Molecular Biology and Evolution 18:1134-1135 (2001)
© 2001 Society for Molecular Biology and Evolution
LETTER |
Coalescent Simulations and Statistical Tests of Neutrality
Department of Ecology and Evolution, University of Chicago
Recently, Depaulis and Veuille (1998)
proposed two new test statistics based on the number and frequency of different haplotypes. In a companion letter, Markovtsova, Marjoram, and Tavaré (2001) point out that Depaulis and Veuille (1998)
do not use the standard implementation for their coalescent simulations. Standard coalescent simulations first produce random genealogies, then place mutations at constant rate 
/2 ( = 4Nµ is the population mutation parameter, where N is the effective population size and µ is the per-locus mutation rate per generation) along each of the branches (Kingman 1982a, 1982b
; Hudson 1990
). Instead, Depaulis and Veuille (1998)
generate distributions for their statistics by first constructing random genealogies, then placing S (the observed number of segregating sites) mutations on each tree. This "fixed S" method has been used before (e.g., Hudson 1993
; Rozas and Rozas 1999
), partly because it is easy to simulate, and partly because S is observed, while must be estimated from the data (see, e.g., Fu 1996
). In fact, it is not clear how to estimate independent of polymorphism data. Although the fixed S scheme does not directly use , Markovtsova, Marjoram, and Tavaré (2001) highlight that the actual distributions of test statistics conditional on S are not independent of . In particular, knowing both and S changes the expected shape of a genealogy. For example, if S is unusually large given , we expect the genealogy to be longer than average. Thus, the critical values in Depaulis and Veuille (1998)
might not be appropriate, since the actual rejection probabilities for their tests are functions of the unknown parameter .
In this letter, we examined the type I errors of various statistical tests by simulation, and we determined what the actual rejection probability was for the fixed S method when data was simulated under standard coalescent assumptions. We considered three different test statistics: K (Strobeck 1987
; Fu 1996
; Depaulis and Veuille 1998
), D (Tajima 1989
), and D* (Fu and Li 1993
). First, we ran 105 replicates under the fixed S method to generate critical values (at the 5% level) for each possible value of S. Then, we ran 105 standard coalescent simulations with fixed mutation parameter . For each trial, acceptance or rejection was determined from the fixed S critical values using the observed value of S. We tabulated what proportion of trials led to significantly low or significantly high values of each of the three test statistics. K is the number of distinct haplotypes in the sample, and D and D* are two commonly used test statistics that determine whether the frequencies of segregating mutations are consistent with the standard neutral model. An excess of low-frequency variants leads to negative D and D* values, while an excess of intermediate-frequency variants leads to positive D and D* values. To more accurately compare nominal and actual rejection probabilities, we used a randomized test (see, e.g., Lehmann 1986
, p. 71). All simulations were run with no recombination.
Table 1
shows results for a sample size of n = 50 and = 3.0, 5.0, 10.0, and 15.0. The actual rejection probabilities for K were near 5%, while those for D and D* were between 5% and 5.4%. In all cases, the fractions of trials rejected on each tail (i.e., the proportions that were significantly too high or significantly too low) were roughly equal. If a nonrandomized test was used (as would be done in practice), the actual rejection probabilities for K and D* were substantially lower (
3.6% for K and 4.4% for D*), while those for D were about the same. To check for the effect of sample size, we reran all of our simulations with n = 20. The actual (randomized) rejection probabilities were about the same as in those table 1
: for 3.0 20.0, rejection probabilities were 5% for K, 5.6% for D*, and 5.4% for D (results not shown).
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For K, we examined the relationship between
and S more systematically by determining how different values of S affect the rejection probabilities when = 5.0. Table 2
shows the results of 106 simulations with n = 50 and S grouped into seven different classes. As expected, the actual rejection probability varied widely for different values of S. When S was near its expected value, the actual rejection probability was less than the nominal 5% level. However, rejection probabilities were far higher when S was very high or very low. When S was small, the underlying genealogy tended to be short; most mutations occurred on separate branches of the genealogy, leading to more haplotypes than expected and high K values. In contrast, when S was large, the oldest branches tended to be extremely long, and many mutations lay on them. This often led to fewer than expected distinct haplotypes (i.e., low K values).
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In summary, we find that for a given value of
, the actual rejection probability (when critical values are determined from fixed S simulations) is close to the nominal rejection level. Thus, the fixed S scheme may still be appropriate, unless a researcher has a prior belief that the actual value of is far from Watterson's (1975)
estimate W (which is based on the observed value of S). In practice, if one knows that W is inaccurate, one may also have independent evidence that the standard neutral model is false and should not be used. Finally, we point out that although the unknown parameter may not prevent the construction of an appropriate statistical test, uncertainty in the population recombination parameter C = 4Nr (where N is the effective population size and r is the per-locus recombination rate per generation) is much more problematic. The distributions of all three test statistics (especially K) are highly dependent on the unknown value of C (Wall 1999
), making it extremely difficult to construct a test for nuclear sequence data with nominal rejection probability equal to the actual rejection probability.
Acknowledgements
We thank Y.-X. Fu, P. Marjoram, M. Przeworski, and an anonymous reviewer for helpful comments and discussions. J.D.W. was supported by NIH grant 5 R01 H610847.
Footnotes
1 Present address: Department of Organismic and Evolutionary Biology, Harvard University. 
2 Keywords: Coalescent theory, neutrality tests
3 Address for correspondence and reprints: Jeffrey D. Wall, Department of Organismic and Evolutionary Biology, Harvard University, 2102 Biological Laboratories, 16 Divinity Avenue, Cambridge, Massachusetts 02138. jwall{at}oeb.harvard.edu
literature cited
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