case, but they are compounded under missing not at random (MNAR). We believe that, especially when the primary analysis assumes MAR, the fit of an MAR model can often be addressed by standard model-checking diagnostics, leaving the sensitivity analysis to MNAR models that deviate from MAR. This approach is suggested in order not to overburden the primary analysis. The discussion in Chapter 4 provides some references for model-checking of MAR models. In addition, with MAR missingness mechanisms that deviate markedly from missing completely at random (MCAR), as in the hypertension example in Chapter 4, analyses with incomplete data are potentially less robust to violations of parametric assumptions than analyses with complete data, so checking them is even more critical.

The data can never rule out an MNAR mechanism, and when the data are potentially MNAR, issues of sensitivity to modeling asumptions are even more serious than under MAR. One approach could be to estimate from the available data the parameters of a model representing an MNAR mechanism. However, the data typically do not contain information on the parameters of the particular model chosen (Jansen et al., 2006).

In fact, different MNAR models may fit the observed data equally well but have quite different implications for the unobserved measurements and hence for the conclusions to be drawn from the respective analyses. Without additional information, one cannot usefully distinguish between such MNAR models based solely on their fit to the observed data, and so goodness-of-fit tools alone do not provide a relevant means of choosing between such models.

These considerations point to the necessity of sensitivity analysis. In a broad sense, one can define a sensitivity analysis as one in which several statistical models are considered simultaneously or in which a statistical model is further scrutinized using specialized tools, such as diagnostic measures. This rather loose and very general definition encompasses a wide variety of useful approaches.

A simple procedure is to fit a selected number of (MNAR) models, all of which are deemed plausible and have equivalent or nearly equivalent fit to the observed data; alternatively, a preferred (primary) analysis can be supplemented with a number of modifications. The degree to which conclusions (inferences) are stable across such analyses provides an indication of the confidence that can be placed in them.

Modifications to a basic model can be constructed in different ways. One obvious strategy is to consider various dependencies of the missing data process on the outcomes or the covariates. One can choose to supplement an analysis within the selection modeling framework, say, with one or several in the pattern mixture modeling framework, which explicitly models the missing responses at any given time given the previously observed responses. Alternatively, the distributional assumptions of the models can be altered.



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