GWAS analyses typically employ the trend test, which effectively fits a multiplicative model. While this may result in model mis-specification (if the model underlying the data is not multiplicative), it will nevertheless pick up some of the association signal. For a given underlying disease model, allele frequency, and ratio of cases to controls in the sample, there will be a characteristic mean value for the additive parameter when fitting the multiplicative model. We refer to this as the effective additive parameter and denote it by β′. It can be calculated numerically by fitting the multiplicative model, using logistic regression, to the theoretical genotype frequencies for cases and controls under the disease model of interest, weighted by the case-control sampling ratio. In other words, we pretend the theoretical frequencies are sample counts. To see why this works, imagine taking a very 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
