  Clinical prediction models (CPMs) are used to predict clinically relevant
outcomes or events. Typically, prognostic CPMs are derived to predict the risk
of a single future outcome. However, with rising emphasis on the prediction of
multi-morbidity, there is growing need for CPMs to simultaneously predict risks
for each of multiple future outcomes. A common approach to multi-outcome risk
prediction is to derive a CPM for each outcome separately, then multiply the
predicted risks. This approach is only valid if the outcomes are conditionally
independent given the covariates, and it fails to exploit the potential
relationships between the outcomes. This paper outlines several approaches that
could be used to develop prognostic CPMs for multiple outcomes. We consider
four methods, ranging in complexity and assumed conditional independence
assumptions: namely, probabilistic classifier chain, multinomial logistic
regression, multivariate logistic regression, and a Bayesian probit model.
These are compared with methods that rely on conditional independence: separate
univariate CPMs and stacked regression. Employing a simulation study and
real-world example via the MIMIC-III database, we illustrate that CPMs for
joint risk prediction of multiple outcomes should only be derived using methods
that model the residual correlation between outcomes. In such a situation, our
results suggest that probabilistic classification chains, multinomial logistic
regression or the Bayesian probit model are all appropriate choices. We call
into question the development of CPMs for each outcome in isolation when
multiple correlated or structurally related outcomes are of interest and
recommend more holistic risk prediction.
