The bingo model of survivorship. II: Statistical aspects of the bingo model of multiplicity 1 with application to hereditary polyposis of the colon

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Abstract
Some Mendelian disorders (Huntington chorea, hereditary polyposis coli) are not manifest at birth show a distribution in the age of onset. Patients at risk fall into three groups. In type I, they are affected when first examined. In type II, they are not affected at one visit, but are at a later visit. Those of type III (who comprise an indistinguishable mixture of those who have, and those who have not, inherited the gene) are never found to be affected. This paper posits a model that the age of onset is logistic. (It is a degenerate bingo model in which competing causes of death may be ignored.) The statistical properties of maximum likelihood estimation (MLE) are explored by Monte Carlo simulation of this logistic function with known arbitrary parameters. Two schemes are used: point‐prevalence (or synchronic) data of types I and III, and piecewise longitudinal (diachronic) data; this allows all three types to be included. Samples of various sizes between 25 and 100 are used. While estimates of the parameters are positively biased (especially with small samples), the estimate of the mean appears to be consistent, almost unbiased, and fairly precise, though somewhat larger than the estimates from the lower bound (a fact that calls for some caution in interpreting actual data). The MLE was applied to 109 patients with the Gardner syndrome (GS); measures of variability found by applying MLE to four random subsets of 25 each were compared against the asymptotic estimates. The analysis was also applied to 36 persons with familial polyposis coli (FPC). The mean age of onset in GS and FPC was similar, and since they are rather earlier than is currently believed, it is recommended that regular supervision be started at not later than 10 years of age.