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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)=1Mi(1pi) The shift of the mean of quantitative trait value, δ, 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 pi and an effect size λi

ITERATION

For each out of the total Q possible subset of locus combination

  • Calculate the probability of observing such particular genotype combination Qi: qi=iobservedpijunobserved(1pj)
  • Calculate the conditional probability Pr(Qi|G=2)=qiPr(G=2)
  • Calculate the effect size for this particular combination γi=iobservedλi

OUTPUT

  • The expectation is given by δ=QiγiPr(Qi|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(Qi|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 zβ=|δ|1mp+1m(1p)+zα/2 Notice that “samples” in this setting means haplotypes and the final sample size should be Nsamples=Nhaplotypes2

Example

Please find more details in this tutorial on analytic power calculation for quantitative traits.