It was found that higher order polynomials than cubic resulted in over fitting, whereas lower order polynomials were insufficient to model the bias over the different minor allele frequencies. Taking the negative logarithm and removing all constants that do not depend on β results in the following negative log likelihood:l(X|β)=∑i=1M12(Σi(4a,4a)Σi(4b,4b)−Σi(4a,4b))×[(xi(4a)−Nf(pi)−μi(4a))2Σi(4b,4b)−2(xi(4a)−Nf(pi)−μi(4a))(xi(4b)−Nf(pi)−μi(4b))Σi(4a,4b)+(xi(4b)+Nf(pi)−μi(4b))2Σi(4a,4a)],where xi(4a) and xi(4b) are the number of case-mother duos (with expected values μi(4a)=Nθi(4a)=Npi2(1−pi) and μi(4b)=Nθi(4b)=Npi(1−pi)2) for cells 4a and 4b, respectively. Newton’s Method is used to fit β in PREMIM by minimizing the negative log-likelihood with (fixed) estimated minor allele frequencies, pi, for each SNP i. The new adjusted cell counts for cells 4a and 4b are then given by xi(4a) − Nf(pi) and xi(4b) + Nf(pi), respectively.