Incorporation of genetic model parameters for cost-effective designs of genetic association studies using DNA pooling

Table 1 The analysis of variance table for a two-stage nested design

ANOVA Table
Source	DF	SS	E(MS)
Case or control (α)	1	$\sum_{i} \sum_{j} \sum_{k} (Y_{i • •} - Y_{• • •})^{2} = J K \sum_{i} (Y_{i • •} - Y_{• • •})^{2}$	$\frac{J K \sum α_{i}^{2}}{I - 1} + (K σ_{\bar{P}}^{2} + σ_{E}^{2})$
Pools nested in case or control (P)	2(J - 1)	$\sum_{i} \sum_{j} \sum_{k} (Y_{i j •} - Y_{i • •})^{2} = K \sum_{i} \sum_{j} (Y_{i j •} - Y_{i • •})^{2}$	$(K σ_{\bar{P}}^{2} + σ_{E}^{2})$
Replicates (E)	IJ(K - 1)	$\sum_{i} \sum_{j} \sum_{k} (Y_{i j k} - Y_{i j •})^{2}$	$σ_{E}^{2}$

Abbreviations for column headings are as noted below.
DF: the degrees of freedom for the respective source row;
SS: the sum of squares for the respective source row;
E(MS): expectation of the mean square for the respective source row.
The sums of squares are based on the following terms: $Y_{i j •} = \frac{\sum_{k = 1}^{K} Y_{i j k}}{K}, Y_{i • •} = \frac{\sum_{j = 1}^{J} \sum_{k = 1}^{K} Y_{i j k}}{J K}, and Y_{• • •} = \frac{\sum_{i = 0}^{1} \sum_{j = 1}^{J} \sum_{k = 1}^{K} Y_{i j k}}{2 J K} .$
The model we consider for individual pooled allele frequency estimates is $Y_{i j k} = μ + α_{i} + P_{j (i)} + σ_{E} E_{i j k} = \frac{A_{i j k}}{A_{i j k} + B_{i j k}},$
where the "group" effect associated with cases or controls is $α_{i} = E (Π_{i}) - E (\frac{Π_{0} + Π_{1}}{2})$ , i = 0,1 subject to the constraint ∑α_i= 0. The random effect associated with the j th pool in either cases or controls is $P_{j (i)} ~ N (0, σ_{P, i}^{2})$ , with $σ_{P, i}^{2} = \frac{J τ_{i}^{2}}{N}$ . Finally, {E_ijk} are independent N(0, 1) random variables and $τ_{i}^{2}$ is the variance of the allele frequency in the i^th group.

ISSN: 1471-2164