We first performed expression quantitative trait linkage (eQTL) analysis using MERLIN regress v 1.0.1 [5, 6]. We zeroed out the genotypes of the child as well as those of the grandparents when a Mendelian inconsistency was detected. All other genotypes were retained. Marker allele frequencies were estimated from the data and single-point linkage analyses of gene expression data were performed for all pairs of 3554 genes and 2819 SNPs, excluding the ones (genes or SNPs or both) on the sex chromosomes. We then stratified the linkage results by the map distance between each SNP and the gene. Gene positions were obtained from Build 36.1 of the UCSC Genome Browser [7]. SNP locations were obtained from Build 126 of dbSNP [8] (on Build 36.1 of the human genome). We used the definition of Morley et al. [4] in which *cis* regulators are the SNPs within 5 Mb of genes and *trans* regulators are the remaining SNPs. Stratum 1 contained *cis* SNPs and stratum 2 contained *trans* SNPs, and the aggregated group included all SNPs ignoring stratification. We used *m* to denote the total number of hypotheses among which *m*_{0} were true nulls, *R* to denote the number of rejections, and *p*_{(i)}, *i* = 1,...,*m* to denote the linkage p-values while *p*_{(1)} ≤ *p*_{(2)} ≤ ... ≤ *p*_{(m) }were the ordered *p*-values. We used superscript (*k*), *k* = 1, 2 as the stratum indicator. Finally, we applied the stratification principle [2] under the following three frameworks.

### Framework I: fixed rejection region

This framework chooses the rejection region in advance, i.e., it rejects all hypotheses with unadjusted *p*-values less than a pre-determined *α* value, e.g., *α* = 0.01%. The corresponding FDR level among the *R* positives can then be estimated by F\widehat{D}R\left(\alpha \right)=\left(m{\widehat{\pi}}_{0}\alpha \right)/R, where *R* = {*p*_{(i) }≤ *α*}, {\widehat{\pi}}_{0} is an estimate of the proportion of null hypotheses, *π*_{0} = *m*_{0}/*m*, e.g., {\widehat{\pi}}_{0}(*λ*) = #{*p*_{
i
}> *λ*}/(*m*(1 - *λ*)) with *λ* = 0.5. The rejection procedure remains the same for the stratified method using the same *α* level (*R*^{(1)} + *R*^{(2)} = *R*). However, the estimates of FDR among *R*^{(1)} and *R*^{(2)} can be considerably different from the aggregated FDR, with one stratum estimate closer to 0 and the other closer to 1. For both cases, one thus obtains more information on the specificity of the results.

### Framework II: fixed FDR

Under this framework, the targeted FDR level is pre-chosen at a *γ* level, e.g., *γ* = 5%. Storey [9] showed that controlling FDR at the *γ* level is equivalent to rejecting all tests with *q*-values ≤ *γ*, and the *q*-values can be estimated by {\widehat{q}}_{(i)}=\mathrm{min}\{{\widehat{\pi}}_{0}m{p}_{(i)}/i,{\widehat{q}}_{(i+1)}\}, and {\widehat{q}}_{(m)}={\widehat{\pi}}_{0}{p}_{(m)}. This method is equivalent to the FDR adjusted *p*-value method [10, 11]. To fairly compare the performance of the stratified FDR method with the aggregated one, we controlled the FDR at the same level for both Strata 1 and 2 and the aggregated group using the above *q*-value method. The objective was to show that the total number of rejections *R*^{(1)} + *R*^{(2)} under stratification is greater than *R* under aggregation.

### Framework III: fixed number of rejections

In this case, the total number of significant results *R* that merits further study is pre-determined based on, for example, the budget and capacity of a particular chip platform. Without stratification, the choice of *R* hypotheses is straightforward, i.e., the *R* tests with the smallest *p*-values: *p*_{(1)},...,*p*_{(R)}. The corresponding FDR level can be estimated by F\widehat{D}R=\left(m{\widehat{\pi}}_{0}{p}_{(R)}\right)/R. Under stratification, one needs to find the optimal configuration of *R*^{(1)} and *R*^{(2)} such that *R*^{(1)} + *R*^{(2)} = *R* and the overall FDR is minimized, where F\widehat{D}{R}_{stra}=\left({m}^{(1)}{\widehat{\pi}}_{0}^{(1)}{p}_{({R}^{(1)})}^{(1)}+{m}^{(2)}{\widehat{\pi}}_{0}^{(2)}{p}_{({R}^{(2)})}^{(2)}\right)/R. The goal is to show that stratification leads to a smaller FDR rate given the same number of positives allowed. More importantly, the configuration of *R*^{(1)} and *R*^{(2)} obtained using stratification can differ markedly from the aggregation case. Without stratification, the distribution of *R* rejections between the two strata is roughly proportional to the number of hypotheses in each stratum; while with stratification, the stratum with smaller *π*_{0} (less noise) and higher power to detect true signals proportionally rejects more hypotheses.