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 Incoming links: Empty groups in a binomial homogeneity test

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 37 / 2 / i / Solution Created 2026-10-03 Updated 2026-10-14

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Treat the counts as independent binomial distributions,

R_{i} \sim Bin (n_{i}, π_{i})

, conditional on the known numbers of operations. The null hypothesis is that the probabilities

π_{i}

are all equal, with no assumed numerical value. Centre 4 has no operations in this period and hence contributes likelihood one, supplying no information about its mortality probability. The empty groups in a binomial homogeneity test principle leaves 11 contributing centres and 10 test degrees of freedom.

Under the common-probability model the maximum-likelihood estimator is

π = \frac{\sum _{i} r _{i}}{\sum _{i} n _{i}} = \frac{283}{1353} = 0.209165.

(1)

Use the two-column contingency table of deaths and survivors, deleting its all-zero row. The Pearson chi-squared test of homogeneity has statistic

X^{2} = \sum_{i : n_{i} > 0} {\frac{( r _{i} - n _{i} π ) ^{2}}{n _{i} π} + \frac{(( n _{i} - r _{i} ) - n _{i} ( 1 - π ) ) ^{2}}{n _{i} ( 1 - π )}} = \sum_{i : n_{i} > 0} \frac{( r _{i} - n _{i} π ) ^{2}}{n _{i} π ( 1 - π )} .

(2)

Under the null, the large-sample reference distribution is

χ_{10}^{2}

. The expected counts are adequate for this approximation here, including at the smallest nonempty centre. Calculating gives

X^{2} = 39.3923, p \approx 2.17 \times 1 0^{- 5} .

(3)

An alternative is the likelihood-ratio test against separate centre probabilities, with binomial deviance

D = 2 \sum_{i : n_{i} > 0} {r_{i} lo g \frac{r _{i}}{n _{i} π} + (n_{i} - r_{i}) lo g \frac{n _{i} - r _{i}}{n _{i} ( 1 - π )}} .

(4)

The unrestricted probabilities are

r_{i} / n_{i}

; subtracting one fitted common probability leaves the same 10 degrees of freedom. Here

D = 39.5033

and

p \approx 2.07 \times 1 0^{- 5}

. Thus there is strong evidence against a common mortality probability. This omnibus test does not identify Centre 1 as the sole source of the heterogeneity. If its comparison with the remaining centres is the primary scientific question, specify a separate planned one-versus-rest comparison; inspection of many pairwise comparisons calls for consideration of multiple testing.

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