We explore the power of the twin study to resolve sources of familial resemblance when the data are measured at the binary or ordinal level. Four components of variance were examined: additive genetic, nonadditive genetic, and common and specific environment. Curves are presented to compare the power of the continuous case with those of threshold models corresponding to different prevalences in the population: 1, 5, 10, 25, and 50%. Approximately three times the sample size is needed for equivalent power to the continuous case when the threshold is at the optimal 50%, and this ratio increases to about 10 times when 10% are above threshold. Some power may be recovered by subdividing those above threshold to form three or more ordered classes, but power is determined largely by the lowest threshold. Non-random ascertainment of twins (i) through affected twins and examining their cotwins or (ii) through ascertainment of all pairs in which at least one twin is affected increases power. In most cases, strategy i is more efficient than strategy ii. Though powerful for the rarer disorders, these methods suffer the disadvantage that they rely on prior knowledge of the population prevalence. Furthermore, sampling from hospital cases may introduce biases, reducing their value. A useful approach may be to assess the population with a screening instrument; the power calculations indicate that sampling all concordant and half of the discordant pairs would be efficient, as along as the cost of screening is not too high.