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### Estimates and Tests Without Covariates

The simplest question we might ask about the multiple event types is whether the type-specific hazard functions are the same for all event types, that is, hj(t) = h(t) for all j. For the leader data, just looking at the frequencies of the three event types suggests that deaths due to natural causes (n=27) are much less likely to occur than the other two types, which have identical frequencies (n=165). In fact, we can easily obtain a formal test of the null hypothesis from these three frequencies. If the null hypothesis of equal hazards is correct, the expected frequencies of the three event types should be equal in any time interval. Thus, we have an expected frequency of 119=(472 - 115)/3 for each of the ...

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