The joint models discussed in Chapter 4 assume that the event times are generated from a single failure type, so are not applicable to studies with multiple failure types or competing risks. Use the scleroderma lung study we have introduced in Chapter 1 as an example. Recall that the primary outcome is forced vital capacity (FVC, as % predicted) determined at 3-month intervals from baseline. The event of interest is time to treatment failure or death. The longitudinal and survival data are possibly correlated, which could introduce nonignorable non-response missing values for %FVC after event times. Dependence between the two endpoints is further complicated by informatively censored events due to dropout during follow-up. Note that both death and dropout could lead to nonignorable missing data in %FVC measurements. The joint model developed by Elashoff, Li, and Li (2008)[61], as described in Section 5.1, has the capacity to handle multiple failure types, or competing risks at the survival endpoint. Parameter estimation and inference of this model via a Bayesian approach can be found in Hu, Li, and Li (2009)[100]. A robust joint model with t-distributed random errors (Li, Elashoff, and Li, 2009[137]) is described in Section 5.2. Li et al. (2010)[138] developed a model for studies with ordinal longitudinal measurements and competing risks, and this approach is discussed in Section 5.3. Section 5.4 extends joint models with competing risks to the scenario where there exists heterogeneity in the random effects covariance across study subjects (Huang, Li, and Elashoff, 2010[103]; Huang, Li, Elashoff, and Pan, 2011[104])