Using path analysis to measure natural selection

We expand current methods for calculating selection coefficients using path analysis and demonstrate how to analyse nonlinear selection. While this incorporation is a straightforward extension of current procedures, the rules for combining these traits to calculate selection coefficients can be complex. We demonstrate our method with an analysis of selection in an experimental population of Arabidopsis thaliana consisting of 289 individuals. Multiple regression analyses found positive directional selection and positive nonlinear selection only for inflorescence height. In contrast, the path analyses also revealed positive directional selection for number of rosette leaves and positive nonlinear selection for leaf number and time of inflorescence initiation. These changes in conclusions came about because indirect selection was converted into direct selection with the change in causal structure. Path analysis has great promise for improving our understanding of natural selection but must be used with caution since coefficient estimates depend on the assumed causal structure.     

Life-history evolution in Australian snakes: a path analysis

I recently attempted to investigate interspecific patterns in ecological traits of Australian snakes using univariate statistical techniques (Shine 1994), but high intercorrelations among variables (especially with mean adult body size) made it difficult to interpret the observed patterns. In the present paper, I attempt to tease apart causal factors using multivariate (path) analysis on the same data set (103 species, based on dissection of >22000 museum specimens). Two separate path analyses were conducted: one that treated each species as an independent unit (and thus, ignored phylogeny) and the other based on independent phylogenetic contrasts. Path coefficients from the two types of analyses were similar in magnitude, and highly correlated with each other, suggesting that most interspecific patterns among traits may reflect functional association rather than phylogenetic conservatism. Path analysis showed that indirect effects of one variable upon another (i.e., mediated via other traits) were often stronger than direct effects. Thus, even when two variables appeared to be uncorrelated in the univariate analysis, this apparent lack of relationship sometimes masked strong but conflicting indirect effects. For example, a tradeoff between clutch size and offspring size tends to mask the direct effect of mean adult body size on clutch size. Path analysis may also suggest original causal hypotheses. For example, interspecific allometry of sexual size dimorphism (as seen in Australian snakes, and many other animal groups) may result from a strong effect of another allometrically-tied trait (offspring size) on growth trajectories of females.     

Reducing environmental bias when measuring natural selection

Abstract.— Crucial to understanding the process of natural selection is characterizing phenotypic selection. Measures of phenotypic selection can be biased by environmental variation among individuals that causes a spurious correlation between a trait and fitness. One solution is analyzing genotypic data, rather than phenotypic data. Genotypic data, however, are difficult to gather, can be gathered from few species, and typically have low statistical power. Environmental correlations may act through traits other than through fitness itself. A path analytic framework, which includes measures of such traits, may reduce environmental bias in estimates of selection coefficients. We tested the efficacy of path analysis to reduce bias by re-analyzing three experiments where both phenotypic and genotypic data were available. All three consisted of plant species (Impatiens capensis, Arabidopsis thaliana , and Raphanus sativus) grown in experimental plots or the greenhouse. We found that selection coefficients estimated by path analysis using phenotypic data were highly correlated with those based on genotypic data with little systematic bias in estimating the strength of selection. Although not a panacea, using path analysis can substantially reduce environmental biases in estimates of selection coefficients. Such confidence in phenotypic selection estimates is critical for progress in the study of natural selection.     

Association analysis of some drought related characters in hexaploid spring wheat (Triticum aestivum L.).

The association among yield components and their direct and indirect influence on the grain yield of wheat were investigated. 24 breeding lines were tested in a randomized complete block experiment design with three replications. According to the results the phenotypic correlation among the traits and their path coefficient were estimated. Positive significant correlation coefficients were obtained for association between survival rate treatment I, III, leaf venation, stomatal frequency, osmotic pressure, flag leaf area and number of tillers per plant with grain yield per plant at both phenotypic and genotypic levels. Negatively significant correlation between hygrophilic colloids and epidermal cell size with grain yield per plant was obtained at phenotypic and genotypic levels. Path coefficient was also computed to estimate the contribution of character to the yield. Path coefficient analysis revealed that flag leaf area, root/shoot ratio and survival rate II had the highest positive direct effects on grain yield, while hygrophilic colloids and osmotic pressure had negative direct effect on grain yield. These results thus obtained suggested that flag leaf area is an important component of yield and hence needs special attention in selection strategies.     

Quantifying the causal pathways contributing to natural selection

The consequences of natural selection can be understood from a purely statistical perspective. In contrast, an explicitly causal approach is required to understand why trait values covary with fitness. In particular, key evolutionary constructs, such as sexual selection, fecundity selection, and so on, are best understood as selection via particular fitness components. To formalize and operationalize these concepts, we must disentangle the various causal pathways contributing to selection. Such decompositions are currently only known for linear models, where they are sometimes referred to as “Wright's rules.” Here, we provide a general framework, based on path analysis, for partitioning selection among its contributing causal pathways. We show how the extended selection gradient—which represents selection arising from a trait's causal effects on fitness—can be decomposed into path-specific selection gradients, which correspond to distinct causal mechanisms of selection. This framework allows for nonlinear effects and nonadditive interactions among variables, which may be estimated using standard statistical methods (e.g., generalized linear [mixed] models or generalized additive models). We thus provide a generalization of Wright's path rules that accommodates the nonlinear and nonadditive mechanisms by which natural selection commonly arises.     

Evolution in heterogeneous environments and the potential of maintenance of genetic variation in traits of adaptive significance

The maintenance of genetic variation in traits of adaptive significance has been a major dilemma of evolutionary biology. Considering the pattern of increased genetic variation associated with environmental clines and heterogeneous environments, selection in heterogeneous environments has been proposed to facilitate the maintenance of genetic variation. Some models examining whether genetic variation can be maintained, in heterogeneous environments are reviewed. Genetic mechanisms that constrain evolution in quantitative genetic traits indicate that genetic variation can be maintained but when is not clear. Furthermore, no comprehensive models have been developed, likely due to the genetic and environmental complexity of this issue. Therefore, I have suggested two empirical approaches to provide insight for future theoretical and empirical research. Traditional path analysis has been a very powerful approach for understanding phenotypic selection. However, it requires substantial information on the biology of the study system to construct a causal model and alternatives. Exploratory path analysis is a data driven approach that uses the statistical relationships in the data to construct a set of models. For example, it can be used for understanding phenotypic selection in different environments, where there is no prior information to develop path models in the different environments. Data from Brassica rapa grown in different nutrients indicated that selection changed in the different environments. Experimental evolutionary studies will provide direct tests as to when genetic variation is maintained.     

Hamilton and Zuk meet heterozygosity? Song repertoire size indicates inbreeding and immunity in song sparrows (Melospiza melodia)

Hamilton and Zuk's influential hypothesis of parasite-mediated sexual selection proposes that exaggerated secondary sexual ornaments indicate a male's addictive genetic immunity to parasites. However, genetic correlated of ornaments and immunity have rarely been explicitly identified. Evidence supporting Hamilton and Zuk's hypothesis has instead been gathered by looking for positive phenotypic correlations between ornamentation and immunity; such correlations are assumed to reflect causal, addictive relationships between these traits. We show that in a song sparrows, Melospiza melodia, male's song repertoire size, a secondary sexual trait, increased with his cell-mediated immune response (CMI) to an experimental challenge. However, this phenotypic correlation could be explained because both repertoire size and CMI declined with a male's inbreeding level. Repertoire size therefore primarily indicated a male's relative heterozygosity, a non-addictive genetic predictor of immunity. Caution may therefore be required when interpreting phenotypic correlations as support for Hamilton and Zuk's addictive model of sexual selection. However, our results suggest that female song sparrows choosing with large repertoires would on average acquire more outbred and therefore more heterozygous mates. Such genetic dominance effects on ornamentation are likely to influence evolutionary trajectories of female choice, and should be explicitly incorporated into genetic models of sexual selection.     

High-Altitude Adaptation: Mechanistic Insights from Integrated Genomics and Physiology

Population genomic analyses of high-altitude humans and other vertebrates have identified numerous candidate genes for hypoxia adaptation, and the physiological pathways implicated by such analyses suggest testable hypotheses about underlying mechanisms. Studies of highland natives that integrate genomic data with experimental measures of physiological performance capacities and subordinate traits are revealing associations between genotypes (e.g., hypoxia-inducible factor gene variants) and hypoxia-responsive phenotypes. The subsequent search for causal mechanisms is complicated by the fact that observed genotypic associations with hypoxia-induced phenotypes may reflect second-order consequences of selection-mediated changes in other (unmeasured) traits that are coupled with the focal trait via feedback regulation. Manipulative experiments to decipher circuits of feedback control and patterns of phenotypic integration can help identify causal relationships that underlie observed genotype–phenotype associations. Such experiments are critical for correct inferences about phenotypic targets of selection and mechanisms of adaptation.     

How to quantify (the response to) sexual selection on traits

Natural selection operates via fitness components like mating success, fecundity, and longevity, which can be understood as intermediaries in the causal process linking traits to fitness. In particular, sexual selection occurs when traits influence mating or fertilization success, which, in turn, influences fitness. We show how to quantify both these steps in a single path analysis, leading to better estimates of the strength of sexual selection. Our model controls for confounding variables, such as body size or condition, when estimating the relationship between mating and reproductive success. Correspondingly, we define the Bateman gradient and the Jones index using partial rather than simple regressions, which better captures how they are commonly interpreted. The model can be applied both to purely phenotypic data and to quantitative genetic parameters estimated using information on relatedness. The phenotypic approach breaks down selection differentials into a sexually selected and a “remainder” component. The quantitative genetic approach decomposes the estimated evolutionary response to selection analogously. We apply our method to analyze sexual selection in male dusky pipefish, Syngnathus floridae, and in two simulated datasets. We highlight conceptual and statistical limitations of previous path‐based approaches, which can lead to substantial misestimation of sexual selection.     

Accounting for the causal basis of collinearity when measuring the effects of habitat loss versus habitat fragmentation

Collinearity among metrics of habitat loss and habitat fragmentation is typically treated as a nuisance in landscape ecology, and it is the norm to use statistical approaches that remove collinear information prior to estimating model parameters. However, collinearity may arise from causal relationships among landscape metrics and may therefore signal the occurrence of indirect effects (where one model predictor influences the response variable by driving changes in another influential predictor). Here we suggest that, far from being merely a statistical nuisance, collinearity may be crucial for accurately quantifying the effects of habitat loss versus habitat fragmentation. We use simulation modelling to create datasets of collinear landscape metrics in which collinearity arose from causal relationships, then test the ability of two statistical approaches to estimate the effects of these metrics on a simulated response variable: 1) multiple regression, which statistically removes collinearity, and was identified in a recent study as the best approach for estimating the effects of collinear landscape metrics (although this study did not account for any indirect effects implied by collinearity among metrics); and 2) path analysis, which accounts for the causal basis of collinearity. In agreement with this previous study, we found that multiple regression gave unbiased estimates of direct effects (effects not mediated by other model predictors). However, it gave biased estimates of total (direct + indirect) effects when indirect effects occurred. In contrast, path analysis reliably identified the causal basis of collinearity and gave unbiased estimates of direct, indirect, and total effects. We suggest that effective research on the impacts of habitat loss versus fragmentation will often require tools that can empirically test whether collinear landscape metrics are causally related, and if so, account for the indirect effects that these causal relationships imply. Path analysis, but not multiple regression, provides such a tool.     

Path analysis of the genetic integration of traits in the sand cricket: a novel use of BLUPs

This study combines path analysis with quantitative genetics to analyse a key life history trade-off in the cricket, Gryllus firmus. We develop a path model connecting five traits associated with the trade-off between flight capability and reproduction and test this model using phenotypic data and estimates of breeding values (best linear unbiased predictors) from a half-sibling experiment. Strong support by both types of data validates our causal model and indicates concordance between the phenotypic and genetic expression of the trade-off. Comparisons of the trade-off between sexes and wing morphs reveal that these discrete phenotypes are not genetically independent and that the evolutionary trajectories of the two wing morphs are more tightly constrained to covary than those of the two sexes. Our results illustrate the benefits of combining a quantitative genetic analysis, which examines statistical correlations between traits, with a path model that focuses upon the causal components of variation.     

矮生菜豆豆荚产量构成因素的通径分析

以8个矮生菜豆品种为材料,研究了单株豆荚产量与其主要构成因素的关系,结果表明,单株豆荚产量及作为豆荚产量构成因素的主茎高、分枝数、花序数、豆荚数、豆荚长和豆荚重在品种间均存在极显著的差异;但这些产量构成因素与单株豆荚产量之间,无论是表型相关系数还是遗传相关系数或是环境相关系数均未达到显著水平。通径分析结果显示,一个与单株豆荚产量关系密切的性状。其对单株豆荚产量的效应总是由于存在一个或多个负向的间接通径系数而被削弱。从而掩盖了该性状对单株豆荚产量的遗传效应,从表型和遗传通径系数看,对单株产量最为重要的是单株结荚数及单株花序数。单株分枝数、主茎高和豆荚重其次。豆荚长则较为次要。根据通径分析结果,就矮生菜豆丰产性育种中各有关性状的选择进行了讨论。     

Components of Resistance to Bacterial Blight Disease of Rice

Studies on the genetic variation, correlation, correlated response and path analysis were conducted on 8 rice cultivars to bring out the association and channelling of the pathway of different components of resistance to Xanthomonas campestris pv. oryzae. High genotypic coefficient of variation coupled with high heritability and genetic gain was observed for lesion size (LS) and the area under disease progress curve (AUDPC) indicating the predominance of additive gene effects. There was a strong association among all the components both at genotypic and phenotypic levels. Genotypic correlations were higher than the corresponding phenotypic correlations indicating the modifying effect of environment on association of components at genotypic level. Maximum correlated response and relative selection efficiency on AUDPC was observed through indirect selection for LS followed by the number of bacteria per unit leaf area (NB). Path analysis revealed highest direct effect of LS on AUDPCboth at genotypic and phenotypic levels. Indirect effects of fairly high magnitude were also exerted by incubation period (ICP) and NB towards AUDPC.     

Path analysis in genetic epidemiology: a critique

Path analysis, a form of general linear structural equation models, is used in studies of human genetics data to discern genetic, environmental, and cultural factors contributing to familial resemblance. It postulates a set of linear and additive parametric relationships between phenotypes and genetic and cultural variables and then essentially uses the assumption of multivariate normality to estimate and perform tests of hypothesis on parameters. Such an approach has been advocated for the analysis of genetic epidemiological data by D. C. Rao, N. Morton, C. R. Cloninger, L. J. Eaves, and W. E. Nance, among others. This paper reviews and evaluates the formulations, assumptions, methodological procedures, interpretations, and applications of path analysis. To give perspective, we begin with a discussion of path analysis as it occurs in the form of general linear causal models in several disciplines of the social sciences. Several specific path analysis models applied to lipoprotein concentrations, IQ, and twin data are then reviewed to keep the presentation self-contained. The bulk of the critical discussion that follows is directed toward the following four facets of path analysis: (1) coherence of model specification and applicability to data; (2) plausibility of modeling assumptions; (3) interpretability and utility of the model; and (4) validity of statistical and computational procedures. In the concluding section, a brief discussion of the problem of appropriate model selection is presented, followed by a number of suggestions of essentially model-free alternative methods of use in the treatment of complex structured data such as occurs in genetic epidemiology.     

Artificial pheromone for path selection by a foraging swarm of robots

Foraging robots involved in a search and retrieval task may create paths to navigate faster in their environment. In this context, a swarm of robots that has found several resources and created different paths may benefit strongly from path selection. Path selection enhances the foraging behavior by allowing the swarm to focus on the most profitable resource with the possibility for unused robots to stop participating in the path maintenance and to switch to another task. In order to achieve path selection, we implement virtual ants that lay artificial pheromone inside a network of robots. Virtual ants are local messages transmitted by robots; they travel along chains of robots and deposit artificial pheromone on the robots that are literally forming the chain and indicating the path. The concentration of artificial pheromone on the robots allows them to decide whether they are part of a selected path. We parameterize the mechanism with a mathematical model and provide an experimental validation using a swarm of 20 real robots. We show that our mechanism favors the selection of the closest resource is able to select a new path if a selected resource becomes unavailable and selects a newly detected and better resource when possible. As robots use very simple messages and behaviors, the system would be particularly well suited for swarms of microrobots with minimal abilities.     

A PATH-ANALYTIC MODEL FOR THE MEASUREMENT OF SELECTION ON MORPHOLOGY

This paper describes a path model for the analysis of phenotypic selection upon continuous morphological characters. The path-analysis model assumes that selection occurs on unmeasured general size and shape allometry factors that summarize linear relations among sets of ontogenetically, phylogenetically, or functionally related traits. An unmeasured factor for general size is considered the only aspect of morphometric covariance matrices for which there is an a priori biological explanation. Consequently, selection coefficients are derived for each measured character by holding constant only a general size factor, rather than by using multiple regression to adjust for the full covariance matrix. Fitness is treated as an unmeasured factor with loadings, representing directional selection coefficients, computed as the covariances of the size-adjusted characters with the measured fitness indicator. The magnitudes and signs of the selection coefficients, combined with biological insight, may suggest hypotheses of selection on one or more shape allometry factors. Hypotheses of selection on general size and shape allometry factors are evaluated through cycles of measurement, analysis, and experimentation, designed to refine the path diagram depicting the covariances among the measured characters, the measured indicator of fitness, and unmeasured factors for morphology and fitness. The path-analysis and multiple-regression models were applied to data from remeasurement of Lande and Arnold's (1983) pentatomid bugs and to Bumpus's (1899) data on house sparrows. The path analysis suggested the hypothesis that variation in bug survivorship was an expression of directional selection on wing loading. Bumpus's data are consistent with a hypothesis of stabilizing selection on general size in females and directional selection for small wing size relative to body size in males.     

MEASURING NATURAL SELECTION ON PHENOTYPIC PLASTICITY

To understand natural selection we need to integrate its measure across environments. We present a method for measuring phenotypic selection that combines the potential for both environmental variation and phenotypic plasticity. The method uses path analysis and a measure of selection that is analogous to selection on breeding values. For individuals growing in alternative environments, paths are created that represent potential changes in the environment. The probabilities for these changes are then multiplied by the path coefficients to calculate selection coefficients. Selection on plasticity is measured as the difference in selection within each environment. We illustrate these methods using data on selection in an experimental population of Arabidopsis thaliana. Individuals from 36 families were grown in one of four environments, a factorial combination of shaded/open and early/late shading. For final height of the inflorescence, there was positive selection in both the open and shaded environments and negative selection on plasticity of height. For bolting time, there was also positive selection in both environments, but no selection on plasticity. We show how to use this information to examine how selection would change with changes in environmental frequencies and their transition probabilities. These methods can be expanded to encompass continuous traits and continuous environments as well as other complexities of natural selection.     

Statistical Genetics of an Annual Plant, Impatiens Capensis. II. Natural Selection

Measurement of natural selection on correlated characters provides valuable information on fitness surfaces, patterns of directional, stabilizing, or disruptive selection, mechanisms of fitness variation operating in nature, and possible spatial variation in selective pressures. We examined effects of seed weight, germination date, plant size, early growth, and late growth on individual fitness. Path analysis showed that most characters had direct or indirect effects on individual fitness, indicating directional selection. For most early life-cycle characters, indirect effects via later characters exceed the direct causal effect on fitness. Selection gradients were uniform across the experimental site. There was no evidence for stabilizing or disruptive selection. We discuss several definitions of stabilizing and disruptive selection. Although early events in the life of an individual have important causal effects on subsequent characters and fitness, there is no detectable genetic variance for most of these characters, so little or no genetic response to natural selection is expected.