# Differences

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 — cc [2016/04/13 18:02] (current) Line 1: Line 1: + ===== 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.

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