Exact conditional tests of quasi-independence for triangular contingency tables: estimating attained significance levels

SUMMARY Some investigations of whether disease is related to birth order result in a triangular table of counts of diseased individuals classified by birth order and sibship size. Here testing for an interaction corresponds to testing for quasi-independence of the classification factors. We propose a Monte Carlo algorithm for estimating the significance level of this test, which should be used when the asymptotic results are suspect. We describe the procedure by using a classic triangular table classifying stroke patients. Unlike other researchers, we conclude that there is moderate evidence for rejecting quasi-independence here. Our main application concerns the relationship between neurosis and birth order. Here there is strong evidence for rejecting quasi-independence in favour of the uniform association model.