A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

Abstract

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.

Appendix

1.1 A.1 Computational Details

The first and second derivatives of the densities involved in the calculation of the E-step of Section 2.2 and the standard errors of Section 2.3 are presented. The computational issues that emerge involve both the log-likelihood of the univariate probabilities of response and that of the latent variables. The first issue considers the first and the second derivative of $\log (\gamma_{it}^{(s)}-\gamma_{it}^{(s-1)})$. It is easily seen that

$$\frac{\partial\log(\gamma_{it}^{(s)}-\gamma_{it}^{(s-1)})}{\partial\theta_k}=\frac{1}{\gamma_{it}^{(s)}-\gamma _{it}^{(s-1)}} \biggl(\frac{\partial\gamma_{it}^{(s)}}{\partial \theta_k}-\frac{\partial\gamma_{it}^{(s-1)}}{\partial\theta_k} \biggr),$$

where

The second derivative of $\log(\gamma_{it}^{(s)}-\gamma _{it}^{(s-1)})$ is,

where

$$\frac{\partial^2\gamma_{it}^{(s)}}{\partial\theta_k\partial\theta _l}= \frac{\partial\gamma_{it}^{(s)}}{\partial\theta_l} \bigl (1-2\gamma_{it}^{(s)}\bigr)\frac{\partial\eta_{it}^{(s)}}{\partial\theta_k}.$$

Regarding the log-likelihood of the latent variables, two cases will be discriminated: the first case involves two time-dependent latent variables and two item random effects (z _t ,z _t′,u _i ,u _i′), when, for example, a pair (y _it ,y _i′t′) is considered; whereas, the other case involves one time-dependent latent variable and two item random effects (z _t ,u _i ,u _i′) corresponding to a pair (y _it ,y _i′t ). The covariance matrix of z _t and z _t′ is

$$\Sigma=\left(\matrix{\phi^{2(t-1)}\sigma_1^2+A_t^* & \phi^{t+t'-2}\sigma _1^2+\phi^{t'-t}A_t^* \cr\noalign{\vspace*{2pt}}\phi^{t+t'-2}\sigma_1^2+\phi^{t'-t}A_t^* &\phi^{2(t'-1)}\sigma_1^2+A_{t'}^*}\right)$$

with

$$A_t=\sum_{k=0}^{t-2}\phi^{2k},\qquad A_t^*=I(t\geq2)A_t.$$

Therefore, the logarithm of the joint density of (z _t ,z _t′,u _i ,u _i′) is

where $\omega_{11}=\phi^{2(t'-1)}\sigma_{1}^{2}+A^{*}_{t'}$, $\omega _{12}=\phi^{t+t'-2}\sigma_{1}^{2}+\phi^{t'-t}A_{t}^{*}$, $\omega_{22}=\phi ^{2(t-1)}\sigma_{1}^{2}+A_{t}^{*}$. With respect to the parameters, ϕ or $\sigma_{1}^{2}$, the derivatives of the logarithm of the joint density, are

$${\partial\log h(z_t,z_{t'},u_i,u_{i'})\over\partial\theta_k}=-{1\over2} {{\partial|\Sigma|\over \partial \theta_k}\over|\Sigma|}-{1\over2} \bigl[K_{\theta_k}^{\omega_{11}}z_t^2 -2K_{\theta_k}^{\omega _{12}}z_tz_{t'}+K_{\theta_k}^{\omega_{22}}z_{t'}^2\bigr],\quad \theta_k=\sigma_1^2,\phi $$

and

$$K^\alpha_{\theta_k}={{\partial\alpha\over\partial \theta_k}|\Sigma|-\alpha{\partial|\Sigma|\over\partial \theta_k}\over|\Sigma|^2},\quad \alpha=\omega_{11},\omega_{12},\omega_{22}.$$

The first derivative with respect to the parameters $\sigma^{2}_{uk}$ is

$${\partial\log h(z_t,z_{t'},u_i,u_{i'})\over\partial\sigma^2_{uk}}-{1\over2\sigma_{uk}^2}-{u_k^2\over2\sigma_{uk}^2},\quad k=i,i'.$$

The second derivative with respect to θ _k and θ _l :

$${\partial^2\log h(z_t,z_{t'},u_i,u_{i'})\over\partial\theta_k\partial\theta_l}=-{1\over2} {{\partial^2|\Sigma|\over\partial\theta_k\partial \theta_l}|\Sigma|-{\partial|\Sigma|\over\partial\theta_k}{\partial|\Sigma|\over\partial\theta_l}\over|\Sigma|^2}-{1\over2} \bigl[L_{\theta _k,\theta_l}^{\omega_{11}} z_t^2-2L_{\theta_k,\theta_l}^{\omega _{12}}z_tz_{t'}+L_{\theta_k,\theta_l}^{\omega_{22}}z_{t'}^2\bigr]$$

where $\theta_{k},\theta_{l}=\sigma_{1}^{2},\phi$ and

$$L^\alpha_{\theta_k,\theta_l}={ ({\partial^2\alpha\over \partial\theta_k\partial\theta_l}|\Sigma|+{\partial \alpha\over\partial\theta_k} {\partial|\Sigma|\over\partial \theta_l}-{\partial\alpha\over\partial\theta_l} {\partial|\Sigma|\over\partial\theta_k}-\alpha{\partial^2|\Sigma|\over\partial\theta_k\partial\theta_l} )|\Sigma|^2-2|\Sigma|{\partial|\Sigma|\over\partial \theta_l} ({\partial\alpha\over\partial\theta_k}|\Sigma|-\alpha{\partial|\Sigma|\over\partial\theta_k})\over|\Sigma|^4}$$

where α=ω ₁₁,ω ₁₂,ω ₂₂. The second partial derivative of the logarithm of the joint density with respect to $\sigma^{2}_{ui}$ or $\sigma^{2}_{ui'}$ is

$${\partial^2\log h(z_t,z_{t'},u_k,u_{k'})\over\partial\sigma^2_{uk}\partial\sigma^2_{uk}}={1\over2\sigma^4_{uk}}+{u_k^2\over2\sigma^4_{uk}},\quad k=i,i'.$$

When the k indices are different, this second derivative is zero.

When the joint density of the triplet (z _t ,u _i ,u _i′) is considered, we set |Σ|=Var(z _t ), and ω ₁₁=1 and ω ₁₂=ω ₂₂=0.