  In the analysis of time-to-event data with multiple causes using a competing
risks Cox model, often the cause of failure is unknown for some of the cases.
The probability of a missing cause is typically assumed to be independent of
the cause given the time of the event and covariates measured before the event
occurred. In practice, however, the underlying missing-at-random assumption
does not necessarily hold. Motivated by colorectal cancer subtype analysis, we
develop semiparametric methods to conduct valid analysis, first when additional
auxiliary variables are available for cases only. We consider a weaker
missing-at-random assumption, with missing pattern depending on the observed
quantities, which include the auxiliary covariates. Overlooking these
covariates will potentially result in biased estimates. We use an informative
likelihood approach that will yield consistent estimates even when the
underlying model for missing cause of failure is misspecified. We then consider
a method to conduct valid statistical analysis when there are no auxiliary
covariates in the not missing-at-random scenario. The superiority of our
methods in finite samples is demonstrated by simulation study results. We
illustrate the use of our method in an analysis of colorectal cancer data from
the Nurses' Health Study cohort, where, apparently, the traditional
missing-at-random assumption fails to hold for particular molecular subtypes.
