Testing for heterogeneity among phenotypic correlations: a comparison of methods using Monte Carlo simulations

Heterogeneous phenotypic correlations may be suggestive of underlying changes in genetic covariance among life-history, morphology, and behavioural traits, and their detection is therefore relevant to many biological studies. Two new statistical tests are proposed and their performances compared with existing methods. Of all tests considered, the existing approximate test of homogeneity of product-moment correlations provides the greatest power to detect heterogeneous correlations, when based on Hotelling's z*-transformation. The use of this transformation and test is recommended under conditions of bivariate normality. A new distribution-free randomisation test of homogeneity of Spearman's rank correlations is described and recommended for use when the bivariate samples are taken from populations with non-normal or unknown distributions. An alternative randomisation test of homogeneity of product-moment correlations is shown to be a useful compromise between the approximate tests and the randomisation tests on Spearman's rank correlations: it is not as sensitive to departures from normality as the approximate tests, but has greater power than the rank correlation test. An example is provided that shows how choice of test will have a considerable influence on the conclusions of a particular study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Data transformations in the analysis of community-level substrate utilization data from microplates

BIOLOG EcoPlates provide one method for determination of functional diversity indices and community-level physiological profiling of microbial populations based on carbon substrate utilization. In this study, the effect of data transformation on BIOLOG EcoPlate data derived from wetland mesocosms and biofiltration systems was examined. Homoscedasticity, normality, and the number of linear correlations between variables were quantified and evaluated for data that had been transformed using either Taylor or logarithmic transforms. Subsequent multivariate analysis was implemented using the untransformed, Taylor transformed and logarithmic transformed data sets. The effect of data transformation and its effect on principle component analysis are presented. The transforms are shown to help increase homogeneity of variance, increase normality of the data, and increase the number of significant linear correlations for the data. Separate principle component analyses and ordinations of the data showed the transforms to be well suited to this type of data and in particular illustrate the ability of the logarithmic transform to reduce the influence of high leverage or outlying observations on the overall analysis and its robustness in terms of treating data from different ecological systems. Although BIOLOG EcoPlates were used in this study to illustrate the use of transformations on multivariate data, the techniques described may be employed on similar microplate data. In addition, if homoscedasticity, normality and the number of linear correlations within a data set are not evaluated and the possibility of transforming the data, using the Taylor, logarithmic or another transform, is not considered, erroneous analysis and misleading conclusions may be attained when performing multivariate analysis on microplate data.     

Probit analysis of Upper Pleistocene hominids

Probit analysis allows the testing of the normality of a sample population where curve fitting and differential weighting occur simultaneously. Probit analysis may be utilized with small samples, and is ideal for use in the study of fossil populations. It is assumed that population which are normally distributed are likely to be in genetic equilibrium, so that graphic and computational probit analysis may be used to discern the effects of evolutionary forces on human populations. Probit analysis of cranial length in a sequence of Athenian population is presented in order to verify its sensitivity to these evolutionary forces. Probit analysis and other tests of homogeneity are applied to similar variables of Upper Pleistocene European Neandertal and modern man.Despite significant differences between variable means of Neandertal and modern man, probit analyses of their combined populations showed no less deviation from normality than either group separately. This suggests that the evolutionary changes in the cranium from the Neandertal stage to the modern stage of man were effected without significantly disrupting the genetic equilibrium of these populations, i.e. that rapid changes in these populations are unlikely to have occurred.     

A nonparametric approach to the discrimination of two cell populations

An approach to the statistical testing of differences between two cell populations the elements of which are characterized by multivariate data is described. The approach is based on the Fisher discriminant and the Kruskal-Wallis test. The method, which is sufficient but not necessary, makes no assumptions about the normality of the data or about the equality of the covariance matrices for each population.     

Frequencies of mtDNA haplogroups in southeastern Europe--Croatians, Bosnians and Herzegovinians, Serbians, Macedonians and Macedonian Romani

Mitochondrial DNA polymorphisms were analyzed in of 1,610 randomly chosen adult men from 11 different regions from southeastern Europe (Croatians, Bosnians and Herzegovinians, Serbians, Macedonians and Macedonian Romani). MtDNA HVS-I region together with RFLP sites diagnostic for main Euroasian and African mtDNA haplogroups were typed to determine haplogroup frequency distribution. The most frequent haplogroup in studied populations was H with the exception of Macedonian Romani among whom the most frequent were South Asian (Indian) specific variants of haplogroup M. The multidimensional scaling plot showed two clusters of populations and two outliers (Macedonian Romani and the most distant from mainland Croatian island of Korcula). The first cluster was formed by populations from three Croatian islands (Hvar, Krk and Brac) and the second cluster was formed by Macedonians, Serbians, Croatians from mainland and coast, Herzegovinians, Bosnians, Slovenians, Poles and Russians. The present analysis does not address a precise evaluation of phylogenetic relations of studied populations although some conclusions about historical migrations could be noticed. More extended conclusions will be possible after deeper phylogenetic and statistical analyses.     

Complementarity of statistical treatments to reconstruct worldwide routes of invasion: the case of the Asian ladybird Harmonia axyridis

Inferences about introduction histories of invasive species remain challenging because of the stochastic demographic processes involved. Approximate Bayesian computation (ABC) can help to overcome these problems, but such method requires a prior understanding of population structure over the study area, necessitating the use of alternative methods and an intense sampling design. In this study, we made inferences about the worldwide invasion history of the ladybird Harmonia axyridis by various population genetics statistical methods, using a large set of sampling sites distributed over most of the species’ native and invaded areas. We evaluated the complementarity of the statistical methods and the consequences of using different sets of site samples for ABC inferences. We found that the H. axyridis invasion has involved two bridgehead invasive populations in North America, which have served as the source populations for at least six independent introductions into other continents. We also identified several situations of genetic admixture between differentiated sources. Our results highlight the importance of coupling ABC methods with more traditional statistical approaches. We found that the choice of site samples could affect the conclusions of ABC analyses comparing possible scenarios. Approaches involving independent ABC analyses on several sample sets constitute a sensible solution, complementary to standard quality controls based on the analysis of pseudo‐observed data sets, to minimize erroneous conclusions. This study provides biologists without expertise in this area with detailed methodological and conceptual guidelines for making inferences about invasion routes when dealing with a large number of sampling sites and complex population genetic structures.     

A Bayesian approach to efficient differential allocation for resampling-based significance testing

Background  

Large-scale statistical analyses have become hallmarks of post-genomic era biological research due to advances in high-throughput assays and the integration of large biological databases. One accompanying issue is the simultaneous estimation of p-values for a large number of hypothesis tests. In many applications, a parametric assumption in the null distribution such as normality may be unreasonable, and resampling-based p-values are the preferred procedure for establishing statistical significance. Using resampling-based procedures for multiple testing is computationally intensive and typically requires large numbers of resamples.     

Number of individuals and molecular markers to use in genetic differentiation studies

Molecular markers are frequently used to study genetic variation among individuals within or between populations. Differences in marker banding patterns can be used to verify if individuals do, or do not, represent distinct groups or populations. Only in 2005, more than 500 studies used molecular markers to group individuals in clusters. Such studies make use of an arbitrary number of molecular markers from each of an arbitrary number of individuals presumed to represent distinct genotypes. However, the greater the genetic variation, the more likely a larger number of individuals and markers will be needed to capture a population's genetic signature. The numbers of both, markers and individuals included thus affect the way in which individuals are organized through cluster analyses, thereby affecting the conclusions drawn. Here we present a method that provides statistical criteria to verify that individual and marker sample sizes are sufficient to accurately depict genetic differentiation among different populations. Our method uses a resampling technique to assess the reproducibility of obtaining a particular grouping pattern for specific data sets. It thus, allows to estimate the robustness of the results obtained without including additional individuals, or markers.     

Elimination of inbreeding depression from a captive population of Speke's gazelle: Validity of the original statistical analysis and confirmation by permutation testing

The Speke's gazelle captive breeding program was designed in the early 1980s to simultaneously maintain the population's genetic diversity while reducing the severity of the inbreeding depression in a situation in which inbreeding could not be avoided. Statistical analyses of the resulting data using both regression techniques and nonparametric exact contingency tests revealed that the inbreeding depression was indeed reduced, and genetic surveys revealed that high levels of nuclear genetic diversity had indeed been maintained. Hence, the twin goals of the breeding program appeared to have been achieved. Recently, several papers have been published that question the validity of the original statistical analyses and resulting biological conclusions. Specifically, these papers raise three major issues: (1) that a small sample correction factor used in the regression analysis represents a statistical “flaw,” (2) that new analyses of the data do not confirm the original conclusion of a significant reduction in the level of inbreeding depression, and (3) that the biological conclusions about the program are not justified. In this paper we show (1) that there is no “flaw” in the small sample correction, (2) that the recent permutational test given by Willis and Wiese seriously violates standard procedures and has no statistical validity, (3) that the regression procedures used by Ballou are inappropriate because the data seriously violate the underlying statistical assumptions and that the statistically valid components of Ballou's work strongly confirm the validity of the Speke's gazelle program, (4) that permutational tests done in accordance with standard statistical practice strongly confirm the results of the original analysis, and (5) that the original biological conclusions are fully justified by multiple types of statistical analyses. Zoo Biol 17:77–94, 1998. © 1998 Wiley-Liss, Inc.     

Ancient DNA applications for wildlife conservation

Ancient DNA analyses of historical, archaeological and paleontological remains can contribute important information for the conservation of populations and species that cannot be obtained any other way. In addition to ancient DNA analyses involving a single or few individuals, population level studies are now possible. Biases inherent in estimating population parameters and history from modern genetic diversity are exaggerated when populations are small or have been heavily impacted by recent events, as is common for many endangered species. Going directly back in time to study past populations removes many of the assumptions that undermine conclusions based only on recent populations. Accurate characterization of historic population size, levels of gene flow and relationships with other populations are fundamental to developing appropriate conservation and management plans. The incorporation of ancient DNA into conservation genetics holds a lot of potential, if it is employed responsibly.     

There Is No Need To Be Normal: Generalized Linear Models of Natural Variation

Both ecological field studies and attempts to extrapolate from laboratory experiments to natural populations generally encounter the high degree of natural variability and chaotic behavior that typify natural ecosystems. Regardless of this variability and non-normal distribution, most statistical models of natural systems use normal error which assumes independence between the variance and mean. However, environmental data are often random or clustered and are better described by probability distributions which have more realistic variance to mean relationships. Until recently statistical software packages modeled only with normal error and researchers had to assume approximate normality on the original or transformed scale of measurement and had to live with the consequences of often incorrectly assuming independence between the variance and mean. Recent developments in statistical software allow researchers to use generalized linear models (GLMs) and analysis can now proceed with probability distributions from the exponential family which more realistically describe natural conditions: binomial (even distribution with variance less than mean), Poisson (random distribution with variance equal mean), negative binomial (clustered distribution with variance greater than mean). GLMs fit parameters on the original scale of measurement and eliminate the need for obfuscating transformations, reduce bias for proportions with unequal sample size, and provide realistic estimates of variance which can increase power of tests. Because GLMs permit modeling according to the non-normal behavior of natural systems and obviate the need for normality assumptions, they will likely become a widely used tool for analyzing toxicity data. To demonstrate the broad-scale utility of GLMs, we present several examples where the use of GLMs improved the statistical power of field and laboratory studies to document the rapid ecological recovery of Prince William Sound following the Exxon Valdez oil spill.     

A biological basis for lichenometry?

Aim Develop a biologically defensible conceptual model to explain lichen population dynamics. Methods This work critically examines the biological basis for several of the key assumptions that are used to interpret lichen-size age data. It then presents a biologically defensible conceptual model that can be used to develop computer based simulations of lichen population dynamics or to design alternative approaches to lichenometry. Main conclusions Most lichenometric ages are not verifiably accurate and are generated by methodologies that do not incorporate widely accepted biological principles. Polymodality in thallus-size distributions is not always the product of discrete events. Polymodality could reflect continuous changes in the availability of inhabitable patches (gaps) in a heterogenous mosaic. Improvement in lichenometry could be realized by investigations that seek to quantify the temporal and perhaps density-dependent onset of departures from statistical normality for certain lichen cohorts or species groupings. The study of lichen communities on substrates of known age may, in the short term provide some indication of the relative timing of age-stage changes in lichen demographic structure.     

Density-dependence by habitat heterogeneity: individual quality versus territory quality

The aim of this comment is to review the ecological issues concerning the role of individual and habitat heterogeneity as possible mechanisms explaining density-dependent fecundity in animal populations. Our intention is to discuss different approaches to determine whether or not studied populations are subjected to density-dependent processes and which mechanisms are involved. We show that because individual quality (e.g. measured in terms of age of breeding individuals) and territory quality can be correlated, untangling both effects on fecundity is frequently a difficult task. We discuss the misuse of statistical methods and other problems related to the specific characteristics of the studied populations that can have a strong influence on the conclusions reached by researchers working in this field.     

Mating Patterns of Plasmodium falciparum

Recent empirical data have enabled a more informed debate over the extent of clonality in Plasmodium falciparum populations. Oocyst heterozygosity data reveal that the mating structure of malaria populations varies according to the transmission intensity. This finding provides a more detailed picture of the malaria mating structure than previous conclusions, which were based on indirect measures of population mating structure, ie. linkage disequilibrium analyses. In this article, Ric Paul and Karen Day discuss aspects of the genetic structure of malaria populations as evidenced by oocyst heterozygosity and linkage disequilibrium data. They address the difficulties of performing genetic analyses of malaria parasite population structure inherent in parasite sampling, why two identical parasites are rarely observed in the field and how features of the epidemiology determine parasite population structure.     

Large standard deviations and logarithmic-normality

While many quantifiable biological phenomena can be described by making use of an assumption of normality in the distribution of individual values, many biological phenomena are not accurately described by the normal distribution. An unquestioned assumption of normality of distribution of possible outcomes can lead to misinterpretation of data, which could have serious consequences. Thus it is extremely important to test the validity of an assumption of normality of possible outcomes. As it turns out, the logarithmic-normal (log-normal) distribution pattern is often far more accurate in describing statistical biological phenomena. Herein I examine large samples of values for circulating blood cell (hemocyte) concentration (CHC) among both wild-type and mutant Drosophila larvae, and demonstrate in both cases that the distribution of individual values does not conform to normality, but does conform to log-normality.     

Landscape genetics of a specialized grasshopper inhabiting highly fragmented habitats: a role for spatial scale

Aim The study of geographical discontinuities in the distribution of genetic variability in natural populations is a central topic in both evolutionary and conservation research. In this study, we aimed to analyse (1) the factors associated with genetic diversity at the landscape spatial scale in the highly specialized grasshopper Mioscirtus wagneri and (2) to identify the relative contribution of alternative factors to the observed patterns of genetic structure in this species. Location La Mancha region, Central Spain. Methods We sampled 28 populations of the grasshopper M. wagneri and genotyped 648 individuals at seven microsatellite loci. We employed a causal modelling approach to identify the most influential variables associated with genetic differentiation within a multiple hypothesis‐testing framework. Results We found that genetic diversity differs among populations located in different river basins and decreases with population isolation. Causal modelling analyses showed variability in the relative influence of the studied landscape features across different spatial scales. When a highly isolated population is considered, the analyses suggested that geographical distance is the only factor explaining the genetic differentiation between populations. When that population is excluded, the causal modelling analysis revealed that elevation and river basins are also relevant factors contributing to explaining genetic differentiation between the studied populations. Main conclusions These results indicate that the spatial scale considered and the inclusion of outlier populations may have important consequences on the inferred contribution of alternative landscape factors on the patterns of genetic differentiation even when all populations are expected to similarly respond to landscape structure. Thus, a multiscale perspective should also be incorporated into the landscape genetics framework to avoid biased conclusions derived from the spatial scale analysed and/or the geographical distribution of the studied populations.     

Multiplicity of inferences in clinical trials: adjustment methods, clinical interpretation issues

Multiplicity of inferences is present in a large majority of clinical trials and conducts to false analyses or interpretation issues. The main risk consists in false positive conclusions. A large number of statistical methods is available for controlling the rate of false positive conclusions. But formal adjustment is not necessary in all cases and depends on the aims of the study.     

Consequences of ignoring spatial variation in population trend when conducting a power analysis

Long‐term, large‐scale monitoring programs are becoming increasingly common to document status and trends of wild populations. A successful program for monitoring population trend hinges on the ability to detect the trend of interest. Power analyses are useful for quantifying the sample size needed for trend detection, given expected variation in the population. Four components of variation (within‐year variation at a given site, interannual variation within a site, variation among sites in the interannual variation, and variation among sites in mean abundance or density) are commonly considered in power analyses for population trend, but a fifth is rarely considered: variation among sites in the local trend. Spatial variation in trend is expected to reduce statistical power, but the magnitude of this reduction has not been fully explored. We used computer simulations to evaluate the consequences of ignoring spatial variation in trend under a variety of sampling designs and wide ranges of other components of variation. The effect of spatial variation in trend on power was minor when other input parameters took extreme values that made the trend either very difficult or very easy to detect. However, at moderate values of the other parameters, spatial variation in trend had a strong effect, reducing statistical power by up to 60%. In some cases, ignoring spatial variation in trend resulted in an 80% probability of a type I error (falsely detecting a trend in a stable population). Spatial variation in trend is therefore an important consideration when designing a long‐term monitoring program for many species, especially those affected by local conditions at sites that are repeatedly surveyed. If variation in trend is ignored, as in many previous power analyses, the recommended sampling design will likely be insufficient to detect the trend of interest and lead to potentially false conclusions of a stable population.     

EVOLUTIONARY INFERENCES FROM THE ANALYSIS OF EXCHANGEABILITY

Evolutionary inferences are usually based on statistical models that compare mean genotypes or phenotypes (or their frequencies) among populations. An alternative is to use the full distribution of genotypes and phenotypes to infer the “exchangeability” of individuals among populations. We illustrate this approach by using discriminant functions on principal components to classify individuals among paired lake and stream populations of threespine stickleback in each of six independent watersheds. Classification based on neutral and nonneutral microsatellite markers was highest to the population of origin and next highest to populations in the same watershed. These patterns are consistent with the influence of historical contingency (separate colonization of each watershed) and subsequent gene flow (within but not between watersheds). In comparison to this low genetic exchangeability, ecological (diet) and morphological (trophic and armor traits) exchangeability was relatively high—particularly among populations from similar habitats. These patterns reflect the role of natural selection in driving parallel adaptive changes when independent populations colonize similar habitats. Importantly, however, substantial nonparallelism was also evident. Our results show that analyses based on exchangeability can confirm inferences based on statistical analyses of means or frequencies, while also refining insights into the drivers of—and constraints on—evolutionary diversification.