Consider the experimental design with an F population consisting of individuals, each of which is measured for a phenotype of interest. A set of individuals from each of the tails of the distribution (low and high) are collected. DNA bulks are prepared by combining equal amounts of tissue/cells from individuals within each bulk followed by DNA extraction, or by extracting DNA from each individual and combining equal amounts. Following preparation of DNA bulks, genomic libraries are prepared and sequenced at average coverage per SNP. Thus for each SNP the data is four allele counts that can be summarized in a table, where is the allele from the high parent (Table 1). The -values in the table are counts of alleles not individuals. The observed allele frequency of in the low bulk is ; that in the high bulk is . If the SNP is close to a QTL with effects in the expected direction (i.e. the ‘high allele’ increases trait values), then we expect .

The counts in Table 1 are determined by two levels of hierarchical of sampling. The first sample is the chromosomes that constitute each bulk (assuming diploid inheritance). Second, there is random variation in the number of reads per allele within each bulk due to the stochastic nature of next-generation sequencing. Let and be the expected (‘true’) frequency of the high allele in each bulk. The realized frequencies (, ) differ from and in each bulk due to binomial sampling:(1)(2)If we assume that sequencing coverage is approximately Poisson, then the conditional distributions of the observed allele counts are:(3)(4)(5)(6)

A natural statistic to characterize the data at each SNP is the standard -statistic:(7)where is the ‘expected value’ for count . The null hypothesis is that there is no QTL close to the focal SNP. This implies the standard expected counts for a contingency table, e.g. . If the null hypothesis is correct, and . If we further assume no segregation distortion and equal (average) sequencing coverage of each bulk, then . See the supplementary materials (Text S1) for a generalization that includes segregation distortion.

However, due to the hierarchical sampling scheme, the usual expectation that follows a distribution (chi-square with 1 d.f.; [12]) does not hold in the present situation. The mean and variance of are inflated relative the even when the null hypothesis is true (i.e. there is no QTL). Based on the arguments in Text S1 we approximate the mean and variance of as:(8)(9)These equations predict convergence on under certain parameter sets. In particular, if , then and , as expected from .

A simulation model was used to test the accuracy of approximate equations (8) and (9). We simulated genetic data for a chromosomal region of 10 cM in recombinational length. Informative markers were uniformly distributed along this chromosome with SNPs per cM. The causal locus (QTL) was located at the center of the chromosome and was thus flanked by SNPs on each side. Alternative homozygotes at the QTL differ by phenotypic units on average (additive gene action) and simulations of the null hypothesis (no QTL) were done with . In each simulation run, we first established the genotypes and phenotypes of the distinct F segregants. Each individual was assigned a QTL genotype according to Mendelian probabilities (0.25, 0.5, 0.25) and the phenotype was assigned as the genotypic value plus a normal deviate. Individuals were then ranked by phenotype and were selected from each tail. The full haplotype of these individuals was then established by working out from each allele at the QTL and allowing recombination to occur probabilistically according to the linkage map. Given the haplotypes in each bulk, we simulated an independent Poisson number for each count of Table 1 for each SNP. These data were used to calculate at each SNP, and also as described below, within windows around each SNP. For the latter we needed to specify a window size in centimorgans. For each parameter set, this entire procedure was repeated 10,000 times. Table 1 in Text S1 reports simulation results for the null hypothesis () for a range of reasonable combinations of and . There is a close correspondence of observed means and variances of with the values predicted by equations (8) and (9). As expected, in these simulations the distribution of is right skewed with a mean and variance exceeding the expectations.

The full distribution of values is depicted for one parameter set (, ) in Figure 1a. The gray histogram shows the distribution of under the null hypothesis () while the overlapping red histogram shows the corresponding distribution in the case of a weak QTL (). Focusing first on the null distribution, because the distribution is right skewed (mean = 1.19, variance = 2.93), if we compare this distribution to critical values of the observed false positive rate is somewhat elevated (6.98% at ; 1.98% at ). However when approaches the mean and variance of far exceed the expectation and type I error rates increase dramatically. Perhaps even more problematic is the inability of to detect a QTL based on the naïve expectation. For the weak QTL case, where the QTL explains 2% of the phenotypic variance, the causal SNP is significant at a in only 34.9% of the simulations, and in only 16.8% of simulations at . The application of the naïve thus suffers from a lack of power.

A substantial source of variation in is the random margin in Table 1, . To deal with this variation we propose the use of a weighted average of across neighboring SNPs. Averaging values across SNPs is sensible because the real signal of divergence in allele frequency between bulks is conserved between closely linked sites but random noise due to variable sequencing read coverage is not. We suggest the following average test statistic for each SNP:(10)where the sum includes all SNPs within the window bracketing the SNP. This type of weighted moving average, where the weights are given by a kernel function, , is also known as Nadaraya-Watson kernel regression [13], [14]. Nadaraya-Watson kernel regression acts as a smoothing function, with the amount of smoothing increasing with larger window size [15]. The simplest scheme for would be to give equal weight to all SNPs within (a rectangular kernel). We opt instead to apply the tri-cube kernel fuction:(11)where is standardized distance, with value 0 at the focal position and value 1 at the edge of the window. is the sum of for all SNPs in . The tri-cube kernel is commonly used in local polynomial regression methods like LOESS [16] and gives greater weight to observations that are close to the focal SNP. Any other weighting kernel that decreases smoothly to 0 as goes to 1 could be used as well. We discuss the choice of the kernel window size, , below.

A methodological issue arises when kernel smoothing is used – at the beginning or end of a data series it can produce a biased estimate because the data included in the kernel bandwidth is asymmetric. The simplest way to deal with this is to append a reflected version of the values that fall within the right half-bandwith (at the beginning of the series) and left half-bandwidth (at the end of the series), run the kernel smoother as normal, and then trim the appended values from the output.

The null expectation of is given by equation (8). The variance of depends on the variance of individual values (equation 9) and the covariance between SNPs within a window. In Text S1 we show that can be approximated as:(12)where indexes all SNPs other than contained within the window.

Figure 1b illustrates the distribution of for the same parameters as Figure 1a (plus window size cM and SNP density per cM). The difference between the null distributions in Figure 1a and 1b is due to the normalizing effect of averaging. The predicted mean and variance of (1.17 and 0.066) are reasonably close to the observed moments (1.18 and 0.056). The distribution of is still right skewed but the right tail can reasonably predicted from log-normal densities with parameters derived from and (Figure S1 and Text S2). The observed false-positive rates (using a log-normal density estimation) are: 5.14% at and 1.86% at ). Unlike the use of the naive -test based on , the type I error does not increase dramatically as approaches . Furthermore, has good power to detect QTLs. For the example illustrated in Figure 1b the causal SNP is significant in 94.3% of the simulations at , and in 88.0% and 77.2% of simulations at and respectively.

In addition to the theoretical expectations discussed above, an empirical estimate of the null distribution of can be derived from the observed data itself. We assume that the observed data, , is a mixture of the null distribution (non-QTL regions) and several contaminating distributions (QTLs). As discussed above, the null distribution of () is right-skewed with a tail density reasonably predicted from a log-normal distribution, . We also assume the contaminating distributions have higher means than the null distribution. Our goal is to estimate and in a manner that is not unduly influenced by the contaminating distributions.

Recall that for a log-normal distribution: and [17]. Thus if we can estimate the median and mode of can use those to estimate and . To do so we propose the folowing steps:

The logic of this procedure is as follows. The median and MAD are robust estimators of location and spread respectively [20]. In the absence of contaminating distributions should be approximately normally distributed, and hence the median and MAD of can be used as robust estimates of the mean and spread of ( for a symmetric distribution). Hampel's rule is a commonly used procedure to identify likely outliers in a set of data based on the median and MAD; if the underlying distribution is normally distributed and this is approximately equivalent to identifying outliers as those observations with -values (we use a one-sided test in the procedure above). When contaminating distributions (QTLs) are present, lies to the right of the true mean of the null distribution. Thus, and are conservative estimators of and . We then use Hampel's procedure to identify observations likely to be drawn from the contaminating distributions and create a trimmed data set, , with those outlying observations removed. From the trimmed data set we estimate and .

For the null simulations in Figure 1b the observed false-positive rate estimated using this non-parametric approach are 3.18% at and 0.76% at . In general, the non-parameteric procedure tends to be slightly more conservative than our proposed parametric estimators but not greatly so. Because this non-parametric approach makes few distributional assumptions (other than approximate log-normality of the null distribution) it might be preferred in cases where one suspects the sampling (either of segregants or alleles) grossly violates the hierarchical model described above.

A weighted moving average is a type of low-pass filter; the larger the window size the lower the frequncy of signals that are rejected by the filter. The choice of smoothing width, , is therefore a tradeoff between filtering out high-frequency deviations in due to variable sequence coverage and SNP density and attenuating the signal of real QTLs. We want to pick a that minimizes noise while maximizing the underlying signal. The matched filter theorem [21] suggests that the filter that maximizes the signal-to-noise ratio of a symmetric signal is one which matches the shape of the signal. A simple measure of the shape of a symmetric signal is the full-width at half maximum (FWHM). The ratio of the width of the kernel to the peak FHWM (‘smoothing ratio’) is a useful metric for quantifying the effects of smoothing [22]. As a rule of thumb, using a smoothing kernel with a smoothing ratio of approximately two provides a good signal-to-noise ratio [22]. However, the matched filter may fail to distinguish multiple peaks when there are two or more signals in the input [23] as we would expect in cases of multiple QTLs with overlapping regions of elevated . Specifically, peaks separated by less than twice the FWHM of the filter will be merged [24]. Therefore, to distinguish overlapping signals requires filters with smoothing ratios significantly smaller, perhaps as small as 0.7.

In Text S2 we derive the expected shape of around a single causal SNP. For the case in which the causal allele is fixed in one bulk and has a frequency of 0.5 in the other bulk, the half-bandwidth () at half-maximum corresponds to 12.42 cM (). More extreme allelic biases between the bulks favor slightly smaller bandwidths, while less extreme differences favor larger bandwidths. SNP density also affects the optimal kernel bandwidth, with higher SNP density favoring narrower bandwidths. In simulations and applied to real data we have found that kernels with smoothing ratios in the range 1–1.5 produce smoothed estimators with good signal-to-noise ratios and which are neither strongly over- or undersmoothed. In terms of mapping distances this corresponds to kernels with in the range 24.8–37.25 cM.

Since recombination rates vary across genomes, a given genetic distance will correspond to a range of physical distances. In terms of the choice of smoothing width, higher recombination rates favor smaller window sizes (in physical distance). If regional recombination rates are known this can be incorporated into the analysis; however the use of average chromosomal or genomic recombination rates to choose a single physical size for the smoothing window should not be problematic unless recombination rates vary widely. In such cases, one can calculate using a range of smoothing widths to explore whether peak estimates are strongly affected by over- or undersmoothing.

The two low samples ( and ) and the two high samples ( and ) represent independent sequencing runs of the same low and high segregant bulks respectively. Using and from a comparison of vs. and vs. we can estimate the impact of sequencing on the variation of these statistics. When the two bulks differ only due to read number variation, there is only one source of variation, and the statistics of should should be approximately with and . By invoking a weighted version of the central limit theorem [29], we find the distribution of should be approximately normal with and where , the sum of the squared kernel weights in the smoothing window ( converges to in the case of a square kernel). As illustrated in Table 2 the observed data for the null-comparisons conform well to the asymptotic expectations.

In addition to tests of the null model, the design of the yeast experiment facilitates a between replicate comparison of and in the presence of QTLs. There are four possible low-vs-high comparisons; here we focus on two of those, -vs- and -vs-. Figure 3 illustrates the relationships for and at each SNP for -vs- and -vs-. The between replicate correlation for is 0.677, while that between is 0.996. This illustrates the ability of the smoothing kernel to act as a low-pass filter on the -statistic, filtering out the high-frequency noise associated with variation in read counts, while preserving the underlying signal of QTLs and increasing the repeatability of the analysis.

Using the false discovery rate approach outline above, we estimated cutoff values for using a FDR of 0.01 based on both our theoretical results (equations 8 and 12) and the corresponding non-parametric estimators. For the parametric estimate we used the following parameter values: , , . The estimated cutoff values are as follows: -vs- : 2.59 [parametric], 3.51 [non-parameteric]; -vs- : 2.58 [parametric], 3.91 [non-parametric].

Using the theoretical cutoff of 2.59 we find 7,845 SNPs have significant values for the -vs- comparison, and 8,011 significant SNPs for the -vs- comparison, representing approximately 17% of the polymorphic sites. Nearly 38% of the significant sites are on chromosome XIII which appears to have multiple overlapping peaks leading to elevated values across much of the chromosome. The number of significant sites shared between the replicates is 7,330. We identified 12 significant regions (QTLs) in the two replicates (Figure 4). The QTLs are nearly identical between the replicates except for a marginal QTL on chromosome 7, where one of the replicates is significant but the other is just short of significance. To assess the variability in QTL location we compared the distance between peaks (using the single largest peak in cases of multiple peaks per chromosome). The mean and median absolute distances between nine comparable QTL peaks from the two comparisons are 5.08 Kb and 4.97 Kb respectively. The root mean square deviation (RMSD) between comparable QTL peaks is 6.7 Kb. Using the RMSD as a measure of spread and applying the 3 rule of thumb, a conservative confidence interval for QTL peak is 20 Kb (7.4 cM) around the observed peak. The size of this confidence interval is a function of read depth and SNP density, and is a measure of variability in peak estimation due to sequencing only. This confidence interval doesn't include variation that would arise from the bulking of segregants.

As will be described elsewhere, candidate genes corresponding to several of the major peaks in this analysis have been functionally validated to affect yeast colony morphology (J. Granek and P. Magwene, unpublished data).