Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 33 / 3 / e

The dispersion-one assumption is strongly contradicted by the preceding calculation. Use the gamma analysis with estimated dispersion, namely the F table. A failure to reject shape three does not establish that the true shape equals exactly three; retaining the estimated dispersion parameter $0.3103711$ is appropriate.

In the analysis of deviance for nested generalized linear models, a deviance reduction for $r$ extra coefficients is divided by $r\hat{\varphi }$ when dispersion is estimated. The approximate null calibration is $F_{r,57}$. For the type term, the null is that the two type contrasts vanish, against at least one nonzero contrast. The statistic iswith approximate null distribution $F_{2,57}$ and p-value $0.02007$. For position, the null is that its coefficient vanishes after accounting for type; the statistic iswith approximate null distribution $F_{1,57}$ and p-value $0.34963$. At 5% there is evidence that component type affects the expected failure time, and no significant additional position effect. The table is sequential: the type comparison is to the intercept-only model, while the position comparison adjusts for type. The individual gamma coefficient tests also suggest that type 3 accounts for the clearer type difference. The inverse link function means its positive contrast corresponds to a lower fitted mean failure time, holding position fixed.