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

Write $\mu =\alpha /\gamma$ and $\varphi =1/\alpha$. With $\theta =-\gamma /\alpha =-1/\mu <0$, the Gamma exponential dispersion family representation iswhereSubstitution gives the original gamma density, so both unknown parameters are included. The exponential-family derivative identities yield $\mu =b^{\prime }(\theta )=-1/\theta$ and $b^{\prime \prime }(\theta )=1/{\theta }^2={\mu }^2$. ThereforeThe exact natural-parameter canonical link function is $g_{\mathrm{can}}(\mu )=-1/\mu$. The usual inverse-link convention is $g(\mu )=1/\mu$, as used by the printed R fits; multiplying the link and coefficients by $-1$ gives the natural-parameter convention. The sign convention changes neither the fitted means nor the weight matrix below.

A generalized linear model specifies independent responses in an exponential dispersion family, their means ${\mu }_i=E(Y_i)$, and a link function connecting those means to a linear predictor. In the common notation,where $a_i$ are known positive weights, $\varphi$ is the common dispersion parameter, and $V({\mu }_i)=b^{\prime \prime }({\theta }_i)$ is the variance function. The systematic part isHere $x_i$ is a row of the known design matrix and $\beta$ contains the unknown regression coefficients. The link function is invertible on the permitted mean domain; the canonical link function takes the mean to the natural parameter. This separates distributional, predictor, and link assumptions: it does not require the response itself to be normally distributed or the mean itself to be linear in the covariates.

Both fits use the same Gamma distribution mean model and inverse link function. Their coefficient estimates, fitted means and hence their weight matrices coincide: the common dispersion parameter only multiplies the likelihood score by a scalar and therefore does not change its zero. The displayed calls likewise show the same family and formula; fixing dispersion in a summary changes the uncertainty calculation, not these coefficient estimates.

For $V(\mu )={\mu }^2$ and $g^{\prime }(\mu )=-{\mu }^{-2}$, the Fisher information weights areThus the common matrix $(X^{T}WX{)}^{-1}$ is multiplied by dispersion $1$ in the exponential case and by $\hat{\varphi }=0.3103711$ in the fitted gamma case. Taking square roots of the diagonal covariances givesFor instance $0.03607\times 0.5571\approx 0.02009$ for the intercept, and the same factor applies to all the coefficients. The standard errors are smaller because the fitted dispersion parameter is below one, not because a different mean function was fitted.

For the gamma variance function, the Pearson dispersion estimator isusing $61-4=57$ residual degrees of freedom. Under a specified null dispersion ${\varphi }_0$, the scaled statistic based on squared Pearson residuals $57\hat{\varphi }/{\varphi }_0$ has an approximate ${\chi }_{57}^2$ distribution under the usual residual approximation.

Thus test1 corresponds to $H_0:\varphi =1$, equivalently gamma shape $\alpha =1$, the exponential distribution. test2 corresponds to $H_0:\varphi =1/3$, equivalently shape $\alpha =3$. The null hypotheses concern dispersion or shape, not whether the regression coefficients vanish.

The code computes lower-tail probabilities. Used as one-sided tests, the alternatives are $\varphi <1$ and $\varphi <1/3$, respectively, equivalently shapes larger than one and three. At the 5% level the first null is rejected because the lower-tail probability is $1.199504\times 10^{-7}$, while the second is not rejected because its probability is $0.3768748$. If the intended alternatives are two-sided, $\varphi \ne {\varphi }_0$, these displayed numbers must not be called two-sided p-values: doubling the smaller tail gives approximately $2.40\times 10^{-7}$ and $0.75375$, with the same decisions. The chi-squared distribution approximation is not an exact finite-sample gamma identity.

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.