Computing power and sample size for the QTL liability threshold model
In this section, we provide the mathematical background for determination of marker locus genotype frequencies conditional on disease status (affected or unaffected) using the Quantitative Trait Locus (QTL) liability threshold model. Fundamentally, the computation reduces to computing the penetrances
and
.
An important observation to make
here is that, unlike penetrances for dichotomous traits, it is typically not
true that 
.
We begin with some notation that will be in force through the remainder of this section. Note that notation is user-specified.
Notation
=Variance of the QTL locus
r = Dominance/Additive ratio
p = Frequency of QTL increaser allele
q = 1- p (Not entered by user)
Given these parameters, we may
compute the means for the three (univariate) normally distributed variables
corresponding to the quantitative phenotype measures for the three genotypes at
the QTL locus. We denote the means by 
. Using the work of (Fisher 1918) (see e.g., (Lynch and Walsh 1998)), we have:
Also, the residual variance, 
, is just 
. Note that we assume that each of the three normally
distributed variables has variance 
.
Computing penetrances
With the information above, we can compute the penetrances. We have:
where 
is the cumulative distribution function of the standard
normal distribution with mean 0 and variance 1. Assuming Hardy-Weinberg
Equilibrium at the QTL, we also have prevalences:
Conditional marker genotype frequencies
We can now compute the marker genotype frequencies conditional on affection status. If the marker locus in linkage disequilibrium with the QTL locus has two alleles labeled 1 and 2, and we assume that the allele 1 is in coupling with the QTL increaser allele, then the conditional genotype frequencies may be written as:
where 
is the haplotype frequency of the haplotype containing the ith
allele at the QTL [d = QTL increaser allele (not to be confused with
dominance term above (Notation); + = QTL decreaser allele] and the jth
allele at marker locus. These frequencies are computed using information on
disequilibrium between the marker locus and the QTL. See PAWE3D03
for more information.
Acknowledgements
The authors gratefully acknowledge Dr. Shaun Purcell, who provided the expert guidance regarding computation of the conditional genotype frequencies for the QTL liability threshold model.
References
Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans Royal Soc Edinburgh 52:399-433
Lynch M, Walsh B (1998) Genetics and Analysis of Quantitative Traits. Sinauer Associates, Inc., Sunderland