RESULTS AND DISCUSSION

Bias: Our simulations suggest that the heterozygote-excess method slightly overestimates the N_eb when using finite samples and Nei’s unbiased estimator of H_exp. When N_eb was 4, 20, and 100, the harmonic mean estimates of N_eb (from 500 simulations) were 4.1, 22.2, and 103.4, respectively, when sampling 30 individuals and 10 loci (five alleles/locus) with triangular allele frequency distributions and a polygamous breeding system (Table 1). When more individuals were sampled, the bias was slightly lower. Harmonic mean estimates of N_eb were nearly identical for loci with two, three, and five alleles and with widely different allele frequency distributions (Table 1; Figure 1, horizontal bars inside box plots). The largest bias occurred under the polygynous mating system (e.g., = Nˆ_eb 5.0 when true N_eb ≅ 4; Figure 1). This bias was not surprising given the assumption of the heterozygote-excess method that there are equal numbers of male and female parents. Still, the bias was small enough not to substantially diminish the usefulness of the heterozygote-excess method.

The harmonic mean N_eb estimates were similar for the monogamous and polygamous mating systems (Nˆ_eb was 4.5 and 4.1, respectively, when N_eb was 4.0; Figure 1). This suggests that there is little impact of an interfamily Wahlund effect on the accuracy of N_eb estimates. When only approximately two to three large families exist (e.g., under monogamy with N_eb equaling 4-6), sampling across families does not generate a large heterozygote deficiency (i.e., Wahlund effect). However, a Wahlund heterozygote deficiency is expected when many families exist (A. Pudovkin, personal communication). Such a deficiency would at least partially cancel the heterozygote excess caused by small N_eb, and thereby cause biased (high) estimates of N_eb. Although monogamy did not cause a large bias, it did substantially increase the variance among N_eb estimates (see below and Figure 1).

Precision: To quantify the precision of the N_eb estimates, we used the Student’s t-distribution to compute a 95% confidence interval for each Nˆ_eb (as in Pudovkinet al. 1996). The confidence interval on Nˆ_eb contained the true N_eb in 92-96% of the simulation estimates of N_eb when using loci with five alleles (Table 1). As expected, approximately half of the confidence intervals that did not contain the true N_eb were too low (L) and half were too high (H). This suggests that the Student’s t-distribution is useful for computing confidence intervals, even though N_eb is not exactly normally distributed. For loci with three alleles or for monogamous mating systems, the confidence intervals also contained the true N_eb ∼92-96% of the time. However, when using loci with only two alleles, the confidence intervals were generally too narrow and contained the true N_eb in only 83-89% of the simulation estimates of N_eb (Table 1). Thus, confidence intervals must be interpreted with caution or computed by alternative methods (e.g., bootstrap resampling) when using loci with only two alleles (e.g., many allozyme loci).

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Figure 2.

—Distributions of 500 (95%) confidence limits on Nˆ_eb computed from 500 independent simulation estimates, as in Figure 1. Dotted horizontal lines represent the true N_eb being estimated (4 and 10 for a-c and d-f, respectively). (b and e) “80%” shows the 80th percentile of the distribution of the upper 95% confidence limits. This distribution was generated from 500 independent simulation estimates of N_eb. In e, 460 is the 95th percentile of the distribution of the upper 95% confidence limits.

Under extreme polygyny (e.g., one male mating with 99 females), the confidence intervals were often too high. For example, when N_eb was four, ∼25% of the 500 simulated confidence intervals were slightly higher than the true N_eb, and none were lower than N_eb. Although the confidence intervals were often too high, they were also much narrower under polygyny than under monogamy or polygamy (Figure 1). This narrowness substantially increases the usefulness of the heterozygote-excess method under polygyny. Thus, under extreme polygamy, the heterozygote-excess method will be useful for detecting a small N_eb but will be less useful for quantifying the exact size of N_eb.

To determine the minimum number of loci and individuals that must be sampled to achieve a high probability of obtaining narrow confidence intervals, we plotted the distribution of the (upper and lower) 95% confidence limits obtained from 500 simulation estimates of N_eb. When the true N_eb is only 4, at least 10 loci (with five alleles) and 30 individuals must be sampled to achieve an 80% probability of obtaining an upper 95% confidence limit < ∼20 (Figure 2b). In other words, the statistical power is 0.80 when testing the null hypothesis that the true N_eb ≠ 20 (and when the true N_eb is actually only 4). The power will be slightly higher when using a one-tailed test and the null hypothesis that true N_eb ≥ 20 (the alternative hypothesis is N_eb < 20).

These results show that the heterozygote-excess method is sufficiently powerful for detecting a small N_eb when sampling reasonable numbers of individuals and loci with five alleles. Such results are important for conservation biology and the management of captive and natural populations. The precision of Nˆ_eb is often increased more by analyzing a larger number of loci than by sampling more individuals. Doubling the number of loci from 10 to 20 (compare the first box plot in Figure 2, b and c) generally reduces confidence intervals more than doubling the number of individuals sampled from 15 to 30 (compare the first two box plots in Figure 2b). However, the benefit from doubling the number of loci depends on the number and frequency of alleles (Figure 1).

When true N_eb is 10, we must sample >20 polymorphic loci and 60 individuals to have an 80% probability of obtaining confidence intervals that are <50 (and to have a 95% probability of obtaining confidence intervals <100; Figure 2e). When the true N_eb is 100, >80% of the confidence intervals include infinity, even when sampling 120 individuals and 20 loci with five alleles (data not shown). Clearly, when N_eb is >10, very large samples of loci and individuals are required to achieve a high probability of obtaining reasonably small confidence intervals. Thus, the main limitation of the usefulness of the heterozygote-excess method is its poor precision, i.e., its wide confidence intervals. The confidence intervals are generally too wide for the method to be useful when using diallelic loci, loci with mostly rare alleles, or when studying strictly monogamous species (Figure 1).

When applied to data from natural populations, the heterozygote-excess method often gave estimates of N_eb equal to infinity. For example, Nˆ_eb was infinity in 5 of 10 cohorts for which the total number of parents was known (or estimated) to be small (i.e., three to a few dozen). Further, only 2 of the 10 estimates gave 95% confidence intervals that did not include infinity as an upper limit (Table 2). This poor precision is not surprising in that only 5-9 polymorphic loci were analyzed, and only 11-25 progeny were sampled. Additional empirical evaluations are needed, but it is extremely difficult to find large data sets containing individuals produced from a known number of parents.

One potential limitation of the method is the requirement for random, representative sampling. For example, if a sample contains only one or few families (due to sampling error) then we could obtain a very low N_eb estimate, even though many families actually exist and N_eb is large. Another obvious limitation is that the method will work only in species with separate sexes. The method will work for haplo-diploid species (e.g., Hymenopterans), but will require the derivation of equations different from those presented here.

Four approaches may help circumvent the problem of poor precision. First, one can compute 80% confidence intervals (in addition to 95% confidence intervals). This will reduce the likelihood that the upper confidence limit will include infinity and be uninformative. Second, one could explore alternative methods for computing confidence intervals (e.g., nonparametric methods such as bootstrap resampling of loci). Third, one could combine estimates of N_eb from several generations or cohorts by computing the harmonic mean of Nˆ_eb over the multiple generations or cohorts. This can reduce the probability of obtaining infinity for Nˆ_eb because, when computing a harmonic mean, the low estimates carry far more weight than high ones (e.g., infinity). Finally, one can potentially combine estimates of N_e obtained from several independent N_e estimators by computing the harmonic mean of the N_e estimates. Other promising N_e estimators include those based on gametic disequilibrium (Hill 1981) and on temporal variance in allele frequencies (Krimbas and Tsaskas 1971; Waples 1991). These two estimators also suffer from low precision (Waples 1989, 1991; Luikart 1997; Luikartet al. 1998). However, two or more of the estimators may be independent (Waples 1991; Pudovkinet al. 1996) and thus could potentially be used simultaneously to achieve a more precise estimate of N_e. More research is needed to evaluate the precision and accuracy that can be achieved by using several N_e estimators simultaneously. Any improvement in our ability to estimate N_e would be significant in light of both the difficulties in assessing N_e and the importance of N_e in population genetics and in conservation biology.