## 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 al1) $z_\beta=\frac{\sqrt{\Delta p^2rm^*-(r+1)\Delta p}-z_\alpha\sqrt{(r+1)\bar{p}\bar{q}}}{\sqrt{rp_{s1}q_{s1}+p_{s2}q_{s2}}}$

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 al2). [Power of different analytic tests yet to be discussed].

### Example

Please find more details in this tutorial on analytic power calculation for case control data.

1) Joseph L. Fleiss, Alex Tytun and Hans K. Ury (1980). A Simple Approximation for Calculating Sample Sizes for Comparing Independent Proportions. Biometrics
2) 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

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