Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 5J (Statistical Modelling)

The comparison is a nested-model F-test of the three time-by-diet interaction coefficients, with null hypothesis that all are zero. The reduced linear regression model has a common time slope and four diet-specific intercepts; the full model permits four slopes. It has eight fitted coefficients and $579-8=571$ residual degrees of freedom.The rounded sums of squares give the displayed approximation; the software reports $F=16.824$ using unrounded values. The quoted p-value $1.744\times 10^{-10}$ gives overwhelming evidence against equal log-weight growth slopes across diets under the normal linear model assumptions. Thus reject the common-slope hypothesis. Because repeated measurements on the same chicken may be correlated, the exact reference distribution requires independent, homoscedastic Gaussian errors; the displayed ordinary regression output by itself does not establish that assumption.

A residual-versus-fitted plot for the untransformed fit places $e_i=y_i-{\hat{y}}_i$ on the vertical axis and ${\hat{y}}_i$ on the horizontal axis. A fan-shaped cloud, with larger residual spread at larger fitted weight, suggests heteroscedasticity and approximately multiplicative errors. A logarithmic transformation may stabilize this spread when the conditional standard deviation is roughly proportional to the mean.

One can also use the scale-location plot of $\sqrt{|r_i|}$ against fitted values, where $r_i$ is a standardized residual. A systematic rise suggests nonconstant error variance. Neither plot proves that taking logarithms is sufficient: diagnostics of the transformed fit should be checked as well, and dependence between repeated chicken measurements is a separate issue.

The dashed curves are contours of constant Cook's distance, labelled 0.5 and 1, in the plot of standardized residual against regression leverage. Writing $h_{ii}$ for the diagonal of the hat matrix, $s^2$ for the residual variance estimate, and $p=8$ for the number of fitted coefficients,Thus a contour $D_i=d$ has $r_i=\pm \sqrt{dp(1-h_{ii})/h_{ii}}$. On the actual PDF plot, observation 579 has leverage about 0.084 and standardized residual about $-8$, lying beyond the $D=0.5$ contour and below $D=1$ in influence magnitude, approximately $D\simeq 0.7$ to $0.8$ from the plotted coordinates. It is unusually light relative to its fitted log-weight and potentially influential because deleting it can materially change fitted coefficients. It warrants checking, not automatic deletion. The solid smooth red curve is distinct from the dashed Cook's distance contours.