Bivariate Mixed Effects Analysis of Clustered Data with Large Cluster Sizes

Abstract

Linear mixed effects models are widely used to analyze a clustered response variable. Motivated by a recent study to examine and compare the hospital length of stay (LOS) between patients undertaking percutaneous coronary intervention (PCI) and coronary artery bypass graft (CABG) from several international clinical trials, we proposed a bivariate linear mixed effects model for the joint modeling of clustered PCI and CABG LOSs where each clinical trial is considered a cluster. Due to the large number of patients in some trials, commonly used commercial statistical software for fitting (bivariate) linear mixed models failed to run since it could not allocate enough memory to invert large dimensional matrices during the optimization process. We consider ways to circumvent the computational problem in the maximum likelihood (ML) inference and restricted maximum likelihood (REML) inference. Particularly, we developed an expected and maximization (EM) algorithm for the REML inference and presented an ML implementation using existing software. The new REML EM algorithm is easy to implement and computationally stable and efficient. With this REML EM algorithm, we could analyze the LOS data and obtained meaningful results.

Appendix: Properties of the REML EM Algorithm

The following theorem states the general EM algorithm for the REML estimation.

Theorem 1

Suppose $f(y|b; \beta , \theta )$ is the conditional probability density function of y given random effects b and $f(b; \theta )$ is the probability density function of the random effects b. Assume the following “REML” likelihood

$$\begin{aligned} L_{R}(\theta ; y) = \int f(y|b; \beta , \theta ) f(b; \theta ) \mathrm{d}\beta \mathrm{d}b \end{aligned}$$

exists, so that $g(\beta , b|y; \theta ) = f(y|b; \beta , \theta ) f(b; \theta )/L_{R}(\theta ; y)$ is a probability density function. Given estimate $\theta ^{(t)}$ at the tth iteration, define the Q-function for the REML algorithm as follows:

$$\begin{aligned} Q_\mathrm{R}(\theta |\theta ^{(t)}) = {E}_\mathrm{g} [ \log \{f(y|b; \beta , \theta ) f(b; \theta ) \}], \end{aligned}$$

where ${E}_\mathrm{g}$ stands for the expectation taken with respect to $g(\beta , b|y; \theta ^{(t)})$. Denote $\theta ^{(t+1)}$ the update of $\theta$ obtained by maximizing $Q_\mathrm{R}(\theta |\theta ^{(t)})$ with respect to $\theta$, then

$$\begin{aligned} \ell _{R}(\theta ^{(t+1)}; y) \ge \ell _{R}(\theta ^{(t)}; y), \quad \text{ for } \text{ all } t, \end{aligned}$$

where $\ell _{R}(\theta ; y) = \log L_{R}(\theta ; y)$.

Proof

By the definition of $L_{R}(\theta ; y)$, we have

$$\begin{aligned} \log \left\{ f(y|b; \beta , \theta ) f(b; \theta ) \right\} = \log g(\beta , b; \theta ) + \ell _{R}(\theta ; y). \end{aligned}$$

Taking expectation with respect to $g(\beta , b|y; \theta ^{(t)})$ in both sides of the above equation leads to

$$\begin{aligned} Q_\mathrm{R}(\theta |\theta ^{(t)}) = {E}_\mathrm{g} \log g(\beta , b|y; \theta ) + \ell _{R}(\theta ; y). \end{aligned}$$

Denote $H(\theta ) = {E}_\mathrm{g} \log g(\beta , b|y; \theta )$. Then

$$\begin{aligned}&\ell _{R}\left( \theta ^{(t+1)}; y\right) - \ell _{R}\left( \theta ^{(t)}; y\right) \\&\quad = Q_\mathrm{R}\left( \theta ^{(t+1)}|\theta ^{(t)}\right) - Q_\mathrm{R}\left( \theta ^{(t)}|\theta ^{(t)}\right) - \left\{ H(\theta ^{(t+1)}) - H(\theta ^{(t)})\right\} . \end{aligned}$$

By the definition of $H(\theta )$, ${E}_\mathrm{g}$ and Jensen’s inequality, we have

$$\begin{aligned} H(\theta ^{(t+1)}) - H(\theta ^{(t)})= & {} \mathrm{E}_{g} \log \left\{ \frac{g(\beta , b|y; \theta ^{(t+)})}{g(\beta , b|y; \theta ^{(t)})} \right\} \\\le & {} \log \mathrm{E}_{g} \left\{ \frac{g(\beta , b|y; \theta ^{(t+)})}{g(\beta , b|y; \theta ^{(t)})} \right\} \\= & {} \log (1) =0. \end{aligned}$$

Since $\theta ^{(t+1)}$ maximizes $Q_\mathrm{R}(\theta |\theta ^{(t)})$, it follows that $Q_\mathrm{R}(\theta ^{(t+1)}|\theta ^{(t)}) \ge Q_\mathrm{R}(\theta ^{(t)}|\theta ^{(t)})$. Therefore

$$\begin{aligned} \ell _{R}(\theta ^{(t+1)}; y) \ge \ell _{R}(\theta ^{(t)}; y), \quad \text{ for } \text{ all } t. \end{aligned}$$

$\square$