TY - JOUR
AU1 - Wall, Jeffrey D.
AB - Abstract Three new estimators of the population recombination rate C = 4Nr are introduced. These estimators summarize the data using the number of distinct haplotypes and the estimated minimum number of recombination events, then calculate the value of C that maximizes the likelihood of obtaining the summarized data. They are compared with a number of previously proposed estimators of the recombination rate. One of the newly proposed estimators is generally better than the others for the parameter values considered here, while the three programs that calculate maximum-likelihood estimates give conflicting results. Introduction Two important scaled parameters in classical panmictic theory without selection are the population mutation parameter 𝛉 = 4Nμ (where N is the effective population size and μ is the per-locus mutation rate per generation) and the population recombination parameter C = 4Nr (where r is the per-locus recombination rate per generation). New mutations are the ultimate source of all variation; under the infinite-sites model (where every new mutation is assumed to occur at a previously unmutated site), the expected number of segregating sites in a sample is directly proportional to 𝛉 (Watterson 1975 ). The recombination rate has a more subtle effect on the expected amount of linkage disequilibrium between segregating sites and the expected correlation between genealogies of nearby sites (e.g., Griffiths 1981 ; Hudson 1983 ; Kaplan and Hudson 1985 ; Pluzhnikov and Donnelly 1996 ). Two main approaches have been employed for estimating C. One method observes the frequency of sequence exchange between distant markers (e.g., Ashburner 1989 ; True, Mercer, and Laurie 1996 ; Bouffard et al. 1997 ; Nagaraja et al. 1997 ), while the other method estimates C from the patterns of sequence variation expected in random population samples (e.g., Hudson and Kaplan 1985 ; Hudson 1987 ; Griffiths and Marjoram 1996 ; Hey and Wakeley 1997 ; Wakeley 1997 ; Kuhner, Yamato, and Felsenstein 1999 ). If there is local variation in recombination rates (e.g., Fullerton et al. 1994 ), then the former method might be quite inaccurate for a particular region of interest. In this paper, I focus on ways of estimating C from sequence variation in random samples. Researchers have used different aspects of the expected patterns of sequence variation to estimate C. For example, Hudson (1987) used the observed variance of the number of pairwise differences to construct an estimator of C (here called Chud). This variance is a measure of the amount of linkage disequilibrium in a sample; as the recombination rate increases, the expected amount of linkage disequilibrium between segregating sites decreases, as does the expected variance of the number of pairwise differences. Wakeley (1997) made two small technical improvements in Chud to obtain a new estimator, which I call Cwak. Although these two estimators are easy to calculate, they ignore most of the available information by essentially describing the data with a single summary statistic. Another concern is that their method of moments approach is probably not optimal; maximum-likelihood methods might be more appropriate. Perhaps the most elegant approach is to condition on the complete data set to obtain a maximum-likelihood estimate of C (Griffiths and Marjoram 1996 ; Kuhner, Yamato, and Felsenstein 1999 ; Nielsen, personal communication). The full likelihood approach has the advantage of using all of the information that is available, but it is extremely computationally intensive. For example, it is not yet computationally feasible to estimate C for many human polymorphism data sets (e.g., Harding et al. 1997 ; Zietkiewicz et al. 1997 ; Nickerson et al. 1998 ) using maximum-likelihood estimation programs provided by R. C. Griffiths and M. Kuhner. One possible compromise is the approach of Hey and Wakeley (1997) . Their estimator γ is close to the average of maximum-likelihood estimates of many small subsets of the data. Another possible compromise (and the one adopted here) is to describe sequence data using summary statistics, but then to use maximum-likelihood methods on these summaries of the data to estimate C (see below). Recently, some researchers have proposed statistical tests based on the observed number of distinct haplotypes in a sample (Strobeck 1987 ; Fu 1996, 1997 ; Depaulis and Veuille 1998 ). The motivation is that some forces tend to decrease the expected number of distinct haplotypes (e.g., population subdivision, balancing selection, or decreasing population size) (Fu 1996 ; Wall 1999 ), while other forces act in the opposite direction (e.g., linkage to a selective sweep or increasing population size) (Fu 1997 ). However, the latter forces are confounded with the effect of intragenic recombination: the expected number of distinct haplotypes increases rapidly as the recombination rate increases. As a result, the above statistical tests are inappropriate when the recombination rate is unknown. Instead, I use the number of distinct haplotypes (denoted hereinafter as H) to estimate C. The general approach is to find what value of C maximizes the probability of obtaining the observed value of H. This method of using maximum likelihood on summary statistics has been used before (e.g., Fu and Li 1997 ; Takahata and Satta 1997 ; Weiss and von Haeseler 1998 ), and it is expected to be useful if data sets have large sample sizes and relatively few segregating sites. These criteria are satisfied by most recent studies of human genetic variation (e.g., Harding et al. 1997 ; Zietkiewicz et al. 1997 ; Nickerson et al. 1998 ), but not by most studies of Drosophila sequence variation. Although exact results are available for the joint distribution of H and S (the observed number of segregating sites) when there is no recombination (Griffiths 1982 ), it is difficult, if not impossible, to obtain analytic results in the presence of intragenic recombination. Therefore, simulations are used to estimate likelihoods (see below). Other summary statistics can also be used for maximum-likelihood estimation. Hudson and Kaplan (1985) proposed using the minimum number of recombination events (RM) to estimate C. Recombination was inferred to have occurred between a pair of segregating sites when all four gametic types were present. This implicitly assumes an infinite-sites model of mutations (i.e., that each new mutation occurs at a previously unmutated site). Similar to the above, I construct an estimator that maximizes the joint probability of obtaining the observed values of both RM and H. The simulations that are used to estimate likelihoods can be run in two different ways. They can be run with 𝛉 as an unknown parameter or they can be run conditional on S (Hudson 1990, 1993 ). Hudson (1993) proposed using the latter method, since S is known, whereas 𝛉 must be estimated. The drawback is that conditioning on S results in a null model that is slightly different from the standard coalescent one. Instead of a constant mutation rate per generation, there is a constant number of mutations regardless of the length of the tree. I run both types of simulations and thus directly test the effect of conditioning on S. The new estimators of C are compared with many previously proposed ones (Hudson 1987 ; Griffiths and Marjoram 1996 ; Hey and Wakeley 1997 ; Wakeley 1997 ; Kuhner, Yamato, and Felsenstein 1999; Nielsen, personal communication). Materials and Methods Coalescent theory (Kingman 1982a, 1982b ) provides an efficient framework for simulating neutral population histories. I assume that there is a large, constant-sized, panmictic population, no selection, and an infinite-sites model. The mutation rate and recombination rate are taken to be constant per base pair (cf. Hudson 1983, 1990 ). Simulations either condition on S or treat 𝛉 as a parameter (see below), and are run using modifications of programs kindly provided by R. R. Hudson. One or more summary statistics are used to characterize the data. The ones considered are S, RM, and H. Most simulations are run conditional on S (cf. Hudson 1993 ). The estimators CH and CHRM are defined as the values of C that maximize the likelihoods L(C|H) and L(C|H, RM), respectively. For each possible value of S, a range of C values from 0.0 to 300.0 is considered, and 105 trials are run for each value of C. Pr(H|C0) and Pr(H, RM|C0) are estimated as the proportion of trials with C = C0 that show a particular configuration of {H} or {H, RM}, respectively. Standard coalescent simulations (cf. Hudson 1990 ) are also run. The estimator CSH is defined as the value of C that maximizes the joint likelihood L(𝛉, C|S, H). A range of parameter values is considered, with 𝛉 varying from 0.5 to 20.0 and C varying from 0.0 to 260.0. For each combination of 𝛉 and C, 105 trials are run, and Pr(S, H|𝛉0, C0) is estimated as the proportion of trials with 𝛉 = 𝛉0 and C = C0 showing a particular configuration of S and H. CSH is very similar in motivation to CH. However, it is much quicker to calculate CH, since simulations are needed for a single value of S instead of for many possible values of 𝛉. Coalescent trials were run with fixed values of 𝛉 and C, and the distributions of estimator values for these trials are compared with each other. The other estimators of C considered were Chud (Hudson 1987 ), Cwak (Wakeley 1997 ), CGM (Griffiths and Marjoram 1996 ; the program recom58 uses Monte Carlo recursion methods to estimate C and was kindly provided by R. C. Griffiths), CKYF (Kuhner, Yamato, and Felsenstein 1999 ; the program RECOMBINE, version 1.0, incorporates a Metropolis-Hastings [Metropolis et al. 1953 ; Hastings 1970] Markov chain Monte Carlo [MCMC] algorithm with a finite-sites model and estimates the likelihood surface using importance sampling), CN1 (R. Nielsen, personal communication; a preliminary version of the program Baysim was kindly provided by R. Nielsen. Baysim also uses Metropolis-Hastings MCMC, but with an infinite- sites model and a Bayesian approach to parameter estimation assuming a uniform prior. CN1 is the maximum-likelihood estimate.), CN2 (R. Nielsen, personal communication; CN2 is the posterior mean for C), and γ (Hey and Wakeley 1997 ; the program SITES is available at http://heylab.rutgers.edu). CH, CSH, Chud, Cwak, and γ were compared using a sample size of n = 50 and C = 𝛉 = 1.0, 2.0, ..., 12.0, C = 2𝛉 = 2.0, 4.0, ..., 24.0, C = 4𝛉 = 4.0, 8.0, ..., 48.0, and 𝛉 = 5 with C = 0.0, 5.0, 10.0, ..., 45.0. Ten thousand trials were run for each estimator and each pair of parameter values. Due to computational constraints, the other estimators (CHRM, CGM, CKYF, CN1, and CN2) were compared only with n = 50 and C = 𝛉 = 3.0. CHRM was calculated for 1,000 different trials, while 100 trials were run for the other four estimators. For each trial, CGM was estimated from 500,000 replicates with a single set of generating parameters (C = 𝛉 = 3.0). Trials for CKYF had 10 short chains (40,000 iterations each), one long chain (600,000 iterations), and starting parameters C = 𝛉 = 𝛉W (Watterson’s [1975] estimate of 𝛉), while those for CN1 and CN2 had a burn-in time of 106 iterations and a single chain of 5 × 106 iterations. Although more replicates would have been desirable, the amount of time required to run these simulations was already substantial (i.e., 5–6 h per trial of CGM or CKYF on a 400-MHz Pentium II processor). Note that the simulations conducted for this study were meant to be exploratory, not exhaustive. The mean and the mean squared error (MSE) are calculated for the estimators’ distributions. These two summaries are inadequate, both because the distributions are not approximately normal (see Results) and because estimators do not always return defined and finite values. Chud, Cwak, and γ are undefined when there is not enough information (i.e., not enough segregating sites) or when the patterns of variation are consistent with free recombination. Similarly, CH, CSH, and CHRM are undefined whenever S < 2 or H ≈ 2S. In addition, sometimes CH, CSH, and CHRM could not be calculated (and were considered undefined) because simulations with high enough values of C were not run. This latter case was always rare. It did not occur in any of the trials shown in table 1 and does not affect the summary statistic described below. I present histograms of the distributions when C = 𝛉 = 3.0, and for the other simulations I summarize the results using where g is the probability that a particular estimator will return a good estimate (defined arbitrarily as the probability that the estimate is within a factor of 2 of the actual value). g might be a more appropriate measure than the bias or MSE for comparing highly skewed distributions. Results figure 1 shows histograms of the distributions of all 10 estimators when n = 50 and C = 𝛉 = 3.0. Summary measures of these distributions are presented in table 1 . As can be seen, there is a wide variation in the shapes of the distributions and the qualities of the estimators. The 95% confidence intervals (i.e., the range between the 2.5th and 97.5th percentiles) suggest that an actual value of C = 3.0 is consistent with a wide range of estimated values. In fact, C = 3.0 is consistent with any possible observed value of Chud, Cwak, CN1, CN2, or γ. The best estimator is CKYF; it has the second smallest bias, the smallest MSE, and the greatest proportion of estimates within a factor of two of the actual value. CN2 is also generally a good estimator, although a few extreme values greatly increase its MSE. On the other end of the spectrum are the other two full likelihood estimators, CGM and CN1; they almost always underestimate the actual value. This variation in the qualities of maximum-likelihood estimators is surprising. In principle, CKYF and CN1 are computing the same quantity, so they should converge to the same value as the number of iterations increases. CGM assumes an outgroup (unlike the other two), so it should converge to a different number. table 2 shows the actual values of CKYF, CGM, and CN1 for the first 20 trials. Although the values of CGM and CN1 are often quite similar to each other, neither seem to be strongly correlated with CKYF. To determine whether this discrepancy was due to CKYF’s initial parameter values, all 100 trials were rerun with starting values of 5C = 𝛉 = 𝛉W. Having a lower starting value of C has very little effect on the final estimate of C (the mean was 2.31, compared with 2.50 in table 1 ). There is also some concern that the choice of generating parameters for CGM might have some influence on the final estimate. The first 20 trials were rerun (with 106 replicates) using the estimated values of 𝛉 and C as new generating parameters. This also had a minor effect on the estimate of CGM (results not shown). Thus, sensitivity to initial parameter values explains very little if any of the difference in the distributions of CKYF versus CGM or CN1. The most disparate trial in table 2 (trial 11) was chosen for further simulations. To ascertain the effect of a limited number of iterations on the results, all estimators were rerun with 10 times the number of iterations. The new estimates from these runs (shown in table 3 ) are quite similar to the old ones. In addition, trial 11 was rerun 20 times (using the same parameters as in table 1 ) with different random number seeds. The mean and variance of the estimated values for trial 11 are shown in table 3 . The variance gives some idea of how far the estimators are from converging to a single estimate (presumably the MLE). table 3 shows that these variances are not that large, which reinforces the observed discrepancies between the different estimators. All 20 of the trials for CGM and CN1 returned values that were less than all but one of the 20 different CKYF values. The three new estimators compare favorably (especially using g) with all of the others except CN2 and CKYF. CHRM seems to be the best of the three, while the properties of CH and CSH seem quite similar. Thus, it does not seem to make much difference whether one conditions on S or not. The raggedness of the distributions of CH and CSH is due to the inherent discreteness of the estimators; there are not that many combinations of S and H that are possible. γ has moderate bias, but its low MSE seems to be an artifact of its downward bias and the consequent artificial truncation at 0. Notice that the estimator with the second highest bias and the highest MSE (Cwak) actually has a higher value of g than does γ. This illustrates how looking only at the mean and the MSE can be misleading; these two summary statistics are highly sensitive to the tail of the distribution. g was calculated for CH, CSH, Chud, Cwak, and γ under a wider range of parameter values. figure 2A shows simulations with C = 𝛉, figure 2B assumes C = 2𝛉, and figure 2C assumes C = 4𝛉. As expected, all estimators improve as 𝛉 increases. As above, CH and CSH are roughly of the same quality, and both usually have higher g values than the other three estimators. γ is poor for small data sets but improves more rapidly with increasing 𝛉 than do the others in figure 2A and B . However, γ is relatively less accurate in figure 2C for higher recombination rates. When the actual recombination rate is so high, γ’s downward bias has a large influence in lowering the value of g. This trend is highlighted in figure 3 , which plots g as a function of C when 𝛉 = 5. While other estimators get better as C increases (presumably because less linkage disequilibrium provides more useful information), γ’s g value starts decreasing when C > 20. In both figures 2 and 3 , Chud and Cwak perform almost identically, with the latter having marginally higher g values. In summary, all of the test statistics are not very good unless both C and 𝛉 are reasonably high. For example, in figure 2A, when C = 𝛉 = 5.0 (and the expected number of segregating sites is roughly 22), all estimators are within a factor of 2 of the true value less than half the time. In contrast, a simple estimator of 𝛉 (based on the observed value of S; cf. Watterson 1975 ) for these simulations is within a factor of 2 of the true value of 𝛉 almost all of the time. Discussion One of the fundamental goals of molecular population genetics is to infer demographic and selective history from the patterns observed in current sequence data. Evolutionary inference requires an accurate estimate of the recombination rate, since the relative likelihoods of the data under various models are a function of C. In particular, the standard assumption of no recombination (C = 0) can lead to misleading conclusions. For example, consider the Runt data set from Drosophila simulans (Labate, Biermann, and Eanes 1999 ). This data set has a sample size of n = 11 and S = 20. Tajima’s (1989)D statistic for this data set is −1.47. If we assume panmixia and no recombination, then P = Pr(D ≤ −1.47) = 0.065 (105 coalescent simulations conditional on S = 20 were run to obtain this probability and the others described below). If, instead, we assume a symmetric island model (with two islands, 4Nm = 0.5, C = 0, and roughly equal sampling from each island), then P = Pr(D ≤ −1.47) = 0.036. Hence, with a no-recombination model, the value of Tajima’s D statistic is negative (but not significantly so) and is slightly less likely under a simple island model than under panmixia. However, this assumption of no recombination is known to be incorrect. Our best guess for the scaled recombination rate of the Runt gene is Ĉ = 84.6; this assumes a conservative value of N = 106 for the population size of D. simulans and estimates the recombination rate from the observed rate of exchange between flanking markers (True, Mercer, and Laurie 1996 ). If null simulations are rerun with C = 84.6, then the resultant probabilities are P = Pr(D ≤ −1.47) = 0.00087 under panmixia and P = Pr(D ≤ −1.47) = 0.00001 under the island model. Not only are both probabilities drastically lower (suggesting that the data are not very consistent with either model), but the ratio of the two probabilities has changed as well. Once recombination has been included, the particular island model considered is much less likely than the panmictic model. In practice, the island model would probably be discarded, both because of the low absolute P value and the low relative P value. However, when C = 0, the island model would remain a plausible option, since it is not much less likely than the panmictic model. Which alternatives are plausible depends on what recombination rate is assumed. Previous researchers have noted that Chud and Cwak are effective only for very large data sets (e.g., C = 𝛉 ≈ 100) (Hudson 1987 ; Wakeley 1997 ). To my knowledge, there are no sequence polymorphism data sets with sample sizes greater than three and 𝛉 > 50. Thus, there is a premium on estimators that are still effective even when 𝛉 is relatively small. The results obtained here suggest that while some estimators are reasonable for intermediate-sized data sets, all of them (with the possible exceptions of CN2 and CKYF) behave poorly for small data sets (e.g., 𝛉 < 5). figure 2 implies that for most of the parameter values considered, all ad hoc estimators use no more information than a simple count of the number of distinct haplotypes that are present (i.e., other estimators are generally not as good as CH). Although more clever ways of summarizing the data will lead to better (yet still easy to calculate) estimators of the population recombination rate, it will always be easier to estimate 𝛉 than to estimate C. This is not only because it is conceptually more difficult to understand the effects of recombination on the patterns of observed sequence variation, but also because given data sets can be consistent with a wide range of possible recombination rates. For example, nearly half of the trials shown in figure 1 and table 1 are consistent with no recombination at all. One open question is whether estimators that summarize the data (e.g., CH, CSH, CHRM, Chud, and Cwak) or estimators that only consider subsets of the data (e.g., γ) are throwing away valuable information. As computational power increases, methods that condition on all of the data will become feasible for a wider range of data sets. In addition, technical advancements will increase the efficiency of existing likelihood algorithms. A program written by P. Fearnhead which incorporates an importance sampling scheme appears to be a substantial improvement over the Griffiths and Marjoram (1996) method (results not shown; P. Fearnhead, personal communication). Although in certain circumstances full maximum-likelihood estimators are expected to be asymptotically optimal, there is no guarantee that they are the best estimators for actual data sets (for which there is dependency and the small sizes are far from the asymptotic limit). It is possible that the MLE is highly biased (see, e.g., CGM and CN1 in tables 1 and 2 ), so other likelihood-based methods like the Bayesian approach of CN2 might be best. In any case, table 2 suggests that there are probably still some methodological problems with more than one of the programs used. Ad hoc estimators will still be necessary for analyzing larger data sets. Of these, compromise approaches that use likelihood methods on summaries of the data (such as CH, CSH, CHRM, and γ) look promising. An estimator of C based on the likelihood of observed pairwise sample configurations appears to be superior to CH (Hudson 1993 ; R. Hudson, personal communication). Of the estimators considered here, CH or CHRM is preferred as long as the sample size is not too small; otherwise, γ is probably the best, although one should keep in mind that γ’s downward bias is large when the recombination rate is high. Stanley Sawyer, Reviewing Editor 1 Keywords: recombination, coalescent simulations, evolutionary inference, statistical tests of neutrality. 2 Address for correspondence and reprints: Jeffrey D. Wall, Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, Illinois 60637. E-mail: jdwall@midway.uchicago.edu. View largeDownload slide Fig. 1.—Histograms of the distributions of all 10 estimators when n = 50 and C = 𝛉 = 3.0. A–J show, in order, CH, CSH, CHRM, Chud, Cwak, CGM, CKYF, CN1, CN2, and γ, each normalized by the actual value of C. The number of independent trials run for each estimator is shown in table 1 . The “*” column displays the frequency of trials that are undefined. View largeDownload slide Fig. 1.—Histograms of the distributions of all 10 estimators when n = 50 and C = 𝛉 = 3.0. A–J show, in order, CH, CSH, CHRM, Chud, Cwak, CGM, CKYF, CN1, CN2, and γ, each normalized by the actual value of C. The number of independent trials run for each estimator is shown in table 1 . The “*” column displays the frequency of trials that are undefined. View largeDownload slide Fig. 2.—The population recombination rate, C, versus g, the proportion of estimated values within a factor of 2 of the true value of C. A assumes C = 𝛉, B assumes C = 2𝛉, and C assumes C = 4𝛉. Ten thousand replicates were run for each estimator and each parameter combination. View largeDownload slide Fig. 2.—The population recombination rate, C, versus g, the proportion of estimated values within a factor of 2 of the true value of C. A assumes C = 𝛉, B assumes C = 2𝛉, and C assumes C = 4𝛉. Ten thousand replicates were run for each estimator and each parameter combination. View largeDownload slide Fig. 3.—The population recombination rate, C, versus g, the proportion of estimated values within a factor of 2 of the true value of C, with 𝛉 = 5. Ten thousand replicates were run for each estimator and each parameter combination. View largeDownload slide Fig. 3.—The population recombination rate, C, versus g, the proportion of estimated values within a factor of 2 of the true value of C, with 𝛉 = 5. Ten thousand replicates were run for each estimator and each parameter combination. table 1 Comparison of Estimators with n 50 and C = 𝛉 = 3.0 View Large table 2 Comparison of Maximum-Likelihood Estimators for the First 20 Trials of table 1 View Large table 3 Comparison of Maximum-Likelihood Estimators for Trial 11 (See Text) View Large I thank R. Hudson, M. S. McPeek, and M. Przeworski for helpful discussions and R. Hudson, M. Przeworski, S. Sawyer, and two anonymous reviewers for comments on an earlier version of this manuscript. P. Fearnhead, B. Griffiths, J. 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TI - A Comparison of Estimators of the Population Recombination Rate
JF - Molecular Biology and Evolution
DO - 10.1093/oxfordjournals.molbev.a026228
DA - 2000-01-01
UR - https://www.deepdyve.com/lp/oxford-university-press/a-comparison-of-estimators-of-the-population-recombination-rate-iiKl0AcZTw
SP - 156
EP - 163
VL - 17
IS - 1
DP - DeepDyve
ER -