Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 44 / 4

A court-based randomized controlled trial would assign eligible offenders between the two sentencing policies using randomization, ideally stratified by court and important prespecified risk factors. With allocation concealment, the next assignment cannot be selected to suit an offender's prognosis. Baseline characteristics, both recorded and unrecorded, are balanced in distribution rather than confounded with judges' sentencing preferences. This makes a comparison of outcomes interpretable as a causal effect of assignment, subject to complete and comparable follow-up, whereas separate historical cohorts can differ in severity, social circumstances or calendar period.

The commission should compare both specified outcomes over the same 360-day period from sentencing, using an intention-to-treat analysis. An offender initially assigned community service remains in that group even if the order is later revoked and prison follows. Revocation is part of the policy's consequences; restricting the comparison to completers would select on a post-assignment outcome and lose the protection of randomization.

Within-court individual randomization also provides an objective allocation procedure and valid uncertainty estimates. It does not guarantee identical characteristics in a particular finite sample or eliminate loss to follow-up, unequal outcome ascertainment or limited generalizability. Eligibility, genuine uncertainty about the two policies, and a feasible authorized allocation mechanism would need agreement before the trial. If whole courts rather than individuals were assigned, that would be a cluster-randomized trial and its correlation structure would require a different sample size calculation.

Under the no-death assumption, use the mixture of the two community-service paths. Of 10,000 initially assigned offenders, 4,000 breach and leave their subsequent 60-day prison term at day 120. Their remaining 240 days of follow-up match the supplied reconviction window. The 6,000 completers finish community service at day 180 and have the supplied 180-day post-completion follow-up.

The expected value of the number reconvicted is consequentlyThus the initial-policy 360-day reconviction probability is $0.4(0.8)+0.6(0.4)=0.56$. This is a mixture calculation for the initially sentenced cohort, not the 40% rate of its selected completers alone. It uses the supplied windows and assumes they capture the reconvictions being counted, rather than adding unspecified reconvictions during the earlier order or custody periods.

Assume equal allocation of $n$ independent offenders to each arm and a two-sided statistical test of significance level $\alpha$. Write $p_0=0.60$ for P30 and $p_1=0.56$ for initial community-service assignment, with $\Delta =p_0-p_1=0.04$ and $\bar{p}=(p_0+p_1)/2=0.58$. The two event counts have independent binomial distributions. Under the stated alternative, the difference of sample proportions is approximately a normal distribution with mean $\Delta$ and varianceThe pooled two-proportion test has approximate upper rejection threshold $z_{1-\alpha /2}\sqrt{2\bar{p}(1-\bar{p})/n}$ at the planning rates, where $z_q$ is the standard normal quantile. To obtain statistical power $0.80$ against a positive difference, place the alternative mean $z_{0.80}$ alternative-standard-errors above that threshold:Solving yields the usual sample size for comparing two proportions:per arm, rounded upwards, with total sample size $2n$. This targets the dominant rejection tail under the planning alternative; an exact power calculation can include the very small opposite-tail probability or use the binomial distributions directly.

For illustration, at $\alpha =0.05$, $z_{0.975}\simeq 1.960$ and $z_{0.80}\simeq 0.842$, giving $n\simeq 2388.8$. ThusFor a prespecified one-sided test, replace $z_{1-\alpha /2}$ by $z_{1-\alpha }$. The calculation assumes complete endpoint ascertainment, independent individual allocation and that the two initial-policy rates are appropriate; losses or clustered allocation would require corresponding design adjustments.

Interpret the quoted daily death rates as piecewise constant hazard functions, and use the custody schedules described in the problem. This means community service itself has the ordinary community hazard, a completer has no prison exposure, and a breacher is imprisoned from day 60 to day 120. Subsequent imprisonment after another conviction is not specified, so it is not added to the exposure schedule. The abbreviated “CS180” denotes the same initial community-service policy.

For a completer, all 360 days have the baseline rate, giving integrated hazardFor a breacher, the four periods are 60 community days, 60 prison days, 30 high-risk days immediately after release, and 210 further community days. ThusThe piecewise-exponential survival model gives death probabilities $1-e^{-H_C}$ and $1-e^{-H_B}$. Average these over the stated 60%/40% mixture of paths. Equivalently, everyone first survives the common initial 60-day community interval, and survivors then take the two paths in those proportions. The expected value among 10,000 initial assignments isThis accounts for removal of people after death. Since the daily hazards and endpoint death probabilities are small, the usual person-time approximation replaces $1-e^{-H}$ by $H$ and givesBoth calculations use the same specified schedules. The latter treats all planned exposure as if contributed and is slightly larger. Additional custody histories would change the exposure-weighted integrated hazard; the provided reconviction proportions alone do not determine those histories.

Mortality requires its own sample size calculation, rather than using the much commoner reconviction endpoint. Under the same specified exposure assumptions, the P30 schedule has 30 prison days, 30 high-risk post-release days, and 300 ordinary days. Its integrated hazard isHence its 360-day death probability is $p_P=1-e^{-0.0145}\simeq 0.0143954$. For initial community-service assignment the preceding mixture gives $p_C\simeq 0.0127180$. The absolute risk difference is therefore only about $0.0016774$, or 16.8 fewer deaths per 10,000 initial assignments. The person-time approximation gives 145 versus 128 deaths per 10,000, a difference of 17.

As an illustrative design target, take two-sided significance level $0.05$, statistical power $0.80$ and equal individual allocation. Apply the sample size for comparing two proportions formula with these mortality probabilities:It gives about 74,600 participants per arm, or 149,200 altogether; using the simpler exposure approximations gives about 73,130 per arm. At 30,000 sentences per year, even if every offender were eligible and enrolled, this requires roughly five years of recruitment and another year to complete the final cohort's follow-up.

For scale, one year's equally allocated cohort has 15,000 per arm, with roughly 216 expected deaths after P30 and 191 after community-service assignment. The standard error of the risk difference is about $0.001335$, so the expected standardized separation is only about $1.26$, below the usual two-sided 5% critical value. A one-year trial would consequently have poor statistical power for this mortality difference.

Thus mortality discrimination is much more demanding than reconviction discrimination. National recruitment and reliable linkage to death records could make a multiyear study feasible, but restricted eligibility, incomplete follow-up, or lower recruitment would lengthen it. A trial sized only for the 4-percentage-point reconviction difference would not provide a precise mortality comparison. The numerical conclusion depends on the supplied daily hazards, custody schedules, and the chosen power and significance targets; the prompt does not specify a mortality design target of its own.