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Population Biology |
Department of Evolution, Ecology, and Organismal Biology, The Ohio State University, 1735 Neil Avenue, Columbus, Ohio 43210-1293 USA
Received for publication May 3, 2001. Accepted for publication August 28, 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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review of the plant literature. We show here that some comparisons may be inappropriate if GST was calculated in a different way than that used by Hamrick and Godt (HG). An alternative method advocated by Nei is mathematically different from the HG technique, occasionally resulting in different GST values. We reviewed 695 studies that appeared between 1990 and September 1999 that cited Hamrick and Godt (1989)
and found that many of these calculated GST according to Nei's method (46%), with the majority of these papers (61%) including comparisons to Hamrick and Godt's review. We suggest that if GST estimates are compared across studies, it is most appropriate to calculate them the same way. However, we found that in most cases, the magnitude of difference in GST values was small, suggesting that qualitative comparisons of GST estimates between most studies are probably valid. Nevertheless, we have identified theoretical and empirical situations in which large differences in GST values are likely to arise. Thus, we advise future investigators to carefully consider which method to use in calculating GST for a given data set.
Key Words: genetic variation • GST • Hamrick and Godt • population differentiation
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; often referred to as 1990) review entitled "Allozyme diversity in plant species." This paper examined the literature to study patterns of genetic variation (e.g., number of alleles per locus, percentage of polymorphic loci, etc.) according to a variety of different traits, including life form, geographic range, and breeding system. Since the review was first published 11 yr ago, it has been cited in increasing numbers as researchers use the data set as a point of reference for their own studies (Fig. 1).
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review is GST, a measure of population differentiation. Values of GST range from zero to one, with low values indicating that little variation is proportioned among populations (high values denote that a large amount of variation is found among populations). The purpose of this paper is to point out that there are two common but different methods of calculating GST, one used by Hamrick and Godt (1989)
and the original method advocated by Nei (1973)
. For clarity, these will be referred to as the HG and Nei methods throughout this paper. We show here that these two methods can give different results in certain cases. Thus, conclusions drawn from a comparison of GST with the Hamrick and Godt review may be inappropriate if GST is calculated using the Nei method. Furthermore, we determine how frequently this miscalculation has occurred in the literature by reviewing 695 studies published in the 11 yr since the review appeared.
As one of Nei's (1973)
genetic diversity statistics, GST is defined as the proportion of genetic diversity that resides among populations. It is equivalent to Wright's (1951)
FST when there are only two alleles at a locus, and, in the case of multiple alleles, GST is equivalent to the weighted average of FST for all alleles (Nei, 1973
). GST is also similar to Weir and Cockerham's (1984)
, except that the latter accounts for effects of uneven sample sizes and number of sampled populations. Although rare, may take on negative values (Weir, 1996
). GST is calculated from the total genetic diversity in the pooled populations (HT) and mean diversity within each population (HS) as:
pi2, where pi is the frequency of a given allele; Nei, 1973
). Unbiased estimates of HS can also be obtained for small sample sizes (Nei, 1978
; Nei and Chesser, 1983
; but see Cockerham and Weir, 1986
) and few populations (Nei, 1986
). Because HS represents a subset of the total diversity found in HT, it can never be more than HT. Using allele frequencies, Nei's (1973)
genetic statistics (HT, HS, DST, GST) are calculated for each locus (Table 1).
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). This can be written as the following, where Np is the number of polymorphic loci and i represents the ith locus:
In the Nei method, HT and HS values are first averaged across all loci, and GST is calculated from these mean values according to Eq. 1. Although both monomorphic and polymorphic loci are usually used in calculations of HT and HS (Nei, 1986
), Nei's GST is unaltered by the inclusion of monomorphic loci because they contribute to both the numerator and denominator; in effect, they cancel each other out (i.e., N is absent in Eq. 14 below). Nei's GST can be rewritten as the following, where N is the number of all loci:
As evident in Eqs. 8 and 14, the HG and Nei methods of calculating GST are not mathematically identical. Both methods will yield the same value in only a few rare cases. First, if all populations are completely differentiated from one another, all of the diversity will lie among populations, rather than within them (HS = 0). If this is true for all loci, both methods yield GST values of one. This can also occur when values of HT are identical over all loci. Second, if the total diversity is contained within each population (HS = HT) for all loci, a GST value of zero will result using both methods. Finally, HG's and Nei's GST will be equivalent if values of (HS/HT) are identical for all loci. If any of the above cases is not true for at least one locus (e.g., 0 < HS < HT), GST values calculated using the Nei and HG methods will differ to some extent from one another.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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by reviewing studies appearing between 1990 and 19 September 1999 that cited the paper. Using the Science Citation Index (Institute for Scientific Information [ISI], Philadelphia, Pennsylvania, USA), 695 studies were retrieved that fit this criterion. For each study, the following were recorded: (1) what, if any, genetic statistic was calculated (e.g., GST, FST, or ) and (2) whether the statistic was compared to GST values presented in Hamrick and Godt (1989)
. For papers that calculated GST, the reported information was then used to determine which method (Nei or HG) was employed. This was accomplished by recalculating GST both ways, using a reported GST loci table and then determining which value matched the reported number. In a few cases, reported GST values had been calculated by the Nei method, but were unbiased for sample size (Nei and Chesser, 1983
) and population number (Nei, 1986
). If a loci table was not given, it was generated using published allele frequencies. In some cases, not enough information was given to accurately determine how GST was calculated (see below).
Of the 695 studies that cited Hamrick and Godt (1989)
, a large number (45%) did not calculate any statistics. Several other papers included various measures of population differentiation, such as FST (15%), (3%), 
ST (<1%), and 
(<1%). Over a third of the total papers (36% or 252 studies) reported a GST value. Of these, 49 studies contained insufficient information so that (1) neither GST value could be recomputed, (2) only one value could be recalculated and this did not match the reported GST, or (3) both values could be recalculated but neither matched the reported number. These studies were not considered further.
Of the 203 remaining studies in which GST could be recalculated, the method (HG or Nei) was confidently determined in 167 papers (82%) because enough data were given to calculate GST using both methods. This set of papers will hereafter be referred to as HGconfident or Neiconfident, depending upon which method could be confidently assigned. In the remaining 36 papers (18%), the method could not be established with certainty because insufficient data made it impossible to calculate GST according to both methods. For example, mean HT and HS values were sometimes reported (allowing computation of Nei's GST), but tables of allele frequencies or GST values across loci were not given (i.e., HG's GST could not be calculated). In these papers, the GST value that could be recalculated matched the reported GST within the scope of rounding error (approximately ±0.01). These studies will henceforth be referred to as HGinfer or Neiinfer, depending upon which method could be inferred.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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61% of these papers contained comparisons to Hamrick and Godt's (1989)
review. These comparisons would be incorrect if empirical HG and Nei GST values were substantially different from one another, as previously indicated by theory. To determine how often these values actually diverged and the magnitude of such a difference, we compared the recalculated HG and Nei GST values in 167 studies in which the method could be confidently determined. Several of these studies contained multiple GST values for different species (considered different data sets), resulting in 227 different pairwise comparisons.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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0.01–0.80). In contrast, papers in which there were no substantial differences between the two values (
0.002) generally had a lower range of GST values across loci (0.0–0.30), although there were some exceptions. In addition, similar GST values were obtained in rare cases in which populations were either completely fixed for different alleles (HS = 0) or had identical allele frequencies (HS = HT) for all loci (two cases noted earlier). A wide range of GST values usually resulted from a mixture of uneven allele frequencies across populations at some loci (resulting in high GST values) and similar allele frequencies across populations at other loci (low GST). Uneven frequencies at individual loci were largely due to fixation of different alleles and/or the loss of a common allele within a few populations, which could occur as a result of a reduction in effective population size and subsequent genetic drift. A low range of GST values typically reflected a similarity of allele frequencies across all populations for all loci.
The HG and Nei GST values may also diverge because of a difference in their underlying mathematical properties. As apparent in Fig. 3, the Nei method gave relatively higher GST values than the HG technique in a number of papers. This is an example of Jensen's inequality, a mathematical property of nonlinear functions (Hansen, 2000
). Essentially, the difference between the HG and Nei methods is how a ratio (HS/HT) is averaged before it is subtracted from one (Eq. 2). In this particular case, Jensen's inequality states that the mean of a ratio (the HG method) will always be less than the ratio of the means (the Nei method), assuming that both the numerator (HS) and denominator (HT) are independent of one another. If true, the HG estimate of GST should be relatively lower than the Nei estimate. In an analysis of ten studies with large GST differences, HT and HS values were less correlated (more independent) with one another (r = 0.61, P = 0.58) than in ten other studies in which there was no difference between Nei and HG values (r = 0.98, P = 0.0001). Thus, there are certain cases in which the Nei method will give relatively higher GST values than the HG method simply because of the way means are calculated.
A closer examination also revealed that the Nei method appears to be more sensitive to interlocus variation in allele frequencies than the HG technique. If HT increases relative to HS for a locus, Nei's GST will increase while HG's value remains the same. For example, the addition of a third population fixed for a third allele to a system results in an increase of Nei's GST, while the HG value remains unchanged (Table 3). As more populations and fixed alleles are added to the first locus, Nei's GST increases, while HG's value remains the same.
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) and the number of populations needed to sample a certain level of genetic variation (i.e., proportion of variation = 1 – GnST; Ceska, Affolter, and Hamrick, 1997
). For example, the second equation is used for sampling and preserving germplasm of rare, threatened, and endangered species. Although the applicability of these approaches has been debated (Bossart and Prowell, 1998
; Whitlock and McCauley, 1999
), they represent important and relatively common applications of GST. To determine whether different GST values would yield differing predictions, we inserted the HG and Nei GST values from the 227 paired comparisons into each equation. Typically, the estimate of gene flow, Nem, was relatively higher when the HG GST was used than when the Nei GST was employed. Although estimates of Nem were significantly different overall using the two methods (Wilcoxon signed rank test, Ts = –6772, P = 0.0001), >75% of the differences were <1.0, and the majority was <0.50 (Fig. 4). Consequently, it is only when the estimates of GST are strongly different that conclusions may be erroneous. The effect on the number of populations to sample was less dramatic. We compared the proportion of variation that would be captured with N = 2, 5, 10, and 12 populations sampled, using either the HG or Nei GST value. Differences only occurred at N = 5 populations with the HG estimate predicting that slightly more of the variation would be captured for any given GST value (Fig. 5).
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review, GST should be calculated using the HG method. If the comparison involves another study that used the Nei method, GST should be calculated according to Nei. However, as we have demonstrated here, in some cases, qualitative comparisons of estimates calculated in different ways are unlikely to lead to erroneous conclusions. Still, major differences in GST values are possible, especially in situations in which conservation issues apply, such as populations that are fragmented or have experienced a bottleneck (when GST is likely to be used). We suggest that researchers should fully explore their data rather than relying solely on mean statistics to draw conclusions and make predictions. In certain cases, investigators should be particularly concerned about comparisons using the two methods. Data sets where we found the greatest difference in GST values also showed higher mean DST values, more loci with a DST >0.05, and a higher proportion of fixed or nearly fixed loci relative to loci with alleles evenly distributed across populations. Consequently, the variance in diversity statistics across loci may be more informative than means of these statistics.
If a comparison to a published study is not intended, an additional factor to consider in choosing an appropriate method is whether one technique is mathematically or biologically more meaningful than the other. Unfortunately, there is no clear answer about which method is best to use. From a mathematical viewpoint, there is no inherent reason to prefer taking the mean of ratios over the ratio of means, but the HG method is more likely to generate lower results than Nei's method (i.e., Jensen's inequality). At present, there is no way of knowing which value best represents population genetic structure in a given situation. The HG method gives equal weight to loci with high and low HT values, which may result in a range of different GST values across loci; the Nei method would be less affected by such loci (J. Hamrick, University of Georgia, personal communication). The Nei method may be more biologically meaningful, as it is more sensitive to variation in allele frequencies across populations (the very factor promoting population differentiation). However, investigators should consider using other genetic measures instead of GST (see below) if a comparison to Hamrick and Godt's (1989)
review is not intended. For example, is advantageous because it has a real biological definition (correlation of uniting games) and is unbiased with respect to sample size and the number of sampled populations (Weir and Cockerham, 1984
).
Based on our review, we have several suggestions for future studies in which GST values are presented. First, the method of calculation should be clearly explained; many studies have cited Nei (1973)
without any further explanation. At the very least, the HG method can be described as "GST values were averaged over polymorphic loci," while the Nei method can be expressed as "the mean values of HT and HS over all loci were used to calculate GST according to Nei (1973)
." Second, GST values should be reported for individual loci (along with HT, HS, and DST) to facilitate calculations of both HG and Nei GST values and to allow an examination of variation in these statistics. Several studies only reported mean values of HT, HS, and GST for the species or population(s) in question. Third, sample sizes should be given in allele frequency tables, as the absence of this information makes it impossible to recalculate GST values. Fourth, scientists should carefully consider whether biased or unbiased estimates of HT and HS are appropriate for the data, especially when studying rare species (see Nei, 1978, 1986
; Nei and Chesser, 1983
; Chakraborty and Danker-Hopfe, 1991
). Finally, there was some confusion in the literature over the use of HT, HS, Hes, and Hep when comparing results to Hamrick and Godt (1989)
. In that review, genetic diversity was calculated over all loci (monomorphic and polymorphic) as Hes at the species level by pooling across all populations, and as Hep at the population level (mean of population values). These statistics are analogous to Nei's (1973)
HT and HS, respectively. However, the HT and HS values reported in Hamrick and Godt (1989)
are calculated over polymorphic loci only, and as such, are not directly comparable with Nei's HT and HS. Thus, it is important that the researcher state whether all loci or only polymorphic loci were used to calculate HT and HS.
In this paper, we have been concerned with GST as a measure of population differentiation because it is commonly used and there was an urgent need to clarify how it is calculated. Whether or not GST is most appropriate as a measure of population structure is yet another issue. Several other statistics do exist (e.g., FST, , RST, ST) and may be more suitable than GST in many situations. For example, GST is dependent on sample sizes and number of populations, in addition to its reliance on Hardy-Weinberg genotype proportions (P. Lewis, University of Connecticut, personal communication), conditions that may be violated when analyzing small isolated populations. Although Nei (1986)
and Nei and Chesser (1983)
suggested that unbiased estimates of GST can be obtained through modifications of the original formula (but see Cockerham and Weir, 1986
), others have argued that is a better estimator (Weir and Cockerham, 1984
; but see Chakraborty and Danke-Hopfe, 1991
). In view of this debate, investigators should include raw genotype counts in published studies so that can be recomputed if necessary for comparison with other studies. Although a discussion of these genetic statistics is beyond the scope of the current paper, the researcher should carefully consider which statistic is best for the system under investigation.
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FOOTNOTES |
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2 Author for reprint requests, current address: Department of Ecology and Evolutionary Biology, University of California-Irvine, Irvine, California 92697-2525 USA (tel: 949-824-1772, FAX: 949-824-2181; tculley{at}uci.edu
)
3 Current address: Department of Ecology and Evolutionary Biology, University of Kansas, 1200 Sunnyside Avenue, Lawrence, Kansas 66045-7534 USA
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Bossart J. L. D. P. Prowell 1998 Genetic estimates of population structure and gene flow: limitations, lessons and new directions. Trends in Ecology and Evolution 13: 202-206
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