Using the filter-forward backward sampling algorithm in second-order Bayesian latent growth modeling
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In educational and social science research, large-scale testing data are frequently collected longitudinally so that researchers can evaluate change over time. Researchers may then wish to assess the impact of various explanatory variables on student growth in achievement outcomes. Use of structural equation modeling allows for the modeling of item-level measurement error and allows growth trajectories to vary by student. If a Bayesian perspective is adopted, one may use psychometric information known a priori about the test items in the estimation process, which may improve ability estimation. In addition, Bayesian estimation procedures, like the Kalman filter, are able to take advantage of the autoregressive structure of time series data to obtain closed-form solutions for ability distributions. In contrast, a structural equation modeling-based approach using likelihood-based estimation would need to rely on iteratively updating proposed model estimates and checking a discrepancy function, which might achieve a local minimum, or fail to converge. Researchers have previously estimated second-order latent growth models with IRT elements, and this work will expand upon that literature in a number of ways. Bayesian research to date has typically relied on use of the WinBUGS software program to estimate these models which does not allow for certain distributional assumptions. For instance, although certain non-informative priors may be specified, it is not possible to use improper non-informative priors with WinBUGS. Also, WinBUGS does not take advantage of the autoregressive structure of a time series analysis to speed up the estimation process, which is possible using the Kalman filter. Because thousands of iterations of calculation and random-number generation are recommended when using a Bayesian Gibbs sampler, the improved computational efficiency of the Kalman filter may make growth models easier to estimate. When time series data are highly correlated, the Kalman filter, theoretically, should improve the rate of convergence for a Gibbs sampler. Furthermore, research on second-order latent growth modeling has not evaluated the use of informative priors for item parameters. The present work will address these limitations. Parameters based on educational psychology research will be used to simulate a dataset which will be analyzed with and without the Kalman filter. Then, convergence diagnostics, including the traceplot, will be assessed to determine whether the Kalman filter improved the rate of convergence. Additionally, both informative and non-informative priors will be used for item parameters, and parameter recovery will be assessed.