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+ | ===== 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. | ||