Generalisations of two tests for the detection of household aggregation of disease

Two statistical tests have been suggested to detect household aggregation of disease. Mathen and Chakraborty [1950] proposed counting the number of households, Z, in which there were no cases of the disease, and Walter [1974] proposed counting the number of pairs, N, of cases within households. As originally formulated, both of these methods suffer from the limitation that it is assumed, in the null case, that all members of the population, who are assumed to be susceptible, are at equal risk to disease. In this paper we generalise these tests to the situation in which different population strata are at different risks to disease. Formulae are given for the first four moments of Z. The revised Walter's test is shown to be a special case of a test of Pike and Smith [1974] which enables the expectation and variance of N to be directly evaluated. A similar relationship is noted between Walter's test and Knox's [1964] test for space-time clustering. A further test statistic is suggested: the number of cases, T, in households containing two or more cases.