= Solution
The dispersion-one assumption is strongly contradicted by the preceding calculation. \b[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 $r\widehat\phi$ 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 is
$$
F=\frac{2.60027/2}{0.3103711}=4.1890,
$$
with 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 is
$$
F=\frac{0.27603}{0.3103711}=0.8894,
$$
with 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.
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