Models involving unobservable latent quantities, such as structural measurement error models and so-called "joint" models for longitudinal and time-to-event data, are widely used in a host of applications. Provided that the model for the latent variable is correctly specified, likelihood-based approaches are appealing because they lead to consistent and efficient inference. However, intuition suggests that misspecification of this model may compromise such inference, although some recent empirical studies have exhibited striking robustness to the assumption on the latent variable. The data analyst faces the difficulty that the extent to which inference may be sensitive to the choice of model for unobservable latent variables is not known in a given problem. Techniques for studying and diagnosing robustness in these models would thus be invaluable.We present a framework for assessing model robustness in the class of structural latent variable models, focusing on the particular subclass of structural measurement error models, and propose practical strategies for diagnosing misspecification of the model for the true predictor, the latent variable for this subclass. The methods are illustrated via several analytic examples and by application to simulated data and a study of coronary heart disease.
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