===== 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.