Specification
of Linear Trend Test non-centrality parameter
In this link, we
present the formula for the non-centrality parameter for The Linear Trend Test (Cochran 1954;
Armitage 1955). Chapman and Nams
work provides the theoretical basis for this formula (Chapman and Nam
1968). We use this
non-centrality parameter in our webtool, PAWE-3D (http://linkage.rockefeller.edu/pawe3d/).
The authors most gratefully acknowledge Wang Kook (Stony Brook University) who
provided the mathematical derivation of the non-centrality parameter.
Definition of Linear Trend Test Statistic
Consider the
parameters as presented in table 1, where the values 
represent the weight given to an observation in the ith
category, and the values 
(respectively,
) represent the observed number of genotypes in ith
category for cases (respectively, controls). For example, when considering SNPs
with alleles A and B, then 
could represent the number of cases with genotype AA, 
could represent the number of cases with genotype AB,
and 
could represent the number of cases with genotype BB.
Observe that, in table 1, we have 
, 
, 
, N = R +S. We comment that some
commonly used weight settings 
are: (0,1,2)
(additive genetic model), (0,0,1) or (1,0,0) (recessive genetic model), and
(1,1,0) or (0,1,1) (dominant genetic
model).
|
|
|
|
|
|
|
Case |
|
|
|
R |
|
Control |
|
|
|
S |
|
Total |
|
|
|
N |
Let
,
where
fixed 
and 
.
We have
= 
= 
.
The
test statistic
approximately follows a standard normal distribution under
the null hypothesis that genotype frequencies for each category i are
equal in cases and controls. From elementary mathematical statistics theory, 
follows a 
distribution under the null hypothesis. We refer to 
as The Linear Trend Test Statistic.
Non-centrality
parameter for Linear Trend Test
In what follows, we restrict our attention to the situation where the number of categories I in table 1 is equal to 3, as would be the case for a di-allelic locus such as a SNP. Based on previous work (Chapman and Nam 1968), it can be shown that the non-centrality parameter for The Linear Trend Test is given by:
,
where
, 
, and R, S, and N are defined as in
table 1. This non-centrality parameter enables us to compute asymptotic power
for any significance level, given values of the weights xi,
sample sizes R, S, and genotype frequencies in cases (
) and controls (
). Equivalently, minimal sample size for a fixed power can be
determined by solving for R, and specifying the ratio S/R of
controls to cases [see (Gordon et al., 2002) for an example with the
genotypic test of association on 2 degrees of freedom].
References
1. Cochran, W.G. (1954) Some methods for strengthening the common chi-squared tests. Biometrics. 10, 417-451.
2. Armitage, P. (1955) Tests for linear trends in proportions and frequencies. Biometrics. 11, 375-386.
3. Chapman, D.G. and Nam, J.M. (1968) Asymptotic power of chi square tests for linear trends in proportions. Biometrics. 24, 315-327.
4. Gordon, D., Finch, S.J., Nothnagel, M. and Ott, J. (2002) Power and sample size calculations for case-control genetic association tests when errors are present: application to single nucleotide polymorphisms. Hum Hered. 54, 22-33.