Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 41 / 1

Screening applies the eligibility criteria for a clinical trial before treatment allocation. It identifies the population to which the comparison pertains and excludes individuals for whom the intervention or measurements are unsuitable. Recording potentially eligible individuals also helps describe recruitment and assess external validity. It should precede randomization so eligibility is not decided using knowledge of the allocated treatment.

Obtain informed consent so that participation in the clinical trial is voluntary and based on an understanding of its procedures, potential benefits and risks, alternatives and withdrawal arrangements. Consent before allocation prevents willingness to participate from being selected according to the known treatment assignment.

Measure baseline covariates before the intervention can change them. These measurements describe the recruited population, permit assessment of covariate balance, and provide prognostic predictors that can improve statistical efficiency. In particular, including prognostic baseline covariates in a linear regression reduces unexplained outcome variation without conditioning on a treatment consequence.

Randomization makes assignment independent of the participants' pre-existing potential outcomes, giving an unbiased comparison of allocation groups under the usual trial assumptions. It therefore protects against baseline confounding, including from unmeasured factors, and supplies an assignment distribution for inference. Allocation concealment protects this step from recruitment decisions based on the upcoming assignment. It does not guarantee exact balance in every realized sample.

Use a baseline-adjusted linear regression for an intention-to-treat analysis:Condition on the observed baseline covariates and assignment. The parameter of primary interest is $\beta$, the mean effect of assignment to the experimental intervention, under this constant-effect conditional mean model. Independent normal distributions with common variance give the usual Gaussian likelihood and regression inference. Prognostic baseline covariates improve precision by explaining outcome variability.

Do not include the actual intervention-use variable $D_i$ as an ordinary adjustment variable for this estimand: it is measured after assignment and can mediate the effect being estimated. Post-randomization adjustment changes a treatment estimand; the resulting coefficient on $Z_i$ would generally represent a different comparison. Participants remain in their assigned groups even when adherence differs.

Let $R_i=1$ indicate an observed outcome. Fitting the conditional outcome model to responders requireswith a positive observation probability over the relevant predictor support. This is conditional missing at random for the predictors used in the regression: after conditioning on them, observation must not further select participants by their unobserved outcome. It gives $f(Y_i\mid x_i,Z_i,R_i=1)=f(Y_i\mid x_i,Z_i)$, the complete-case regression under conditional missing at random condition. Correct specification of the conditional mean model is also required. Missing completely at random is sufficient but stronger; simply assuming missingness depends only on a post-randomization variable excluded from this model would not suffice.

Write ${\mu }_i=\alpha +\beta z_i+{\gamma }^{\mathsf{T}}{x}_i$ and let $f_i(y)={\sigma }^{-1}\varphi ((y-{\mu }_i)/\sigma )$ be the normal distribution density. Set $q_j(y)=\mathrm{expit}({\eta }_j+\theta y)$, where $q_j$ is the probability that attempt $j$ is unsuccessful conditional on the preceding attempts being unsuccessful. This is a sequential response selection model. For first-attempt success, the observed outcome and attempt count contributeThis likelihood conditions on baseline and assignment; the displayed response probabilities do not use $d_i$. No factor for random allocation is needed when conditioning on its realized value.

Four unsuccessful attempts have conditional probability ${\prod }_{j=1}^4{q}_j(y)$. The outcome is then unobserved, so the observed-data likelihood integrates over it:This is a probability, whereas a responder contributes a joint density in the observed outcome. The sequential response selection model keeps outcome-dependent nonresponse in the likelihood; evaluating the product at an imputed mean would not be this likelihood.

Within the displayed sequential response selection model, $\theta =0$ makes every attempt's nonresponse probability independent of $Y$, and hence gives the outcome-independence needed for complete-case analysis. The estimate $\hat{\theta }=0.38$ suggests that larger outcomes are associated with greater odds of an unsuccessful attempt. Per one-unit outcome increase, the fitted odds ratio is $e^{0.38}\approx 1.46$, with the reported interval transformed to $(e^{-0.06},e^{0.82})\approx (0.94,2.27)$.

The confidence interval for $\theta$ includes zero, so there is no rejection of outcome-independent missingness at the five-percent level within this model. This does not establish that the missing at random assumption is true: the interval also allows appreciable outcome dependence, and inference depends on the chosen selection model. The intercepts describe response levels, not independence from the outcome. Nor can the respondents' mean and standard deviation establish the missing outcomes' distribution. There are $300-211=89$ unobserved outcomes, about $29.7\%$, so conclusions about the full trial depend materially on the missingness assumptions.