645 4.1024 4 .0480 .76 Reading Grade 4.7358 - .1557 — .0469 — .0746 4 .1127 .56 Ko. of fixations 4.5393 y .6568 -.1055 4 .1931 —.0129 .77 Duration of pix. 4.6021 4 .2789 4.1335 — .2606 — .0590 .53 perceptual Tune 4*8615 4 .6947 — .0262 —.0770 .0965 .93 Re gre s s i ve Mo vement s 4.4180 4 ,6eoi [4 .0039 - .1565 4 .2052 .69
Table XI Factor Weightings, Analysis VI
Analys is Variable number I II III V VI Variation Accounted -fan. yew Stanford, Paragraph 1 - .•8368 .8730 .8911 .8654 ,76 pew Stanford, word 2 .8802 .8794 .8739 .8772 .77 yew Stanford, Total 3 ,8681 .8899 .8894 .78 Chapman-Cook, Speed 4 .8458 .8428 .8485 .71 Monroe Rat e 5 .8326 .8067 .7072 ,8015 .62 Monroe COmprehens ion 6 .8933 .8795 .8656 .77 pintner yon-pang. Mental 7 .2645 .4130 .2524 .10 Burgess Scale 8 .8218 .8312 .8156 .8155 . 67 yenmon-yel son Mental . 9 .7969 .8043 .8045 .8709 .7913 .65 ITelson, Vocabulary 10 .3863 .8942 O 8835 .79 ye Ison, ceneral Significance 11 .9333 .9195 .9107 .85 ye Ison, Details 12 .9198 .9094 .8973 .83 yelson, predict Outcome 13 .9295 .9137 .9082 .85 Melson, paragraph 14 .8840 .9291 .9224 .83 yelson, mo tai 15 .9739 .9691 .9573 .93 Composite, paragraph 16 .8796 .8770 .8654 .76 Compos ite, Word 17 .8715 .8839 .8748 .76 Composite, Bate 18 .7926 .8117 .8033 .65 Composite, Comp re hens ion 19 .8682 .8664 .8532 .74 Teachers» Beading Grade 20 .7358 .54 number of pixat ions 21 .5398 .29 Duration of Fixations 22 .6021 .36 perceptual Time 23 .6615 .44 yo. Regressive Movements 24 .4180 .17 ■%- fill SlGcNS Are +
Table XIII Weightings of Factor A*
Ho. variable Weighting we ight ing Variable NO. 3 9 17 Few Stanford, Total Henman-Nelson, Mental Composite, word Analys ;3349 .30 48 .2514 LS I ;2175 .1933 .1809 .1508 .1373 Kelson, Significance Kelson, Outcomes Kelson, Details Compos ite, Rate Burgess scale ' 11 13 12 18 8 9 1 2 10 Henman-ReIson, Menta1 Few Stanford, Paragraph Rew Stanford, word Melson, Word Analys: .3092 .2885 .2293 .1536 is JI .3043 .2123 .1934 .1858 Monroe, Rate Burgess Scale Monroe, Comprehension Chapman-Cook, pate 5 8 6 4 1 3 9 7 2 17 Few Stanford, paragraph pew Stanford, Total Henman-^e1s on, Mental won-Language, Rental Few Stanford, Word Composite, word Analys .3934 .3749 .3217 .3133 .2572 .1524 is III .2653 .2553 .2312 .2202 .2190 .1971 .1800 .1536 Monroe, Rate Composite, Rate Chapman-Cook, pate Kelson, Significance Kelson, paragraph Kelson, Details Kelson, Outcomes Burgess scale 5 18 4 11 14 12 13 8 H H C\J ( Few Stanford, Total rew Stanford, paragraph Henman-poison, Mental Ron-Language, Mental pew Stanford, word composite, word Kelson, word Humber of Fixations ■ ——— Analys phird facte rev .3917 .3471 .3152 .2939 .2729 .1784 .1345 .1055 is VI Dr with s srsed) .2455 .2375 .2310 i .2283 .2198 .1925 .1816 .1544 .1335 .1224 igns composite, Rate Qhapman-cook, Rate Kelson, Significance Kelson,, paragraph Kelson, Details Kelson, Outcomes Burgess scale Monroe, Rate Duration of fixations Monroe, Comprehension 18 4 11 14 12 13 8 5 22 6
Table XIV COMPARATIVE WEIGHTINGS, FACTOR "B"
J$o. Variable Weighting __ Variable No. Analysis I (Sians reversed') 18♦ Composite, Rate .2437 .1892 Nelson Outcomes 13 8. Burgess Scale .1981 .1510 Composite Paragraph 16 17. Composite, Word .1008 .1485 Nelson, Total 15 .1334 Nelson, Significance 11 .1327 Nelson, Details 12_ Analysis II 6. Monroe, Cornprehension (Signs r .2722 aversed) .1744 Chapman-Cook, Rate 4 5. Monroe, Rate • 1459 .1732 Burgess Scale 8 2. New Stanford, Word .1264 • 1525 N. Stanford, Paragraph 14 Analysis III ' 2, New Stanford, Word ,.2823 .2190 Nelson, Paragraph 14 17. Composite, Word .2516 .2076 Nelson, Details 12 5® Monroe, Rate .2024 .2027 Nelson, Outcomes 13 18* Composite, Rate .1980 .1798 Non-Language Mental 7 10. Nelson, Word .1844 ,1710 Composite, Paragraph 16 6. Monroe, Comprehension .1770 • 1403 Nelson, Significance 11 Analysis VI (Fourth factor) J 5. Monroe, Rate .4621 .2606 Duration of Fixations 22 6. Monroe, Comprehension .2797 .1762 Nelson, Paragraph 14 21. Number of Fixations .1931 • 1433 Nelson, Outcomes 13 24. Number of Regressions .1565 .1395 Nelson, Significance 11 8. Burgess Scale .1264 .1237 Nelson, Details 12
Table XV COMPARATIVE WEIGHTINGS, FACTOR "C"
Analysis I Analysis II Analysis III Analysis IV Analysis V Analysis VI Factors Fa ctors Factors Factors factors Factors Variable Numb* 3? 1 2 _3 4 1 2 3 4 1 2 3 4 1 2 3 4 ' 1 2 3 4 1 2 3 4 5 New Stanford, Paragraph 1 \8368 +.2885 +.1525 1.0559 +.8730 4.3934 71144 +.0791 4.8911 4.0228 -.0829 4.1109 + 8654 -.1274 73471 ".0893 + .0935 New Stanford, Word 2 4.8802 4.2293 71243 +. 0503 4.8794 4.2572 1.2823 1.1095 +.8739 -’.2420 -.2371 4.0219 + 8772 70782 -.2729 + .0546 -.1929 New Stanford, Total 3 4 -.8681 -H 3349 +.0099 —.0072 * +.8899 t.3749 +.0870 +.2332 + 8292 -.3315 1.0394 -f.0769 t8894 7063C 73917 -’.0194 '.0053 Chapman-Co ok, Speed 4 \8458 71858 tl628 1.8428 -.2312 '.0963 ~.O657 1.8485 70521 ■’’.2375 4.1124 -.0786 Monroe Rate 5 4.8326 '-.3043 ~.1459 +.0504 t.8067 -.2653 ".1396 4.7072 -.3984 +.1298 7 0340 +7313 —.1405 70625 -.3495 to 015 >0551 4.1544 4.4621 -*-.2563 Monroe Comprehens ion 6 1-8933 71934 -.2722 +.0313 18795 71264 tl77O -.0773 +8656 71671 *1224 *.2797 -.0683 Pintner Non-Lang. Mental 7 1.2645 +.3133 '.1798 -.2988 1.4130 4. 4249 4.0907 4.0070 12739 -.2055 +. 2368 +. 2125 + 2524 70949 '.0843 70222 Burgess Scale 8 +8218 ”.1373 -.1981 +.1330 18312 72123 +.1732 71225 1.8156 --.1536 7 0086 -.0502 70227 4.1816 *.1264 -+.0166 Henmon-Nelson Mental 9 .7969 +.3048 -'.0122 4.1105 +.8043 +.3092 -.0029 72.687 +. 8045 +.3217 70747 ".1754 +.8709 + 1298 71287 ”1986 4.7520 ” .4299 1.7913 71869 “.3052 '.0565 70466 Nelson, Vocabulary 10 1.8863 +.3.536 70191 +.0911 >8942 4.0866 tl844 ".0616 +.0835 71338 ~.1345 '.0523 71070 Nelson, General Significance 11 t .9333 72175 +.1334 —.1406 1.9195 -.2202 ”71403 +.1524 19107 71.319 +.2310 71395 + .1473 Nelson, Note Details 12 + .9198 —.1809 1 + .0203 4.9094 -.1971 72076 4.0102 1*3973 71304 +.2198 71237 +. 0941 Nelson, Predict Outcome 13 .9295 -.1933 -t.1892 —.0269 4*9137 72017 1.1657 19082 7 0947 *.1925 71433 +.1768 Nelson, Paragraph 14 +.8840 ■70972 + 0677 71056 19291 -.2190 7 2244 +.1516 71184 72283 71762 +.1974 Nelson, Total 15 +.97'39 -.0563 +1485 ~.135O 19691 -.1200 70176 4.1216 *o402 73742 -.0315 '.1938 19573 -.1625 *.0692 71192 +“.1171 Composite, Paragraph 16 -t.8796 4.0823 +.1510 4.0476 4.8770 4.0845 “1710 1.0745 "18654 71502 70716 70943 +.1280 Composite, Word 17 +.8715 +.2514 -’.1008 —.2320 1-8839 4.1524 +. 2516 i.0780 ’8748 71144 71784 + 1046 “".0641 Composite, Rate 18 t.7926 ".1508 "72437 +.0368 48117 '.2553 +.1980 71717 I8033 7 0814 *.2554 +.0166 -.5336 Composite, Comprehension 19 4 .8682 -.0489 -.0625 f.1240 48664 -.0256 -.0388 +.0265 18532 -.1367 t.0645 +.1024 +.0480 Teachers * Reading Grade 20 17358 71557 7 0469 70746 +1127 Number of Fixations 21 4 + 7041 +5924 + . 0341 -1183 15393 t.6568 ■'■.1931 70129 Duration of Fixations 22 t6372 *.0018 75461 7 0421 16021 72789 +.1335 ?2.6o6 -0590 Perceptual Time 23 24 18185 *.4102 73221 368 +6615 +6947 70262 70770 To 965 No. Regressive Movements 15950 +■5812 41355 71073 14180 + 6801 +. 0039 11565 +2052
TABLE XVI RESULTS FROM THE SIX ANALYSES
CHAPTER VIII SUMMARY
This study has consisted in the analysis of certain measures of silent reading ability by means of the multiple factor analysis technique. This technique has been shown to be a useful supplement to the other statistical devices, such as mean, deviation, norms, and correlation, which are customarily used in describing and analyzing abilities in terms of test scores. The utility of the technique lies in its ability to yield a simple pattern of relationships between the measures used, and, hence, to make for clearer concepts concerning the nature and organization of the processes involved.
The history of the development of the factor theory, from its first formulation by Spearman in 1904 down to the present, has shown the recent growth of interest in the theory, and the significant advances made in techniques for determining the factors necessary to account for a table of intercorrelations, and for determining
the weightings of tests with the various factors. The recent developments in the technique, point to a more general application of the theory of factors to many of the important psychological problems now engaging the attention investigators.
While due credit must be given to Spearman for his early formation of th< theory of factors, and for his labors through the years in the proof and application of the theory, attention is called to the fact that Thurstone has been instrumental in generalizing the theory and in developing a technique which is readily applicable to many types of psychological problems. The technique developed by Thurstone has been recently used in two studies of ’’lnterest J* and two studies of "Personality.” The application of the technique to school subjects has been suggested in a comparative study of the grades of children in the school subjects. The present study extends the application of the factor analysis method to the study of a single school subject, and presents the results of several analyses of certain measures of silent reading ability.
The subjects contributing the data for this study include 210 sixth grade boys and girls. Scores were obtained on a battery of tests including the New Stanford Reading Test, the Nelson Silent Reading Test, the Burgess Scale, the Conroe Standardized Silent Reading Test, the Chapman-Cook Speed of Reading Test, the Henmon-Nelson Test of kgntal Ability, and the Pintner Non-Language Test. The eye-movements of the subjects were also photographed, and the teachers 1 reading grades were secured.
The intercorrelations of the twenty-four scores obtained from these measures constituted the primary data of the study. An examination of the intercorrelations revealed a variety of information concerning the relet ionships existing between the tests, and indicated the need of further analysis.
The factor analysis technique was applied to six combinations of the tests and the findings may be summarized as f ollows:
1. A high degree of consistency was found in the weightings of each variable in the different analyses.
2. The analyses indicate that three independent factors were operative in the tests employed.
3. These three factors, ”A”, ‘*B n , and “C”, are capable of describing the reading tests to the extent of accounting for eighty-five per cent of the average variations.
4. The three factors are capable of describing ninety-five per cent of the variation on the New Stanford Reading Test, end as much as seventy per cent of the variation on each of the reading tests employed in this study.
5. The three factors may be characterized as follows: (1) Factor H A H , a general factor operative in all of the tests, but in varying degrees in the different tests; (2) Factor H B”, a factor differentiating reading rate from a group of measures, including word meaning, mental ability, and the New Stanford Test; and (3) Factor ”0”, which makes for similarity between measures of reading rate and measures of word meaning, and which contrasts these measures with the comprehension scores on the Nelson Reeding Test.
6. The factor patterns indicate that any general concept of silent reading ability should include at least three elements, as represented by factors ”A”, ”B”, and ”C H , the nature of which may be described at present in terms of their relationship to the different tests, A more detailed study of the nature of these factors should prove helpful in describing more adequately the elements, or abilities, which are essential in the reading process.
BIBLIOGRAPHY A*
Burgess, M. A., A Scale for Measuring Ability in Silent Reading. Russel Sage Foundation, ITewTorFTltst.
Buswell, G. T., "Fundamental Reading Habits; A Study of Their Development." Supplementary tional Monograph, No. 21, universityof fhi c ago 7 1922.
Burt, C., ’’Experimental Tests of General Intelligence.” British Journal of Psychology, 1909, Vol. 3, pp.
Carter, H. D., Pyles, M. K., and Bretnell, E. P., ”A Comparative Study of Factors in Vocational Interest Scores of High School Boys.” Journal of Educational Psychology, 1935, Vol. 2?", pp.'6l-38.
Chapman, J. C. and Cook, Chapman-Cook Speed of Reading Test, Form A, Educatlonal Test Eureau, Inc., Philadelphia, 1924.
Dodd, S. C., R The Theory of Factors.” Psychological Review. 1928, Vol. 35, pp. 211-234, 281- 275.
Eurich, Alvin C., ‘‘Additional Data on the Reliability and Validity of Photographic Eye-Movement Records.*’ Journal of Educational Psychology. 1933, —
Flanagan, J. 0., Factor Analysis in the Study of Personality. Stanford University Press, Stanford University, 1935.
Garnett, J. C. ’’General Ability, Cleverness, and Purpose." British Journal of Psychology, 1919, pp. 345-366? ' ’
Garnett, J. C. M., ”0n Certain Independent Factors in Mental Measurements. 0 Proceeding of the Royal SocietyVof London. 1919, 9 6a, pp. 31-111.
Garrett, Henry C., Statistics in Psychology and Education. Longmans, Green and Company, New Y0rk,1926
♦ Bibliography A includes references cited in the study.
Gates, A., “An Experimental and Statistical Study of Reading and Reading Tests,” Journal of Educational Psychology, 1921, Vol. 30’5-514,
Gray, C. T., Deficiencies in Reading Ability, D. C. He ath and Company, Chicago, 1022.
Gray, C. T., “Types of Reading Ability Exhibited Through Tests and Laboratory Experiments,” Supplementary Education Monograph No. 5, University or dhicago, *
Hart, 8., and Spearman, C., “General Ability, Its Existence and Nature.” British Journal of Psychology. 1912, Vol. 5, pp. 51-84. ‘
Henmon, V. A. C., and Nelson, M. J., The Henmon-Neleon Test of cental Ability, Houghton Mifflin do., New York, 1931.
Johnson, Constance, “Case of Cut Flowers,” in Far and Near, A Fourth Reader, Johnson Publishing Co,, New York, 1928.
Kelley, T. L., Crossroads in the Mind of Man,-Stanford University Press,TTtanford IJnTversity, 1928.
Kelley, T. L., Ruc£, 0. M., and Tenman, L. U., The New Stanford Reading Test, World Book Company, Yonkers-on-Hudson, I^3o.
Krueger, F., and Spearman, C. “Dii Korrelation zwischen verschiedenen geistigen Leistungsfahigkeiten.“ Zeitschrift fur Psychologie, 1906, Vol. 44, pp. 50-114.
Line, W. H., and Hedman, H. 8., “Simplified Statement of the Two Facts Theory,” Journal of Educational Psychology, 1933, Vol. 24, pp. Is
Line, W., Rogers, K. H., and Kgplon, E., “Factoranalysis Techniques Applied to Public School Problems.” Journal of Educational Psychology, 1934, Vol. 25, pp. 55^65".
Monroe, Walter S., Monroe 1 b Standardized Silent Reading Test, (Revised School "Pub 1 ishing do., Bloomington, Ohio, 1321.
Nelson, M. J., The Nelson Silent Reading Tegr, Houghton Miffliii 3o77^ ew York, iSSu.
Perry, R. C., A Group Factor Analysis of the Adjustment University ofSouthern California Press, Los Angeles, 1934.
Pintner, R., Intelligence Testing, Henry Holt and Co., New York,
Pintner, R., Non-Language Mental Tests, College Book Comp any, Columbus, Ohio,
Pintner, R., "Results Obtained with the Non-Language Group Tests,” Journal of Educational Psychology, 1924, Vol. 15, pp ♦ 47 Spearman, C., "Autobiography" in History of Psychology in Autobiography, Clark University £ress, forchester, kass., 1930.
Spearman, C., "Correlations Between Arrays in a Table of Correlations,” Proceedings of Royal Society of London, 1922, Vol. 101 a, pp. 04-100.
Spearman, C., "General Intelligence Objectively Determined and Measured.” American Journal, of Psychology, 1904, Vol. 157 pp .
Spearman, C., "Manifold Sub-theories of the Two Factors.” Psychological Review, 1920, Vol. 27, pp. 175-190.
Spearman, C., and Holzinger, K., ’’Notes on the Sampling Error of Tetrad Differences,” British Journal off Psychology, 1925, Vol. 16, pp. 86-B§. '
Spearman, C. ’’Recent Contributions to the Theory of Two Factors,” British Journal of Psychology. 1922, Vol. 13, pp.' Sg-W.
Spearman, C.,"Some Comments on Mr. Thomson’fe Paper* 1 , British Journal of Psychology, 1916, Vol. 8, pp. .
Spearmen, C., The Abilities of Man, The Macmillan Company, Sfew York,
Spearman, 0., "The Factor School of Psychology,” Psychologies of 1930, Edited by C. Murchison. dlark University Press, Worchester, Mass., 1930.
Spearmanp C., and Holsinger, K., "The Sampling Error in the Theory of Two Factors." British Journal of Psychology, 1924, Vol. 15, pp. 17-22.
Spearman, C., "The Theory of Two Factors." Psychological Review, 1914, Vol. 21, pp. 101-115
Spearman, C., "What the Theory of Factors is Not." Journal of Educational Psychology, 1931, Vol. 22, pp. 112—117.
Thomson, G. H., "A Hierarchy Without a General Factor. British Journal of Psychology. 1916, Vol. 8, pp?
Thomson, G. H., "The Hierarchy of Abilities." British Journal of Psychology, 1919, Vol. 9, pp. TST-344.
Thomson, G. H., "The Proof or Disproof of the Existence of General Ability." British Journal of Psychology. 1919, Vol. $, pp. Kl-330.
Thuretone, L. L., "A Multiple Factor Study of Vocational Interests.” Personnel Journal, 1931, Vol. 10, pp. 198-205.
Thurstone, L. L., A Simplified Multiple Factor Method and an Outline of the Somput atione. ffnlversity ofChicago 6bok store, Chicago, 1933.
Thurstone, L. L., ’’Multiple Factor A nalysis.” Psychological Review, 1931, Vol. 38, pp. 406-427.
Thuretone, L. L., The Theory of Multiple Factors. Edwards Bros., Inc 7, Ann Arbor, iSTT.
Woodworth, R. S., Combining the Results of Several Teste." Psychological Review, 1912, Vol. 19, pp. 97-12 J. —
BIBLIOGRAPHY B*
Adams, H. F., "A Non-Intellectual "G" Factor." Journal of Educational Psychology, 1932, Vol. 23, "173-178.
Cureton, E. E., "Errors of Measurement and Correlation." Archives of Psychology, 1932, Vol. 19, No. 125.
Cureton, E. E. and Dunlap, J. W., ’’Some Effects of Heterogeneity on the Theory of Factors.” American Journal of Psychology, 1930, Vol. 42, pp. 608-620.
Dodd, S. C., ’’Coefficient of Equiproportion.” Journal of Educational Psychology, 1928, Vol. 19, pp. 217-225.
Garnett, J. C. M., "Further Notes in the Single General Factor in Mental Measurements." British Journal of Psychology, 1931, Vol. 22, pp. 464-372.
Holzinger, K., ”0n Tetrad Differences of Overlapping Variables.” Journal of Educational Psychology, 1929, Vol. 20, pp. Sl-S?.
Holsinger, K., ’’Reply to Kelley.” Journal of Educational Psychology, 1931, pp. 455-457.
Holzinger, K., M Thorndike * s C. A. V. D. is full of H G f ’. Journal of Educational Psychology, 1931, Vol. 22, pp. i&L-iSS.
Holzinger, K., Statist!cal Resume of the Spearman twofact o r theory 7 Universif yof Chicago Press, 1930.
Holzinger, K. and Swineford, F., ’’Uniqueness of Factor Patterns.” J diurnal of Educational Psychology. 1932, Vol. 23, pp. 547-SSS\
Hotelling, H., “Analysis of a Complex of Statistical Variables in principal Components.' 1 Journal of Educational Psychology, 1934, Vol. 24, pp. , ’ - <O.
Irwin, J. 0., ”On the Uniqueness of Factor G for General Intelligence 11 . British Journal of Psychology, 1332, Vol. 22, pp. 360-363.
♦References consulted but not cited in this study.
Kelley, T. L., Interpretation of Education Measurements. World Book Company, Yonkers-on- Hudson, 1927.
Kelley, T. L., "What is meant by a 0 Factor?" Journal of Educational Psychology, 1931, Vol. pp. "364-366.
Mackie, J., "Sampling Theory as a Variant of Twofactor Theory." Journal of Educational Psychology, 1928, V 01.1 9, pp. 613-626.
Piaggis, H. T. H., "Three sets of conditions necessary for the existence of a "G" that is Real and Unique except in Sign." British Journal of Psychology, 1933, Vol. 24, pp.
Slocombe, C. S., "Constancy of G." British Journal of Psychology, 1926, Vol. 17, pp.
Slocombe, C. S., "Kelly Measures Mental Traits." Journal of Educational Psychology, Vol. 19, pp. YS V-lS) IT ’’
Spearman, 0., "Disturbers of Tetrad Differences." Journal of Educational Psychology, 1930, —
Spearman, C., "Further Note on the Theory of Two Factors." British Journal of Psychology, 1922, Vol. 13, pp. 266-270.
Spearman, C., "Pearson’s Contribution to the Theory of two Factors." British Journal.of Psychology, 1928, V 01.1 9, pp. 1 95-101.
Spearman, C., "Response to Kelly." Journal of Educational Psychology, 1929, Vol. 20, pp, 561-566. ' -
Spearman, C., and Holsinger, K., "The Average Value for the Probable Error of Tetrad Differences." British Journal of Psychology, 1929, Vol. 20, ppT'o-370.
Spearman, 0., “The Factor Theory and its Troubles—■ 11, Garbling the Evidence," Journal of Educational Psychology, 1933, Vol. $4, pp.
Spearman, C., ’’The Factor Theory and its Troubles — 111, misrepresentation of the Theory.” Journal o_f Educational Psychology, 1933, Vol. 24, pp. 591-801.
Spearman, C., "The Factor Theory and its Troubles— IV, The Uniqueness of G." Journal of Educational Psychology, 1934, pp. 142-153.
Spearman, C., "The Factor Theory and its Troubles— V, Adequacy of Proof." Journal of Educational Psychology. 1934, Vol. 257 PP* 516-515?
Spearman, 0., "The Factor Theory and its Troubles— Conclusion, Scientific Value.” Journal o_f Educational Psychology, 1934, Vol. 25 pp. 38 3-591.
Spearman, 0., "The Uniqueness of "G". Journal of Educational Psychology, 1929, Vol. 20, pp. 212-216.
Spearman, C., "The Uniqueness and Exactness of G." British Journal of Psychology, 1933, Vol. 24, pp. 166-168.
Thomson, G. H., "A worked out example of the possible Linkages of Four correlated variables on the Sampling Theory." British Journal of Psychology, ~1927, Vol. ic.'pp.'
Thomson, G. H., "General Factor Fallacy in Psychology," British Journal of Psychology, 1920, Vol. 10, pp.
Thomson, G. H., ’’Hotelling’s Method Modified to give Spearman’s ”G”.” Journal of Educational 1934, Vol. 25, pp. 366-374.
Thomson, G. H. and Garnett, J. 0. M., "Joint note on ’The Hierarchy of abilities*." British Journal of Psychology, 1919, Vol? pp. 387-361,
Thomson, G. H., "Note on ‘Hierarchical Order’ among correlation coefficients.” British Journal of Psychology, 1923, Vol. 14, pp.
Thomson, G. H., ’’The General Factor Fallacy in Psychology,” British Journal of Psychology, IS3O, Vol. 10, pp. 313-326.
Thomson, G. H., ’’The Tetrad-Difference Criterion.” British Journal of Psychology, 1927, vol. 17, pp. 235-2531
Thomson, G. H., “The Role of interference Factors in Producing Correlations.” British Journal of Psychology, 1920, Vol. 10, pp. ST-ISST
Thurstone, L. L., “Multiple factor Analyses.” National Research Council, 1931, pp. 7-18. ~
Tryon, R. C., “Factor Theory and its Troubles.” ’’Misrepresentation of a criticism of the theory.” Journal of Educational Psychology, 1934, Vol. 25, pp.^23 2-2331
Wilson, E. 8., ’’Comment on Professor Spearman’s Note.” Journal of Educational Psychology, 1929, Vol. 20f pp. 2171'223.“” "
Wishart, J., "Sampling Errors in the Theory of two Factors.” British Journal of Psychology, 1928, Vol, I3TPP• —