Maximum Likelihood Estimation of Multilevel Structural Equation Models with Random Slopes for Latent Covariates

Abstract

A maximum likelihood estimation routine for two-level structural equation models with random slopes for latent covariates is presented. Because the likelihood function does not typically have a closed-form solution, numerical integration over the random effects is required. The routine relies upon a method proposed by du Toit and Cudeck (Psychometrika 74(1):65–82, 2009) for reformulating the likelihood function so that an often large subset of the random effects can be integrated analytically, reducing the computational burden of high-dimensional numerical integration. The method is demonstrated and assessed using a small-scale simulation study and an empirical example.

Appendix

In this section, methods for adapting the estimation routine to allow for data missing at random are sketched. As in Sect. 2, suppose there are k level-2 variables ${\mathbf {z}}_j$ and p level-1 variables ${\mathbf {y}}_{ij}$. However, now consider that one or more elements within these vectors for a given j or ij may be missing. Suppose cluster j has $k_j$ non-missing elements of ${\mathbf {z}}_j$ and individual i in cluster j has $p_{ij}$ non-missing elements in ${\mathbf {y}}_{ij}$.

Define ${\mathbf {K}}_j$ ($k_j \times k$) and ${\mathbf {M}}_{ij}$ ($p_{ij} \times p$) to be zero-one matrices that select the non-missing elements of ${\mathbf {z}}_{j}$ and ${\mathbf {y}}_{ij}$, respectively. For example, suppose k is 4 and cluster $j'$ is missing the third element of ${\mathbf {z}}_{j'}$, so that

$$\begin{aligned} {\mathbf {K}}_{j'} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{pmatrix} \end{aligned}$$

(A1)

can be used to select the non-missing subset of ${\mathbf {z}}_{j'}$:

$$\begin{aligned} {\mathbf {z}}_{j'}^* = {\mathbf {K}}_{j'}{\mathbf {z}}_{j'} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{pmatrix} \begin{pmatrix} z_{1j'} \\ z_{2j'} \\ - \\ z_{4j'} \end{pmatrix} = \begin{pmatrix} z_{1j'} \\ z_{2j'} \\ z_{4j'} \end{pmatrix}. \end{aligned}$$

(A2)

The matrix ${\mathbf {M}}_{ij}$ performs the same role as ${\mathbf {K}}_j$, except it is used to select non-missing elements of ${\mathbf {y}}_{ij}$ rather than ${\mathbf {z}}_j$. Thus, ${\mathbf {z}}_j^* = {\mathbf {K}}_{j}{\mathbf {z}}_{j}$ will be used in place of ${\mathbf {z}}_j$ and ${\mathbf {y}}_j^* = {\mathbf {M}}_{ij}{\mathbf {y}}_{ij}$ will be used in place of ${\mathbf {y}}_{ij}$.

By premultiplying some of the other model matrices within the likelihood calculation by ${\mathbf {K}}_j$ or ${\mathbf {M}}_{ij}$, the estimation routine can be adapted to account for the missing elements within ${\mathbf {z}}_j$ and ${\mathbf {y}}_{ij}$. Specifically, for Eqs. (24)–(25), replace ${\varvec{\mu }}_{{\mathbf {z}}_j}$ and ${\varvec{\mu }}_{{\mathbf {y}}_{ij}}$ with ${\mathbf {K}}_j{\varvec{\mu }}_{{\mathbf {z}}_j}$ and ${\mathbf {M}}_{ij}{\varvec{\mu }}_{{\mathbf {y}}_{ij}}$, respectively. Within Eqs. (26)–(30) replace $\tilde{{\mathbf {G}}}$ and $\tilde{{\mathbf {Q}}}^*_{ij}$ with ${\mathbf {K}}_j\tilde{{\mathbf {G}}}$ and ${\mathbf {M}}_{ij}\tilde{{\mathbf {Q}}}^*_{ij}$, and replace ${\varvec{\Sigma }}_{W}^*$ with ${\varvec{\Sigma }}_{Wij}^* = {\mathbf {M}}_{ij}{\varvec{\Sigma }}_{W}^*{\mathbf {M}}_{ij}'$. Lastly, replace ${\mathbf {I}}_{n_j} \otimes {\varvec{\Sigma }}_W^*$ in Eq. (26) with

$$\begin{aligned} \bigoplus _{i = 1}^{n_j} {\varvec{\Sigma }}_{Wij}^*, \end{aligned}$$

(A3)

where $\oplus$ is the direct sum. Using these replacements, the simplified expressions for $|{\varvec{\Sigma }}_{{\mathbf {d}}_j}|$ and ${\varvec{\epsilon }}_{{\mathbf {d}}_j}'{\varvec{\Sigma }}_{{\mathbf {d}}_j}^{-1}{\varvec{\epsilon }}_{{\mathbf {d}}_j}$ in the new conditional log-likelihood

$$\begin{aligned} f({\mathbf {d}}_j | {\tilde{{\varvec{\beta }}}}_{Wj}) = (2\pi )^{-( \sum _i p_{ij} + k_j)/2}|{\varvec{\Sigma }}_{{\mathbf {d}}_j}|^{-1/2}\text {exp} \bigg \{-\frac{1}{2}{\varvec{\epsilon }}_{{\mathbf {d}}_j}' {\varvec{\Sigma }}_{{\mathbf {d}}_j}^{-1}{\varvec{\epsilon }}_{{\mathbf {d}}_j} \bigg \} \end{aligned}$$

(A4)

are

$$\begin{aligned} |{\varvec{\Sigma }}_{{\mathbf {d}}_j}| = \bigg \{ \prod _{i = 1}^{n_j} |{\varvec{\Sigma }}_{Wij}^*|\bigg \} |{\varvec{\Sigma }}_{{\varvec{\xi }}\bullet {\varvec{\beta }}_W}||{\varvec{\Sigma }}^{-1}_{{\varvec{\xi }}\bullet {\varvec{\beta }}_W} + {\mathbf {A}}_j| |{\varvec{\Sigma }}_{zz.y}|, \end{aligned}$$

(A5)

and

$$\begin{aligned} {\varvec{\epsilon }}_{{\mathbf {d}}_j}'{\varvec{\Sigma }}_{{\mathbf {d}}_j}^{-1}{\varvec{\epsilon }}_{{\mathbf {d}}_j} =&\sum _{i = 1}^{n_j} {\varvec{\epsilon }}_{{\mathbf {y}}_{ij}}'{\varvec{\Sigma }}_{Wij}^{*-1}{\varvec{\epsilon }}_{{\mathbf {y}}_{ij}} + {\mathbf {p}}_j'{\mathbf {H}}_j {\mathbf {p}}_j \nonumber \\&- 2{\mathbf {p}}_j'{\mathbf {C}}_j'{\varvec{\Sigma }}_{{\varvec{\xi }}\bullet {\varvec{\beta }}_W}\tilde{{\mathbf {G}}}'{\varvec{\Sigma }}_{zz.y}^{-1}{\varvec{\epsilon }}_{{\mathbf {z}}_j} \nonumber \\&+ {\varvec{\epsilon }}_{{\mathbf {z}}_j}'{\varvec{\Sigma }}_{zz.y}^{-1}{\varvec{\epsilon }}_{{\mathbf {z}}_j}. \end{aligned}$$

(A6)