There are two distributions that are often assigned to the residuals to transform the underlying liability to the probability of being affected or unaffected, the standard normal distribution (probit model) and the logistic distribution (logistic regression). Given the logistic probability function of f(li) = exp(li)/[1 + exp(li)] with li being the liability of the ith individual, the odds ratio (OR) for a SNP j in a multiple-SNP analysis is exp(bj), with bj being the log(OR) in a joint analysis, and is exp(βj) for a single-SNP model L = xjβj + e, with βj being the log(OR) in a single-SNP analysis. Even though the residuals follow a logistic distribution, the least-squares estimates of effect sizes are unbiased, because the least-squares approach does not rely on the assumption of normality. Hence, we can apply the same methods as described above for a quantitative trait to a case-control study, as long as the effect sizes and standard errors are expressed on the log(OR) scale.
