## 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 this tutorial on analytic power calculation for quantitative traits.