A robust score statistic to map quantitative trait loci (QTL) using extended pedigrees

For notational simplicity, we suppress the index for each family. Let Y denote the phenotype for the members of a pedigree. Let ν _ij (t) denote the number of alleles IBD at locus t between individual i and j, centered to have expected value 0. Let A _ν (t) be the IBD matrix with [A _ν (t)] _ij = ν _ij (t). Define Σ to be the phenotypic covariance matrix. Assuming no dominant genetic effect, then according to Tang and Siegmund [2], the conditional covariance matrix

Σ _A = Cov(Y, Y | A _ν (τ)) = Σ + αA _α (τ),

where α ≥ 0 denotes the additive genetic effect.

From the working assumption that at a trait locus τ, conditional on A _ν (τ), Y follows a multivariate normal distribution, one can derive a robust score statistic for testing whether there is an additive genetic effect at τ [2] in the form Z(τ) = l _α (τ)/[E₀$l_{\alpha }^2$(τ)]^1/2. Here,l _α (τ) = 2^-1∑[-trΣ^-1A _ν + trΣ^-1A _ν Σ^-1YY'].

In practice, unobserved values of the IBDs in l _α (t) are replaced by their conditional expectation given the genotypic data, while their variances are estimated from multipoint genotypic data. To make the test robust to the normality assumption of the traits, we use Z(τ) = l _α (τ)/[E₀($l_{\alpha }^2$(τ) | Y)]^1/2.

Generation-specific effects in extended pedigrees were allowed in estimating the mean and variance of each trait, while the phenotypic correlations ρ₁ for grandparent-grandchild and ρ₂ for sibs were estimated by maximum-likelihood estimation (MLE) with the genetically natural constraint ρ₁ ≤ ρ₂/2. The sex-average genetic map provided to Genetic Analysis Workshop 15 by Sung et al. [5] was used. The expected IBD counts were computed by MERLIN [6] using all 2819 SNP markers and full pedigrees. The score statistic was then computed at marker locations using the estimated IBD counts. Let Z _n (t) be the score statistic at marker t for the n^th phenotype. We defined the genome scan statistic to be Z _n = max _t Z _n (t) over all marker loci for trait n. The genome-wide p-value for each Z _n (i.e., for each of the 3554 traits) was then computed using the skewness correction described in the Appendix.

Our theoretical calculation shows that, for these 14 pedigrees, using only sibships causes a loss of power equal to roughly 35% of the sample size. Here, we compare linkage scores for sibships and for entire pedigrees, and then we use the more powerful pedigree-based tests for analysis of the effects of our correction procedures.

Control FDR based on the empirical null distribution

One useful method for addressing the multiple testing problem is to control the FDR [7]. For our problem, a successful FDR procedure requires 1) accurate evaluation of the genome-wide p-value for each trait, and 2) adjustment for the correlations among the genome scan test statistics. Here, we correct FDR using Efron's method to estimate the empirical null distribution [4]. For trait n, we computed the genome scan statistic Z _n and approximated the genome-wide p-value p _n using the skewness-correction method described in the Appendix (accuracy is checked using a Monte Carlo simulation). We then transformed p _n to the normal quantile q _n = Φ^-1(1 - p _n ) and applied Efron's method on {q _n } to estimate the empirical null distribution N(μ, σ²) The expected number of false positives for threshold q is 3554Φ((q - μ)/σ) and the FDR is estimated as FDR = 3554Φ((q - μ)/σ)/#{q _n > q}. Here, we have implicitly assumed that the proportion of traits without linkage signals is close to one.

Control FDR using permutations

To validate the FDR results obtained by correcting for skewness and for the empirical null distribution, we used 1000 permutations to determine the number of false positives and hence the FDR following Efron's method [3]. For permutation n, we computed the genome scan statistics $Z_1^n,\cdots ,Z_{3554}^n$, and computed $Y_0^n=\#\{Z_k^n<3.5\}$ and $Y_1^n(b)=\#\{Z_k^n>b\}$ for b > 3.5, where 3.5 is the median value of Z. The correlation among the genome scan statistics causes $Y_0^n,Y_1^n(b)$ to be correlated. So we can fit a linear regression model Y₁(b) = a₁ + a₂Y₀ + ε to the 1000 pairs of ($Y_0^n,Y_1^n(b)$). For the observed data, we computed the genome scan statistics, Y₀ and Y₁, then computed the expected number of false positives among the Y₁(b) positive findings as a₁ + a₂Y₀. The estimated FDR for threshold b was then (a₁ + a₂Y₀)/Y₁. The permutation-based FDR procedure does not require accurate evaluation of genome-wide p-values or appropriate correction for the correlations among tests, but it is computationally intensive.

Search for cis-regulated genes

If most linkages prove to be in cis regions, then the power to detect these linkages could be increased by considering only the markers within 5 Mb of that gene, because genome-wide p-values would have to be corrected only for this small proportion of markers for each expression trait. We searched the location of the 3554 gene names on http://sky.bsd.uchicago.edu/genequery.html. The markers within 5 Mb of each target gene were identified and the scan statistic was obtained as the maximum score for these markers. Because the number of markers and the genetic lengths in the cis region are highly variable, we evaluated the region-wide p-values empirically. We ran 8000 permutations and fit a quadratic curve (log p _n = α_n,0 + α_n,1b + α_n,2b²) to the results to predict the region-wide p-value for the n^th trait. The form of the p-value is suggested by the formula in Appendix. We then transformed p _n to normal quantile q _n and estimated the empirical null distribution based on the quantitles. The expected number of false positives and the FDR are based on the estimated empirical null distribution.

Genome-wide p-value

To evaluate the validity of the skewness correction procedure described in the Appendix, we used that procedure to estimate the genome-wide p-value associated with the LOD score threshold of 5.3 (Z = 4.94) assumed by Morley et al. to have a genome-wide p-value of 0.001 based on Gaussian process theory [1]. Our skewness-correction procedure, however, determined that Z = 4.94 has a genome-wide p-value of 0.016. We then carried out 1600 Monte Carlo simulations (assuming no linkage) of genotypes for 2800 SNP markers (with minor allele frequencies from 0.25 to 0.5 assigned randomly) at 1.2-cM spacing, in 14 families with eight siblings and two parents per family. The genome-wide p-value for Z = 4.94 was found to be 0.0195 (SD = 0.0035), in close agreement with the skewness-corrected theoretical result, which supports the validity of the correction. Note that in the remainder of our analyses, we applied an FDR threshold of 10% (Z = 6.4) rather than a p-value threshold. When a large number of true positive results is expected, assessing significance by FDR might have more practical value than a family-wise error rate.

Analysis of sibships vs. pedigrees

Figure 1 illustrates differences between linkage results for sibships and for pedigrees. In the two panels to the left, the score statistic results are shown for two expressed genes (ZP3 and TM7SF3) for the chromosome on which the gene is located, with the gene location in these two cases being directly under the peak score. Larger scores are observed for full pedigree analysis. In the panel to the right, the score based on sibship data is plotted against the score based on full pedigree data for 3554 traits. The largest (most significant) scores are larger using the full pedigree data.

Corrected vs. uncorrected linkage results

Following Efron [4], we estimated the empirical null to be N(0.25, 1.05²). The Z _n threshold of 6.4 for FDR = 10% was determined as shown in Figure 2. Using this threshold, we observed 22 gene expression levels with significant evidence of linkage (Tables 1 and 2). Among those genes, three were mapped to trans loci and 19 to cis loci. Using only sibships, we applied the same procedure and found evidence of linkage for only six genes at FDR = 10%. Similar results were obtained using permutation-based FDR method: 19 gene expression levels have significant evidence of linkage, among which 2 were mapped to trans loci and 17 to cis loci.

Table 1 shows, for comparison, the results of the uncorrected analysis reported by Morley et al. [1]. Using a Z _n threshold of 4.94 (LOD = 5.3 using regression statistics and an assumption of a Gaussian distribution), they reported 142 significant linkages, most of them trans, and calculated FDR = 2.5%. However, using the method described above, we expect 50 false-positive results at a threshold of 4.94 after correcting only for skewness, and 110 such results after further correcting for the empirical null. Therefore, we estimate FDR = 77.5% for the results reported by Morley et al. [1].

Finally, Figure 3 and Table 1 summarize the results of the corrected analysis limited to cis regions (within 5 Mb of each gene). Because this procedure maximizes Z _n over a smaller number of tests, it detected 46 significant linkages at FDR = 10%.