Ruth G. Shaw

Regression analysis of natural selection statistical inference and biological interpretation

Recent theoretical work in quantitative genetics has fueled interest in measuring natural selection in the wild. We discuss statistical and biological issues that may arise in applications of Lande and Arnolds (1983) multiple‐regression approach to measuring selection. We review assumptions involved in estimation and hypothesis testing in regression problems, and we note difficulties that frequently arise as a result of violation of these assumptions. In particular, multicollinearity (extreme intercorrelation of characters) and extrinsic, unmeasured factors affecting fitness may seriously complicate inference regarding selection. Further, violation of the assumption that residuals are normally distributed vitiates tests of significance. For this situation, we suggest applications of recently developed jackknife tests of significance. While fitness regression permits direct assessment of selection in a form suitable for predicting selection response, we suggest that the aim of inferring causal relationships about the effects of phenotypic characters on fitness is greatly facilitated by manipulative experiments. Finally, we discuss alternative definitions of stabilizing and disruptive selection.

Anova for Unbalanced Data: An Overview

Ecological studies typically involve comparison of biological responses among a variety of environmental conditions. When the response variables have continuous distributions and the conditions are discrete, whether inherently or by design, then it is appropriate to analyze the data using analysis of variance (ANOVA). When data conform to a complete, balanced design (equal numbers of observations in each experimental treatment), it is straightforward to conduct an ANOVA, particularly with the aid of the numerous statistical computing packages that are available. Interpretation of an ANOVA of balanced data is also unambiguous. Unfortunately, for a variety of reasons, it is rare that a practicing ecologist embarks on an analysis of data that are completely balanced. Regardless of its cause, lack of balance necessitates care in the analysis and interpretation. In this paper, our aims is to provide an overview of the consequences of lack of balance and to give some guidelines to analyzing unbalanced data for models involving fixed effects. Our treatment is necessarily cursory and will not substitute for training available from a sequence of courses in mathematical statistics and linear models. It is intended to introduce the reader to the main issues and to the extensive statistical literature that deals with them.

MAXIMUM-LIKELIHOOD APPROACHES APPLIED TO QUANTITATIVE GENETICS OF NATURAL POPULATIONS

Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least‐squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F‐ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within‐generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative‐genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.

Locomotor performance of hatchling fence lizards sceloporus occidentalis quantitative genetics and morphometric correlates

SummaryWe examined heritabilities and correlations among measures of locomotor performance (speed, stamina) and among possible morphometric determinants of performance (hindlimb span, tail length) in families of hatchling lizards (Sceloporus occidentalis). We were particularly interested in determining whether these traits were heritable and thus might potentially respond genetically to selection. Moreover, we wished to determine whether speed and stamina are negatively genetically correlated, as suggested bya priori physiological and empirical considerations. All four traits appeared to be significantly heritable. Broadsense heritabilities were 0.33–0.36 for speed, 0.35–0.36 for stamina, 0.45–0.51 for hindlimb span, and 0.46–0.47 for tail length. Contrary to expectations, speed and stamina were not negatively genetically correlated. Hindlimb span and tail length, however, were negatively genetically correlated (but not phenotypically correlated). Hindlimb span and stamina were positively phenotypically correlated. Thus, for example, selection for longer hindlimb span could potentially result in shorter tails, contrary to evolutionary predictions based only on phenotypic correlations.

The comparison of quantitative genetic parameters between populations

A statistical method for comparing matrices of genetic variation and covariation between groups (e.g., species, populations, a single population grown in distinct environments) is proposed. This maximum‐likelihood method provides a test of the overall null hypothesis that two covariance component matrices are identical. Moreover, when the overall null hypothesis is rejected, the method provides a framework for isolating the particular components that differ significantly between the groups. Simulation studies reveal that discouragingly large experiments are necessary to obtain acceptable power for comparing genetic covariance component matrices. For example, even in cases of a single trait measured on 900 individuals in a nested design of 100 sires and three dams per sire in each population, the power was only about 0.5 when additive genetic variance differed by a factor of 2.5. Nevertheless, this flexible method makes valid comparison of covariance component matrices possible.

Response to density in a wild population of the perennial herb Salvia lyrata: variation among families.

In order to determine the potential for natural selection to promote genetic specialization to different environments, this study quantified genetic variation for response to conspecific density and to other aspects of the environment that vary spatially. The progeny of a random collection of Salvia lyrata, a perennial herb, were planted into the source field in a range of densities and into several closely neighboring locations. The highest density and particular locations induced significantly greater mortality relative to the remaining densities and locations, indicating major effects of density and spatial location on fitness. Over the duration of the entire study, there was also significant variation among families in mortality. The survival data gave no indication of variation among families in their responses to the range of environments. Conversely, results based on growth and size indicated that different families were favored in different densities and locations, in support of the hypothesis that environment‐dependent selection promotes specialization to different environments in this species. The correlation among families between leaf number in high and low density was small and positive, indicating near‐independence of performance in different densities.

Density-dependence in Salvia lyrata, a herbaceous perennial: the effects of experimental alteration of seed densities.

(1) A demographic study of seedling populations of the perennial herb, Salvia lyrata L., was conducted from June 1979 to May 1983. The number of seedlings establishing varied widely from year to year. The half-life of the seedling population was about 10 months and seedling mortality rate declined with time. Correlation of seedling mortality with population density was low. (2) Field experiments showed that established S. lyrata individuals significantly reduced the proportion of seedlings emerging and surviving. Seed density had no demonstrable effect on the proportion of seedlings emerging over a range of densities (0-1440 seeds dm-2). There was thus no evidence of regulation by limitation of safe sites in this species in nature. (3) Seedling mortality depended strongly on seed and seedling density. Fewer individuals survived in densities above 720 seeds dm-2 compared with densities of 240 and 480 seeds dm2. (4) Seedling growth was negatively density-dependent. Growth reduction, suffered at initially high densities, persisted after a large fraction of individuals had died. (5) Regulation of numbers of S. lyrata seedlings occurs primarily through the negative effects of adults. The densities of seedlings and year-old individuals are sufficient to reduce survival and growth only in infrequent clumps.

Environmental and genetic constraints on adaptive population differentiation in Anthoxanthum odoratum

Maximum‐likelihood estimates of environmental and broad sense genetic (co)variance components were obtained for the growth and reproductive output of clones of the grass Anthoxanthum odoratum. The clones were transplanted between a mesic and a xeric field site and across‐environment genetic correlations were used to estimate the strength of genotype‐environment interaction. Significant across‐environment clonal covariance matrices were found for several traits, including lifetime reproductive output in one population. None of the matrices differed significantly between populations. Significant within‐site clonal variation was found, but there was no significant across‐environment clonal covariation. Most broad sense heritability estimates of character states within sites were small (median = 0.12), suggesting that only a slow response to selection is possible. All significant within‐site clonal correlations between growth and reproductive output were positive, although the pattern of negative clonal correlations suggests that there may be a cost to first year reproduction, which might constrain future selection response.

SEED SET AND SEED MASS IN IPOMOPSIS AGGREGATA: VARIANCE PARTITIONING AND INFERENCES ABOUT POSTPOLLINATION SELECTION

Events that follow pollination, such as pollen‐tube growth and seed maturation, comprise an important phase of angiosperm reproduction. Differential success during this “postpollination” phase may represent phenotypic selection, including sexual selection, or interaction between parents caused, for example, by their genetic similarity. By providing a detailed partitioning of variance in success, diallel crossing designs offer great potential to determine which processes are occurring and their relative magnitudes. We performed three partial diallels with the montane herb Ipomopsis aggregata, using a large sample of parental plants (69 total). Embedded in the designs were crossing‐distance treatments of 1 m, 10 m, and 100 m, reflecting a range of parental genetic similarity. We partitioned phenotypic variance in seed set per fruit into six components using restricted maximum‐likelihood (REML) analysis. For one diallel, we also partitioned variance in seed mass into five components, and estimated two components of covariance between seed set and mass. Variance caused by maternal effects (Vmat) comprised 12%–35% of total variance in seed set and 62% of variance in seed mass, and there was a significant negative environmental covariance between seed set and seed mass. Parental interaction made no detectable contribution to phenotypic variance in either of our measures of postpollination success, although crossing distance did contribute slightly but significantly to fit of the model in some cases. Finally, there was no detectable paternal variance (Vpat) in seed set or seed mass. These results are in keeping with reports from other studies of natural plant populations. The finding of little or no paternal variance in particular suggests little scope for postpollination sexual selection through the male function of cosexual plants such as I. aggregata.

Density-dependence in Salvia lyrata: experimental alteration of densities of established plants

(1) Two experiments were conducted in a natural population of Salvia lyrata to investigate the effects of conspecific density on survival, fecundity and growth of established individuals. In both experiments, the density around monitored Salvia individuals was altered, either by removal of surrounding Salvia individuals within a specified distance, or by planting Salvia in as neighbours. (2) In both experiments, survival, fecundity and growth showed small responses to the density manipulations. The responses varied with individual size. In large individuals these traits tended to decrease with increasing density, whereas small plants were in several respects favoured by more crowded conditions. (3) The weakness of the responses to neighbours suggests that the population of mature Salvia is not strongly self-regulating at typical natural densities or that marked effects of crowding appear only after a delay of more than two years. Mortality at the seedling stage as well as density-independent determinants of growth appear to maintain the adult population below the threshold at which neighbour interactions have strong effects.