Modeling Associations Among Multivariate Longitudinal Categorical Variables in Survey Data: A Semiparametric Bayesian Approach

Abstract

This paper proposes a semiparametric Bayesian framework for the analysis of associations among multivariate longitudinal categorical variables in high-dimensional data settings. This type of data is frequent, especially in the social and behavioral sciences. A semiparametric hierarchical factor analysis model is developed in which the distributions of the factors are modeled nonparametrically through a dynamic hierarchical Dirichlet process prior. A Markov chain Monte Carlo algorithm is developed for fitting the model, and the methodology is exemplified through a study of the dynamics of public attitudes toward science and technology in the United States over the period 1992–2001.

Appendices

Appendix A

Estimation proceeds through the following steps:

Step 1

Sample z _tigj , t=1,…,T, i=1,…,n _t , and g=1,…,K, from

where TN denotes a truncated normal distribution.

Step 2

Sample $\tilde{\pi}_{t}$, t=1,…,T−1, and $\tilde{\omega}_{tl}$, t=1,…,T, l=1,…,L, from

Step 3

Sample m _ti and c _ti , t=1,…,T, i=1,…,n _t , from a multinomial distribution with

where N _r (x;μ;Σ) denotes the probability density function of an r-dimensional vector having a multivariate normal distribution with mean vector μ and covariance matrix Σ, evaluated at x.

Step 4

Sample the component parameters $\{ \varLambda_{l}^{*}\}_{l = 1}^{L}$ from

Step 5

Sample each ψ _tig , t=1,…,T, i=1,…,n _t , and g=1,…,K, from

$$\psi_{\mathit{tig}}|A_{g},B_{g},\varLambda_{ti},z_{\mathit{tig}}\sim N_{q_{g}}\bigl(W(A_{g}'z_{\mathit{tig}} + \varOmega_{g}^{ - 1}B{}_{g}\varLambda_{ti}),W\bigr), \quad W = \bigl(A_{g}'A_{g} + \varOmega_{g}^{ - 1}\bigr)^{ - 1}. $$

Step 6

Sample μ _gj , g=1,…,K, j=1,…,p _g from

$$\mu_{gj}|z_{tgj},\psi_{tg},A_{g}\sim N\bigl(n + \sigma_{g0}^{ - 2}\bigr)^{ - 1}\Biggl( \sigma_{g0}^{ - 2}\mu_{g0} + \sum _{t = 1}^{T} \sum_{i = 1}^{n_{t}} (z_{\mathit{tig}j} - a_{g,j}\psi_{\mathit{tig}})^{2} ,\bigl(n + \sigma_{g0}^{ - 2}\bigr)^{ - 1}\Biggr). $$

Step 7

Sample a _g,j , the jth row A _g , g=1,…,K, and $j = 2, \ldots,p_{g_{1}} + p_{g_{2}}$, from

Step 8

Sample b _g,j , the jth diagonal element of B _g , g=2,…,K and j=1,…,q _g , from

Step 9

Sample $\sigma_{g,j}^{ - 2}$, g=1,…,K and j=1,…,q _g , from

Step 10

Sample the thresholds ς _gj,r , $g = 1,\ldots,K, j = 1, \ldots,p_{g_{2}}$, and r=1,…,ν _j , from the full conditional

which, following Albert and Chib (1993), is a uniform distribution on the interval

$$\bigl[\max\bigl\{ \max\{ z_{\mathit{tig}j}:y_{\mathit{tig}j} = r\} ,\varsigma_{gj,r - 1}\bigr\} ,\min\bigl\{ \min\{ z_{\mathit{tig}j}:y_{\mathit{tig}j} = r + 1\} ,\varsigma_{gj,r + 1}\bigr\} \bigr]. $$

Appendix B

Table B.1. National Science Foundation Surveys of Public’s Understanding of S&T data.