Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 13J (Statistical Modelling)

Let $Y_i$ be the first team's goal count in match $i$, let $x_i^T$ be row $i$ of the design matrix, and let $\beta$ be the regression coefficient vector. The fitted Poisson regression assumes that the $Y_i$ are independent random variables withThe likelihood function maximized by glm is therefore

None of the displayed one-coefficient Wald tests for team1_form rejects at the $5\%$ significance level. That does not establish that formation has no effect: the relevant null hypothesis sets all nine displayed team1_form coefficients to zero simultaneously, and several individually imprecise estimates may still be jointly significant. One should compare the fitted model with the nested model omitting team1_form by an analysis of deviance for nested generalized linear models, equivalently a likelihood-ratio test of those joint restrictions.

Writing $Y_i$ for team1_goal and $x_i$ for team1_xg, the first hypothesis concerns only the expected value:The second specifies the entire conditional probability distribution:with independence between matches as assumed in the question. The Poisson limit theorem, often called the law of small numbers in this setting, suggests a Poisson distribution when a goal count is the sum of many approximately independent rare scoring opportunities.

Under $H_1$, the slope in this through-origin linear regression is $1$. Its observed Wald statistic isThe corresponding two-sided p-value is about $0.07$, so the fit does not give evidence against $H_1$ at the $5\%$ significance level.

The usual least-squares standard error assumes constant conditional variance. Goal counts are counts, and under the natural Poisson distribution model their variance equals their mean $x_i$, so observations with different expected goals are heteroscedastic. The reported homoscedastic standard error is therefore doubtful; a heteroscedasticity-robust calculation or a suitable count model would be more credible.

The fitted Poisson regression hasso that ${\mu }_i=e^{{\beta }_0}{x}_i^{{\beta }_1}$. Hypothesis $H_2$ requires both restrictionsas well as the adequacy of the assumed Poisson distribution. The displayed interval addresses only the single slope restriction ${\beta }_1=1$. Even if the separate interval for the intercept also contains zero, separate one-parameter intervals do not implement the relevant joint hypothesis test, because the two coefficient estimators can be correlated. One should test $({\beta }_0,{\beta }_1)=(0,1)$ jointly, for example by a likelihood-ratio test, and separately assess the Poisson goodness of fit. Thus the slope interval alone cannot decide whether $H_2$ is rejected at the $5\%$ level.