Issues with the expected information matrix of linear mixed models provided by popular statistical packages under missingness at random dropout.

Likelihood-based methods ignoring missingness at random (MAR) produce consistent estimates provided that the whole likelihood model is correct. However, the expected information matrix (EIM) depends on the missingness mechanism. It has been shown that calculating the EIM by considering the missing data pattern as fixed (naive EIM) is incorrect under MAR, but the observed information matrix (OIM) is valid under any MAR missingness mechanism. In longitudinal studies, linear mixed models (LMMs) are routinely applied, often without any reference to missingness. However, most popular statistical packages currently provide precision measures for the fixed effects by inverting only the corresponding submatrix of the OIM (naive OIM), which is effectively equivalent to the naive EIM. In this paper, we analytically derive the correct form of the EIM of LMMs under MAR dropout to compare its differences with the naive EIM, which clarifies why the naive EIM fails under MAR. The asymptotic coverage rate of the naive EIM is numerically calculated for two parameters (population slope and slope difference between two groups) under various dropout mechanisms. The naive EIM can severely underestimate the true variance, especially when the degree of MAR dropout is high. Similar trends emerge under misspecified covariance structure, where, even the full OIM may lead to incorrect inferences and sandwich/bootstrap estimators are generally required. Results from simulation studies and application to real data led to similar conclusions. In LMMs, the full OIM should be preferred to the naive EIM/OIM, though if misspecified covariance structure is suspected, robust estimators should be used.

A note on computing Louis' observed information matrix identity for IRT and cognitive diagnostic models.

Using Louis' formula, it is possible to obtain the observed information matrix and the corresponding large-sample standard error estimates after the expectation-maximization (EM) algorithm has converged. However, Louis' formula is commonly de-emphasized due to its relatively complex integration representation, particularly when studying latent variable models. This paper provides a holistic overview that demonstrates how Louis' formula can be applied efficiently to item response theory (IRT) models and other popular latent variable models, such as cognitive diagnostic models (CDMs). After presenting the algebraic components required for Louis' formula, two real data analyses, with accompanying numerical illustrations, are presented. Next, a Monte Carlo simulation is presented to compare the computational efficiency of Louis' formula with previously existing methods. Results from these presentations suggest that Louis' formula should be adopted as a standard method when computing the observed information matrix for IRT models and CDMs fitted with the EM algorithm due to its computational efficiency and flexibility.

Information matrix estimation procedures for cognitive diagnostic models.

Two new methods to estimate the asymptotic covariance matrix for marginal maximum likelihood estimation of cognitive diagnosis models (CDMs), the inverse of the observed information matrix and the sandwich-type estimator, are introduced. Unlike several previous covariance matrix estimators, the new methods take into account both the item and structural parameters. The relationships between the observed information matrix, the empirical cross-product information matrix, the sandwich-type covariance matrix and the two approaches proposed by de la Torre (2009, J. Educ. Behav. Stat., 34, 115) are discussed. Simulation results show that, for a correctly specified CDM and Q-matrix or with a slightly misspecified probability model, the observed information matrix and the sandwich-type covariance matrix exhibit good performance with respect to providing consistent standard errors of item parameter estimates. However, with substantial model misspecification only the sandwich-type covariance matrix exhibits robust performance.

Standard error estimation using the EM algorithm for the joint modeling of survival and longitudinal data.

Joint modeling of survival and longitudinal data has been studied extensively in the recent literature. The likelihood approach is one of the most popular estimation methods employed within the joint modeling framework. Typically, the parameters are estimated using maximum likelihood, with computation performed by the expectation maximization (EM) algorithm. However, one drawback of this approach is that standard error (SE) estimates are not automatically produced when using the EM algorithm. Many different procedures have been proposed to obtain the asymptotic covariance matrix for the parameters when the number of parameters is typically small. In the joint modeling context, however, there may be an infinite-dimensional parameter, the baseline hazard function, which greatly complicates the problem, so that the existing methods cannot be readily applied. The profile likelihood and the bootstrap methods overcome the difficulty to some extent; however, they can be computationally intensive. In this paper, we propose two new methods for SE estimation using the EM algorithm that allow for more efficient computation of the SE of a subset of parametric components in a semiparametric or high-dimensional parametric model. The precision and computation time are evaluated through a thorough simulation study. We conclude with an application of our SE estimation method to analyze an HIV clinical trial dataset.