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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 Stable URL: http://www.jstor.org/stable/2408904 . Accessed: 27/05/2014 08:12 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] . Society for the Study of Evolution is collaborating with JSTOR to digitize, preserve and extend access to Evolution. http://www.jstor.org This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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 This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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 This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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, This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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." This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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) = This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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, This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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. This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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). This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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- This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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 This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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 This content downloaded from 130.241.163.102 on Tue, 27 May 2014 08:12:44 AM All use subject to JSTOR Terms and Conditions 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. 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