Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 13J (Statistical Modelling)

Let $Y_{go}$ be the count for treatment(Control, LD, or SD) and outcome(Better or Worse), and write ${\mu }_{go}=\mathbb{E}{Y}_{go}$. Both fits take the six cell counts to be independent Poisson variables.

The additive log-linear model fitted by fit1 iswith reference constraints ${\alpha }_C=0$ and ${\beta }_B=0$. Thus treatment changes the overall group count but not the relative frequencies of the two outcomes.

The interaction model fitted by fit2 iswhere, under reference coding,and the two free interactions are ${\gamma }_{LW}$ and ${\gamma }_{SW}$. This is the saturated log-linear model for the $3\times 2$ table.

Letand condition on $N_g=n_g$. The corresponding model for each treatment group isFor fit1, the additive Poisson means implyso the outcome probabilities are common to all three treatments. For fit2,so each treatment may have its own outcome probabilities.

The Poisson trick is Poisson-multinomial conditioning: independent Poisson cell counts, conditional on their total, are multinomial with probabilities proportional to their means. With free treatment main effects, profiling the Poisson likelihood over the group totals gives the multinomial likelihood up to a parameter-independent factor. The Poisson regressions therefore fit these multinomial models and give the same likelihood-ratio comparisons for the outcome parameters.

The additive model has four free coefficients: an intercept, two treatment contrasts, and one outcome contrast. The interaction model addstreatment-by-outcome interaction term, making six parameters for six cells. It is saturated, so the residual degrees of freedom fall from two to zero and the likelihood-ratio test has

The deviance reduction is $44.48$, with p-value $2.194\times 10^{-10}$. The analysis of deviance for nested generalized linear models therefore gives overwhelming evidence against a common outcome distribution across the three groups. The observed proportions and the negative treatment--Worse interactions show that both doses reduce the chance of worsening relative to Control, with a larger reduction for LD.

Because the second model is saturated, its fitted worsening probabilities are the observed proportions:Thus the drug efficacy as a risk reduction isandThese are reasonably similar to the published estimates of $90.0\%$ and $62.1\%$.

The interaction coefficients provide the same quick comparison on the odds scale:They are treatment-versus-control odds ratio. Since worsening is rare in every group, each odds ratio is close to its risk ratio, giving the approximate efficacies $87.5\%$ and $65.3\%$.

Equal LD and SD efficacy is equivalent tobecause both efficacy definitions use the same control probability. In this two-outcome table that is also equivalent to equality of the LD and SD log-odds interactions,The summary gives the two estimates and their individual standard error, but a test of their difference needsThe required covariance is absent from the displayed table, so the two separate coefficient p-values cannot test equality.

Instead, fit the equal-efficacy Poisson log-linear model. For example, create one indicator for a Worse outcome in either treated group:The reduced model has one common treated-versus-control outcome interaction, while retaining separate LD and SD main effects for their different group totals. The full model has two such interactions. Their analysis of deviance for nested generalized linear models is therefore a one-degree-of-freedom likelihood-ratio test of equal efficacy.