Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 4

The administrative data include only encounters with $M=1$. Conditioning on the stop indicator selects on the colliderwhich opens the noncausal path $D\leftrightarrow U\to Y$ and creates collider bias. The equal entries estimate only the selected risks $\mathbb{P}(Y=1\mid D=d,M=1)$. They ignore racial differences in the probability of being stopped and therefore do not identify the total causal effect of race on violence. This is also selection bias, because encounters with $M=0$ never enter the dataset.

Let $Y(d,m)$ be the potential violence outcome if race were set to $d$ and stop status to $m$. The substantive assumption is the structural zeroEquivalently, with the natural stop status $M(d)$, $M(d)=0$ implies $Y(d)=Y(d,M(d))=0$.

The graph has no cause of $D$, so $D$ is independent of its potential outcomes. By consistency of potential outcomes, the g-formula therefore givesCondition on $M$ and use the structural zero from part ii:Consequently the causal risk ratio isApplying Bayes theorem to the second factor giveswhich proves the stated formula. The unmeasured common cause $U$ of $M$ and $Y$ does not obstruct this total-effect argument because no mediator effect is being identified.

The stop records estimatebut they cannot estimate the population encounter probabilities $\mathbb{P}(D=1)$ and $\mathbb{P}(D=0)$ because encounters without a stop are absent. Equivalently, they do not determine the race-specific stop-probability ratio.

The scientist needs a representative denominator for all police-civilian encounters, including those with $M=0$. Suitable sources could include a carefully designed population or travel survey, systematic street and traffic observation, dispatch or body-camera sampling that records non-stop encounters, or an external administrative source measuring exposure to police by race. Combining its estimate of the population race odds with the estimable stop-data terms identifies the displayed causal risk ratio, provided the external sample targets the same city, period, and encounter population.