Derivation of non-centrality
parameter for PAWE method
In this section, we show how we compute the non-centrality parameter in formula (3) of Gordon et al. (Gordon et al. 2002). While it was not presented in our paper, we also compute here the non-centrality parameter for the allelic test of association. In what follows, we shall use the same notation that Mitra (Mitra 1958) used. We shall also use the following notation from our paper:
Count parameters:
NA = number of cases
NU = number of controls
Probability parameters:
pA = allele frequency of SNP marker 1 allele in the case group (affecteds)
pU = allele frequency of SNP marker 1 allele in the control group (unaffecteds)
= frequency of
SNP marker genotype j in the case group (j=0 for 11 genotype, j=1
for 12 genotype, j= 2 for 22 genotype) (see note directly below)
= frequency of SNP marker genotype j in the
control group (j=0 for 11 genotype, j=1 for 12 genotype, j=
2 for 22 genotype) (see note directly below)
Note: We used the notation 
(respectively, 
) in our paper to indicate the conditional genotype
frequencies in the case (respectively, control) populations. However, Mitra uses
this notation for something else. Therefore, we use the notation 
and 
.
Mitra Calculations
For either test of association, Mitra (Mitra 1958) shows (page 1230) that the non-centrality parameter may be written as:
(1)
To make sense of
expression (1), we must define each of the terms. Let Ni
represents the number of samples collected in the ith
population (i=1 or 2), and let 
. Then 
, and the terms Ci are chosen to satisfy
the equation:
(2)
where S is an arbitrary constant chosen by the researcher (for Mitra’s application, it was the total cost of collecting all samples).
The value r in the summand of
expression (1) refers to the number of columns in a 2 ´ r contingency
table. The terms 
(
) are defined by the expression:
(3)
where 
, the proportion of observations in the ijth
cell (vij is the number of observations in the ijth cell),
and 
is the jth column mean.
For our purposes, S
= 1, and it follows from equation (2) and the definition of the terms Qi that 
(
). Using equation (3), we have
.
Non-centrality parameter for allelic test of association
We are now ready
to apply derive the non-centrality parameters for the allelic and genotypic
tests of association. For the allelic test,
(the unit of measure
is the allele), and so the list of parameters becomes:
(4a)
It follows from the values of the cell
parameters pij that
for j =1, 2. Substituting the values (4a) into the
equation (1) gives formula:
Non-centrality parameter for genotypic test of association
For the
genotypic test,
(the unit of measure
is the genotype (or equivalently, the individual)), and so the list of
parameters becomes:
(4b)
As above, substituting the values (4b) into equation (1) gives non-centrality parameter for the genotypic test of association [formula (3) in our paper (Gordon et al. 2002)].
References
Gordon D, Finch SJ, Nothnagel M, Ott J (2002) Power and sample size calculations for case-control genetic association tests when errors are present: application to single nucleotide polymorphisms. Human Heredity 54:22-33
Mitra SK (1958) On the limiting power function of the frequency chi-square test. Annals of Mathematical Statistics 29:1221-1233