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 — qt [2016/04/13 18:02] (current) Line 1: Line 1: + ===== Power for Quantitative Traits ===== + ==== Model ==== + The key to power and sample size calculation for quantitative traits is on the comparison of quantitative trait values between groups. Consider a two-sample $t$ test framework with unequal sample size, where one group of sample consists of //wildtype (wt)// haplotype, and the other group //​none-wildtype (nwt)// haplotypes. The probability of falling into the //nwt// group is $Pr(G=2) = 1 - \prod_i^M(1-p_i)$ The shift of the mean of quantitative trait value, $\delta$, under our current modeling, is the expected effect size of the //nwt// group, which I calculate numerically using the algorithm described below: + + === INPUT === + *  A multi-site genotype having $M$ sites, each associated with a MAF $p_i$ and an effect size $\lambda_i$ + + === ITERATION === + For each out of the total $Q$ possible subset of locus combination + + *  Calculate the probability of observing such particular genotype combination $Q_i$: $q_i=\prod_{i\in observed}p_i \prod_{j\in unobserved}(1-p_j)$ + *  Calculate the conditional probability $Pr(Q_i|G=2) = \frac{q_i}{Pr(G=2)}$ + *  Calculate the effect size for this particular combination $\gamma_i = \sum_{i\in observed}\lambda_i$ + + === OUTPUT === + *  The expectation is given by $\delta = \sum_i^Q \gamma_i Pr(Q_i|G=2)$ + + === Low-order approximation === + The method described above is exact, but can be very slow for long genomic regions due to the huge number of possible subset of locus combination. A low-order approximation is used in this program to only consider a maximum of up to 2 or 3 loci in a genotype combination,​ ignoring the contributions from all other high order possibilities. For genes having variant sites smaller than 8, 3 order approximation is applied; for larger genes 2 order approximation is applied. $Pr(Q_i|G=2)$ will be adjusted accordingly such that they still sum up to $1$. + + ==== Power and Sample Size Calculation ==== + Power and sample size estimations can be performed under a two-sample $t$ test framework $z_\beta = \frac{|\delta|}{\sqrt{\frac{1}{mp}+\frac{1}{m(1-p)}}}+z_{\alpha/​2}$ Notice that "​samples"​ in this setting means **haplotypes** and the final sample size should be $N_{samples} = \frac{N_{haplotypes}}{2}$ + + ==== Example ==== + Please find more details in [[http://​bioinformatics.org/​spower/​analytic-tutorial|this tutorial]] on analytic power calculation for quantitative traits.