A Bayesian Modeling Approach for Generalized Semiparametric Structural Equation Models

Abstract

In behavioral, biomedical, and psychological studies, structural equation models (SEMs) have been widely used for assessing relationships between latent variables. Regression-type structural models based on parametric functions are often used for such purposes. In many applications, however, parametric SEMs are not adequate to capture subtle patterns in the functions over the entire range of the predictor variable. A different but equally important limitation of traditional parametric SEMs is that they are not designed to handle mixed data types—continuous, count, ordered, and unordered categorical. This paper develops a generalized semiparametric SEM that is able to handle mixed data types and to simultaneously model different functional relationships among latent variables. A structural equation of the proposed SEM is formulated using a series of unspecified smooth functions. The Bayesian P-splines approach and Markov chain Monte Carlo methods are developed to estimate the smooth functions and the unknown parameters. Moreover, we examine the relative benefits of semiparametric modeling over parametric modeling using a Bayesian model-comparison statistic, called the complete deviance information criterion (DIC). The performance of the developed methodology is evaluated using a simulation study. To illustrate the method, we used a data set derived from the National Longitudinal Survey of Youth.

Appendix:  The Full Conditional Distributions

The full conditional distributions involved in the Gibbs sampler are given as follows.

(1) The full conditional distribution of Ω: $p(\boldsymbol{\varOmega} |\mathbf{D},\mathbf{Z}^{*},\mathbf{W},\boldsymbol{\theta}_{*})= \prod_{i=1}^{n} p(\boldsymbol{\omega}_{i}|\mathbf{d}_{i},\mathbf {z}_{i}^{*},\mathbf{w}_{i},\boldsymbol{\theta}_{*})$, and

(A.1)

(2) The full conditional distribution of W: $p(\mathbf {W}|\mathbf{D} ,\mathbf{Z}^{*},\boldsymbol{\varOmega},\boldsymbol{\theta}_{*})=\prod_{i=1}^{n} \prod_{j=r_{3}+1}^{r_{4}} p(\mathbf{w}_{ij}|u_{ij},\allowbreak\boldsymbol{\omega }_{i}, \boldsymbol{\theta}_{*})$, and

$$ p(\mathbf{w}_{ij}|u_{ij}=l,\boldsymbol{\omega}_i,\boldsymbol{\theta} _*) \stackrel{D}{=}N \bigl[\boldsymbol{\mu}_j+\mathbf{1}_{L-1}\boldsymbol{\varLambda}_j' \boldsymbol{\omega}_i, \mathbf{I}_{L-1}\bigr]\mathrm{I}(\mathbf {w}_{ij} \in \mathbf{R}_l), $$

(A.2)

where I _L−1 is an identity matrix with order L−1, and

The distribution of p(w _ij |u _ij ,ω _i ,θ _∗) is a truncated multivariate normal distribution. Following Song et al. (2007), we use the following partitioning of variables to simulate observations {w _ij,1,…,w _ij,L−1} via Gibbs sampler. Let w _ij,−l be w _ij with w _ij,l deleted, the distribution of w _ij,l given u _ij , w _ij,−l , ω _i , and θ _∗ is a univariate truncated normal distribution as follows:

(A.3)

(3) The full distribution of b _l is

$$ p(\mathbf{b}_l|\cdot) \stackrel{D}{=}N\bigl[\mathbf{b}_l^*, \boldsymbol{\varSigma}_{bl}^*\bigr], \quad l=1,\ldots,m, $$

(A.4)

with the constraint $\mathbf{1}_{n}'\mathbf{B}_{l}^{c} \mathbf{b}_{l}=0$, where $\boldsymbol{\varSigma}_{bl}^{*}=({\mathbf{B}_{l}^{c}}'\mathbf {B}_{l}^{c}/\psi_{\delta}+ \mathbf{M}_{bl}/\tau_{bl})^{-1}$, $\mathbf{b}_{l}^{*}=\boldsymbol{\varSigma}_{bl}^{*}{\mathbf {B}_{l}^{c}}'\boldsymbol{\eta}_{c}^{*}/\psi_{\delta}$, and $\boldsymbol{\eta}_{c}^{*}=(\eta_{c1}^{*}, \ldots,\eta_{cn}^{*})'$ with

$$\eta_{ci}^*=\eta_i-\sum_{j \neq l} \sum_{k=1}^{K_{bl}} b_{jk}B_{jk}^c(c_{ij})- \sum_{j=1}^{q_2} \sum _{k=1}^{K_l} \gamma_{jk}B_{jk} \bigl(\varPhi^*(\xi_{ij})\bigr). $$

To sample an observation b _l from its full conditional distribution with the constraint, we can sample an observation $\mathbf{b}_{l}^{(\mathrm{new})}$ from $N[\mathbf{b}_{l}^{*},\boldsymbol{\varSigma}_{bl}^{*}]$, then transform $\mathbf{b}_{l}^{(\mathrm{new})}$ to b _l as follows:

$$\mathbf{b}_l=\mathbf{b}_l^{(\mathrm{new})}-\boldsymbol{\varSigma}_{bl}^*{ \mathbf{B}_{l}^c}'\mathbf{1}_n\bigl( \mathbf{1}_n'\mathbf{B}_{l}^c \boldsymbol{\varSigma}_{bl}^*{\mathbf{B}_{l}^c}' \mathbf{1}_n\bigr)^{-1} \mathbf{1}_n' \mathbf{B}_{l}^c\mathbf{b}_l^{(\mathrm{new})}. $$

(4) The full distribution of γ _l is

$$ p(\boldsymbol{\gamma}_l|\cdot) \stackrel{D}{=}N\bigl[\boldsymbol {\gamma}_l^*, \boldsymbol{\varSigma}_{\gamma l}^*\bigr], \quad l=1,\ldots,q_2, $$

(A.5)

with the constraint $\mathbf{1}_{n}'\mathbf{B}_{l} \boldsymbol{\gamma}_{l}=0$, where $\boldsymbol{\varSigma}_{\gamma l}^{*}=(\mathbf{B}_{l}'\mathbf {B}_{l}/\psi_{\delta}+ \mathbf{M}_{\gamma l}/\tau_{\gamma l})^{-1}$, $\boldsymbol{\gamma}_{l}^{*}=\boldsymbol{\varSigma}_{\gamma l}^{*}\mathbf {B}_{l}'\boldsymbol{\eta}^{*}/\psi_{\delta}$, and $\boldsymbol{\eta}^{*}=(\eta_{1}^{*}, \ldots,\eta_{n}^{*})'$ with

$$\eta_{i}^*=\eta_i-\sum_{j=1}^m \sum_{k=1}^{K_{bl}} b_{jk}B_{jk}^c(c_{ij})- \sum_{j \neq l} \sum_{k=1}^{K_l} \gamma_{jk}B_{jk}\bigl(\varPhi^*(\xi_{ij})\bigr). $$

Similar to b _l , we can sample an observation $\boldsymbol {\gamma}_{l}^{( \mathrm{new})}$ from $N[\boldsymbol{\gamma}_{l}^{*},\boldsymbol {\varSigma}_{\gamma l}^{*}]$, then transform $\boldsymbol{\gamma}_{l}^{(\mathrm{new})}$ to γ _l as follows:

$$\boldsymbol{\gamma}_l=\boldsymbol{\gamma}_l^{(\mathrm{new})}- \boldsymbol{\varSigma}_{\gamma l}^*\mathbf{B}_{l}' \mathbf{1}_n \bigl(\mathbf{1}_n'\mathbf{B}_{l}\boldsymbol{\varSigma}_{\gamma l}^* \mathbf{B}_{l}'\mathbf{1}_n\bigr)^{-1} \mathbf{1}_n'\mathbf{B}_{l}\boldsymbol{\gamma}_l^{(\mathrm{new})}. $$

(5) The full conditional distributions of τ _b and τ _γ are as follows:

$$ \begin{aligned}[c] & p\bigl(\tau_{bl}^{-1}\bigl|\cdot \bigr) \stackrel{D}{=}\mathrm{Gamma} \bigl[\alpha_{b0}+K_{bl}^*/2, \beta_{b0}+\mathbf{b}_l'\mathbf{M}_{bl} \mathbf{b}_l/2\bigr], \quad l=1,\ldots,m, \\ & p\bigl(\tau_{\gamma l}^{-1}\bigl|\cdot\bigr) \stackrel {D}{=}\mathrm{Gamma}\bigl[ \alpha_{\gamma 0}+K_{l}^*/2, \beta_{\gamma0} + \boldsymbol{\gamma}_l'\mathbf{M}_{\gamma l}\boldsymbol{\gamma}_l/2 \bigr], \quad l=1,\ldots,q_2. \end{aligned} $$

(A.6)

(6) The full conditional distribution of μ is given as follows:

(A.7)

(A.8)

(A.9)

where $\sigma_{j}^{*}=(\sigma_{j0}^{-1}+n \psi_{j}^{-1})^{-1}$, $\mu_{j}^{*}=\sigma_{j}^{*}(n \bar{z}_{j}^{*}/\psi_{j}+\mu_{j0}/\sigma_{j0})$, with $\bar{z}_{j}^{*}=n^{-1}\sum _{i=1}^{n} (z_{ij}^{*}-\boldsymbol{\varLambda}_{j}'\boldsymbol{\omega}_{i})$, $\mu_{j}^{**}=\sigma_{j}^{*}(n \bar{y}_{j}/\psi_{j}+\mu_{j0}/\sigma_{j0})$ with $\bar{y}_{j}=n^{-1}\sum_{i=1}^{n} (y_{ij}-\boldsymbol{\varLambda}_{j}'\boldsymbol{\omega}_{i})$, $\boldsymbol{\varSigma}_{\mu j}=(n\mathbf{I}_{L-1}+\mathbf{H}_{\mu j0}^{-1})^{-1}$, $\tilde {\boldsymbol{\mu}}_{j}=\boldsymbol{\varSigma}_{\mu j}(n\bar{\mathbf{w}}_{j}+\mathbf{H}_{\mu j0}^{-1}\boldsymbol{\mu }_{j0})$ with $\bar{\mathbf{w}}_{j}=n^{-1} \sum_{i=1}^{n} (\mathbf{w}_{ij}-\mathbf{1}_{L-1}\boldsymbol {\varLambda}_{j}'\boldsymbol{\omega}_{i})$.

(7) The full conditional distributions of Λ and Ψ are given as follows:

(A.10)

(A.11)

(A.12)

(A.13)

where $\mathbf{H}_{j}^{*}=(\mathbf{H}_{j0}^{-1}+\boldsymbol{\varOmega }\boldsymbol{\varOmega}')^{-1}$, $\boldsymbol{\varLambda}_{j}^{*}=\mathbf{H}_{j}^{*} [\mathbf{H}_{j0}^{-1}\boldsymbol{\varLambda}_{j0}+\boldsymbol {\varOmega}\widetilde{\mathbf{Z}}^{*}_{j}]$, $\boldsymbol{\varLambda}_{j}^{**}=\mathbf{H}_{j}^{*} [\mathbf{H}_{j0}^{-1}\boldsymbol{\varLambda}_{j0}+\boldsymbol {\varOmega}\widetilde{\mathbf{Y}}_{j}]$, $\beta_{j}^{*}=\beta_{j0}+2^{-1}[\widetilde{\mathbf{Z}}_{j}^{*'} \widetilde{\mathbf{Z}}^{*}_{j}-{\boldsymbol{\varLambda}^{*}_{j}}'\mathbf {H}_{j}^{*-1}{\boldsymbol{\varLambda}^{*}_{j}} +\boldsymbol{\varLambda}_{j0}'\mathbf{H}_{j0}^{-1}\boldsymbol {\varLambda}_{j0}]$, $\beta_{j}^{**}=\beta_{j0}+2^{-1}[\widetilde{\mathbf{Y}}_{j}' \widetilde{\mathbf{Y}}_{j}-{\boldsymbol{\varLambda}^{**}_{j}}'\mathbf {H}_{j}^{*-1} {\boldsymbol{\varLambda}^{**}_{j}}+\boldsymbol{\varLambda }_{j0}'\mathbf{H}_{j0}^{-1}\boldsymbol{\varLambda}_{j0}]$, with $\widetilde{\mathbf{Z}}_{j}^{*}=(z_{1j}^{*}-\mu_{j},\ldots ,z_{nj}^{*}-\mu_{j})'$ and $\widetilde{\mathbf{Y}}_{j}=(y_{1j}-\mu_{j},\ldots,y_{nj}-\mu_{j})'$. Finally, $\mathbf{H}_{j}^{**}=(\mathbf{H}_{j0}^{-1}+(L-1)\boldsymbol {\varOmega}\boldsymbol{\varOmega}')^{-1}$ and $\boldsymbol{\varLambda}_{j}^{***}=\mathbf{H}_{j}^{**} [\mathbf{H}_{j0}^{-1}\boldsymbol{\varLambda}_{j0}+\boldsymbol {\varOmega}\widetilde{\mathbf{W}}_{j}]$ with $\widetilde{\mathbf{W}}_{j}=(\mathbf{1}_{L-1}' (\mathbf {w}_{1j}-\nobreak\boldsymbol{\mu}_{j} ),\ldots, \mathbf{1}_{L-1}' (\mathbf{w}_{nj}-\boldsymbol{\mu}_{j}) )'$.

(8) The full conditional distributions of ψ _δ and Φ are as follows:

$$ p\bigl(\psi_{\delta}^{-1}\bigl|\cdot\bigr) \stackrel {D}{=}\mathrm{Gamma}\bigl[n/2+ \alpha_{\delta0},\beta_{\delta}^*\bigr], \qquad p\bigl( \boldsymbol{\varPhi}^{-1}\bigl|\cdot\bigr) \stackrel{D}{=}\mathrm {Wishart}\bigl[\mathbf{R}^*, n/2+ \rho_0\bigr], $$

(A.14)

where $\beta_{\delta}^{*}=\beta_{\delta0}+2^{-1} [\eta_{i}-\sum_{l=1}^{m} \sum_{k=1}^{K_{bl}} b_{lk} B_{lk}^{c}(c_{il})-\sum_{l=1}^{q_{2}} \sum_{k=1}^{K_{l}} \gamma_{lk} B_{lk}(\varPhi^{*}(\xi_{i})) ]^{2}$, and $\mathbf{R}^{*}=\mathbf{R}_{0}+\sum_{i=1}^{n} \boldsymbol{\xi}_{i} \boldsymbol{\xi}_{i}'$.

(9) The full conditional distributions of α and Z ^∗ are given as follows. For j=1,…,r ₁, let $\mathbf{Z}_{j}^{*}=(z_{1j}^{*},\ldots,z_{nj}^{*})'$, Z _j =(z _1j ,…,z _nj )′ we have $p(\boldsymbol{\alpha},\mathbf{Z}^{*}|\mathbf{D},\mathbf {W},\boldsymbol{\varOmega},\boldsymbol{\theta}_{*})=\prod_{j=1}^{r_{1}} p(\boldsymbol{\alpha}_{j},\mathbf{Z}_{j}^{*}|\mathbf{Z}_{j},\allowbreak \boldsymbol {\varOmega}, \boldsymbol{\theta}_{*})$, and $p(\boldsymbol{\alpha}_{j},\mathbf{Z}^{*}_{j}|\mathbf {Z}_{j},\boldsymbol{\varOmega},\boldsymbol{\theta}_{*})= p(\boldsymbol{\alpha}_{j}|\mathbf{Z}_{j},\boldsymbol{\varOmega },\boldsymbol{\theta}_{*}) p(\mathbf{Z}^{*}_{j}|\boldsymbol{\alpha}_{j}, \mathbf {Z}_{j},\boldsymbol{\varOmega},\boldsymbol{\theta}_{*})$, with

(A.15)

Finally, $p(\mathbf{Z}^{*}_{j}|\boldsymbol{\alpha}_{j},\mathbf {Z}_{j},\boldsymbol{\varOmega},\boldsymbol{\theta}_{*})= \prod_{i=1}^{n} p(z^{*}_{ij}|\boldsymbol{\alpha}_{j},z_{ij}, \boldsymbol {\omega}_{i}, \boldsymbol{\theta}_{*})$, and

$$ p\bigl(z^*_{ij}\bigl|\boldsymbol{\alpha}_{j},z_{ij}, \boldsymbol{\omega}_i, \boldsymbol{\theta}_*\bigr) \stackrel {D}{=}N\bigl[\mu_{j}+ \boldsymbol{\varLambda}_{j}'\boldsymbol{\omega}_i, \psi_{j} \bigr] \mathrm{I}_{[\alpha_{j,z_{ij}},\alpha_{j,z_{ij}+1})}\bigl (z^*_{ij}\bigr). $$

(A.16)