Pseudoreplication in genomic-scale data sets

Abstract

In genomic-scale data sets, loci are closely packed within chromosomes and hence provide correlated information. Averaging across loci as if they were independent creates pseudoreplication, which reduces the effective degrees of freedom (df') compared to the nominal degrees of freedom, df. This issue has been known for some time, but consequences have not been systematically quantified across the entire genome. Here, we measured pseudoreplication (quantified by the ratio df'/df) for a common metric of genetic differentiation (F_ST ) and a common measure of linkage disequilibrium between pairs of loci (r² ). Based on data simulated using models (SLiM and msprime) that allow efficient forward-in-time and coalescent simulations while precisely controlling population pedigrees, we estimated df' and df'/df by measuring the rate of decline in the variance of mean F_ST and mean r² as more loci were used. For both indices, df' increases with N_e and genome size, as expected. However, even for large N_e and large genomes, df' for mean r² plateaus after a few thousand loci, and a variance components analysis indicates that the limiting factor is uncertainty associated with sampling individuals rather than genes. Pseudoreplication is less extreme for F_ST , but df'/df ≤0.01 can occur in data sets using tens of thousands of loci. Commonly-used block-jackknife methods consistently overestimated var (F_ST ), producing very conservative confidence intervals. Predicting df' based on our modelling results as a function of N_e , L, S, and genome size provides a robust way to quantify precision associated with genomic-scale data sets.

Keywords: FST; Ne; degrees of freedom; genome size; jackknife variance; linkage disequilibrium; simulations.

Figure 1.

Experimental design for simulations. For each evolutionary scenario (combination of N_e and genome size), four ancestral populations (AP1–AP4) were simulated to ensure coalescence (10N_e generations), at which point each ancestral population split into four daughter populations (D1-D4). The 4x4 = 16 daughter populations then evolved independently under isolation for t = 0.2N_e generations. Subsequently, the model differed slightly for the F_ST and LD analyses. In the latter (as depicted in the figure), for each daughter population, eight mutational replicates (different set of loci) were generated based on the same pedigree, producing a total of 128 replicates for each evolutionary scenario. For F_ST, each set of four daughter populations allowed six pairwise comparisons of populations, and for each two-population pedigree six mutational replicates were generated (Figure S4).

Figure 2.

Effective number of loci (L’) for mean r² as a function of the number of diallelic (SNP) loci, L. Top: Influence of number of chromosomes (C), with N_e = 200 and S = 50. Bottom: Influence of N_e, with C = 16 and S = 25. Mean r² was calculated across all n = L(L-1)/2 pairs of loci. Figure S8 (Supplementary Information) shows these same results except the Y axis is plotted as the effective number of locus pairs (n’).

Figure 3.

Effective number of loci (L’) for mean r² as a function of the sample size of individuals (S = 25-800) and the number of diallelic (SNP) loci, L. Results are for N_e = 800, C = 16, and using all pairwise comparisons of loci.

Figure 4.

Variance components analysis for mean r². As depicted in Table 3, V₁ is the variance of mean r² for the same individuals assayed for different, non-overlapping sets of loci, and V₂ is the variance of mean r² for different (potentially overlapping) sets of individuals assayed for the same loci. “Sum” = V₁+V₂ and “Observed” is the total observed variance of mean r². Results are for N_e = 200, C = 16, S = 50, and using all pairwise comparisons of loci.

Figure 5.

Comparison of parametric and actual 90% confidence intervals for ${\widehat{N}}_e$ based on LD (top) and ${\widehat{F}}_{ST(L)}^{Hudson}$ (bottom). Parametric CIs use the nominal degrees of freedom (L = the number of diallelic (SNP) loci for ${\widehat{F}}_{ST(L)}^{Hudson}$; n = L(L-1)/2 for LD); actual CIs use the effective degrees of freedom calculated in this study (L’ and n’). Results are for simulations with N_e = 200, C = 16, and S = 50. Note the different X-axis scales in the two panels.

Figure 6.

Rate of decline in the variance of multilocus ${\widehat{F}}_{ST(L)}$ as more diallelic loci (SNPs, L) were used in the analysis. Results are for simulations with N_e = 200, C = 4, and S = 25 and are shown for the estimators of Nei $({\widehat{F}}_{ST}^{Nei})$ and Hudson $({\widehat{F}}_{ST}^{Hudson})$. Figure S15 shows comparable results for another scenario with different values of N_e, C, and S.

Figure 7.

Influence of N_e (top panel, with number of chromosomes, C, fixed at 16) and C (bottom panel, with N_e fixed at 200) on the effective degrees of freedom (L’) for ${\widehat{F}}_{ST(L)}^{Nei}$ computed between pairs of populations. Black dotted line represents L’ = L = the number of SNPs.

Figure 8.

Effects of pedigree on variation in mean ${\widehat{F}}_{ST(L)}^{Hudson}$. For each of two, 2-population pedigrees, 8 replicate samples (demarcated by vertical lines) were taken of S = 100 individuals. These results are for simulations with N_e = 200 and 4 chromosomes. Sampled individuals were drawn hypergeometrically from the N_e individuals in the final generation. For each sample, six mutational replicates generated non-overlapping sets of L = 5000 SNP loci that were used to compute mean ${\widehat{F}}_{ST(L)}$. Solid horizontal lines (“Pedigree F_ST”) represent mean ${\widehat{F}}_{ST}^{Hudson}$ across all 8x6 = 48 replicates within each pedigree. The first set of samples shows results for comparison of daughter populations 1 and 2 and the second set of samples shows results for comparison of daughter populations 3 and 4, all derived from the same ancestral population.

Figure 9.

Coverage of 90% confidence intervals (CIs) around F_ST estimators for the population pedigrees and samples shown in Figure 8 (N_e = 200; C = 4; L = 5000; S = 100). Top: CIs generated from block-jackknife estimates of $\text{var}({\widehat{F}}_{ST}^{Hudson})$. Bottom: CIs generated based on L’ for ${\widehat{F}}_{ST}^{Nei}$ estimated from this study. CI coverage is evaluated with respect to mean ${\widehat{F}}_{ST}^{Hudson}$ or mean ${\widehat{F}}_{ST}^{Nei}$ across all replicates within each pedigree (“Pedigree F_ST”, horizontal lines). The black X symbols indicate an upper (or lower) bound that was below (or above) the mean pedigree F_ST.