## Abstract

There are a number of polymorphism-based statistical tests of neutrality, but most of them focus on either the amount or the pattern of polymorphism. In this article, a new test called the two-dimensional (2D) test is developed. This test evaluates a pair of summary statistics in a two-dimentional field. One statistic should summarize the pattern of polymorphism, while the other could be a measure of the level of polymorphism. For the latter summary statistic, the polymorphism-divergence ratio is used following the idea of the Hudson–Kreitman–Aguadé (HKA) test. To incorporate the HKA test in the 2D test, a summary statistic-based version of the HKA test is developed such that the polymorphism–divergence ratio at a particular region of interest is examined if it is consistent with the average of those in other independent regions.

SINCE the development of the coalescent theory (Kingman 1982; Hudson 1983; Tajima 1983), a number of polymorphism-based statistical tests have been developed to examine a neutral null model (*i.e.*, neutrality tests). With increasing intraspecific variation data in various species, these tests have been ubiquitous tools in molecular population genetic analysis (Kreitman 2000).

Neutrality tests include the following two major categories, although there are other types of tests available such as haplotype tests (Hudson *et al.* 1994; Fu 1996; Sabeti *et al.* 2002) (see Innan *et al.* 2005, for a recent review of haplotype tests). The first category focuses on the amount of polymorphism. Balancing selection increases the level of polymorphism because multiple alleles are likely maintained for a long time (Hudson and Kaplan 1988), while the level of polymorphism is reduced shortly after a fixation of adaptive mutation. This event is called a selective sweep because the fixation of a beneficial allele could sweep out the variation in the surrounding region of the selection target site by the hitchhiking effect (Kaplan *et al.* 1989). The Hudson–Kreitman–Aguadé (HKA) test (Hudson *et al.* 1987) focuses on this effect of selection by comparing the levels of polymorphism and divergence from an outgroup (see also Wright and Charlesworth 2004).

The second major category of neutrality tests examines whether the observed frequency spectrum of nucleotide polymorphism is consistent with the neutral expectation. Tajima (1989) has devised a simple method that compares and , two unbiased estimators of θ, the population mutation rate. is identical to the average number of pairwise nucleotide differences, which can be a direct estimator of θ (Tajima 1983). is an estimator based on the number of segregating sites, *S* (Watterson 1975). Tajima's *D* is defined as , and its expectation under the standard neutral model of a constant-size population is ∼0. Balancing selection creates an excess of alleles in intermediate frequencies so that Tajima's *D* is likely positive, while Tajima's *D* tends to be negative in a region shortly after a selective sweep or under the pressure of purifying selection due to an excess of variation in low frequencies. A number of tests similar to Tajima's *D* have been developed (Fu and Li 1993; Simonsen *et al.* 1995; Fay and Wu 2000).

Thus, most neutrality tests use either the amount or the allele frequency spectrum of polymorphism. That is, those tests do not use part of the important information, which could result in a loss of power to detect selection. For example, consider a gene that experienced a recent selective sweep so that no polymorphism is observed. The HKA test could work, but the second category of tests cannot be performed when the number of segregating sites is zero. This article introduces a simple algorithm to examine both the amount and the freqeuncy spectrum of polymorphism simultaneously, which is referred to as the two-dimensional (2D) test because it examines a pair of test statistics in a two-dimensional field.

To incorporate the first category of tests into the 2D test, the HKA test is modified. The original HKA test examines the null hypothesis that the ratio of the level of polymorphism to divergence is the same for multiple regions (Hudson *et al.* 1987). The most common application of the HKA test is to a pair of regions. The test examines whether the ratios of polymorphism to divergence in the two regions are consistent with each other. A possible understanding of the rejection of the neutral hypothesis is that one region might be subject to selection, but it is difficult to determine which one is under selection. The application of the HKA test to more than two regions has a similar problem. Here, the HKA test is modified to test only a particular region of interest using information from other multiple reference regions as a control, which are supposed to be neutral. Therefore, one can test whether the ratio of polymorphism to divergence in a region of interest is consistent with the average of other reference regions. This design of the HKA test may be useful when polymorphism data from multiple regions are becoming available in many species.

## MODIFIED HKA TEST: A FOCAL REGION *VS.* MULTIPLE REFERENCE REGIONS

While the original version of the HKA test uses a contingency table of polymorphism and divergence (Hudson *et al.* 1987), the modified version of the HKA test uses a summary statistic, *r*, which is the ratio of the amount of polymorphism to the level of divergence. Suppose we are interested in a particular region, for which polymorphism data of species A and a sequence from outgroup species B are available. Let *L*_{f} be the nucleotide length of this focal region. Assume that the sample size of the polymorphism data is *n*_{f}. The observed levels of polymorphism and divergence are denoted by *p*_{f} and *d*_{f}, respectively. There are also polymorphism data of species A and outgroup sequences of species B for *m* independent reference regions available. We want to know whether the ratio of the level of polymorphism to divergence in the focal region (*r*_{f} = *p*_{f}/*d*_{f}) is consistent with *r*_{ave}, the average *r* of the reference regions. This modified version of the HKA test consists of two steps:

The divergence time (

*T*) between species A and B is estimated from the*m*reference regions.The null distribution of

*r*is obtained conditional on the estimated*T*, from which the statistical significance of*r*_{f}is evaluated.

For step I, a rejection-sampling method (*e.g.*, Tavaré *et al.* 1997; Pritchard *et al.* 1999; Beaumont *et al.* 2002; Marjoram *et al.* 2003) can be useful, which produces a sample from the posterior distribution of *T* conditional on *r*_{ave}. Following the standard framework of the HKA test (Hudson *et al.* 1987), it is assumed that the present population of species A and the ancestral population have the same constant diploid effective population size, *N*, and that the two species split *T* × 2*N* generations ago. The population mutation and recombination rates are assumed to be θ = 4*N*μ and ρ = 4*N*γ, where μ and γ are the mutation and recombination rates per site per generation. It is assumed that θ and ρ are constant across loci.

Let be the polymorphism and divergence data for the *m* independent reference regions. δ_{i} consists of the sample size (*n _{i}*), the nucleotide length (

*L*), the level of polymorphism (

_{i}*p*), and the divergence from an outgroup sequence (

_{i}*d*) for the

_{i}*i*th regions. From ,

*r*

_{ave}can be calculated as(1)

Then, the posterior distribution of *T* conditional on *r*_{ave} is obtained by a rejection-sampling method (*e.g.*, Marjoram *et al.* 2003), which is implemented basically as follows.

Generate a random value of

*T*from its prior distribution.Simulate polymorphism and divergence data for

*m*independent regions using a coalescent simulation (*e.g.*, Hudson 2002). The nucleotide length and sample size for the*i*th region are*L*and_{i}*n*, respectively. Estimates of θ and ρ from the_{i}*m*reference regions may be used in the coalescent simulation.Calculate

*r*_{ave}using (1) in the simulated data, which is denoted by*r*′_{ave}. Accept*T*if |*r*_{ave}−*r*′_{ave}| < δ, where δ is a constant; otherwise, discard*T*and go to step 1.

This process is continued until a sufficient number of accepted values of *T* are obtained. The choice of δ involves a tradeoff between computational time and accuracy (see below). The prior distribution of *T* can be determined according to the prior knowledge of the divergence time. One of the possible prior distributions is a uniform distribution from 0 to a sufficiently large value, *T*_{max}. *T*_{max} is set such that the acceptance probability is nearly zero for *T* > *T*_{max}. When a uniform distribution is used as the prior distribution, the posterior distribution is approximately proportional to the likelihood distribution. Another possible choice would be a normal distribution, which could reduce the computational effort. The mean and variance of the normal distribution could be determined according to the observed distribution of *r* across the reference regions. For example, *T* can be roughly estimated as 1/*r* − 1 for each reference region. Then, the mean and variance of the estimated *T* may be plugged in the prior distribution. Although this is a biased estimate, it might serve as an efficient prior distribution for *T*. The posterior distribution would be much narrower than the prior distribution because the acceptance of *T* is based on *r*_{ave}.

The modified HKA test uses this posterior distribution of *T* to determine the null distribution of *r*_{f} in step II. In practice, a coalescent simulation under the standard framework of the HKA test (see above) is performed, in which *T* is a randomly chosen value from the list of the accepted *T* in step I. From this simulation, the null distribution of *r*_{f} is obtained conditional on *r*_{ave}, making it possible to test whether *r*_{f} is consistent with the average of the reference regions.

This process of the modified HKA test is demonstrated under a simple condition: all reference regions have the same sample sizes and nucleotide lengths, and the average *r* in the reference regions is *r*_{ave} = 0.1. θ = ρ = 0.01, *n* = 50, and *L* = 1 kb are assumed for all reference regions. To investigate the effect of *m* on the posterior distribution of *T*, *m* = 3, 5, 10, 20, 50, and 100 are considered. For step I, the prior distribution of *T* is set to a uniform distribution from 0 to 30, and the average number of pairwise nucleotide differences, π_{i}, is used as a measure of the level of polymorphism, *p _{i}*. δ is set to 0.01 because preliminary simulations demonstrated that δ = 0.01 is sufficiently small so that δ < 0.01 only slightly improved the accuracy. Polymorphism and divergence are simulated using the “ms” software (Hudson 2002). It is found that the variance of the estimate of

*T*decreases as

*m*increases, making the posterior distribution of

*T*narrower. To visualize this effect, Figure 1 shows the three posterior distributions of

*T*(

*m*= 5, 20, and 100). Almost the same results are obtained for (

*n*,

*L*) = (20, 1 kb), (20, 5 kb), and (50, 1 kb) (results not shown).

The computation time for step I increases with *m*. However, if *m* is sufficiently large, the posterior distribution of *T* can be approximately obtained by a bootstrap method (Efron 1982). That is, choose a random set of *m* regions with replacement and calculate *r*_{ave}, from which an estimate of *T* is approximately given by 1/*r*_{ave} − 1. Although this estimate is biased as mentioned above, the bias is negligible when *m* is very large. Repeating this process produces an approximate posterior distribution of *T* in significantly less computational time.

As the null distribution of *r*_{f} is conditional on the posterior distribution of *T*, it is expected that more reliable estimates of *T* (*i.e.*, narrower posterior distribution of *T*) could make the test more accurate and powerful. This effect of the posterior distribution of *T* on the power of the modified HKA test is investigated by coalescent simulations. θ = ρ = 0.01 is assumed for the focal region, as well as for the reference regions. It is also assumed that *n* and *L* for the focal region are the same as those for the reference regions. The null distribution of *r* for the focal region (*r*_{f}) for each parameter set is determined as described above. Then, the effect of *m* on the power to detect selection is investigated for two modes of selection: biallelic symmetric balancing selection and recent selective sweep.

The power is evaluated by simulating a number of patterns of polymorphism under selection models. For balancing selection, the “sarg” software (Nordborg and Innan 2003) is used. The backward mutation rate between two alleles is set to be α = 0.01 and 0.02 (for details, see Nordborg and Innan 2003), which determines the expectation of the age of the alleles. Table 1 summarizes the results, where the power is measured as the number of replications of simulation that reject the null neutral model at the 5% level. As expected, the power to reject the null model increases with increasing *m*, the number of reference regions. It seems that the power is nearly saturated for large *m*. The results for the two sample sizes (*n* = 20 and 50) are similar. The power is higher for the smaller region (1 kb) because the signature of balancing selection does not likely extend far from the target site (Hudson and Kaplan 1988; Schierup *et al.* 2001; Navarro and Barton 2002; Nordborg and Innan 2003).

For recent selective sweeps, patterns of polymorphism are simulated by using the “sw” software (Kim and Stephan 2002). The parameters to determine the selection intensity (2*Ns*) are assumed to be 100 and 1000 (for details, see Kim and Stephan 2002). τ is the time to the completion of the selective sweep in units of 2*N* generations. The results are similar to those of balancing selection: the power increases with increasing *m* (Table 1). When 2*Ns* = 1000, the power is higher for the wider region (5 kb) because the signature of a strong selective sweep extends much longer than that of balancing selection, although the relationship between the power and the region length may be complicated when selection is relatively weak (Kaplan *et al.* 1989; Braverman *et al.* 1995; Kim and Stephan 2002; Przeworski 2002).

## 2D TEST

The modified HKA test using a summary statistic *r* is ready to be incorporated in the 2D test. The 2D test requires another summary statistic, which should use information that the HKA test does not use, such as Tajima's *D* (Tajima 1989) and Fu and Li's *D** (Fu and Li 1993). The basic idea is that a pair of summary statistics is evaluated in a two-dimensional field. As an example, the two-dimensional density distribution of Tajima's *D* and *r* (*r*_{f}) is considered when θ = ρ = 0.01, *n* = 50, and *L* = 1 kb. For reference regions, it is assumed that *m* = 5, *n* = 50, and *L* = 1 kb. Suppose *r*_{ave} = 0.1; therefore, the posterior distribution of *T* is that shown with solid squares in Figure 1. Under the standard HKA model of a constant population size, the two-dimensional density distribution of Tajima's *D* and *r* is obtained from 10^{6} replications of a coalescent simulation (Figure 2). In the simulation, the posterior distribution of *T* is incorporated in the same way as the modified HKA test: in each replication, one of the accepted *T* is randomly picked up.

From the two-dimensional distribution of *D* and *r*, the 95% confidence region can be determined as follows. The number of replications of simulation is denoted by *U*. The simulation results (*U* = 10^{6} pairs of *D* and *r*) are binned into two-dimensional grids. Let *a* and *b* be the grid sizes for *D* and *r*, respectively. For this example, *a* and *b* are set to be 0.01 and 0.001. This binned two-dimensional density distribution is denoted by *G*(*x*, *y*), which represents the number of pairs of *D* and *r* in the grid *x* − *a*/2 < *D* ≤ *x* + *a*/2 and *y* − *b*/2 < *r* ≤ *y* + *b*/2.

Then, the 95% confidence region of *D* and *r* may be defined as the region with(2)where *p* is defined such that *p* satisfies(3)and(4)

Note that this method produces a somewhat discrete confidence region unless the number of replications of a coalescent simulation is extremely large. Therefore, in practice, it is recommended that *G*(*x*, *y*) is first smoothed, which is denoted by *G*′(*x*, *y*). For example, *G*′(*x*, *y*) could be the average over (2*x*_{s} + 1) × (2*y*_{s} + 1) neighbor grids:(5)

Then, the 95% confidence region may be approximately obtained from (2)–(4) by replacing *G*(*x*, *y*) with *G*′(*x*, *y*). However, this method with a smoothed distribution may sometimes produce a biased confidence region when *x*_{s} and *y*_{s} are not very small. This bias may be corrected if the shape of the 95% confidence region is determined by *G*′(*x*, *y*) and the bias is adjusted by using *G*(*x*, *y*). That is, the 95% confidence region is approximately given as the region with *G*′(*x*, *y*) > *p*′, where *p*′ satisfies(6)and(7)

A C-program to determine the 95% confidence region is available on request. This procedure with *x*_{s} = *y*_{s} = 5 is applied to the distribution in Figure 2, and the obtained 95% confidence region is shown in Figure 3A.

The effect of selection on *D* and *r* is also visually demonstrated in Figure 3A. The shaded circles and solid squares represent simulated pairs of *D* and *r* in a region under balancing selection (α = 0.01) and after a recent selective sweep (2*Ns* = 1000 and τ = 0.2), respectively. Most shaded circles are in a region of high *D* and *r*, while the solid squares make a cluster in a region of low values of *D* and *r*. In addition to Tajima's *D*, it is possible to use other summary statistics such as Fu and Li's *D** (Figure 3B).

The power of the two 2D tests (*r vs.* Tajima's *D* and Fu and Li's *D**) is quantitatively evaluated by coalescent simulations as described above. For balancing selection, it is demonstrated that the power of the 2D tests is generally higher than that of the three single tests (modified HKA, Tajima's *D*, and Fu and Li's *D**) (Figure 4) and that the 2D test with *r* and Tajima's *D* may be the most powerful. For a selective sweep, the modified HKA test may be the most powerful. The 2D tests are not as powerful as the modified HKA test especially when *L* is short, probably because the 2D test might share the weakness of Tajima's *D* and Fu and Li's *D** tests, that is, low power when the number of segregating sites is small.

The 2D test with Tajima's *D* and *r* is applied to the *GD2-A* and *GD2-B* genes in *Arabidopsis thaliana*. It is considered that these two genes were duplicated recently. Moore and Purugganan (2003) showed that the levels of polymorphism in these duplicated genes are generally lower than those in six single-copy loci, suggesting the fixation processes of duplicated genes might have occurred in a short time, likely by adaptive selection. Here, these six single-copy loci are used as reference regions and the 2D test is applied to each of the duplicated genes. Note that interlocus gene conversion might be active in young duplicated genes, which could elevate the level of polymorphism (Innan 2003). However, because the maximum-parsimony haplotype network of this pair of genes exhibits no evidence for gene conversion (see Figure 3B of Moore and Purugganan 2003), the standard coalescent model for a single-copy gene is employed (see Innan 2003 for a coalescent simulation of duplicated genes). First, the posterior distribution of *T* is obtained from the six reference genes assuming θ = 0.02, which is roughly in agreement with the average of estimates of θ over the reference genes. It is important to note that the regions of interest (*i.e.*, *GD2-A* and *GD2-B*) should not be included in the estimation of θ. The population recombination parameter is assumed to be × θ according to Hagenblad and Nordborg (2002). Then, the 2D test is performed for the focal regions (*L* ≈ 500 bp) and the result is shown in Figure 5. The 95% confidence region is smoothed with *x*_{s} = *y*_{s} = 5. The two observed pairs of *D* and *r* are within the 95% confidence region, although they are close to the 95% boundary.

Note that the boundary of the 95% confidence region is not as smooth as those in Figure 3 because of the low θ*L* used to determine the 2D null distribution. Finite numbers of polymorphic sites and sampled sequences make the distributions of *D* and *r*_{f} somewhat discrete, and this effect may not be negligible when θ*L* and *n* are not sufficiently large, suggesting some limitation of the application of the 2D test to data in a short region.

## DISCUSSION

The HKA test (Hudson *et al.* 1987) examines the null hypothesis that the polymorphism–divergence ratio is constant across regions. When neutrality is rejected, however, the test does not determine which region is likely under selection. This article introduces a new design of the HKA test such that it tests whether the polymorphism–divergence ratio in a region of interest (*r*_{f}) is consistent with the average over multiple reference regions (*r*_{ave}). This design of the modified HKA test may be useful when polymorphism data from multiple regions are becoming available in many species. Coalescent simulations show that the power of the test increases with increasing the number of reference regions (see Table 1).

There are a number of polymorphism-based statistical tests of neutrality, but most of them focus on either the amount or the pattern (*e.g.*, allele frequency spectrum) of polymorphism. Because selection affects both, it may be more powerful to detect selection if information from both is used. This article introduces the 2D test, which evaluates a pair of statistics that summarize the amount and the pattern of polymorphism in a two-dimensional field. Following the original idea of the HKA test (Hudson *et al.* 1987), the polymorphism–divergence ratio is used for the one that summarizes the amount of polymorphism. There might be several candidates for a statistic that summarizes the pattern of polymorphism such as Tajima's *D* (Tajima 1989) and Fu and Li's *D** (Fu and Li 1993). As shown in Figure 4, the 2D tests are generally more powerful than the commonly used single tests for detecting balancing selection, while the modified HKA may be more powerful than the 2D tests for detecting selective sweeps.

Using the 2D test might be one of the solutions to a multiple-testing problem. Suppose that two statistical tests of neutrality (*e.g.*, the HKA and Tajima's *D* tests) are applied to a single-polymorphism data set and that one rejects neutrality but the other does not. As these two tests are not independent because they are applied to the same data, it may be difficult to evaluate the joint result of the two tests. For the 2D test, such a problem could be somewhat relaxed although similar problems could arise when more than two tests are used. It may be possible to evaluate more than two summary statistics in a multidimensional field, but the computational effort would be huge.

There are many difficulties in testing neutrality from polymorphism data (*e.g*., Kreitman 2000; Nielsen 2005) and the 2D and modified HKA tests are not exceptions. The most serious one could be that demography also affects the amount and the pattern of polymorphism. In other words, the effects of selection and demography are confounded. To demonstrate the effect of demography, the 95% confidence region of the 2D test is investigated under two demographic models, recent expansion following a bottleneck event and structured population. The demographic parameters are adjusted such that the expectation of *r* ∼ 0.1. In the bottleneck-expansion model, in which the expectation of Tajima's *D* is negative, the 95% confidence region shifts left (Figure 6). On the other hand, the 95% confidence region shifts right in the structured population, in which Tajima's *D* tends to be positive. Demography also plays an important role to determine the variance of the coalescent time. This directly affects the 95% confidence interval of the modified HKA test as shown in Figure 6. In the bottleneck-expansion model, in which the variance of the coalescent time is smaller than that in the standard constant-size population model, the 95% confidence region is narrower, while in the structured population model, the 95% confidence region is (slightly) wider because of large variance of the coalescent time. A similar effect is also seen in the 95% confidence regions of the 2D test (Figure 6).

To evaluate the effect of selection alone, coalescent simulations to determine the null distribution of a test statistic should be carried out under a demographic model that is consistent with the history of the population (*e.g.*, Innan and Stephan 2000), rather than using the standard constant-size population model. Reference regions required by the two tests are useful to obtain information on the demographic history of the population (*e.g.*, Weiss and von Haeseler 1998; Pritchard *et al.* 2000; Wakeley *et al.* 2001; Adams and Hudson 2004).

In a similar sense, one of the advantages of the 2D and the modified HKA tests is that the null distribution is determined with θ and ρ, which could be estimated from the reference regions. This strategy works as long as θ and ρ are constant across the genome. It is obvious that more reference regions provide better estimates with low variances and consequently better statistical results. See Wall and Hudson (2001) and Innan *et al.* (2005) for the effect of the uncertainty about θ and ρ on neutrality tests, especially when these parameters are estimated from the region to which neutrality tests are applied. In practice, however, the mutation and recombination rates might be very difficult to estimate even with large amounts of polymorphism data, because they are not constant across the chromosome (Andolfatto 2001; Daly *et al.* 2001; Jeffreys *et al.* 2001; Crawford *et al.* 2004; McVean *et al.* 2004). Other independent information could be helpful, such as recombination rate estimates based on physical maps.

Very important caveats must be taken into consideration when applying the 2D and modified HKA tests to data. First, reference regions have to be a random independent sample from the genome. Currently, polymorphism data for multiple regions are being accumulated in several model species such as humans (Hinds *et al.* 2005; International HapMap Consortium 2005), *Drosophila melanogaster* (Glinka *et al.* 2003), and *A. thaliana* (Nordborg *et al.* 2005), and the genome projects of their close relatives are underway. Such genomewide polymorphism data are suitable for reference regions. Although there might be ascertainment bias due to nonrandom sampling of investigated individuals and/or regions (especially in genotyping data in humans), simple bias may be corrected as long as the sampling strategy is known (*e.g.*, Nielsen and Signorovitch 2003). The 2D and modified HKA tests work best for such model species for which genomewide data of polymorphism and divergence are available. Once researchers find a region of their special interest, the 2D and modified HKA tests can be readily applied to the region of interest using the available genomewide polymorphism data as the reference regions. The genomewide polymorphism data are also suitable for estimating demographic parameters and mutation and recombination rates, which can be incorporated to determine the null distributions of the two tests.

A more important caveat is the choice of the focal region, which has to be selected without any prior knowledge of polymorphism. That is, the focal region should be chosen on the basis of information independent of polymorphism such as phenotypes. It is not appropriate to choose one region from a multilocus polymorphism data set as the focal region after looking at its pattern of polymorphism. Suppose that in such a multilocus data set, one locus seems unusual in some way (*e.g.*, very high level of polymorphism). If this “unusual” locus is used as the focal region and the rest are used for the reference regions, then it is not surprising that the *P*-value for the focal region is very low. In other words, this *P*-value for the focal region is not the rejection probability of neutrality because of the prior knowledge of polymorphism (*i.e.*, ascertainment bias in the choice of the focal region).

Then, can the 2D and modified HKA tests be applied to such a multilocus polymorphism data set? I recommend the following methods:

The focal and reference regions are chosen before producing or looking at the polymorphism data. This strategy is fair, but may not agree with the purpose of multilocus polymorphism data, that is, to look for outliers with unusual patterns of polymorphism, which could be candidate regions for selection. In such a case, the second approach should be used.

All regions are used as the reference regions, and the

*P*-value is determined for each region. The obtained*P*-values can be used as a measure of the “unusualness” (but cannot be considered as the rejection probabilities of neutrality as mentioned above). To understand how unusual they are statistically, the*q*-values (Storey and Tibshirani 2003) could be suitable, which is a modified version of the false discovery rate (Benjamini and Hochberg 1995). The*q*-value, which can be computed from the obtained list of the*P*-values, represents the likelihood for a significant test to be false positive; therefore, we can measure the relative responsibilities of selection to the unusualness.

## Acknowledgments

The author thanks R. R. Hudson, Y. Kim, and M. Nordborg for programs; R. Moore for data; and the two anonymous reviewers and S. A. Barton for comments. H.I. is supported by grants from the University of Texas at Houston.

## Footnotes

Communicating editor: J. B. Walsh

- Received January 25, 2006.
- Accepted April 11, 2006.

- Copyright © 2006 by the Genetics Society of America