It is possible to summarise the steps involved in drawing inference from incomplete data as (Daniels and Hogan (2008)):

Specification of a full data model for the response and missingness indicators \(f(y,r)\)

Specification of the prior distribution (within a Bayesian approach)

Sampling from the posterior distribution of full data parameters, given the observed data \(Y_{obs}\) and the missingness indicators \(R\)

Identification of a full data model, particularly the part involving the missing data \(Y_{mis}\), requires making unverifiable assumptions about the full data model \(f(y,r)\). Under the assumption of the ignorability of the missingness mechanism, the model can be identified using only the information from the observed data. When ignorability is not believed to be a suitable assumption, one can use a more general class of models that allows missing data indicators to depend on missing responses themselves. These models allow to parameterise the conditional dependence between \(R\) and \(Y_{mis}\), given \(Y_{obs}\). Without the benefit of untestable assumptions, this association structure cannot be identified from the observed data and therefore inference depends on some combination of two elements:

Unverifiable parametric assumptions

Informative prior distributions (under a Bayesian approach)

We show some simple examples about how these *nonignorable* models can be constructed, identified and applied. In this section, we specifically focus on the class of nonignorable models known as *Shared Parameter Models*(SPM).

## Conlcusions

To summarise, shared parameter models are very useful for characterizing joint distributions of repeated measures and event times, and can be particularly useful as a method of data reduction when the dimension of \(Y\) is high. Nonetheless, their application to the problem of making full data inference from incomplete longitudinal data should be made with caution and with an eye toward justifying the required assumptions. Sensitivity analysis is an open area of research for these models.

# References

*Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis*. Chapman; Hall/CRC.

*Biometrics*, 939–55.

*Biometrics*, 175–88.