Mutlivariate and Joint Models for Longitudinal Data

Longitudinal studies are commonly designed in many research fields in order to see changes over a time interval shared by all participants. Joint modeling consists of two interlinked sub-models with any type of outcome (continuous, binomial, etc). One of the most commonly used is longitudinal sub-model is the linear mixed effect model and a cox proportional hazard with time-to-event sub-model.

Joint modeling reduces the bias of parameter estimates by accounting for the association between the longitudinal and time-to-event data. In clinical trials this often leads to more efficient estimation of the treatment effect on both sub-model outcomes.

Let's assume Y_i1 and Y_i2 are two outcomes measured on subject i. We can attempt to specify a joint density f(y_i1, y_i2), but this is only feasible if we assume certain things about the marginal association among the longitudinally measured elements within each of the outcome vectors. This is easier when both outcomes and conditional models are of the same type, but this is not a requirement. When Y_i1 and Y_i2 are of different types, or in the case of unbalanced data, this becomes cumbersome. Thus, extensions to higher dimensional data involves considerable challenges as this would require assumption on larger covariance structure and higher order associations.

Conditional Models

We can avoid direct specification of the joint density f(y_i1, y_i2) are and reduce the modeling tasks by specifying a model for each outcome separately:
         f(y_i1, y_i2)   =  f(y_i1 | y_i2) f(y_i2)
                            = f(y_i2 | y_i1) f(y_i1)

The main drawback is that this requires reflection about the plausible association between Y_i1 and Y_i2 that may be inappropriate in some settings. For example, in a clinical trial with two main post-randomization outcomes conditioning on one of the outcomes may attenuate (make smaller) the treatment effect on the other. Also, it is difficult to implement in high-dimension data.

Shared-Parameter Models

In previous chapters we've seen how random effects can be used to generate an association structure between repeated measurements of a specific outcome.  We can use a similar approach to generate association between two outcomes.

\( f(y_{i1}, y_{i2}) = \int f(y_{i1}, y_{i2} | b) f(b) db = \int f(y_{i1}, y_{i2} | b) f(y_{i1}, y_{i2} | b) f(b) db \)