Use the binomial distribution for independent outcomes in each nonoverlapping batch, and let the CUSUM accumulate the log-likelihood ratio favoring a specified change. The two numbered calculations below treat the batch score and the initial-response design separately.

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Let

p_{1}

be the alternative death probability. The odds ratio specifies

\frac{p _{1}}{1 - p _{1}} = k \frac{p _{0}}{1 - p _{0}}, p_{1} = \frac{k p _{0}}{1 - p _{0} + k p _{0}}, 1 - p_{1} = \frac{1 - p _{0}}{1 - p _{0} + k p _{0}} .

(1)

Assume

0 < p_{0} < 1

and

k > 0

. For observed batch count

y_{t}

, the common combinatorial factor in the two binomial likelihoods cancels, giving the CUSUM increment

W_{t} = lo g \frac{( y _{t} n ) p _{1}^{y_{t}} ( 1 - p _{1} ) ^{n - y_{t}}}{( y _{t} n ) p _{0}^{y_{t}} ( 1 - p _{0} ) ^{n - y_{t}}} = y_{t} lo g (p_{1} / p_{0}) + (n - y_{t}) lo g [(1 - p_{1}) / (1 - p_{0})] .

(2)

Therefore

W_{t} = y_{t} lo g k - n lo g {1 + (k - 1) p_{0}} .

(3)

For

k > 1

large death counts add positive evidence of deterioration; for

k < 1

unusually small death counts add evidence for improvement. The score has nonpositive expectation under the null and nonnegative expectation under the specified alternative, as follows from the nonnegativity of Kullback-Leibler divergence. Independent, nonoverlapping batches avoid counting the same patient's outcome repeatedly.

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A classical sequential probability ratio test addresses a fixed-start choice between two simple statistical hypotheses. It sums log-likelihood ratio increments until an upper boundary accepts the alternative or a lower boundary accepts the null; it then terminates that test. A CUSUM instead monitors indefinitely for a change at an unknown time. Its reset at zero discards sustained evidence from a previously satisfactory period, so a late change need not first undo all earlier negative scores. An upper boundary

h

signals when the accumulated change evidence is sufficiently large.

A fast initial response CUSUM takes

0 < X_{0} < h

, often

X_{0} = h /2

, in place of a zero start. Until its first reset or signal, it acts like a fixed-start sequential probability ratio test with a lower boundary at

- X_{0}

and an upper boundary at

h - X_{0}

for the subsequent cumulative increments. Upon reaching the lower boundary it resets and continues as an ordinary CUSUM. The head start gives faster detection if a change is already present when monitoring begins or restarts. It is useful after an intervention, a shutdown or a previous alarm when immediate renewed surveillance is desired. Its initial false-alarm behavior differs from the zero-start scheme, so the head start and signalling boundary must be calibrated together; the preceding numerical choice is a design convention, not a universal optimum.

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Four distinct objections, each with a corresponding improvement, are as follows.

Different patients imply different baseline risks. Raw death rates confound performance with case mix, including urgency, disease severity and comorbidity. Use a prespecified, validated risk-adjusted provider comparison, standardizing to comparable patients or comparing observed deaths with $\sum_{i} p_{i}$ . Check overlap and model calibration rather than assuming adjustment removes all confounding.
Small annual volumes produce unstable rankings. A few deaths can move a surgeon far up or down a league table; zero deaths need not indicate zero risk. Display counts and suitably calibrated confidence intervals for binomial proportions, preferably in funnel plots against volume. Pool suitable periods or use a hierarchical Bayesian model to stabilize estimates, while allowing genuine time changes.
Many simultaneous comparisons generate chance outliers. With $m$ independent null comparisons, the probability of at least one false positive is $1 - 0.9 5^{m}$ , and ranks amplify selection of extremes. Use multiple testing adjustments or simultaneous control limits and seek confirmation before interpreting an apparent outlier. Dependence through the national comparator must also be accounted for.
The endpoint and unit of attribution can mislead. In-hospital death depends on discharge and transfer policy, and care is delivered by a team; follow-up after discharge is omitted. Use an independently audited fixed-horizon endpoint, such as a prespecified 30-day mortality outcome, linked across hospitals and to death registrations, and report an appropriate team or institutional comparison alongside any surgeon-level estimate. Reliable attribution and consistent inclusion criteria reduce incentives to manipulate coding or case selection.

A confidence interval containing the national rate does not demonstrate equivalence, and a raw ranking does not establish a causal performance difference. The improved presentation should convey precision and uncertainty as well as the estimated outcome.

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 3

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 3

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