Quantile-based permutation thresholds for quantitative trait loci hotspots

Abstract

Quantitative trait loci (QTL) hotspots (genomic locations affecting many traits) are a common feature in genetical genomics studies and are biologically interesting since they may harbor critical regulators. Therefore, statistical procedures to assess the significance of hotspots are of key importance. One approach, randomly allocating observed QTL across the genomic locations separately by trait, implicitly assumes all traits are uncorrelated. Recently, an empirical test for QTL hotspots was proposed on the basis of the number of traits that exceed a predetermined LOD value, such as the standard permutation LOD threshold. The permutation null distribution of the maximum number of traits across all genomic locations preserves the correlation structure among the phenotypes, avoiding the detection of spurious hotspots due to nongenetic correlation induced by uncontrolled environmental factors and unmeasured variables. However, by considering only the number of traits above a threshold, without accounting for the magnitude of the LOD scores, relevant information is lost. In particular, biologically interesting hotspots composed of a moderate to small number of traits with strong LOD scores may be neglected as nonsignificant. In this article we propose a quantile-based permutation approach that simultaneously accounts for the number and the LOD scores of traits within the hotspots. By considering a sliding scale of mapping thresholds, our method can assess the statistical significance of both small and large hotspots. Although the proposed approach can be applied to any type of heritable high-volume "omic" data set, we restrict our attention to expression (e)QTL analysis. We assess and compare the performances of these three methods in simulations and we illustrate how our approach can effectively assess the significance of moderate and small hotspots with strong LOD scores in a yeast expression data set.

Figure 1

N- and Q-method analyses for simulated example 1. (A) Inferred hotspot architecture using a single-trait permutation threshold of 3.65 corresponding to a GWER of 5% of falsely detecting at least one QTL somewhere in the genome. The blue line at count 560 corresponds to the hotspot size expected by chance at a GWER of 5% according to the N-method permutation test. The red line at count 7 corresponds to the Q-method’s 5% significance threshold. The hotspots on chromosomes 5, 7, 8, and 15 have sizes 50, 500, 125, and 280, respectively. (B) N-methods permutation null distribution of the maximum genome-wide hotspot size. The blue line corresponds to the hotspot size 560 expected by chance at a GWER of 5%. (C) Q-methods permutation null distribution of the maximum genome-wide hotspot size. The red line at 7 shows the 5% threshold. Results are based on 1000 permutations.

Figure 2

NL-method analysis for simulated example 1. (A–D) Hotspot architecture inferred using different quantile-based permutation thresholds; i.e., for each genomic location it shows the number of traits that mapped there with a LOD threshold higher than the quantile-based permutation threshold. (A) Hotspot architecture inferred using a permutation LOD threshold of 7.07 corresponding to the LOD threshold that controls the probability of falsely detecting at least a single linkage for any of the traits somewhere in the genome under the null hypothesis that none of the traits have a QTL anywhere in the genome, at an error rate of 5%. (B, C, and D) Hotspot architectures computed using QTL mapping LOD thresholds of 4.93, 4.21, and 3.72 that aim to control GWER at a 5% error rate for spurious eQTL hotspots of sizes 50, 200, and 500, respectively.

Figure 3

Hotspot size significance profile derived with the NL-method for simulated example 1. For each genomic location (i.e., x-axis position) the hotspot sizes at which the hotspot was significant are shown, that is, at which the hotspot locus had more traits mapping to it with a LOD score higher than the threshold on the right, than expected by chance. The scale on the left shows the range of spurious hotspot sizes investigated by our approach. The scale on the right shows the respective LOD thresholds associated with the spurious hotspot sizes on the left. The range is from 7.07, the conservative empirical LOD threshold associated with a spurious “hotspot of size 1,” to 3.65, the single-trait empirical threshold, associated with a spurious hotspot of size 560. All permutation thresholds were computed targeting GWER ≤ 0.05, for n = 1, … , 560.

Figure 4

N- and Q-method analyses for simulated example 2. (A) Inferred hotspot architecture using a single-trait permutation threshold of 3.65 corresponding to a GWER of 5% of falsely detecting at least one QTL somewhere in the genome. The blue line at count 19 corresponds to the hotspot size expected by chance at a GWER of 5% according to the N-method permutation test. The red line at count 8 corresponds to the Q-method’s 5% significance threshold. The hotspots on chromosomes 5, 7, and 15 have sizes 50, 464, and 220, respectively. (B) The N-method’s permutation null distribution of the maximum genome-wide hotspot size. The blue line at 19 corresponds to the hotspot size expected by chance at a GWER of 5%. (C) The Q-method’s permutation null distribution of the maximum genome-wide hotspot size. The red line at 8 shows the 5% threshold. Results are based on 1000 permutations.

Figure 5

Observed GWER for the Q-, N-, and NL-methods under varying strengths of phenotype correlation. Black lines show the targeted error rates. Red curves show the observed GWER. (A–C) Results for uncorrelated phenotypes. (D–F) Results for weakly correlated phenotypes generated using a latent variable effect of 0.25. (G–I) Simulation results for highly correlated phenotypes generated using latent effect set to 1. The left, middle, and right columns show the results for the Q-, N-, and NL-methods, respectively. Note the different y-axis scales for the Q-method panels. The red curves on the NL-method panels show the observed GWER for hotspot sizes ranging from 1 to N, where N is the median N-method threshold for α = 0.10.

Figure 6

N- and Q-method analyses for the yeast data. (A) Inferred hotspot architecture using a single-trait permutation threshold of 3.44 corresponding to a GWER of 5% of falsely detecting at least one QTL somewhere in the genome. The blue and red lines at counts 96 and 28 correspond to the hotspot size expected by chance at a GWER of 5% according to the N- and the Q-method permutation tests, respectively. (B and C) The permutation null distributions of the maximum genome-wide hotspot size based on 1000 permutations. The blue and red lines at 96 and 28 correspond, respectively, to the hotspot size expected by chance at a GWER of 5% for the N- and Q-methods.

Figure 7

Hotspot size significance profile derived with the NL-method. The range is from 7.40, the conservative empirical LOD threshold associated with a spurious “hotspot of size 1,” to 3.45, the single-trait empirical threshold, associated with a spurious hotspot of size 96. All permutation thresholds were computed targeting GWER ≤ 0.05, for n = 1, … , 96.