Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 (Statistics in Medicine)

Vitamin D is a causal determinant of mortality risk if an intervention that changes vitamin-D status changes the distribution of the corresponding potential outcome for mortality. This is a claim about a causal effect, rather than merely an observed association.

First, an observational study can suffer from confounding or reverse causality. Ill health may both lower circulating vitamin D and raise mortality, while lifestyle, socioeconomic status, and comorbidity may affect both variables. Randomization breaks these baseline associations in expectation.

Second, the interventions answer different questions. Supplementation may be too small, too late, too short, or poorly adhered to, and an average effect can be nearly zero when benefit is confined to people with severe deficiency. Such heterogeneous treatment effects can coexist with a strong observational gradient.

The plots test the plausibility of the instrumental-variable independence and exclusion restriction assumptions. Both candidate instruments are strongly associated with 25(OH)D, supporting instrument relevance. The focused instrument has estimates near zero for the other measured traits. The polygenic instrument is strongly associated with LDL cholesterol and triglycerides, suggesting horizontal pleiotropy: it may influence mortality through lipid pathways that do not pass through vitamin D.

I would therefore prefer the focused instrument. Its use of fewer biologically understood variants may reduce precision, but its cleaner associations make the causal assumptions more credible.

Let $G$ be the genetic instrument, $X$ vitamin-D concentration, $U$ an unmeasured cause of vitamin D and mortality, and $Y$ mortality. The causal directed acyclic graph containsAlthough $G$ and $U$ are marginally independent, $X$ is a collider. Conditioning on $X$ opens the path $G\leftrightarrow U\to Y$, violating instrumental-variable independence within the resulting strata. Residual exposure stratification instead removes the component of $X$ predicted by $G$ before stratification; under the additive first-stage model, the stratifying variable is no longer caused by $G$.

The overall odds ratio is $0.99$ with 95% interval $(0.95,1.02)$, so there is little evidence for an average effect. The estimate changes sharply across residual-vitamin-D strata: it is $0.69$ $(0.59,0.80)$ in the deficient group, $0.94$ $(0.89,0.99)$ in the insufficient group, and close to one in the two higher groups. The pattern is consistent with effect modification: raising vitamin D may reduce mortality among deficient people but offer little benefit once concentration is adequate. The claim still depends on the Mendelian randomization assumptions and should account for the fact that several subgroup estimates were examined.

One check is to use measured risk factors as negative control outcomes. A valid instrument should not predict traits that cannot plausibly be downstream consequences of vitamin D; persistent associations would expose horizontal pleiotropy or population structure.

A second check is to repeat the analysis with separate biologically motivated variants or gene-region scores and compare their ratio estimates. Agreement across instruments with distinct biological pathways supports the common vitamin-D mechanism, whereas excess between-instrument heterogeneity suggests direct effects. The same data can also support sensitivity analyses that adjust for measured pleiotropic pathways.

Ignoring factors that do not depend on the parameters, the product of the two binomial likelihoods isIncluding the sampling constants multiplies this by $\left(\genfrac{}{}{0ex}{}{200}{120}\right)\left(\genfrac{}{}{0ex}{}{220}{110}\right)$.

The maximum-likelihood estimators are the success proportionsThe invariance property of maximum likelihood estimation therefore gives

A complete-case analysis estimates each success probability among clinic attenders. If attendance depends on the unobserved outcome even after conditioning on treatment, the observed success proportions differ systematically from those in the randomized groups. This outcome-dependent dropout causes selection bias and can bias their difference.

For a patient whose $Y$ is missing, summing the Bernoulli likelihood contribution over its two possible values givesConsequently the observed-data likelihood for all 490 patients isup to constants. The 70 missing outcomes contribute no information about ${\varphi }_0$ or ${\varphi }_1$ in this marginal model, so the estimators and $\hat{\alpha }=-1/10$ are unchanged. This likelihood calculation alone does not make an outcome-dependent missingness mechanism ignorable.

The data have a monotone missing-data pattern: $Z$ is always observed; a missing $Y$ always entails a missing later $W$; and $W$ may be missing after an observed $Y$.

Under missing at random, dropout before the one-month visit may depend on observed treatment $Z$ but, conditional on $Z$, not on the unseen $Y$ or $W$. Dropout between the one- and six-month visits may depend on the observed history $(Z,Y)$ but, conditional on that history, not on the unseen $W$.

The saturated first imputation model reproduces the observed conditional proportions. Thus

For $Z=0$, the fitted conditional success probabilities for $W$ areAmong all 250 subjects with $Z=0$, multiple imputation asymptotically assigns $100$ to $Y=0$ and $150$ to $Y=1$. HenceUsing the supplied limit for ${\hat{\psi }}_1$ gives

The SI model has two compartments and one transition,where $S$ and $I$ are the numbers susceptible and infectious and $\beta >0$ is the per-infective transmission-rate parameter under frequency-dependent homogeneous mixing.

The ordinary differential equations areThe conservation of population identity $S+I=N$ follows by adding the equations.

Using $S=N-I$ reduces the system to the logistic differential equationSeparation of variables and the initial values giveDifferentiation yields the incidence

Writing $x=\theta e^{-\beta t}$ gives $\Lambda =N\beta x/(1+x{)}^2$. Its maximum occurs at $x=1$, orIf time is restricted to $t\ge 0$ and $\theta <1$, the unconstrained maximizer precedes the initial time, so the maximum on the observed interval is instead at $t=0$.

Set $\tau =t-t_{\ast }$. The Logistic solution of the SI model becomesIt is a symmetric, unimodal bell-shaped curve about the incidence peak $\tau =0$.

The removal rate is $\dot{R}=\gamma I$. Because $\gamma$ is constant, its interior peak occurs when $I$ is maximal. At such a point,and $I>0$, so

Dividing the $I$ equation by the $S$ equation givesThereforeis a first integral. Define the basic reproduction number by $R_0=\beta S(0)/\gamma$. At the removal peak, $S=S(0)/R_0$, and henceand, using $S+I+R=N$,

The same first integral, evaluated initially and after the epidemic when $I(\infty )=0$, givesWhen $I(0)$ is negligible and $R(0)=0$, conservation of population gives $S(\infty )\simeq S(0)-R(\infty )$. Since $\beta /\gamma =R_0/S(0)$, this is equivalent to the final size relation for an epidemic

At event time $t_j$, let $n_{Aj},n_{Bj}$ be the two risk set sizes, $n_j=n_{Aj}+n_{Bj}$, and let $d_{Aj},d_{Bj}$ be the event counts with $d_j=d_{Aj}+d_{Bj}$. Under the null hypothesis of equal hazards, conditioning on the risk set and total number of events gives a hypergeometric distribution, soandThe log-rank statistic and its estimated null variance areUnder the null, $U/\sqrt{V}$ is asymptotically standard normal, or $U^2/V$ is asymptotically chi-squared with one degree of freedom.

The expected Treatment A deaths are $1/2$ at month 1, $1/2$ at month 2, and $3(2/6)=1$ at the three tied deaths at month 5. ThusThis is not half of the five deaths because right censoring changes the treatment proportions in successive risk sets. For Treatment B, $O_B=2$ and $E_B=3$. One observed-to-expected relative-risk estimate is therefore

In a constant hazard survival model, the maximum-likelihood estimator is the number of observed events divided by total person-time at risk. Treatment A contributes $3$ deaths in $1+2+3+3+5=14$ months, while Treatment B contributes $2$ deaths in $1+5+5+7+10=28$ months. Hence

The Nelson–Aalen estimator isFor tied events, $d_j/r_j$ uses the number of events sharing time $t_j$ and the risk-set size just before that time.

At month 5,Their ratio is $29/10=2.9$.

The three estimates are $9/4=2.25$, $3$, and $2.9$. They use different weightings of follow-up time and event times, but all indicate a substantially greater mortality hazard under Treatment A; the exponential and Nelson–Aalen estimates are particularly close.

Conditioning on $T_1=t$ and using independent random variables giveswhere the notation in the paper uses $F_2$ for the survival function.

With common fixed censoring time $c$, the ordering is known exactly when the earlier event occurs by $c$; continuity makes ties have probability zero. Thus the informative event is $\{min(T_1,T_2)\le c\}$ and

For independent exponential distributions, the numerator from part a(ii) iswhile the denominator is the expression in braces. Their ratio is

Condition on the independently generated random censoring information. For every realized common censoring horizon, part b(i) gives the same conditional probability ${\lambda }_1/({\lambda }_1+{\lambda }_2)$. The law of total probability therefore gives that ratio after averaging over the censoring distribution as well.

Since $T_i$ has cumulative hazard function ${\varphi }_i{H}_0(t)$,Thus the cumulative-hazard time change $U_i=H_0(T_i)$ has an exponential distribution of rate ${\varphi }_i$.

The increasing time change $H_0$ preserves the ordering of event times and transforms each censoring time by the same rule. Applying part b to $U_1,U_2$ therefore givesThis is the pairwise race probability for a proportional hazards family.

In a competing risks model, the latent times to different event types form such a race: only the smallest time and its cause are observed. With proportional cause-specific hazards ${\varphi }_i{h}_0(t)$, the probability that cause $i$ wins is ${\varphi }_i/{\sum }_j{\varphi }_j$, independently of the baseline hazard and under independent censoring. This is the same cancellation derived above.

A proportional frailty model specifiesIts population survival function is the Laplace transform of the frailty density,If $m=\mathbb{E}U<\infty$, replacing $U$ by $U/m$ and $h_0$ by $m{h}_0$ leaves their product, and hence the model, unchanged. The normalization $\mathbb{E}U=1$ identifies this otherwise arbitrary scale and makes $h_0(0)$ the initial population hazard when $H_0(0)=0$.

For $g(u)=e^{-u}$,Therefore the population hazard function is

By Bayes theorem, the frailty distribution among survivors has densityIt is an exponential distribution with rate $1+H_0(t)$.

The conditional mean frailty among survivors isBecause the cumulative hazard function is nondecreasing, this mean is nonincreasing: high-frailty individuals tend to experience the event earlier, leaving a progressively more robust surviving population.

Conditionally on survival, average the individual hazard $U{h}_0(t)$:which agrees with part a.