===== Power for Case Control Studies =====
==== Cumulative Minor Allele Frequency in Cases and Controls ====
To compare the difference between cases/ctrls for cumulative MAF of all variants across a gene analytically, the case/ctrl group specific MAF have to be calculated given the MAF in population and effect size of variant sites. Consider a multi-site genotype having \(M\) sites, each with a MAF (population MAF) and an odds ratio associated with the site. It is then possible to calculate the corresponding MAF in //case// group under Bayesian arguments. Specifically \(Pr(genotype)\) is a function of MAF (\(p_{AA}=q^2, p_{aa} = (1-q)^2, p_{Aa} = 2q(1-q)\) under Hardy-Weinberg Equilibrium assumptions), \(Pr(status|genotype)\) is genotype penetrance, \(Pr(status)\) is prevalence; \(Pr(genotype|status)\) can thus be calculated and can be translated into MAF in cases \[Pr(g|s)=\frac{Pr(g)Pr(s|g)}{Pr(s)}\] and cumulative MAF \[p_s = 1 - \prod^M_i(1-p_{is})\]

It is also possible to evaluate power and sample size for common variant analysis (e.g., in GWAS) using SEQPower. This is just a special scenario of \(M=1\) and the cumulative MAF is the locus MAF.

==== Analytic Power for Comparing Difference in Cumulative MAF ====
Power and sample size can be evaluated using a test for binomial proportions as described by Fleiss et al((Joseph L. Fleiss, Alex Tytun and Hans K. Ury (1980). **A Simple Approximation for Calculating Sample Sizes for Comparing Independent Proportions**. //Biometrics//)) \[z_\beta=\frac{\sqrt{\Delta p^2rm^*-(r+1)\Delta p}-z_\alpha\sqrt{(r+1)\bar{p}\bar{q}}}{\sqrt{rp_{s1}q_{s1}+p_{s2}q_{s2}}}\]

Sample size can be obtained via inverting the equation above. Please refer to Fless et al. 1980 for notations for the power function.

Other statistical tests can also be applied, e.g, Casagrande et al((J. T. Casagrande, M. C. Pike and P. G. Smith (1978). **An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions**. //Biometrics//)). [Power of different analytic tests yet to be discussed].

==== Example ====
Please find more details in [[http://bioinformatics.org/spower/analytic-tutorial|this tutorial]] on analytic power calculation for case control data.