large case-control sample: the resulting estimate of β′ will be very close to its mean, and the genotype counts will closely match the underlying genotype frequency distribution. In the logistic regression fit, point estimates only depend on relative frequencies of the different genotype/phenotype classes (although estimates of uncertainty will also depend on the absolute counts). Specifically, increasing the counts but keeping the relative ratios the same is equivalent to scaling the log-likelihood by a constant—it will make it more peaked but not change the location of the mode.
