On generalized Simes critical constants

We consider the problem treated by Simes of testing the overall null hypothesis formed by the intersection of a set of elementary null hypotheses based on ordered p‐values of the associated test statistics. The Simes test uses critical constants that do not need tabulation. Cai and Sarkar gave a method to compute generalized Simes critical constants which improve upon the power of the Simes test when more than a few hypotheses are false. The Simes constants can be viewed as the first order (requiring solution of a linear equation) and the Cai‐Sarkar constants as the second order (requiring solution of a quadratic equation) constants. We extend the method to third order (requiring solution of a cubic equation) constants, and also offer an extension to an arbitrary kth order. We show by simulation that the third order constants are more powerful than the second order constants for testing the overall null hypothesis in most cases. However, there are some drawbacks associated with these higher order constants especially for , which limits their practical usefulness.     

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.     

Trimmed Weighted Simes' Test for Two One‐Sided Hypotheses With Arbitrarily Correlated Test Statistics

The two‐sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one‐sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non‐negative. In this article, we introduce a trimmed version of the one‐sided weighted Simes test for two hypotheses which rejects if (i) the one‐sided weighted Simes test rejects and (ii) both p‐values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level α if (i) the common distribution of test statistics is point symmetric and (ii) the two‐sided weighted Simes test at level 2α controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two‐sided p‐values of one‐ or two‐sample t‐tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one‐sided p‐values if the correlation is non‐negative. The Simes test for such types of bivariate t‐tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one‐sided bivariate t‐test with arbitrary correlation.     

A coherent approach for analysis of the Illumina HumanMethylation450 BeadChip improves data quality and performance in epigenome-wide association studies

DNA methylation plays a fundamental role in the regulation of the genome, but the optimal strategy for analysis of genome-wide DNA methylation data remains to be determined. We developed a comprehensive analysis pipeline for epigenome-wide association studies (EWAS) using the Illumina Infinium HumanMethylation450 BeadChip, based on 2,687 individuals, with 36 samples measured in duplicate. We propose new approaches to quality control, data normalisation and batch correction through control-probe adjustment and establish a null hypothesis for EWAS using permutation testing. Our analysis pipeline outperforms existing approaches, enabling accurate identification of methylation quantitative trait loci for hypothesis driven follow-up experiments.

Electronic supplementary material

The online version of this article (doi:10.1186/s13059-015-0600-x) contains supplementary material, which is available to authorized users.     

Applications of the Wei-Lachin Multivariate One-Sided Test for Multiple Outcomes on Possibly Different Scales

Many studies aim to assess whether a therapy has a beneficial effect on multiple outcomes simultaneously relative to a control. Often the joint null hypothesis of no difference for the set of outcomes is tested using separate tests with a correction for multiple tests, or using a multivariate T ²-like MANOVA or global test. However, a more powerful test in this case is a multivariate one-sided or one-directional test directed at detecting a simultaneous beneficial treatment effect on each outcome, though not necessarily of the same magnitude. The Wei-Lachin test is a simple 1 df test obtained from a simple sum of the component statistics that was originally described in the context of a multivariate rank analysis. Under mild conditions this test provides a maximin efficient test of the null hypothesis of no difference between treatment groups for all outcomes versus the alternative hypothesis that the experimental treatment is better than control for some or all of the component outcomes, and not worse for any. Herein applications are described to a simultaneous test for multiple differences in means, proportions or life-times, and combinations thereof, all on potentially different scales. The evaluation of sample size and power for such analyses is also described. For a test of means of two outcomes with a common unit variance and correlation 0.5, the sample size needed to provide 90% power for two separate one-sided tests at the 0.025 level is 64% greater than that needed for the single Wei-Lachin multivariate one-directional test at the 0.05 level. Thus, a Wei-Lachin test with these operating characteristics is 39% more efficient than two separate tests. Likewise, compared to a T ²-like omnibus test on 2 df, the Wei-Lachin test is 32% more efficient. An example is provided in which the Wei-Lachin test of multiple components has superior power to a test of a composite outcome.     

Truncated Simes tests

It is known that the one‐sided Simes’ test controls the error rate if the underlying distribution is multivariate totally positive of order 2 (MTP2), but not in general. The two‐sided test also controls the error rate when the coordinate absolute value has an MTP2 distribution, which holds more generally. We prove mathematically that when the coordinate absolute value controls the error rate at level 2α, then certain kinds of truncated Simes’ tests also control the one‐sided error rate at level α. We also compare the closure of the truncated tests with the Holms, Hochberg, and Hommel procedures in many scenarios when the test statistics are multivariate normal.     

The relationship between functional dispersion of mixed‐species leaf litter mixtures and species' interactions during decomposition

We tested the hypothesis that interactions between plant species during the process of mixed‐species leaf litter decomposition increases with increasing functional diversity of leaves within the mixtures; specifically, there is a positive correlation between functional dispersion and the deviations from Grime's biomass‐ratio hypothesis, with a null intercept. We measured decomposition rates (mg g^?1 d^?1) of mixed‐species leaf litter from two experimental designs: 1) a microcosm experiment with litterbags of species mixtures combining six tree species, alone and in 42 combinations, and 2) an in situ litterbag experiment with all possible mixture combinations of four herb species (from one to four species). Interaction strengths and directions were measured as deviations from community‐weighted means (CWM) of monoculture decomposition values, following the biomass‐ratio hypothesis (BRH). Functional diversity was measured as Laliberté and Legendre's functional dispersion (FDis), using leaf dry matter content (LDMC), leaf nitrogen and carbon contents, and proportions of water soluble compounds, cellulose, hemicellulose and lignin. Correlations between FDis and deviations from BRH varied strongly, depending upon the combination of functional traits, the plant type or the environmental conditions, and the way in which prediction error was expressed (absolute or actual deviation). For tree species, FDis that was based on a combination of water soluble compounds, hemicellulose concentration, and LDMC was negatively correlated with interaction strength but positively with its absolute value. For herbs, interaction strength (absolute or actual) decreased as FDis of the mixtures increased, based on cellulose and lignin contents. There was no positive correlation between functional dispersion and the deviations from Grime's biomass‐ratio hypothesis, with a null intercept. Despite a relationship between litter interactions and functional divergence, this relationship was not generalisable. Other functional traits that were missing in our study might have played an important role.     

Duffy blood group and hemoglobin variants

Summary Data on a sample of 809 Afro-Americans indicated that there is no association between Duffy null (a^-, b^-) blood type and sickle cell trait. The results further rule out close linkage as an alternative hypothesis to explain the reported association between these loci in areas where falciparum and vivax malaria are endemic and indicate that, even if the two loci are independent or loosely linked, direct evidence of the selection favoring AS Fy^-Fy^- individuals must come from populations where mixed malaria infections occur. Stratification, as an explanation for the reported association, is also discussed.     

A powerful method for pleiotropic analysis under composite null hypothesis identifies novel shared loci between Type 2 Diabetes and Prostate Cancer

There is increasing evidence that pleiotropy, the association of multiple traits with the same genetic variants/loci, is a very common phenomenon. Cross-phenotype association tests are often used to jointly analyze multiple traits from a genome-wide association study (GWAS). The underlying methods, however, are often designed to test the global null hypothesis that there is no association of a genetic variant with any of the traits, the rejection of which does not implicate pleiotropy. In this article, we propose a new statistical approach, PLACO, for specifically detecting pleiotropic loci between two traits by considering an underlying composite null hypothesis that a variant is associated with none or only one of the traits. We propose testing the null hypothesis based on the product of the Z-statistics of the genetic variants across two studies and derive a null distribution of the test statistic in the form of a mixture distribution that allows for fractions of variants to be associated with none or only one of the traits. We borrow approaches from the statistical literature on mediation analysis that allow asymptotic approximation of the null distribution avoiding estimation of nuisance parameters related to mixture proportions and variance components. Simulation studies demonstrate that the proposed method can maintain type I error and can achieve major power gain over alternative simpler methods that are typically used for testing pleiotropy. PLACO allows correlation in summary statistics between studies that may arise due to sharing of controls between disease traits. Application of PLACO to publicly available summary data from two large case-control GWAS of Type 2 Diabetes and of Prostate Cancer implicated a number of novel shared genetic regions: 3q23 (ZBTB38), 6q25.3 (RGS17), 9p22.1 (HAUS6), 9p13.3 (UBAP2), 11p11.2 (RAPSN), 14q12 (AKAP6), 15q15 (KNL1) and 18q23 (ZNF236).     

Simultaneous Statistical Inference for Epigenetic Data

Epigenetic research leads to complex data structures. Since parametric model assumptions for the distribution of epigenetic data are hard to verify we introduce in the present work a nonparametric statistical framework for two-group comparisons. Furthermore, epigenetic analyses are often performed at various genetic loci simultaneously. Hence, in order to be able to draw valid conclusions for specific loci, an appropriate multiple testing correction is necessary. Finally, with technologies available for the simultaneous assessment of many interrelated biological parameters (such as gene arrays), statistical approaches also need to deal with a possibly unknown dependency structure in the data. Our statistical approach to the nonparametric comparison of two samples with independent multivariate observables is based on recently developed multivariate multiple permutation tests. We adapt their theory in order to cope with families of hypotheses regarding relative effects. Our results indicate that the multivariate multiple permutation test keeps the pre-assigned type I error level for the global null hypothesis. In combination with the closure principle, the family-wise error rate for the simultaneous test of the corresponding locus/parameter-specific null hypotheses can be controlled. In applications we demonstrate that group differences in epigenetic data can be detected reliably with our methodology.     

CROSS-MODAL ADDITIVITY OF TASTE AND SMELL

Subjects were simultaneously given subthreshold levels of taste and odor stimuli, delivered orally, for both a commonly paired and an uncommonly paired taste–odor combination. Results indicate cross‐modal summation of subthreshold concentrations of both taste–odor pairs when the olfactory stimulus is delivered orally. Results of control studies suggest that the summation was indeed across modalities, and not due to the taste of the odor compound or the smell of the taste compounds. Furthermore, results indicate that regardless of taste–odor pair commonness, taste and smell can combine in a completely additive fashion (i.e., at threshold detectability when both stimuli are presented simultaneously at 50% threshold level) if the taste–odor pair is presented orally. In several instances, but not all, measured probabilities exceeded those predicted by probability summation, indicating that hyperadditive mixing often occurs, but there do seem to be individual differences. Cross‐modal summation, regardless of taste–odor pair commonness, has broader implications for the development of foods, beverages and pharmaceuticals, especially in masking undesirable tastes and smells.     

Testing for Treatment Effects on Subsets of Endpoints

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p‐value adjustment methods, such as single‐step, step‐up, or step‐down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step‐by‐step closed strategy. This paper proposes a global‐based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single‐step p‐value‐based statistic based on the distribution of the smallest p‐value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.     

Can vector summation describe the orientation system of juvenile ospreys and honey buzzards? – An analysis of ring recoveries and satellite tracking

Juvenile bird migrants are generally believed to use a clock‐and‐compass migratory orientation strategy. According to such a strategy migrants accomplish their migration by flying a number of successive flight steps with direction and number of steps controlled by an endogenous programme. One powerful way of testing this is by comparing predictions from a model of such a strategy with observed patterns. We used data from ringing and satellite‐based radio telemetry to investigate the orientation system of juvenile ospreys (Pandion haliaetus) and honey buzzards (Pernis apivorus) migrating from Sweden to tropical west Africa. The ring recoveries showed a much larger scatter in the orientation of ospreys than of honey buzzards, but there was only a slight such difference in the satellite tracks. These tracks of individuals of both species were rather straight with a high directional concentration per step. The honey buzzard data showed a close fit to a simple vector summation model, which is expected if birds follow a clock‐and‐compass strategy. However, the osprey data did not fit such a simple model, as ring recoveries showed a significantly greater deviation at short distances than predicted on the basis of long distance data. Satellite tracking also indicated less concentrated orientation on short distances. The pattern observed for the osprey can generally be explained by an extended vector summation model, including an important element of pre‐migration dispersal. The existence of extensive dispersal in the osprey stands in contrast to the apparent absence of such dispersal in the honey buzzard. The explanation for this difference between the species is unclear. The model of orientation by vector summation is very sensitive to the existence of differences in mean direction between individuals. Assuming such differences, as tentatively indicated by the satellite tracking data, makes simple compass orientation by vector summation inconsistent with the distribution of ring recoveries at long distances, with a high proportion of misoriented birds falling outside the normal winter range.     

Testing biogeographical hypotheses in space and time: faunal relationships of the putative ancient Lake Egˇirdir in Asia Minor

Aim The aims of this study are to establish a multi‐locus phylogeny‐based hypothesis for the biogeographical relationship of gastropods from the putative ancient Lake Egˇirdir, to test the respective null hypothesis, to estimate the timing of biogeographical events based on independent molecular clock approaches, and to interpret the data with respect to the putative ancient character of Lake Egˇirdir. Location Lake Egˇirdir, western Taurus Lake District, Turkey. Methods DNA sequences from the putatively only extant endemic taxon of Lake Egˇirdir, Falsipyrgula pfeiferi, as well as representatives of other pyrgulinid genera from Europe and western Asia are used for phylogenetic analyses based on Bayesian inference. The respective null hypothesis is tested utilizing parametric bootstrapping. The timing of evolutionary events is estimated based on two independent molecular clock approaches, which involve the modelling of judicious errors associated with branch‐length calculations and calibration points. Results Bayesian inference indicates a very close relationship between the Lake Egˇirdir and Ponto‐Caspian taxa. Parametric bootstrapping rejects the null hypothesis that these taxa are not monophyletic (P ≤ 0.01). The alternative hypothesis, namely monophyly of the Ponto‐Caspian and Egˇirdir taxa, can therefore be accepted. The two independent molecular clock approaches show diversion times for the Ponto‐Caspian/Egˇirdir taxa of 0.42 ± 0.18 and 0.43 ± 0.63 Ma. Main conclusions The present study shows that there is no close biogeographical affiliation between the probably only remaining endemic taxon of Lake Egˇirdir and taxa from central Europe or the Balkan region. Instead, there is a very close and relatively young (i.e. late Pleistocene) biogeographical relationship with the Ponto‐Caspian pyrgulinids. However, fossil and comparative data from other invertebrates indicate that biogeographical connections between Lake Egˇirdir and the Ponto‐Caspian region existed during various time periods, i.e. the Miocene/Pliocene, early Pleistocene, and late Pleistocene. Acknowledging the still‐restricted knowledge of the evolutionary history of the lake, the data presented here do not reject the putative ancient status of Lake Egˇirdir. Future studies utilizing endemic taxa of other lakes in the region need to show whether the western Taurus Lake District represents a melting pot of Pleistocene refuge biodiversity from different regions, and whether the admixture of divergent lineages has created a genetically distinct set of taxa that would warrant the designation of the area as a unique biogeographical subregion.     

Time scale matters: genetic analysis does not support adaptation‐by‐time as the mechanism for adaptive seasonal declines in kokanee reproductive life span

Seasonal declines of fitness‐related traits are often attributed to environmental effects or individual‐level decisions about reproductive timing and effort, but genetic variation may also play a role. In populations of Pacific salmon (Oncorhynchus spp.), seasonal declines in reproductive life span have been attributed to adaptation‐by‐time, in which divergent selection for different traits occurs among reproductively isolated temporal components of a population. We evaluated this hypothesis in kokanee (freshwater obligate Oncorhynchus nerka) by testing for temporal genetic structure in neutral and circadian‐linked loci. We detected no genetic differences in presumably neutral loci among kokanee with different arrival and maturation dates within a spawning season. Similarly, we detected no temporal genetic structure in OtsClock1b, Omy1009uw, or OmyFbxw11, candidate loci associated with circadian function. The genetic evidence from this study and others indicates a lack of support for adaptation‐by‐time as an important evolutionary mechanism underlying seasonal declines in reproductive life span and a need for greater consideration of other mechanisms such as time‐dependent, adaptive adjustment of reproductive effort.     

Comparative 3D quantitative analyses of trapeziometacarpal joint surface curvatures among living catarrhines and fossil hominins

Comparisons of joint surface curvature at the base of the thumb have long been made to discern differences among living and fossil primates in functional capabilities of the hand. However, the complex shape of this joint makes it difficult to quantify differences among taxa. The purpose of this study is to determine whether significant differences in curvature exist among selected catarrhine genera and to compare these genera with hominin¹ fossils in trapeziometacarpal curvature. Two 3D approaches are used to quantify curvatures of the trapezial and metacarpal joint surfaces: (1) stereophotogrammetry with nonuniform rational B‐spline (NURBS) calculation of joint curvature to compare modern humans with captive chimpanzees and (2) laser scanning with a quadric‐based calculation of curvature to compare modern humans and wild‐caught Pan, Gorilla, Pongo, and Papio. Both approaches show that Homo has significantly lower curvature of the joint surfaces than does Pan. The second approach shows that Gorilla has significantly more curvature than modern humans, while Pongo overlaps with humans and African apes. The surfaces in Papio are more cylindrical and flatter than in Homo. Australopithecus afarensis resembles African apes more than modern humans in curvatures, whereas the Homo habilis trapezial metacarpal surface is flatter than in all genera except Papio. Neandertals fall at one end of the modern human range of variation, with smaller dorsovolar curvature. Modern human topography appears to be derived relative to great apes and Australopithecus and contributes to the distinctive human morphology that facilitates forceful precision and power gripping, fundamental to human manipulative activities. Am J Phys Anthropol, 2010. © 2009 Wiley‐Liss, Inc. ¹ The term “hominin” refers to members of the tribe Hominini, which includes modern humans and fossil species that are related more closely to modern humans than to extant species of chimpanzees, Wood and Lonergan (2008). Hominins are in the family Hominidae with great apes.     

Loss of crossbridge inhibition drives pathological cardiac hypertrophy in patients harboring the TPM1 E192K mutation

Hypertrophic cardiomyopathy (HCM) is an inherited disorder caused primarily by mutations to thick and thinfilament proteins. Although thin filament mutations are less prevalent than their oft-studied thick filament counterparts, they are frequently associated with severe patient phenotypes and can offer important insight into fundamental disease mechanisms. We have performed a detailed study of tropomyosin (TPM1) E192K, a variant of uncertain significance associated with HCM. Molecular dynamics revealed that E192K results in a more flexible TPM1 molecule, which could affect its ability to regulate crossbridges. In vitro motility assays of regulated actin filaments containing TPM1 E192K showed an overall loss of Ca²⁺ sensitivity. To understand these effects, we used multiscale computational models that suggested a subtle phenotype in which E192K leads to an inability to completely inhibit actin–myosin crossbridge activity at low Ca²⁺. To assess the physiological impact of the mutation, we generated patient-derived engineered heart tissues expressing E192K. These tissues showed disease features similar to those of the patients, including cellular hypertrophy, hypercontractility, and diastolic dysfunction. We hypothesized that excess residual crossbridge activity could be triggering cellular hypertrophy, even if the overall Ca²⁺ sensitivity was reduced by E192K. To test this hypothesis, the cardiac myosin–specific inhibitor mavacamten was applied to patient-derived engineered heart tissues for 4 d followed by 24 h of washout. Chronic mavacamten treatment abolished contractile differences between control and TPM1 E192K engineered heart tissues and reversed hypertrophy in cardiomyocytes. These results suggest that the TPM1 E192K mutation triggers cardiomyocyte hypertrophy by permitting excess residual crossbridge activity. These studies also provide direct evidence that myosin inhibition by mavacamten can counteract the hypertrophic effects of mutant tropomyosin.     

Detecting disease gene in DNA haplotype sequences by nonparametric dissimilarity test

Association studies for complex diseases based on haplotype data have received increasing attention in the last few years. A commonly used nonparametric method, which takes haplotype structure into consideration, is to use the U-statistic to compare the similarities between genetic compositions in the case and control populations. Although the method and its variants are convenient to use in practice, there are some areas where the tests cannot detect even large differences between cases and controls. To overcome this problem and enhance the power, we propose a new form of the weighted U-statistic, which directly compares the dissimilarity between the haplotype structures in the case and control populations. We show that this test statistic is asymptotically a linear combination of the absolute values of normal random variables under the null hypothesis, and shifts strictly toward the right under the alternative, and therefore has no blind areas of detection. Simulation studies indicate that our test statistic overcomes the weakness of the existing ones and is robust and powerful as well.     

Eight new microsatellite markers in Atlantic cod (Gadus morhua L.) derived from an enriched genomic library

One hundred and fifty random clones from an enriched genomic library of Atlantic cod were sequenced. Primer pairs were designed for 15 microsatellites containing perfect di‐, tri‐, tetra‐ and hexanucleotide repeats and characterized in 96 unrelated fish. Eight markers were successfully amplified, with the number of alleles ranging from two to nine per locus and observed heterozygosity ranging from 0.341 to 0.977. Loci Gmo‐G13 and Gmo‐G14 had a significant excess of homozygotes. All loci conformed to the Hardy–Weinberg equilibrium. Genetic linkage disequilibrium analysis between all pairs of the loci showed no significant departure from the null hypothesis between any of the loci.     

INTEGRATING LANDSCAPE GENOMICS AND SPATIALLY EXPLICIT APPROACHES TO DETECT LOCI UNDER SELECTION IN CLINAL POPULATIONS

Uncovering the genetic basis of adaptation hinges on the ability to detect loci under selection. However, population genomics outlier approaches to detect selected loci may be inappropriate for clinal populations or those with unclear population structure because they require that individuals be clustered into populations. An alternate approach, landscape genomics, uses individual‐based approaches to detect loci under selection and reveal potential environmental drivers of selection. We tested four landscape genomics methods on a simulated clinal population to determine their effectiveness at identifying a locus under varying selection strengths along an environmental gradient. We found all methods produced very low type I error rates across all selection strengths, but elevated type II error rates under “weak” selection. We then applied these methods to an AFLP genome scan of an alpine plant, Campanula barbata, and identified five highly supported candidate loci associated with precipitation variables. These loci also showed spatial autocorrelation and cline patterns indicative of selection along a precipitation gradient. Our results suggest that landscape genomics in combination with other spatial analyses provides a powerful approach for identifying loci potentially under selection and explaining spatially complex interactions between species and their environment.