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Transcript
Estimating the Form of Natural Selection on a Quantitative Trait
Author(s): Dolph Schluter
Source: Evolution, Vol. 42, No. 5 (Sep., 1988), pp. 849-861
Published by: Society for the Study of Evolution
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EVOLUTION
INTERNA TIONA L JO URNA L OF OR GA NIC E VOL UTION
PUBLISHED
THE SOCIETY
Vol. 42
BY
FOR THE STUDY
OF EVOLUTION
September,1988
No. 5
Evolution,42(5), 1988,pp. 849-861
ESTIMATING THE FORM OF NATURAL SELECTION ON A
QUANTITATIVE TRAIT
DOLPH SCHLUTER
The Ecology Group,Depa;9ment of Zoology, University
ofBritishColumbia,
6270 University
Boulevard, Vancouver,BC V6T 2A9, CANADA
Abstract.-The fitnessfunctionfrelatesfitnessof individualsto the quantitativetraitundernatural
selection.The functionis usefulin predictingfitnessdifferences
among individualsand in revealing
whetheran optimum is presentwithinthe rangeof phenotypesin the population. It may also be
thoughtof as describingthe ecological environmentin termsof the trait.Quadratic regressionwill
approximatethe fitnessfunctionfromdata (e.g., Lande and Arnold, 1983), but the method does
not reliablyindicate featuresoff such as the presence of modes (stabilizingselection) or dips
(disruptiveselection).I employan alternativeprocedurerequiringno a priorimodel forthefunction.
The method is usefulin two ways: it provides a nonparametricestimateoff of interestby itself,
and it can be used to suggestan appropriateparametricmodel. I also discuss measuresof selection
intensitybased on the fitnessfunction.Analysis of six data sets yields a varietyof formsoff and
providesnew insightsforsome familiarcases. Low amounts of variationand a low densityof data
pointsnear the tails of manyphenotypedistributionsemergeas limitationsto gaininginformation
on fitnessfunctions.An experimentalapproach in which the distributionof a quantitativetraitis
broadened throughmanipulationwould minimize these problems.
Received August 25, 1987. Accepted March 4, 1988
sureson individuals.For example, individuals may differentially
surviveaccordingto
body size, in whichcase W is survival(1 or
0) and f(z) is the probabilityof survival as
a functionof size. An estimate off would
thus allow the quantitative prediction of
survival probabilities of individuals, the
comparisonofprobabilitiesamong individin size, and the assessmentof
ual.sdiffering
whetheran optimumbody size existswithin
the rangeof phenotypespresentin the population.
A knowledgeoff over a broaderrangeof
phenotypesthan is actually found in any
one population also leads to an estimateof
the "adaptive landscape," describingmean
fitnessin the population WV
as a functionof
W = f(z) + random error
z, the mean value of the phenotypictrait
(Pearson, 1903; Simpson, 1953; Lande, underselection(Wright,1932; Lande, 1979).
1979; Lande and Arnold, 1983).
When thetraitis heritableand uncorrelated
The fitnessfunctionis of interestbecause with other traitsunder selection,then the
itis a completedescriptionofselectionpres- directionof changein z leadingto the maxMeasurementsofnaturalselectionare invaluable in studies of adaptation. Recent
theoreticalworkin thisarea has successfully
addressed two main goals: first,to develop
coefficients
of selectionintensityforquantitativetraits,disentanglingdirectfromindirect effects(Lande and Arnold, 1983;
Manly, 1985), and second, to predictevolutionary response to observed selection
events (Lande, 1979). Data on natural selectioncan also be used fora thirdpurpose:
to estimate the selection surfaceor fitness
function,the unknown functionf relating
survivaland/orreproductivesuccess of individuals (JW)to the phenotypiccharacter
z under selection:
849
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850
DOLPH SCHLUTER
imum increase in mean fitnesswill usually
be the directionof evolutionarychange in
the population.
The fitnessfunctionis additionallyvaluable in other research.For example, optimalitytheoryis employedin manyfieldsto
predictfeaturesof a fitnesssurfacefromhypotheses about how a given trait may be
advantageous in a particularenvironment.
DeAngelis et al. (1985) predictedfitnessin
bivalves as a functionof growthstrategy,
as
determined by predation. Schluter and
Grant(1984) estimatedadaptive landscapes
formean beak size in Gal'apagos finchpopulations on the hypothesisthat these are
determinedby seed supply. Powerfultests
of such adaptive hypothesescould be made
by quantitativelycomparingthe predicted
formof the selectionsurfacewithindependent estimatesbased on survival or reproductive success.
Similarly,ifthe ecological significanceof
a traitunder selection is known, then the
formof the fitnessfunctionwill provide a
usefuldescriptionoftheecological environment in relevant units of fitness.For example, beak size in populations of Galapagos ground finches determines the
withwhichfoods of different
efficiency
size
and hardness can be consumed and, consequently,determinesdiet. Multiple modes
in themean fitnesslandscapesformean beak
size describe a distributionof alternative
feedingniches along a correspondingmean
seed size/hardnessaxis (Schluterand Grant,
1984; Schluteret al., 1985).
A varietyof regressionmethods may be
used to estimatefor featuresoff fromdata
on natural selection. Lande and Arnold
(1983) suggestedtheuse ofquadraticregression to approximatethefitnessfunction,the
coefficientsof which correspondto direct
measuresof selectionintensity.This method can give misleading results,such as in
indicatingthe presence of a dip or mode
(see below). A moreflexibleapproachwould
be to compare several regressionmodels
(e.g., Manly, 1976, 1985; Mitchell-Oldsand
Shaw, 1987). Choosing the best model will
oftenbe difficult,particularlywhen error
distributionsare nonnormal(e.g., survival
data). Also, one cannot be assured thatthe
appropriatemodel is among those initially
fitted.
Here, I explore an alternativemethod to
estimatetheindividualfitnessfunctionfrom
data on naturalselection,one not requiring
that the parametricformoff be specified.
The method has two uses: first,it provides
a nonparametricestimateoff,ofinterestby
itself,and second, it can suggestan appropriateparametricmodel if one is desired. I
use the method to analyze several familiar
data sets involving selection on a single
variable and show where new insightsare
based
gained.Estimatesofselectionintensity
on nonparametricestimatesof fitnessfunctions are also discussed. Finally,I consider
how to measure the samplingvariabilityof
the estimatef The more difficultproblem
ofestimatingsurfacesinvolvingtwo or more
charactersis ignored here but will be discussed in a forthcoming
paper.
Definitions
In thegeneralmodel fornaturalselection,
survival and/orreproductivesuccess of individuals is related to the phenotypez as
W = ftz) + E,where e is a random error.I
definedftz) as individual fitness,because it
measures average survival or reproductive
success of all individuals having the phenotypez (e.g.,Endler,1986). Wis oftenalso
referred
to as "individual fitness"(Falconer,
1981; Lande and Arnold, 1983; Endler,
1986), but I will refrainfromdoing so here
in orderto preventconfusion.Relative fitness is fitnessscaled to have a population
mean of 1:ft(z) = ftz)/W,where W is mean
survival and/orreproductivesuccess.
Natural selectionis definedas variability
among phenotypesin fitness.Three typesof
univariate selection are commonly recognized, on the basis of the formoff over the
rangeofphenotypesin thepopulation(Endler, 1986 pp. 16-18). In directionalselection, individuals on one side of the population mean are favoredover those on the
other.Directionalselectionmay occur with
other types,but in pure directionalselection,fitnessis eithernondecreasingor nonincreasingovertherangeofphenotypes(i.e.,
f is monotonic).In stabilizingselection,intermediatephenotypesare favoredover extremes-the functionhas a mode, or optimum. In disruptive selection, fitness of
extremeindividuals is greaterthan that of
intermediates-the surface has a dip, or
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THE FORM OF NATURAL SELECTION
minimum.Lande and Arnold (1983) redefinedthese threetypesof selection(see below), but I use the more traditionalmeanings here.
and the
SelectionCoefficients
FitnessFunction
851
12-
a)
9
a)
The mostcommon procedurein theanalysis of data on natural selection is to calculate coefficientsof selection intensity.
Here, I brieflyreviewthe most usefulcoef0
ficients,those of Lande and Arnold (1983),
in order to clarifytheirrelationshipto the
selection surface,f; and to show how their
incautious application can mislead.
-3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
The fitnessfunctionf can be thoughtof
Phenotype
as a featureof the environment,describing FIG. 1. Misleading
approximationof a relativefitthe fitnessesof alternativephenotypesbut ness functionf,usingquadraticregression.Bell-shaped
existingindependentlyof their actual dis- curveand shaded area indicatetruncationselectionon
tribution.In contrast,selectioncoefficients, a normallydistributedtraitz withmean 0 and variance
such as those of Lande and Arnold (1983), 1: all individuals above z = 1.0 survive,and all individuals below z = 1.0 die. The regressionsurface(inmeasure the effectsoff on the distribution vertedparabola) erroneouslysuggests
the presence of
of phenotypesin the population. To sim- disruptiveselection.
plify calculations, assume that the single
characterz under selectionhas mean u = 0
of di- while individuals withz < a do not. Selecand variance o2 = 1. The coefficient
rectionalselectionis then
tionis purelydirectional,butitcan be shown
that the coefficienty is nonzero whenever
A = Cov(w, z)
a is not the mean phenotype:y = ae-a2'2/
where w = W/W, and WV
is mean survival q 2w,whereq is thefractionofindividuals
and/orreproductivesuccess. If,in addition, in thepopulationsurviving.The truncation
z is normallydistributed,then
point in Figure 1 is a = 1.0, and y is large
and negative,as indicatedby the dip in the
Z2)
7 = Cov(w,
best-fitquadratic regressionof w on z. A
is the "stabilizing" (y < 0) or "disruptive" similarsituationoccurs in the multivariate
The value of y case, where pure directional selection can
( > 0) selectioncoefficient.
betweenthe vari- resultin a significant"correlational" selecis equal to the difference
ances of z before and afterselection,cor- tioncoefficient
(cf.Lande and Arnold,1983).
rectedby p2, the reductionin variance reThe value of y fails to indicate the pressultingfromdirectionalselection alone: y ence ofa mode or dip infbecause thequan+
32
=
The constantsA and tityf2 does not correctforchangesin variU2after ?2before
y/2are the linearand quadratic coefficients ance resulting
fromdirectionalselection.For
from the quadratic regressionof w on z, example,ifz is normallydistributedas p(z),
when z is normallydistributed(Lande and with ,u = 0 and U2 = 1, and if the fitness
Arnold, 1983).
functionf is smooth (i.e., is twice-differeny does not nec- tiable [']), thenthe totalchange in variance
However, the coefficient
essarilyindicatestabilizingor disruptivese- caused by selection, directional or otherlection, as these terms are definedabove, wise, is actually
when directionalselection occurs (see also
2 + E
a2after - a 2before =
fw"(z)]
Mitchell-Olds and Shaw [1987]). For example, consider truncationselection in a wherefw(z)=ftz)/W. The quantityE[fw'(z)]
normallydistributedpopulationwithmean = f fw"(z)p(z) dz measures mean curvature
0 and variance 1 (Fig. 1). All individuals of the surfacefwover the distributionof
having a phenotypicvalue z > a survive, phenotypes(Lande and Arnold,1983). Thus,
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852
DOLPH SCHLUTER
f2 will measurethechangein varianceresultingfromdirectionalselectiononlyifthe
purelydirectional(monotonic) fitnesssurfacef, is withoutmean curvature(e.g., is
linear).Many directionalfitnesssurfacesare
curved,such as those havingan asymptote,
and in such cases y will be nonzero even
thoughstabilizingand disruptiveselection
are absent.
The absence of a straightforward
correspondencebetweenselectioncoefficients
and
featuresof the selection surface does not
detractfromthe importanceof the Lande
and Arnold(1983) results.Correctlyapplied
and interpreted,theircoefficientscan successfullymeasure directand indirectselection and can relate phenotypicpatternsto
evolutionarychange; knowledgeoff is neither necessary nor sufficientto carry out
these goals. In orderto avoid confusion,y
should merelybe renamed(e.g., as thecoefficientof nonlinearor quadratic selection).
However,theproblemsunderscoretheview
stressedhere,thatidentificationof stabilizing and disruptive(and correlational)selection and identificationof otherfeaturesof
f should always be made with referenceto
a directestimateof the fitnessfunction.In
the followingsections,I apply a nonparametrictechniqueto make such an estimate.
MATERIALS AND METHODS
NonparametricEstimation ofFitness
Functions
The Cubic Spline.-In order to estimate
the fitnessfunctionf fromindividual data
on survival and/or reproduction(W) and
phenotypicmeasurements(z), a reasonable
approach is to begin with maximum-likelihood. Select an estimatef fromthe set of
all possible continuous fitnesssurfacesto
maximize the log likelihood
n
Q(f) =
i=1
ll(Wi; zi,f)
wherethe sum is over the n individuals in
thesample.The terml( Wi;zi,f) is theloge
probabilitythat W = WJwhen z = zi, under
the particularchoice of f In the case that
W is normallydistributedaroundftz) with
constantvariance,l(Wi;zi,f) is proportional
to-[ WJ-ftzi)]2, and thefthatmaximizes
Q(f) is the familiarleast-squares estimate.
Unfortunately,
any functionconnectingall
n data pointswill maximize the likelihood.
Such a curvewould be extremelyrough,zigzaggingrepeatedlyamong the different
values of W, and would have low predictive
value. It would also offendour prejudice
thatthetruefitnesssurfaceis smootherand
simpler.
The failureof the above method has led
to themodifiedtechniqueofpenalized maximumlikelihood(Good and Gaskins, 1971).
We select the functionf to maximize the
penalized log likelihood
n
Q(f)
=
i=l1
l(W; zi, f)
-
nXJ(f) (1)
whereX is a nonnegativeconstantand J(f)
is the summed (squared) curvatureofJfa
measure of "roughness":
J(f)
=
f
[f"!(Z)]2
dZ.
(2)
Note that the functionsconsidered in the
maximization of (1) cannot have sharp
creases, or the operation within square
bracketsin (2) (double differentiation)
will
not be possible.
Assume forthe moment that W is normallydistributedaroundftz) withconstant
variance (nonnormal errorsare discussed
below). For a given X,thef thatmaximizes
(1) is a cubic spline,a functioncomprising
n + 1 cubic polynomialsjoined seamlessly
at pointscorrespondingto the n phenotypic
values. The value of X controlsthe importance of the roughnesspenaltyJ(f) to the
sum in (1) and, thereby,determinesthe exact form of the estimatef A value for X
mustthereforebe chosen wisely.When Xis
small,littleprice is paid forroughness,and
the best function[the one that maximizes
(1)] willnearlyfittheindividualdata points.
The penaltytermassumes greaterweightas
Xgetslarger,and so to compensate,thebest
fmust be smoother.This effectis illustrated
using Houde's (1987) data on sexual selection in guppies (Fig. 2), assuming normal
errors.In thelimit,as Xapproachesinfinity,
the best functionwill be the linear regression of Won z. The constantXis oftencalled
the "smoothingparameter."
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THE FORM OF NATURAL SELECTION
853
0.8 An excellentchoice forXis one providing
maximal predictivepowerin thegivendata co)
+
set, computed using the method of cross- co
2
++
validation. For a given Xand an individual Q 0.6
+
i, letf7be the functionthat maximizes (1),
+
butwithindividuali excludedfromthesum. 3
+
+
/,4+
The functionf' is computed forall n indi- C
=H 0.4 /
viduals in the sample, yieldingthe sum of
E
+
squared predictionerrors2(fP(zj) )2.
4
Short-cutmethodsapproximatethe sum of a)
0.2
squared predictionerrorsand yieldthegeneralized cross-validation(GCV) score (Cra- co
ven and Wahba, 1979). The value forXthat
minimizestheGCV score is knownto min0.0I
I
imize approximatelythesum ofsquared de0
9
18
27
viationsbetweenthe estimatefand thetrue
Amount of orange
fitnesssurfacefThe GCV scoreis therefore FIG. 2. Effectof X on the estimateof the fitness
an appropriate criterionfor estimatingf function.
Data aremating
successofmaleguppies(PoeSuch a GCV estimateis shown in Figure 2 ciliareticulata;
fromdataon femalebehavior)
inferred
oftheirbodyarea thatis colored
(solid line) and indicates a possible mode and thepercentage
withorange(takenfromfigure
3 in Houde [1987];N
in the sexual-selectionsurface.
= 32). Alternative
X = 0.002
estimates represent
The cubic-splineapproach is flexible,al- (dotted),X = 8.2 (solid),andoff
X = 150 (dashed).The
lowing the estimationof a wide varietyof solidlineminimizestheGCV score.
fitnesssurfaces.Standard linear,quadratic,
and higher-order
polynomialfunctions(e.g.,
cubic regression) are themselves cubic apex of a narrowpeak). Gu (1987) suggests
splines,and such functionswill resultifthe a bootstrapprocedureto correctpartlyfor
data warrants,but the methodis in no way bias. Estimatedbias was very small in the
restricted
to theseforms.Intuitionas to how data sets that I analyzed, and so I do not
the cubic spline actually yields a smooth correctforit in my results.
NonnormalErrors.-Measures ofsurvivestimate of f is provided by Silverman
(1984), who shows that the method is ap- al and reproductivesuccess are oftennot
proximately equivalent to the weighted normallydistributed.For example survival
movingaverage,wherethenumberofpoints is binomial and takes on values 0 or 1. The
includedin theaveragevaries withthelocal distributionofnumberofoffspring
may,for
a given value of z, be approximatelyPoisdensityof data points.
The relationshipto the moving average son. The main problem with these alterclarifiessome of the propertiesof the cubic nativedistributionsis thatfitnessis restrictspline. The spline provides a more local es- ed (O < ftz) < 1 in the binomial case, and
timate of the regressionsurfacethan stan- 1(z) ' 0 fora Poisson distribution),and that
dard polynomialregression,which yields a the variance aE2of the errorsaroundfis not
global fitto the data. For example, both constantbut depends on ftz): UE2= ftz)(1 methodsmay be applied to a given data set /(z)) in the binomial case and aE2 = f(z) in
to stabilize
involvinga clusterof (JW,z)values. If a sin- Poisson. Data transformations
gle outlyingpoint is then added and the the variance are not the answer; forexamregressionsare recomputed,the formof the ple, the loge-transformation
cannot be used
regression within the original cluster of on zeros.
These problemscan be solved usingstanpoints may be substantiallyaltered in the
polynomial case, but not in the spline. A dard iterativemethodsofgeneralizedlinear
second, less desirable, propertyis that the models (McCullagh and Nelder, 1983) in
spline, like the moving average, can be combination with the cubic spline (O'Sulbiased in that the estimateoff tends to be livan et al., 1986). In thiscase, the modified
too smooth wherethe slope of the truefit- goal is to maximize the penalized log likeness functionchanges rapidly (e.g., at the lihood Q(g) ratherthan Q(f), where g(z) =
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854
DOLPH SCHLUTER
loge[f(z)/{ 1 - ftz)}] forbinomial data and
g(z) = loge[f(z)] forPoisson data. The method can be thoughtof as a nonparametric
equivalent to logisticand log-linearregression. Note thatno regressiontechniquefor
estimatingf includingthe cubic spline,requires that the distributionof the phenotypictraitz be normal.
The ability to handle nonnormal errors
also allows themethodto be used for"crosssectional" or other data involvingtwo independentsamples.For example,Hagen and
Gilbertson (1973) made a collection of
sticklebackswhenindividualswereaged 21/2
months,and anotherwhen theywere one
year old. Estimates off can be made by
scoringindividualsin the firstsample as W
= 0 and those in the second as W = 1 and
regressingthis variable on the trait of interest.The regressionwill not estimatethe
actual fitnessfunction,but a functionh that
is monotonicallyrelatedto fitness(survival
probability)and has all of the same features
(loge[h(z)/{l - h(z)}] will differfromg(z)
= loge[f(z)/{ 1 -ftz)}]
by a constant).Only
traitsnot subjectto growthcan be analyzed
in this fashion.
Computer Routines. -Efficient FORTRAN77 routinesforcomputingfwithnormal, binomial and Poisson data werekindly
provided by Douglas Nychka (Department
of Statistics,North Carolina State University, Raleigh). An easy-to-use,interactive
programemployinghis routinesis available
fromme on request. The programalso includes an optionto generatebootstrapsamples randomly,which is usefulin assessing
accuracy of f (see below). The programis
designedto runon a PC-compatible microcomputer,but it should work on any system.
Otherprogramsalso exist.The most general,all-purposeroutinesare includedin the
public package PGLMPACK (Yandell,
1987), obtainable fromBrian Yandell (Departmentof Statistics,Universityof Wis-
consin, Madison). These will handle multivariatedata,and splinesotherthanthecubic.
However, theyare inefficient
in the univariate case and have limiteduse on a microcomputer.All analysisin the presentpaper
was done on an IBM-XT using Nychka's
program.
RESULTS
Here I use theabove methodsto estimate
the fitnessfunctionin several other cases.
The purposeoftheanalysisis to see whether
new informationmightbe obtained from
data thathave previouslybeen analyzed in
otherways and to indicate the diversityof
functionsthat may be observed in studies
of natural selection. I also describe measures of selection intensitybased on estimates off and a way to assess the accuracy
of an estimate.
Fitness Functions
House Sparrows.-Bumpus (1899) recorded survival in relationto morphology
in 49 female and 87 male house sparrows
(Passer domesticus)duringa New England
winterstorm. Nine traitswere measured,
including total body length,wing length,
body mass, head length,humerus length,
femurlength,tibio-tarsus
length,skullwidth,
and sternumlength.Bumpus published his
raw data, and as a consequence theyhave
been frequentlyanalyzed (Grant, 1972;
Johnston et al., 1972; O'Donald, 1973;
Lande and Arnold,1983; Manly, 1985). Resultshave suggesteddirectionalselectionin
males and stabilizingselectionin females.
An estimate of the fitnessfunctionf in
relationto the firstprincipalcomponentin
femalehouse sparrows(generalsize, based
on the covariance matrixof all nine traits,
loge-transformed)
confirmsthe presence of
stabilizingselection (Fig. 3A). Probability
of survival was maximal at the mean phenotype(z = 0), decliningfrom0.5 to about
0.1 at thetailsofthephenotypedistribution.
FIG. 3. Nonparametricestimatesof fitnessfunctions
f in six data sets. The "+" symbolsindicate raw data
values, except in 3F, where sample size is very large and averages of sequential groups of 20 individuals are
given.A) Survivaloffemalehouse sparrows(Bumpus, 1899; N= 49) in relationto standardizedPCl. B) Survival
in the Galapagos finchGeospiza fortis(Boag and Grant, 1981; N = 642) in relationto beak size. The dashed
curve is the estimateof the surfaceusing quadratic regression.C) Overwintersurvival in juvenile female song
sparrows(N = 145) as a functionof tarsuslength.Data are fromfourwinterscombined (Schluterand Smith,
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855
THE FORM OF NATURAL SELECTION
A 1.0 -
-
++F
++ ++
B 1.0-
+F
0.6-
0.8
//
0.5-
a
|
/
\
?f0.2/
0.1
00
?--+
-2.7
++lI
111-+1
-1.8
-0.9
+*
0.0
++++1
0.9
+I+ I
1.8
00
I
0.0
2.7
-=
7.1
8.0
General size, PC1
+
0.8
1llll11111
8.9
9.8
10.7
11.6
Beak depth (mm)
1A-F
(D +3-
>?0.6
+++
2
+
a)~~~~~~~~~~~~~C
0.4\
&
~~~~~~~~0
6
0.2
z
+
0.0-
17.7
1.0-
E
18.6
4_
I
19.5
_
20.4
A1
+
17.7
F
A
+ -- 1 IiEiIiiiiiii iiiii 1111 +
.~
~~~
~
--*
++ 111111114++
19.5
+
20.4
21.3
Tarsus length (mm)
-
~~~~~I
0.8
0.4-
0.2-
0.2
-2.7
*++
18.6
1.0-
0.4-
,,,
+
?
+
_
A--I4F-++-IFI--WFH+--H
+A
A
+ A
A-A-
A-
>~~~0.6
0.6-.,.--
?0?- I+
--
IA
IO-
21.3
Tarsus length (mm)
0.8
>
1
Il++WII0IIIIII
-1.8
-0.9
til*
0.0
PC2
0.9
1.8
1
+0.
2.7
+J
??
1
I
3
I
5
I
7
I
I
9
11
Weight (lb)
1986). D) Reproductivesuccess in femalesong sparrowsin 1979 (N = 57) as a functionoftarsuslength(Schluter
and Smith, 1986). E) Overwintersurvivalin juvenile male song sparrows(N = 152) in relationto standardized
PC2. Data are from four winterscombined (Schluter and Smith, 1986). Dashed curves indicate ?1 SE of
predictedvalues ftz) from200 bootstrapreplicatesof the fitnessfunction.F) Survival in male human infants
(Kam and Penrose, 1951; N = 7,037) as a functionof birthweight.
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856
DOLPH SCHLUTER
The functionis roughlyparabolicand might
be approximatedbya quadratic(e.g.,Lande
and Arnold, 1983) or a normal model (e.g.,
O'Donald, 1973).
Galapagos Finches. -Boag and Grant
(1981, 1984) describednonrandomsurvival in relationto various beak and body dimensionsin 642 individuals (males and females combined) of the Galapagos ground
finch,Geospizafortis,over a prolongeddry
season. Price et al. (1984), using the methods of Lande and Arnold (1983), showed
that directional selection was particularly
strongon beak depth,beak width,and body
mass. Schluteret al. (1985), also using the
methods of Lande and Arnold (1983), suggested that some disruptiveselection was
presentin addition to directionalselection,
as shown by a quadratic regressionof survival againstbeak depth (Fig. 3B).
The splineestimateofthefitnessfunction
in G. fortisshowed probabilityof survival
increasing exponentially with increasing
beak depth over the entire range of phenotypes(Fig. 3B). Thus, selection was apparentlypurelydirectional,and disruptive
selectionwas absent,in contrastto the earlier result.The spline is probablycloser to
the true function:inspection of the data
shows that the quadratic curve turns upward at small beak sizes, even thoughthe
very smallest individuals did not survive.
The upturnis a good illustrationof the fact
that a quadratic regressionmay provide a
reasonable overall fitto the data, but its
formis not necessarilyresponsive to local
features(see also Fig. 1).
The revisedestimateoffin G. fortisdoes
not alter a main conclusion of Schluteret
al. (1985), that the small and large seeds
beak sizes in the population.
favordifferent
Large size was favored, but some smallbeaked individuals survived solely by efficientlyexploitingsmall seeds (Price, 1987).
Data fromotherlifestages(Price and Grant,
1984) and years (Gibbs and Grant, 1987)
confirmthat selection may at other times
favorsmall size instead of large size. However,the conclusion of disruptiveselection
in Schluteret al. (1985) is not supportedby
reanalysis.
Song Sparrows. -Schluter and Smith
(1986) describednaturalselectionat differentlifestagesover a five-yearperiod fort-he
residentsong sparrow(Melospizamelodia)
of Mandarte Island, British Columbia.
Strongselectionwas foundin females,particularlyfortarsuslengthand beak length.
Estimatesare given in Figure 3C and D of
the relationbetween fitness(probabilityof
juvenile recruitmentand mean number of
young produced) and tarsus lengthin females. Recruitmentwas a binomial variable, while the numberof youngproduced
by a given female was assumed to have a
Poisson distribution.The two curves show
the opposing relationsbetween fitnessand
lifestages(Schluterand
thetraitat different
Smith,1986). They also show theenormous
differencesin fitnessaccompanying slight
in tarsuslength.Over a 3.6-mm
differences
range,estimatedprobabilityof recruitment
declined fromnearly 1.0 to 0.2, while estimatedreproductivesuccessincreasedmore
than seven-fold.Selectionwas purelydirectional in both cases, but f was not quite
linear,particularlyin survival,wherean asymptotewas approached at low values of
tarsuslength(Fig. 3C).
Analysis of male song sparrowshad suggested the presence of stabilizingselection
on the second principal component (PC2)
in associationwithjuvenile survival(Schluter and Smith, 1986). PC2 was obtained
fromthe covariance matrixof six external
morphological traits and indicated wing
length,tarsuslength,and body mass relative
to length,depth,and width of the beak. A
nonparametricestimateoff (Fig. 3E) confirmsthe presenceof a mode to the leftof
the mean of the phenotypedistribution(2
= 0.0). However,thefunctionwas not symmetricbut slowlyequilibratedto an intermediate value of fitnessat the large end of
the phenotypedistribution.A tiny second
mode, probably not real, was also present
at large values of PC2.
Human Infants.-Karn and Penrose
(1951) presenteddata on survivalto 28 days
in newbornhumansand itsrelationto weight
at birth.They fittedthe data to a logistic
regressionmodel with a quadratic term,
which indicated an optimum birthweight
at about 8 lb (3.6 kg) in both males and
females.Their resultis a classic example of
stabilizingselectionin our species. Similar
studiesare reviewedbyVan Valen and Mellin (1967) and Ulizzi and Terrenato(1987).
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857
THE FORM OF NATURAL SELECTION
The relationbetween probabilityof survival and weightat birthin 7,037 male infantswas estimatedusing the cubic spline
(Fig. 3F). I was initiallyunable to obtain a
sensibleresultusingthis method-no minimum value for the GCV score was observed. The problem probably resultedin
part fromthe small numberof unique values forweight(23), since Karn and Penrose
(1951) presentedtheirobservationsrounded to the nearest half pound. Rounding
seems to renderthe methodespeciallysensitive to values at the ends of the distribution, wheredata are sparse and, as a result
of rounding,separatedfromthe remaining
data bygaps.The problemdisappearedwhen
I groupedthelargestthreeinfants(11.5, 11.5,
and 13 lb) with those in the 1.0-lb class.
The nonparametricestimate off agreed
with the earlieranalysis and indicated stabilizing selection (Fig. 3F). However, the
mode in f was verybroad, and probability
of survival exceeded 0.90 over the range
5.0-1 0.0 lb. Estimatedfitnessdeclined only
slightlyat largerinfantsizes, in contrastto
very low probabilitiesof survival accompanyingthe smallest birth weights.Thus,
most selection appeared to be directional.
Maximum probabilityofsurvival(0.97) was
observedat 7.5 lb, but thismustbe considereda verytentativeestimate,giventheflatness of the mode.
Selection Intensity
The present paper is mainly concerned
with estimatingthe selection function,but
such an estimatecan be employed further.
For example it can be used to estimateselection "intensity,"a useful quantity for
selectionepisodes. Secomparingdifferent
lectionintensityis usuallydefinedas a function of intrapopulationvariation in fitness
(Haldane, 1954; Van Valen, 1965; O'Donald, 1970; Manly, 1977) or of covariation
between fitnessand the phenotypictraits
(Lande and Arnold, 1983). Intensityis not
a propertyof the fitnessfunctionalone, but
of both the functionand the distributionof
phenotypesin the population.
Natural selectionis definedas variation
in fitness,
and hence,a straightforward
measure of total selection intensityis variance
in relativefitnessitself,
V = Var(fj(z)).
1. Estimated selection intensityV in six episodes (cf. Fig. 3).
TABLE
Population
Female house sparrows
Galapagos finches
Female song sparrows
Female song sparrows
Male song sparrows
Male human infants
Fitness measure
Intensity
survival
survival
survival
reproduction
survival
survival
0.054
0.453
0.049
0.114
0.031
0.012
Variance in relativefitnessas definedhere
should not be confused with Var(w), the
variancein relativesurvivalor reproductive
success, known as the opportunityfor selection (Arnold and Wade, 1984). V is estimated as the squared coefficientof variation among individuals in the predicted
values/(z). The measurewill workfornonparametricas well as parametricformsoff
provided that the functionis based on the
entirepopulation or a random sample. Vis
similarto Manly's (1977) index, but is not
wedded to a particularselectionmodel and
is measured directlyon J(z) and not on a
transformedscale. The measure does not
reflectselectionon specificparameters(moments)of the phenotypedistribution,as do
those of Lande and Arnold (1983), and
hence, it cannot be used to predictevolutionarychangein theparameters.Rather,it
reflectsselectionon all the momentsof the
phenotype distributioncombined and is
purelydescriptive.
Estimated selection intensities for the
populationsillustratedin Figure3 are given
in Table 1. Values show weak selection in
human infantsand male song sparrowsand
relativelyintense selection associated with
survival in Gal'apagos finchesand with reproductionin female song sparrows.I did
not calculate V forthe guppydata (Fig. 2),
sincethedistributionofthephenotypictrait
in the males tested differedfromthat in a
random sample of males (Houde, 1987).
It is furtherpossible to calculate an intensityof stabilizing(or disruptive) selection by measurinzgthe contributionof a
mode (or dip) in f to the total selectionexperiencedby a population.The simplestapproach is to compare total selectionintensity V with variance in relative fitness
recomputedafterthemode is removed( VO).
The difference
between V and VOwill mea-
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858
DOLPH SCHLUTER
2. The intensity
ofstabilizing
selectionVsin
The above method can be generalizedto
twopopulations
(cf.Figs.3A,F).Totalselection
inten- other measures of selection and other feasityV is providedforcomparison.
tures
For example, the intensityof seTABLE
Measure
House sparrows
Human infants
Vs
0.027
0.056
0.000
0.012
V
off
lection associated with a second of two
modes can be estimatedby fitting
a function
constrainedto have only one. The amount
of "quadratic" selection in the population
-y = Cov(w,
Z2)
Cov(fW, Z2)
(Lande and
surethe fractionof thetotalvariance in rel- Arnold, 1983) attributableto the presence
ative fitnessamong individuals thatresults of a mode can similarlybe computed.
directlyfromthe mode:
Accuracy
VS= v- VO.
Finally, it is desirable to consider the
As with total selection intensity,the mea- samplingvariabilityin a nonparametricessureis purelydescriptiveand cannotbe used timateof the fitnessfunction.A simple way
to predictevolutionarychange.
tojudge variabilityis by usingthebootstrap
One way to remove a mode in f is to (Efron,1982; Efronand Tibshirani, 1986),
increasethevalue of the smoothingparam- whereregressionsurfacesare repeatedlyeseterX [Eq. (1)] untilthe observed mode has timatedon data resampledfromtheoriginal
barely disappeared. An advantage of this set.The performanceofthebootstrapin the
methodis thatifa mode is symmetricabout case of splineshas not yetbeen testedusing
2, theconstrainedfwillbe a horizontalline, simulation.Hence, bootstrapresultsshould
and Vs= V. A disadvantageis thatthevari- be consideredapproximateand interpreted
ances are not additive: V0 may exceed V cautiously.
and give negative values of Vs. I used an
Two hundredbootstrapregressionswere
alternativeapproach, takingthe predicted calculatedusingsurvivalin male song sparvalues 1(z) fromthe GCV estimateoff and rows(Fig. 3E). Each repetitioninvolvedtwo
the maximum-likelihoodmonotonic steps:i) therandomgenerationof a new WV
fitting
functionusingthe"pool adjacent violators" (0 or 1)^foreach observed z1,where Pr(W,
algorithmofBarlow et al. (1972) (of thetwo = 1) = J(zi) and whereJ(z1)is the predicted
monotonic functionsthat can be fit,non- value in theoriginalestimateoff ii) a search
decreasing and nonincreasing,choose the for the cubic spline minimizingthe GCV
one yieldinggreatervariance in predicted score. Standarderrorsforthe 200 resulting
relative fitness).One disadvantage of this predictedvalues for each z, are shown in
methodis thatifa mode is symmetricabout Figure 3E. Standard errors are not sym2 the constrainedf will not be flat,and Vs metricabout f since they were computed
< V (in practice, Vswill be roughlyV/2). on the logisticscale loge[fz)/{1 - J(z)}].
I calculated Vsin femalehouse sparrows
The bootstrapresultsillustratetwopoints.
and male human infants,which show al- First,estimatesoff may be quite variable,
ternativeextremeeffectsof a mode in f on althoughthis depends on sample size and
a population (Table 2). In house sparrows, theproximityofj(z) to theendpoints(0 and
the mode is nearlysymmetricabout 2 (Fig. 1). Nonparametric estimates have more
3A), and Vs is large relative to V (i.e., is samplingvariationthan parametricregresV/2),indicatingthat variabilityamong in- sion surfaces,and this is the price paid for
dividuals in fitnessresultedprimarilyfrom not knowinga priori the parametricform
stabilizingselection.In contrast,the mode of f Second, uncertaintyis greatestat the
in f is broad in human infants,and fitness ends of the phenotypedistribution,a conshowsa substantialdeclineonlyat thelower sequenceofa reduceddensityofdata points.
end of the phenotypedistribution(Fig. 3F). This is not a limitationspecificto the nonThus, Vs is small relative to V, indicating parametricmethod: in all regressions,the
thatstabilizingselectionwas veryweak and data are most informativewhere they are
that most variabilityin fitnessin the pop- most abundant. However, it is a featureof
ulation was the resultof directionalselec- many naturalphenotypedistributionsthat
tion.
the fitnessfunctioncan be estimatedwith
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THE FORM OF NATURAL SELECTION
accuracy over only a limited phenotypic
range.
The bootstrap can also be used to test
hypothesesconcerningfeaturesoff For example, thepresenceof a mode can be tested
by countingthe frequencyof repetitionsin
which the mode is present.In carryingout
such tests,it is wisest to restrictthe mode
searchto a predefinedregionof z values, as
modes may occasionally appear as a result
of samplingerrorelsewhere,particularlyin
the tails of the phenotypedistribution.For
example, I carried out 200 bootstrap replications of the guppydata (Fig. 2), recording the numberof modes infappearing betweenz = 10 and z = 26. The frequencyof
replicatesin which no mode appeared was
30%, suggestingthat a null hypothesisof
purely directional selection would not be
rejectedby these data.
When testingforaccuracyof functionsof
f and the data (e.g., variance in relativefitness), an alternativeresamplingprocedure
fromtheone suggestedabove shouldbe employed,one whichresamplesalso the z values, in orderto account forsamplingvariabilityin z.
DISCUSSION
Many questions about the workingsof
natural selection on quantitativetraitsare
directlyconcernedwith featuresof the fitness function (Lande and Arnold, 1983;
Endler, 1986). Do modes in fitnesscommonlyoccurwithintherangeofphenotypes
in the population?How frequentis disruptive selection?Are multiplepeaks present?
What is therangein fitnessin a typicalpopulation? Do functionsvary between life
stages,years,or geographiclocalities?I have
methodwhich
summarizeda nonparametric
can be generallyused to estimate fitness
functionsand, thus,could be used to answer
the above questions.
Regressionusingsplinescan be highlyrevealing, as demonstratedby insightsinto
some familiarand well analyzed cases of
naturalselection.The Galapagos finchGeospizafortiswas previouslythoughtto represent one of the few cases of disruptive
selection in nature (Schluter et al., 1985;
Endler, 1986), but the spline suggeststhat
selection was purely directional (monotonic).Analysisofthehuman infantdata of
859
Karn and Penrose (1951) gave the unexpected resultthat very littlestabilizingselectionwas presentand thatthe majorityof
variance in fitnessis attributableto directional selection.Other known cases of disruptive and stabilizingselection mightbe
reevaluatedin this light.Anotherinsightis
the range of fitnesseswithin populations.
The selectionevent in G. fortismay be the
most intense example from a vertebrate
(Boag and Grant, 1981; e.g., Table 1), yet
the estimated range in probabilityof survival was as high or higherin some other
species (Fig. 3). Strongselectionin G. fortis
resultedmainlyfroma lower mean fitness.
In theory,fitnessfunctionsmay assume
a varietyof shapes, and surfacespredicted
from ecological or biomechanical considerations often show a complex patternof
peaksand valleys(Schluterand Grant,1984;
DeAngelis et al., 1985). An advantageofthe
nonparametricmethod is that estimatesof
the functionare not restricteda priorito a
particularmodel ofselection-one thatusuallyassumes thepresenceofat mosta single
mode. If functionsin facthave a complex
patternof peaks and valleys, then a nonparametricapproach is the best way to reveal it. However, the presentmethod will
also estimate simple formsoff and could
be used to suggesta parametricmodel for
thefunction,ifone is desired.Several ofthe
estimatesoff shown in Figure3 mightsubsequentlybe fitto a simpleparametricmodel (e.g., normal, exponential, logistic, or
quadratic).
Though the splineis flexible,estimatesof
fitnessfunctionsbased on measurementsof
natural selectionwere usually simple (Fig.
3), involving only directional and/or stabilizing selection. This simplicityof observed formsof f, when complex patterns
are possible and even expected,may result
fromsmall sample size in some cases. It is
simplynotpossible to reveala complexpatternwhen the densityof data points available to estimatea local featureoff is small.
Also, if the fitnessfunctionis indeed complex and interesting,the range of phenotypesin the population may be too narrow
to detectit. Most of my example data sets
of variaare frombirds,where coefficients
tion for morphologicaltraitsare typically
less than 5%. Frequency of individuals is
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DOLPH SCHLUTER
860
generallylow toward the ends of the phenotype distribution,and this furtherrestrictsthe range of phenotypevalues over
whichaccurateinformationmightbe gained
about the formoff (Fig. 3E).
If the formoff is of interestand if levels
ofphenotypicvariationare low, researchers
might consider alternative strategies,including deliberateexperimentalmanipulationsof the mean and variance of thequantitative trait. Examples of this approach
include manipulations of clutch size (e.g.,
Lessels, 1986; Boyce and Perrins, 1987;
Finke et al., 1987) and tail length(Andersson, 1982) in birdsand ofleafarea in plants
(Willsonand Price,1980). A populationwith
inflated phenotypicvariance would be a
usefulexperimental"probe" ofthe selective
environment,allowingestimationoff over
a broad range. It would also allow calculationofmean fitnessWas a functionofmean
value of the character,usefulin predicting
theoutcomeofselectionand evolution.Manipulateddistributionsshould be relatively
flat(uniform)ratherthan normal, so that
equally reliable informationcan be gained
over mostoftherangeofz. Some instability
at the tails of the distributionwill nonetheless be present,owing to the necessarydecline in the numberof points fallingwithin
any fixedregionof z as the edge is included.
This effectcan be minimizedby furtherinflatingthe frequencyof individuals at the
tails.
The above strategywould assume that
natural selection mightreasonably be expected to occur within the timespan of a
typicalecological study.The large number
of cases of natural selection that have alreadybeen observedinvolvingonlynatural
levelsofphenotypicvariation(Endler,1986)
is highlyencouraging.Indeed, the realization that detectable natural selection is a
frequentevent in ecological time may herald a new era of controlled experimental
studies of the process. In this context,the
presentmethods to estimate fitnessfunctions should prove especiallyuseful.
ACKNOWLEDGMENTS
Thanks to N. Heckman and T. Price for
manyhoursofdiscussionand to D. Nychka
and B. Yandell forprovidingthe computer
programs.P. Grantand J.Smithkindlypro-
vided the data on Galhapagosfinchesand
song sparrows. T. Price suggestedthe example in Figure1. Thanks also to S. Arnold,
P. Boag, P. Grant,N. Heckman, T. Mitchell-Olds, D. Nychka, T. Price, G. Pogson,
J. Smith,and C. Wehrhahnforcomments
on the manuscript.This workwas supported by the Natural Sciences and Engineering
Research Council of Canada (#U0459).
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