chifish: a computer program testing for genetic heterogeneity at multiple loci using chi‐square and Fisher's exact test

chifish is a 32‐bit Windows/DOS program evaluating divergence at multiple gene loci. It tests the hypothesis of no difference at any locus both by means of Pearson's traditional chi‐square and by using Fisher's method of combining P values obtained by Fisher's exact test. Input data are read from a file formatted for genepop . Commonly used population genetics software do not perform chi‐square tests, and the simultaneous application of both techniques aids in situations where poor power of the ‘exact approach’ may prevent detection of true differentiation (e.g. few populations and few alleles per locus).     

Statistical power when testing for genetic differentiation

A variety of statistical procedures are commonly employed when testing for genetic differentiation. In a typical situation two or more samples of individuals have been genotyped at several gene loci by molecular or biochemical means, and in a first step a statistical test for allele frequency homogeneity is performed at each locus separately, using, e.g. the contingency chi-square test, Fisher's exact test, or some modification thereof. In a second step the results from the separate tests are combined for evaluation of the joint null hypothesis that there is no allele frequency difference at any locus, corresponding to the important case where the samples would be regarded as drawn from the same statistical and, hence, biological population. Presently, there are two conceptually different strategies in use for testing the joint null hypothesis of no difference at any locus. One approach is based on the summation of chi-square statistics over loci. Another method is employed by investigators applying the Bonferroni technique (adjusting the P-value required for rejection to account for the elevated alpha errors when performing multiple tests simultaneously) to test if the heterogeneity observed at any particular locus can be regarded significant when considered separately. Under this approach the joint null hypothesis is rejected if one or more of the component single locus tests is considered significant under the Bonferroni criterion. We used computer simulations to evaluate the statistical power and realized alpha errors of these strategies when evaluating the joint hypothesis after scoring multiple loci. We find that the 'extended' Bonferroni approach generally is associated with low statistical power and should not be applied in the current setting. Further, and contrary to what might be expected, we find that 'exact' tests typically behave poorly when combined in existing procedures for joint hypothesis testing. Thus, while exact tests are generally to be preferred over approximate ones when testing each particular locus, approximate tests such as the traditional chi-square seem preferable when addressing the joint hypothesis.     

Power for detecting genetic divergence: differences between statistical methods and marker loci

Information on statistical power is critical when planning investigations and evaluating empirical data, but actual power estimates are rarely presented in population genetic studies. We used computer simulations to assess and evaluate power when testing for genetic differentiation at multiple loci through combining test statistics or P values obtained by four different statistical approaches, viz. Pearson's chi-square, the log-likelihood ratio G-test, Fisher's exact test, and an F(ST)-based permutation test. Factors considered in the comparisons include the number of samples, their size, and the number and type of genetic marker loci. It is shown that power for detecting divergence may be substantial for frequently used sample sizes and sets of markers, also at quite low levels of differentiation. The choice of statistical method may be critical, though. For multi-allelic loci such as microsatellites, combining exact P values using Fisher's method is robust and generally provides a high resolving power. In contrast, for few-allele loci (e.g. allozymes and single nucleotide polymorphisms) and when making pairwise sample comparisons, this approach may yield a remarkably low power. In such situations chi-square typically represents a better alternative. The G-test without Williams's correction frequently tends to provide an unduly high proportion of false significances, and results from this test should be interpreted with great care. Our results are not confined to population genetic analyses but applicable to contingency testing in general.     

Letter to the Editor

Article http://dx.doi.org/10.1002/bimj.200710382 Authors reply http://dx.doi.org/10.1002/bimj.200810480     

A note on P-values for two-sided tests

Based on uniformly most powerful unbiased (UMPU) tests for two-sided hypotheses and a short note in Lehmann (1959) on critical levels for randomized tests, Meulepas (1998, 1999) proposed (two-tailed) P -values taking into account the randomization constant(s) of the UMPU-tests. While UMPU-tests need an extra uniform observation if randomization is required, the P -values proposed by Meulepas need no extra uniform observation. At first glance, his idea looks very promising in order to define a suitable and powerful P -value. Unfortunately, such P -values are generally too conservative.     

Exact Test of Uniform Association in a 2 × 3 Table

In a similar approach to Fisher's exact test of independence (null association) for a 2 × 2 table, this note gives an exact test of uniform association for a 2 × 3 table.     

Hierarchical analyses of genetic differentiation in a hybrid zone of Sorex araneus (Insectivora: Soricidae)

Microsatellites are used to unravel the fine-scale genetic structure of a hybrid zone between chromosome races Valais and Cordon of the common shrew ( Sorex araneus ) located in the French Alps. A total of 269 individuals collected between 1992 and 1995 was typed for seven microsatellite loci. A modified version of the classical multiple correspondence analysis is carried out. This analysis clearly shows the dichotomy between the two races. Several approaches are used to study genetic structuring. Gene flow is clearly reduced between these chromosome races and is estimated at one migrant every two generations using R -statistics and one migrant per generation using F -statistics. Hierarchical F - and R -statistics are compared and their efficiency to detect inter- and intraracial patterns of divergence is discussed. Within-race genetic structuring is significant, but remains weak. F _ST displays similar values on both sides of the hybrid zone, although no environmental barriers are found on the Cordon side, whereas the Valais side is divided by several mountain rivers. We introduce the exact G -test to microsatellite data which proved to be a powerful test to detect genetic differentiation within as well as among races. The genetic background of karyotypic hybrids was compared with the genetic background of pure parental forms using a CRT–MCA. Our results indicate that, without knowledge of the karyotypes, we would not have been able to distinguish these hybrids from karyotypically pure samples.     

spagedi: a versatile computer program to analyse spatial genetic structure at the individual or population levels

spag e d i version 1.0 is a software primarily designed to characterize the spatial genetic structure of mapped individuals or populations using genotype data of codominant markers. It computes various statistics describing genetic relatedness or differentiation between individuals or populations by pairwise comparisons and tests their significance by appropriate numerical resampling. spag e d i is useful for: (i) detecting isolation by distance within or among populations and estimating gene dispersal parameters; (ii) assessing genetic relatedness between individuals and its actual variance, a parameter of interest for marker based inferences of quantitative inheritance; (iii) assessing genetic differentiation among populations, including the case of haploids or autopolyploids.     

RST Calc: a collection of computer programs for calculating estimates of genetic differentiation from microsatellite data and determining their significance

R _ST, an analogue of F _ST, provides a convenient approach for estimating levels of genetic differentiation from microsatellite data. This paper examines current approaches for calculating estimates of R _ST and suggests a weighting scheme based on the transformation of allele sizes at loci across data sets. Combined within an analysis of variance framework this scheme yields an estimator of R _ST analogous to the θ estimator of F _ST. Software for the IBM-PC is described which carries out such calculations and assesses the significance of R _ST or Nm estimates using bootstrap and permutation tests.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

optibin: a computer program to bin alleles similarly distributed across populations

Genetic studies of fish and wildlife populations are increasingly based on highly polymorphic DNA loci. The large number of alleles, and particularly the large number of alleles that are observed in low frequency, have led many researchers to bin alleles. However, most strategies for binning alleles are subjective and may lead to loss of discriminatory power. A computer program that implements the allele‐binning algorithm of Bromaghin and Crape (2005 ) is described.     

wisepair: a computer program for individual matching in genetic tracking studies

Individual‐based data sets tracking organisms over space and time are fundamental to answering broad questions in ecology and evolution. A ‘permanent’ genetic tag circumvents a need to invasively mark or tag animals, especially if there are little phenotypic differences among individuals. However, genetic tracking of individuals does not come without its limits; correctly matching genotypes and error rates associated with laboratory work can make it difficult to parse out matched individuals. In addition, defining a sampling design that effectively matches individuals in the wild can be a challenge for researchers. Here, we combine the two objectives of defining sampling design and reducing genotyping error through an efficient Python‐based computer‐modelling program, wisepair . We describe the methods used to develop the computer program and assess its effectiveness through three empirical data sets, with and without reference genotypes. Our results show that wisepair outperformed similar genotype matching programs using previously published from reference genotype data of diurnal poison frogs (Allobates femoralis) and without‐reference (faecal) genotype sample data sets of harbour seals (Phoca vitulina) and Eurasian otters (Lutra lutra). In addition, due to limited sampling effort in the harbour seal data, we present optimal sampling designs for future projects. wisepair allows for minimal sacrifice in the available methods as it incorporates sample rerun error data, allelic pairwise comparisons and probabilistic simulations to determine matching thresholds. Our program is the lone tool available to researchers to define parameters a priori for genetic tracking studies.     

Nonparametric Two-Sample Tests for General Clinical Data

Starting from the discussion of a practical example a unifying concept for the derivation of meaningfully interpretable nonparametric tests for the two-sample case is developed which may well be adapted for other designs, too. This methodology covers other well-known procedures, e.g. Fisher's exact test, the Wilcoxon-Mann-Whitney and Gehan's tests, and may furthermore be extended to all situations sharing the same fundamental structural property of the sample space, namely its strict order induced by the substantial problem under study. The resulting test procedure is discussed for a randomization argument, exact and approximate, as well as for the general specific test problem. A numerical example is provided.     

Species co‐occurrence analysis: pairwise versus matrix‐level approaches

Veech (2013, Global Ecology and Biogeography, 22 , 252–260) introduced a formula to calculate the probability of two species co‐occurring in various sites under the assumption of statistical independence between the two distributional patterns. He presented his model as a new procedure, a ‘pairwise approach’, different from analyses of whole presence–absence matrices to examine patterns of co‐occurrence. Here I show that: (1) Veech's method is identical to Fisher's exact test, a standard procedure for measuring the statistical association between two discrete variables; (2) in a broad sense, the pairwise approach is very similar to early analyses of spatial association, such as the one advanced by Forbes in 1907; (3) implicit in Veech's formula is a sampling scheme that is indistinguishable from well‐known matrix‐level null models that randomize the distribution of species among equiprobable sites; (4) pairwise co‐occurrence patterns can be analysed using any matrix‐level null model, so pairwise comparisons are not limited to using Veech's formula. The methodological distinction that Veech proposed between pairwise and matrix‐level approaches does not in fact exist, although the conceptual distinction between the two approaches is still a debated topic.     

Additive genetic variance for lifetime fitness and the capacity for adaptation in an annual plant

The immediate capacity for adaptation under current environmental conditions is directly proportional to the additive genetic variance for fitness, V_A(W). Mean absolute fitness, , is predicted to change at the rate , according to Fisher's Fundamental Theorem of Natural Selection. Despite ample research evaluating degree of local adaptation, direct assessment of V_A(W) and the capacity for ongoing adaptation is exceedingly rare. We estimated V_A(W) and in three pedigreed populations of annual Chamaecrista fasciculata, over three years in the wild. Contrasting with common expectations, we found significant V_A(W) in all populations and years, predicting increased mean fitness in subsequent generations (0.83 to 6.12 seeds per individual). Further, we detected two cases predicting “evolutionary rescue,” where selection on standing V_A(W) was expected to increase fitness of declining populations (< 1.0) to levels consistent with population sustainability and growth. Within populations, inter‐annual differences in genetic expression of fitness were striking. Significant genotype‐by‐year interactions reflected modest correlations between breeding values across years, indicating temporally variable selection at the genotypic level that could contribute to maintaining V_A(W). By directly estimating V_A(W) and total lifetime , our study presents an experimental approach for studies of adaptive capacity in the wild.     

Genetic similarity of polyploids: a new version of the computer program POPDIST (version 1.2.0) considers intraspecific genetic differentiation

For evolutionary studies of polyploid species estimates of the genetic identity between species with different degrees of ploidy are particularly required because gene counting in samples of polyploid individuals often cannot be done, e.g., in triploids the phenotype AB can be genotypically either ABB or AAB. We recently suggested a genetic distance measure that is based on phenotype counting and made available the computer program POPDIST. The program provides maximum-likelihood estimates of the genetic identities and distances between polyploid populations, but this approach is not informative for populations within species that only differ in their allele frequencies. We now close this gap by applying the frequencies of shared 'bands' in both populations to Nei's identity measure. Our simulation study demonstrates the close correlation between the band-sharing identity and the genetic identity calculated on the basis of gene frequencies for any degree of ploidy. The new extended version of POPDIST (version 1.2.0) provides the option of choosing either the maximum-likelihood estimator or the band-sharing measure.     

An approach to the conditional error rate principle with nuisance parameters

In the presence of nuisance parameters, the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters. To use the conditional error rate principle with nuisance parameters, we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters. We show that the optimization problem that defines such a test can be solved efficiently by existing algorithms.     

Evolution of phenotypic plasticity: Genetic differentiation and additive genetic variation for induced plant defence in wild arugula Eruca sativa

Phenotypic plasticity is the primary mechanism of organismal resilience to abiotic and biotic stress, and genetic differentiation in plasticity can evolve if stresses differ among populations. Inducible defence is a common form of adaptive phenotypic plasticity, and long‐standing theory predicts that its evolution is shaped by costs of the defensive traits, costs of plasticity and a trade‐off in allocation to constitutive versus induced traits. We used a common garden to study the evolution of defence in two native populations of wild arugula Eruca sativa (Brassicaceae) from contrasting desert and Mediterranean habitats that differ in attack by caterpillars and aphids. We report genetic differentiation and additive genetic variance for phenology, growth and three defensive traits (toxic glucosinolates, anti‐nutritive protease inhibitors and physical trichome barriers) as well their inducibility in response to the plant hormone jasmonic acid. The two populations were strongly differentiated for plasticity in nearly all traits. There was little evidence for costs of defence or plasticity, but constitutive and induced traits showed a consistent additive genetic trade‐off within each population for the three defensive traits. We conclude that these populations have evolutionarily diverged in inducible defence and retain ample potential for the future evolution of phenotypic plasticity in defence.