Robustness in Latent Variable Models

Abstract

Statistical models involving latent variables are widely used in many areas of applications, such as biomedical science and social science. When likelihood-based parametric inferential methods are used to make statistical inference, certain distributional assumptions on the latent variables are often invoked. As latent variables are not observable, parametric assumptions on the latent variables cannot be verified directly using observed data. Even though semiparametric and nonparametric approaches have been developed to avoid making strong assumptions on the latent variables, parametric inferential approaches are still more appealing in many situations in terms of consistency and efficiency in estimation and computation burden. The goals of our study are to gain insight into the sensitivity of statistical inference to model assumptions on latent variables, and to develop methods for diagnosing latent-model misspecification to enable one to reveal whether the parametric inference is robust under certain latent-model assumptions. We refer to such robustness as latent-model robustness.

We start with a simple class of latent variable models, the structural measurement error models, to first tackle the problem. We define theoretical conditions under which a certain degree of latent-model robustness is achieved and study some special structural measurement error models analytically to gain insight into the sensitivity of inference to latent-model assumptions under these specific contexts. Then we borrow the idea of simulation-extrapolation (SIMEX), or remeasurement method, introduced by Cook and Stefanski (1994) to develop an empirical diagnostic tool that is able to reveal graphically whether or not robustness is attained under the imposed latent-variable assumptions. Testing procedures are proposed as a numerical supplement to the graphical diagnostic tool. These methods are then generalized and refined to adapt to a more complex class of latent variable models called joint models. For this generalization we focus on joint models that link a primary response, which can be a simple response or a censored time-to-event, to an error-prone longitudinal process. The performances of the proposed methods are demonstrated through application to simulated data and data from medical studies.