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Unifying Life History Analyses for Inference of Fitness and

Population Growth

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Ruth G. Shaw

Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul,

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Minnesota 55108

rshaw@superb.ecology.umn.edu

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Charles J. Geyer

School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455

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charlie@stat.umn.edu

Stuart Wagenius

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Institute for Plant Conservation Biology, Chicago Botanic Garden, Glencoe, Illinois 60022

swagenius@chicagobotanic.org

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Helen H. Hangelbroek

Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul,

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Minnesota 55108

helen hangelbroek@hotmail.com

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and

Julie R. Etterson

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Biology Department, University of Minnesota-Duluth, Duluth, Minnesota 55812

jetterso@d.umn.edu

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Received; accepted

Prepared with AASTEX— Type of submission: Article

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ABSTRACT

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Lifetime fitness of individuals is the basis for population dynamics, and vari-

ation in fitness results in evolutionary change. Though the dual importance of

individual fitness is well understood, empirical datasets of fitness records gen-

erally violate the assumptions of standard statistical approaches. This problem

has plagued comprehensive study of fitness and impeded empirical unification of

numerical and genetic dynamics of populations. Recently developed aster mod-

els address this problem by explicitly modeling the dependence of later expressed

components of fitness (e.g. fecundity) on those expressed earlier (e.g. survival to

reproductive age). Moreover, aster models allow different sampling distributions

for different components of fitness, as appropriate (e.g. binomial for survival over

a given interval and Poisson for fecundity). The analysis is conducted by max-

imum likelihood, and the resulting compound distributions for lifetime fitness

closely approximate the observed data. To illustrate the breadth of aster’s util-

ity, we provide three examples: a comparison of mean fitness among genotypic

groups, a phenotypic selection analysis, and estimation of finite rates of increase.

Aster models offer a unified approach to addressing the wide range of questions

in evolution and ecology for which life history data are gathered.

Subject headings: Chamaecrista fasciculata, community genetics, demography,

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Echinacea angustifolia, fitness components, Uroleucon rudbeckiae

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The fitness of an individual is well understood as its contribution, in offspring,

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to the next generation. Fitness has both evolutionary significance, as an individual’s

contribution to a population’s subsequent genetic composition, and ecological importance,

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as an individual’s numerical contribution to a population’s growth. The simplicity of these

closely linked ideas belies serious complications that arise in empirical studies. Because

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lifetime fitness comprises multiple components of fitness expressed over one to many

seasons or stages, its distribution is typically multimodal and highly skewed in shape

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and thus corresponds to no known parametric distribution. This problem has long been

acknowledged (Mitchell-Olds and Shaw 1987; Stanton and Thiede 2005), yet to date there is

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no single rigorously justified approach for jointly analyzing components of fitness measured

sequentially throughout the lifetime of an individual. This limitation severely undermines

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efforts to link ecological and evolutionary inference.

Here we present applications of a new statistical approach, aster, for analyzing

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life-history data with the goal of making inferences about lifetime fitness or population

growth. Within a single analysis, aster permits different fitness components to be modeled

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with different statistical distributions, as appropriate. It also accounts for the dependence of

fitness components expressed later in the life-span on those expressed earlier, as is necessarily

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the case with sequential measurements of aspects of the life-history, e.g. reproduction

depending on survival to reproductive age. Geyer et al. (2007) present the statistical theory

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for aster models. Our goal here is twofold. First, we illustrate the problem by reviewing the

limitations of approaches that have previously been employed in empirical studies. Second,

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we describe how aster models resolve these problems, illustrating these points with three

empirical examples.

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1. The problem and previous efforts to address it

Individual fitness realized over a lifespan typically does not conform to any well known

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distribution that is amenable to parametric statistical analysis. In contrast, components

of fitness over a given interval, such as survival to age x, reproduction at that age, and

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the number of young produced by a reproductive individual of that age, generally conform

much more closely to simple parametric distributions. For this reason, components of

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fitness are sometimes analyzed separately to circumvent the distributional problem of

lifetime fitness. For example, in a study of genetic variation within a population of Salvia

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lyrata in its response to conspecific density, Shaw (1986) provided separate analyses of

two components of fitness, survival over two time intervals and size of the survivors, as a

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proxy for future reproductive capacity in this perennial plant. This approach considers size,

or in other cases fecundity, conditional on survival. It has the appeal that the statistical

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assumptions underlying the analyses tend to be satisfied, but it offers no way to combine the

analyses to yield inferences about overall fitness. Separate analyses of fitness components

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cannot substitute for an analysis of overall fitness, particularly considering the possibility

of tradeoffs between components.

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A common approach to analyzing fitness as survival and reproduction jointly is to use

fecundity as the index of fitness and retain fecundity values of zero for individuals that

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died prior to reproduction. When observations are available for replicate individuals, a

variant of this approach is to use as the measure of fitness the product of the proportion

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surviving and the mean fecundity of survivors (e.g. Belaoussoff and Shore 1995; Galloway

and Etterson 2007). In either case, the resultant distributions typically have at least two

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modes (one at zero) and are highly skewed, such that no data transformation yields a

distribution that is suitable for parametric statistical analysis. Authors frequently remark

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on the awkwardness of these distributions in their studies (e.g. Etterson 2004), but rarely

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publish fitness distributions. Antonovics and Ellstrand (1984), however, presented the

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distribution of lifetime reproductive output (their Fig. 2) in their experimental studies of

frequency-dependent selection in the perennial grass, Anthoxanthum odoratum, noting its

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extreme skewness. Finding no transformation that yielded a normal distribution suitable

for analysis of variance, they assessed the robustness of their inferences by applying three

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distinct analyses (categorical analysis of discrete fecundity classes, ANOVA of means, and

nonparametric analysis). In this study, results of the three analyses were largely consistent,

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but, in general, results are likely to differ.

Others have noted the importance of complete accounting of life-history in inferring

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fitness or population growth rate, as well as evaluation of its sampling variation, and

have presented methods to accomplish this. Caswell (1989) and Morris and Doak (2002)

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explain how to obtain population projection matrices from life-history records and, from

them, to estimate population growth rate. They also describe methods for evaluating its

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sampling variation and acknowledge statistically problematic aspects of these methods

(their Chapters 8 and 7, respectively). Lenski and Service (1982) considered the complete

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life-history record of individuals as the unit of observation and used jackknife resampling to

estimate population growth rate and its sampling variance. Recent efforts to evaluate the

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nature of selection have likewise taken a comprehensive demographic approach. McGraw

and Caswell (1996) considered individual life-histories but chose the maximum eigenvalue

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of an individual’s Leslie matrix (λ) as its fitness measure. They regressed λ on the fitness

components, age at reproduction and lifetime reproductive output to estimate selection on

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them. Van Tienderen (2000) advocated an alternative approach for studying phenotypic

selection. This approach involves evaluating the relationships between each component

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of fitness and the phenotypic traits of interest via separate multiple regression analyses

(Lande and Arnold 1983) to obtain the selection gradients in different episodes of selection.

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These selection gradients are then weighted by the elasticities (Caswell 1989) of each

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component of fitness obtained from analysis of the appropriate population projection

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matrix. Beyond these approaches linking demography and fitness, methods targeting the

problem of “zero-inflated” data (i.e. many observations of zero distorting a distribution)

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have been proposed (Cheng et al. 2000; Dagne 2004). Each method has liabilities, however.

For example, elasticities do not take into account sampling variation in the life-history,

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and violations of distributional assumptions remain a problem (McGraw and Caswell 1996;

Coulson et al. 2003). Moreover, none of these methods generalize readily for inference in

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the wide range of contexts that life-history data can, in principle, address.

2. Inference of individual fitness with aster

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We present aster models (Geyer et al. 2007) for rigorous statistical analysis of

life-history records as a general approach for addressing diverse questions in evolution

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and ecology. As noted above, two standard properties of life-history data are central to

the statistical challenges that aster addresses. First, the expression of an individual’s

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life-history at one stage depends on its life-history status at earlier stages. For example,

observation of an individual’s fecundity at one stage is contingent on its survival to that

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stage. Second, no single parametric distribution is generally suitable for modeling various

components of fitness, e.g. survival and fecundity. Aster analysis models fitness components

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observed through a sequence of intervals bounded by the times at which individuals are

observed. The intervals could be days or years, and need not all be the same duration.

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The components of fitness are modeled jointly over successive intervals by explicitly taking

into account the inherent dependence of each stage on previous stages, e.g. that only

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survivors reproduce. We represent the life-history and, in particular, the dependence of one

component of the life-history on another, graphically as in Fig. 1.

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EDITOR: PLACE FIGURE 1 HERE.

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The aster approach models the joint distribution of a set of variables (fitness

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components). We say an arrow in the graphical model points from a variable to its successor

or, going backwards along the arrow, from a variable to its predecessor (Geyer et al. 2007

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used “parent” and “child” instead of “predecessor” and “successor” but this is confusing

in biology). The theory underlying the aster approach requires modeling the conditional

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distribution of each variable given its predecessor variable as an exponential family of

distributions (Barndorff-Nielsen 1978; Geyer et al. 2007) with the predecessor variable

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playing the role of sample size. This requirement retains considerable flexibility, because

many distributions are exponential families, including Bernoulli, Poisson, geometric, normal,

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and negative binomial.

If the predecessor is zero then so is the successor. If the predecessor has the value

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n > 0, then the successor is the sum of n independent and identically distributed variables

having the named distribution. For example, the binary outcome of an individual’s survival

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over a given interval is modeled as a Bernoulli variable. Likewise, given that an individual

survived to this point, whether or not it reproduced is considered Bernoulli. Fecundity,

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given that it reproduced in this interval, may be modeled according to a zero-truncated

Poisson distribution (i.e. a Poisson random variable conditioned on being greater than 0).

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Aster analysis yields estimates of unconditional (lifetime) fitness that account for

the expression of all the fitness components. The modeling of each single component of

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fitness with an appropriate probability distribution leads to a sampling distribution for the

joint expression of the fitness components as lifetime fitness that approximates the actual

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distribution, as we show in Example 2 below.

Modeling the joint distribution of fitness components establishes a proper foundation

for sound analysis, but another key idea is needed. Restriction of the choice of

conditional distributions for fitness components to exponential families results in a joint

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distribution of the components that is a multivariate exponential family. Its canonical

parameterization, described by equation (5) in Geyer et al. (2007), is called the unconditional

parameterization of the aster model. Let Xidenote the variables (fitness components) and

ϕithe corresponding canonical parameters of a model. When overall fitness is considered

a linear combination?

expectation is directly controlled by a single parameter βkif the regression part of the

iaiXi, where ai are known constants, then its unconditional

model has the form

ϕi= aiβk+ other terms not containing βk

by equation (22) in Geyer et al. (2007); increasing βkincreases the unconditional expectation

of?

βkaddress overall fitness directly. For this reason, the unconditional parameterization is

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iaiXiother betas being held fixed. Thus confidence intervals and hypothesis tests for

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used for the example in Geyer et al. (2007) and Examples 1 and 2 below. This situation

most often arises when the linear combination is a simple sum (so the aiare zero or one),

e.g. some of the Xiare counts of offspring in one year and?

offspring observed in all years.

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iaiXiis the total number of

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The unconditional parameterization is somewhat counterintuitive because terms in the

regression model that nominally refer to a single component of fitness (affect its ϕionly)

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directly influence the unconditional expectation of overall fitness by affecting not only its

distribution but also the distributions of all components before it in the graphical model

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(its predecessor, predecessor of predecessor, etc.) This makes it somewhat difficult, but not

impossible (see our Example 1), to see the role played by a single component of fitness.

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This is, however, an unavoidable consequence of being able to address overall fitness.

The analysis employs the principle of maximum-likelihood, developed by Fisher (1922)

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and now widely applied as a rigorous, general approach to any statistical problem (Kendall

and Stuart 1977). Software for conducting the analysis, is a contributed package “aster”

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in the R statistical language (R Development Core Team 2006) and is freely available

(http://www.r-project.org).

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We demonstrate the value and versatility of the aster approach with three examples.

In the first, we apply aster to compare mean fitness among groups. Specifically, we

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quantify effects of inbreeding on fitness of Echinacea angustifolia, a long-lived composite

plant. In our second example, we reanalyze data of Etterson (2004) to evaluate phenotypic

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selection on the annual legume, Chamaecrista fasciculata. In the last example, we illustrate

inference of population growth rate via aster. We consider a small dataset that Lenski and

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Service (1982) used to demonstrate their nonparametric method for inferring population

growth rate from a set of individual life-histories of the aphid, Uroleucon rudbeckiae.

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The datasets for our examples are in the aster package for R. Complete analyses for our

examples are given in a technical report (Shaw, et al. 2007) available at the aster website

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http://www.stat.umn.edu/geyer/aster and are reproducible by anyone who has R (see

Chapter 1 of the technical report).

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3. Example 1: Comparison of fitness among groups

In this example, we illustrate use of aster models to compare mean fitnesses of defined

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groups, here, genotypic classes. Specifically, we investigate how parental relatedness

affects progeny fitness in the perennial plant, Echinacea angustifolia (narrow-leaved purple

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coneflower), a common species in the N. American tallgrass prairie and Great Plains. The

plant is self-incompatible, and Wagenius (2000) detected no deviation from random mating

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in large populations. However, following the abrupt conversion of land to agriculture and

urbanization that started about a century ago, the once extensive populations now persist

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in small patches of remnant prairie. In this context of fragmented habitat, we expect

that matings between close relatives in the same remnant, and perhaps also long distance

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matings, have become more common.

To evaluate the effects of different mating regimes on the fitness of resulting progeny,

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formal crosses were conducted in the field to produce progeny of matings between plants

a) from different remnants, b) chosen at random from the same remnant, and c) known

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to share the maternal parent. The resulting seeds were germinated, and the plants were

grown in a growth chamber for three months, after which they were transplanted into an

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experimental field plot. In this example, we focus on pre-reproductive components of fitness,

survival and plant size. Survival of each seedling was assessed in the growth chamber on

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three dates. The seedlings were then transplanted into an experimental field plot, and their

survival was monitored annually 2001–2005. The number of rosettes (basal leaf clusters,

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1–7) per plant was also counted annually 2003–2005. Rosette count reflects plant vigor and

is likely related to eventual reproductive output.

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Mortality of many plants as seedlings and juveniles resulted in a distribution of rosette

count in 2005 having many structural zeros. We modeled survival through each of eight

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observation intervals as Bernoulli, conditional on surviving through the preceding stage;

we modeled rosette count in each of three field seasons, given survival to that season, as

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zero-truncated Poisson (Fig. 1A). To account for spatial and temporal heterogeneity, we

also included in the models the factors a) year of crossing (1999 or 2000), b) planting tray

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during the period in the growth chamber, c) spatial location (row and position within row)

in the field.

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In addition to evaluating the effects of mating treatments on overall fitness, we

developed models to test for differences in the timing and duration of the mating treatment

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effects on fitness. At the earliest stages, in the benign conditions of the growth chamber,

effects of the mating treatments may be negligible. Alternatively, it may be that the

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effects of mating treatment at the earliest stages largely account for their overall effects on

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fitness. These scenarios differ in their implications concerning the inbreeding load expected

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in standing populations (Husband and Schemske 1996). We developed four aster models,

named “chamber,” “field,” “sub,” and “super.” Each was a joint aster analysis of all 11

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bouts of selection (survival over eight intervals, rosette count at three times). The “field”

model includes explicit mating treatment effects only on the final rosette count (variable

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r05 in Figure 1A), but because of the unconditional parameterization of aster models

(section 2, above) these effects propagate back to earlier stages. The “chamber” model

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includes explicit mating treatment effects only on the final survival before transplanting

(variable lds3 in Figure 1A), but, again, these effects propagate through all preceding

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bouts of survival. The “sub” model is the greatest common submodel of “chamber” and

“field,” and the “super” model is their least common supermodel (i. e. “sub” includes no

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effects of mating treatment on any aspect of fitness, whereas “super” includes effects of

mating treatment on both survival up to transplanting and on final rosette count).

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The aster analysis revealed clear differences among the mating treatments in overall

progeny fitness through the end of the available set of records, (model “field” compared to

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“sub”, (P = 1.1 × 10−5). The unconditional expected rosette count for each cross type is

the best estimate for the expected rosette count in 2005 for every seed that germinated in

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2001. The fitness disadvantage of progeny resulting from sib-mating relative to the other

treatments is a 35%–42% reduction in rosette count (Fig. 2).

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EDITOR: PLACE FIGURE 2 HERE.

Because of the aforementioned propagation of effects back to earlier stages, the effects of

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mating treatment in the “field” model directly subsume overall fitness expressed over the

course of the experiment. Though this analysis suffices for inferring the overall effects

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of mating treatment on fitness, we investigated further the timing and duration of these

effects using the additional models described above. The comparison of the “sub” and

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“chamber” models shows that survival before transplanting differs among mating treatments

(P = 0.012). However, the comparison of the “chamber” and “field” models with the

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“super” model shows that “super” fits no better than “field” (P = 0.34) but does fit better

than “chamber” (P = 3.1 × 10−4). Hence the “field” model fully accounts for differences

in expressed fitness. The terms of the “chamber” model that quantify the effect of mating

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treatment on survival up to transplanting are not needed to fit the data, because the

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aforementioned back propagation of effects subsumes the effects of mating treatment in the

growth chamber. This does not mean there are no effects of mating treatment on fitness

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before transplanting. The comparison of “sub” and “chamber” confirms they exist, and

Fig. 2 clearly shows them. The fitness disadvantage of progeny resulting from sib-mating

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relative to the other treatments is clear in the 7%–10% reduced survival up to the time of

transplanting but the overall fitness disadvantage of inbreds is considerably greater (Fig. 2).

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4. Example 2: Phenotypic selection analysis

Lande and Arnold (1983) proposed multiple regression of fitness on a set of quantitative

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traits as a method for quantifying natural selection directly on each trait. In practice,

these analyses have generally employed measures of components of fitness as the response

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variable, rather than overall fitness (see e.g. examples in Lande and Arnold 1983). As a

result, the estimated selection gradients, the partial regression coefficients, reflect selection

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on a trait through a single ’episode of selection’, rather than selection over multiple episodes

or, ideally, over a cohort’s lifespan. Focusing on this limitation, Arnold and Wade (1984a)

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considered partitioning the overall selection gradient into parts attributable to distinct

episodes of selection, and Arnold and Wade (1984b) illustrated the approach with examples.

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Wade and Kalisz (1989) modified this approach to allow for change in phenotypic variance

among selection episodes. Whereas these developments were intended to account for

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the multiple stages of selection, they do not directly account for the dependence of later

components of fitness on ones expressed earlier, an issue that also applies to van Tienderen

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(2000).

Apart from the problem of dependence among fitness components, Mitchell-Olds and

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Shaw (1987), among others, have noted that statistical testing of the selection gradients

is compromised, in many cases, by the failure of the analysis to satisfy the assumption of

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normality of the fitness measure, given the predictors. This concern applies to McGraw and

Caswell’s (1996) approach to phenotypic selection analysis, which integrates observations

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from the full life-history. To address this problem for the case of dichotomous fitness

outcomes, such as survival, Janzen and Stern (1998) recommended the use of logistic

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regression for testing selection on traits and showed how the estimates resulting from

logistic regression could be transformed to obtain selection gradients. Schluter (1988) and

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Schluter and Nychka (1994) suggested estimating fitness functions as a cubic spline to allow

for general form, but this method requires a parametric error distribution, whether normal,

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binomial, or Poisson.

Aster explicitly models the dependence of components of fitness on those expressed

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earlier. Moreover, unconditional aster analysis estimates the relationship between overall

fitness and the traits directly in a single, unified analysis. It thus serves as a basis for

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statistically valid inference about phenotypic selection, unlike other methods whose required

assumptions are often seriously violated. We illustrate this use of aster with a reanalysis

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of Etterson’s (2004) study of phenotypic selection on three traits in three populations of

the annual legume, Chamaecrista fasciculata, reciprocally transplanted into three sites.

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The three traits, measured when the plants were 8–9 weeks old, are leaf number (LN, log

transformed), leaf thickness (measured as specific leaf area, SLA, the ratio of a leaf’s area

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to its dry weight, log transformed) and reproductive stage (RS, scored in 6 categories,

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increasing values denote greater reproductive advancement). Here, for simplicity, we

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consider only the data for the three populations grown in the Minnesota site.

C. fasciculata grows with a strictly annual life-history. In this experiment, fitness

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was assessed as 1) survival to flowering, 2) flowering, given that the plant survived, 3) the

number of fruits a plant produced, and 4) the number of seeds per fruit in a sample of three

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fruits, the last two contingent on the plant having flowered. Preliminary analyses revealed

that nearly all survivors flowered and fruited, so these fitness components were collapsed

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to a single one, modeled as Bernoulli (reprod). Consequently, overall fitness was modeled

based on survival and reproduction, the number of fruits per plant, and the number of

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seeds per fruit, (termed reprod, fruit, and seed, Fig. 1B). Preliminary analyses assessed

the fit of truncated Poisson and truncated negative binomial distributions to the data for

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both fecundity components. On this basis, the fecundity components were modeled with

truncated negative binomial distributions. In addition to the traits of interest, the model

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included the spatial block in which individuals were planted.

To illustrate phenotypic selection analysis most straightforwardly, we begin by

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analyzing two of the fitness components, reprod and fruit in relation to two of the traits,

leaf number (LN) and leaf thickness (SLA). This model detected strong dependence of

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fitness on both traits such that selection is toward more, (P < 10−6) thinner (P = 0.007)

leaves.

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Extending this analysis to assess curvature in the bivariate fitness function, we detected

highly significant negative curvature for both traits, suggestive of stabilizing selection,

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(P < 5.7 × 10−43). The plot of the fitness function together with the observed phenotypes

(Fig 3) reveals that the fitness optimum lies very near the edge of the distribution of leaf

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number.

EDITOR: PLACE FIGURE 3 HERE.

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