Copyright 0 1992 by the Society of America

Comparing and Variability of Quantitative Traits

David Houle’ Department of , North Carolina State University, Raleigh, North Carolina27695, and Department of Ecology and , University of Chicago, Chicago, Illinois 60637 Manuscript received February 20, 1991 Accepted for publication September 12, 199 1

ABSTRACT There are two distinct reasons for making comparisons of for quantitative characters. The first is to compare evolvabilities, or ability to respond to selection, and the second is to make inferences about the forces that maintain . Measures of variation that are standardized by the trait mean, such as the additive genetic coefficient of variation, are appropriate for both purposes. Variation has usually been compared as narrow sense , but this is almost always an inappropriate comparative measure of evolvability and variability. Coefficients of variation were calculated from 842 estimates of trait means,variances and heritabilities in the literature. Traits closely related to fitness have higher additive genetic and nongenetic variability by the coefficient of variation criterion than characters under weak selection. This is the reverse of the accepted conclusion based on comparisons of . The low heritability of fitness components is best explained by their high residual variation. The high additive genetic and residual variability of fitness traits might be explained by the great number of genetic and environmental events they are affected by, or by a lack of stabilizing selection to reduce their phenotypic variance. Over one-third of the papers reviewed did not report trait means or variances. Researchers should always report these statistics, so that measures of variation appropriate to a variety of situations may be calculated.

VOLUTIONARY and ecological geneticists use single generation, and R is the response to selection E relative measures of genetic variation to gain in the following generation (FALCONER1989; TURELLI insight into two sorts of questions. First, we would and BARTON 1990). like to be able to predict the ability of a to Heritability has been widely used to compare the respond to natural or artificial selection; this I will genetic variation of life history traits relative to traits term “evolvability.” Second, wewish to gain insight presumably subject to less strong selection (GUSTAFS- into the strength of the forces which maintain and SON 1986; ROFFand MOUSSEAU1987; MOUSSEAUand depletethe genetic variation on which evolvability ROFF 1987; FALCONER1989). The robust result of depends. For this, we need measures of the relative such comparisons is that life history traits have lower “variability” of traits. heritabilities. The conclusion drawn from such com- At the phenotypic level, we want to know about the parisons has typically been that life history traits in potential for evolutionof the mean . In the some absolute sense possess less genetic variation,and short term, this depends on the additive geneticvari- will evolve less readily than other traits. ance. An essential step in quantitative genetics is the In this paper I will show that trait means, rather partitioning of trait variance, Vp, into additive genetic than Vp are appropriate for standardizing geneticvar- variance, VA, and a remainder, V,, consisting of vari- iances for comparative purposes. To investigate some ance due to other genetic and environmental causes of the implications of mean-standardized comparative (FALCONER1989). Narrow-sense heritability, h2 = studies, I have reviewed over 200 quantitative genetic VA/Vp, is almost always used as a summary measureof studies of animal . When mean-standard- this partitioning. In part, this is because h2 plays an ized measures are compared, traits closely related to important role in determining the response to selec- fitness are more variable than morphological traits, tion as shown by the familiar equation the opposite conclusion to studies based on heritabil- ity. This has important implications for theoriesof the R = h2S, (1) maintenance of geneticvariation. Unfortunately, a very high proportion of quantitative genetic studies where S is the selection differential, the difference in report h2, but not trait means or variances. No sum- population mean before and after selection within a mary measure of variation, h2 or a mean-standardized ’ Address after March 1, 1992: Department of Biology, University of , can substitute for basic statistics about astudy Oregon, Eugene, Oregon 97403. population.

(;rnetics 130 195-204 (January, 1992) 196 D. Houle EVOLVABILITY A good measure of evolvability is the expected response to selection. Looking at Equation 1, it is I is related to the phenotypiccoefficient of variation, natural to interpreth2 as a sufficient descriptor of the CVP = lOO&/x. Application of Equation 1 yields a state of a population for comparing evolvabilities. It version of FISHER’S“fundamental theorem of natural fails to fulfill this purpose for two reasons. First, while selection (FISHER1930; PRICE1972; FALCONER1989): h‘ is a dimensionless quantity R is not. Since, for comparative purposes, we are most often interested Rw = vA/w. (8) in proportional, rather than absolute change (HAL- Standardizing this response to selection by the mean DANE 1949), the potentialresponse needs to be stand- yields an appropriate measure of the relative evolva- ardized. Onestraightforward way of doing so is using bility of fitness the population mean. Second, the selection differential S is not solely a function of the fitness function,but also contains informationabout the selected population.For ex- ample, in the case of truncation selection on a nor- At equilibrium, the predicted gains of Equation 8 mally distributed character, S = i6,where i is the would be offset by processes, such as and “intensity” of selection, determined only by the pro- segregation, which restore both theoriginal mean and portion of the population selected for breeding (FAL- genetic variance. In this case ZA is also an appropriate indicator of the variability of fitness, as large values CONER 1989). For a given i, the higher the standard deviation of the selected population, the more the must be the result of powerful forces maintaining selected individuals may differ from the original pop- genetic variance. ulation mean. An appropriate mean-standardized pre- Equations 8 and 9 also hold for all multiplicative components of fitness (O’DONALD 1970; ARNOLDand dictor ofevolvability under truncation selection of WADE 1984), although forsuch characters the inter- unknown intensity is therefore pretation is complicated by the potential for correla- tions among them. Large values of ZA could also be the result of negative genetic correlationsamong Here, and throughout this paper, x represents the traits, which attenuate the effects of selection pre- population mean before selection. Ingeneral, for dicted for each character separately. Negative corre- fitness function W(X)and trait distribution F(X), lations are, however, not necessarily to be expected m (CHARLESWORTH 1990; HOULE 1991),and the empir- s = IT-’ J (X - Z)W(X)F(X)dX (3) ical evidence fortheir generality is weak (HOULE “m 1991). (KIMURAand CROW 1978) where A second case where h2 is a misleading indicator of evolvability is that of weak gaussian optimizing selec- w = lIW(X)F(X) dX, (4) tion the mean fitness of the population. If W(X) can be approximated as a polynomial of order n, S will be a function of the first n + 1 momentsof F(X).One must where u2is a parameter inversely proportional to the specify theform that selection will takebefore an strength of selection for the optimum 8. Assuming appropriate index of evolvability can be chosen. that X is normally distributed, In two important cases h2 is a particularly mislead- VP VAVP ing estimator of evolvability because S is a function of R=(O-X)-- vp+ u2 vp Vp. One such case is when X is fitness (W(X) = X), making (BULMER1980, p. 150).What data is available on the strength of stabilizing selection suggests thatfre- Jmm (X2 - XX)F(X) dX S= quently u2>> Vp for morphological characters (TUR- J-2XF(X) dX ELLI 1984), making the response to changes in 0 approximately independent of Vp.Then the standard- ized response to selection is

Standardizing by the mean yields a quantity CROW (1 958) called the “opportunity forselection”: Com paring Genetic Variabilities Genetic Comparing 197 A good measure of the evolvability of the population h2, will decline to zero (CHARLESWORTH1987). How- in response to arbitrary changes in 0 is then VA/x. ever, fitness components do in general possess VA While (6 - x)/(Vp+ 0‘) is clearly important in deter- (ISTOCK1983; ROFFand MOUSSEAU1987; MOUSSEAU mining the actual response to selection, its future and ROFF1987), which is not too surprising given the value is not estimable and of no predictive value. variety of potentialperturbing forces (CHARLES- In both of these important cases, directional and WORTH 1987; MAYO, BURGERand LEACH1990). In weak stabilizing selection, a variance to mean ratio is the presence of additive genetic variance for fitness, the most appropriate measureof evolvability of a trait. the fundamental theorem makes no prediction about While IA and VA/xare distinct quantities, they are h2, as FISHERand others have made clear (FISHER correlated, as they share the same components. This 195 1 ; BARTONand TURELLI1989; PRICEand SCHLU- argues that quantities related to the additive genetic TER 1991). FISHER’Sfundamental theorem shows that coefficient of variation, CV, = 100Jva;/, are useful the phenotypic variance of fitness itself is irrelevant for comparing evolvability of some traits, although to its response to selection, and thus thetheorem certainly not all. In contrast, even in the special case makes no prediction concerning residual variance, VR. of truncation selection, h2 is not a good predictor of Reviews of the literaturehave convincingly confirmed the response to selection (Equation 2). the fact that heritabilities of fitness components are lower than those of other traits (GUSTAFSSON1986; VARIABILITY ROFF and MOUSSEAU 1987; MOUSSEAUand ROFF 1987; FALCONER1989), but this can be explained The evolution of genetic variance must be the result either by relatively low VA for fitness components or of a balance between evolutionary processes, but in by high VR, or a combination of the two. Coefficients general we are quite ignorant of the relative impor- of variation provide a reasonable scale on which this tance of such processes, as evidenced the debate by question may be addressed. over the maintenance of quantitative variation (BAR- The value of coefficients of variation for measuring TON and TURELLI1989; HOULE1989; BARTON1990). variability depends on the degree to which they cor- Therefore, it is not possible to derive precisely satis- rect for whatever relationships exist between means fying measures of variability analogous to those for and variances. Previous studies of phenotypic var- evolvabilities. Indeed, comparisons of variability data iation show that there is frequently a residual rela- can be used to gain insight into these processes, by tionship between means and coefficients of varia- comparing populations in which selection, migration, tion (FISHER1937; WRIGHT1968; YABLOKOV1974; and drift differ in importance. When we casually judge a character to be phenotyp- ROHLF,GILMARTIN and HART1983). Therefore, scale ically variable, we implicitly standardize by the mean. effects must be corrected forin analysesof coefficients Therefore CVA seems a reasonable dimensionless cri- of variation, as in any comparative analysis of varia- terion for comparing genetic variabilities. CHARLES- bility. Three potential causes for such relationships WORTH (1 984, 1987) has previously used CVAto com- have been identified. First, ROHLF,GILMARTIN and pare genetic variabilities of Drosophila populations HART(1983) show that characters with small means for this reason. Note, however, that a difference in tend to be measured withless accuracy than those coefficients of variation between traits may be due to with large means. This causes a negative relationship an enormous diversity of factors which may be very between means and CVs based on . Second, difficult to disentangle. For example, high standard- ROHLF,GILMARTIN and HARTalso show that CVs of ized phenotypic variability might suggest a lackof meristic traits are negatively correlated with the num- strong stabilizing selection. However, it is also possible ber of categories in the population, quite independent that a highly variable character is by its nature subject of any measurement errors. This predicts that both to moreenvironmental variance. A further set of genetic and phenotypic CVs will be negatively corre- potential complications applies to genetic variability, lated with the means of meristic traits. where it is possible thatunderlying factors which The other known source of relationships between increase or maintain variance, such as mutation or means and coefficients of variation is the relationship balancing selection, are correspondingly strong. of the variances of partsand wholes. If theparts An example of the sort of question that coefficients combine multiplicatively to produce thewhole, LANDE of variation (CVs) can shed light on concerns the low (1977) has shown thatthe covariance of the parts heritability of fitness components. Many evolutionary determines the sign of the covariance between means geneticists have assumed that this is predicted by and coefficients of variation. Positively correlated FISHER’Sfundamental theorem (GUSTAFSSON1986; parts cause positive correlations of means and coeffi- ROFF and MOUSSEAU 1987; MOUSSEAU and ROFF cients of variation. LANDE points out that since mor- 1987; FALCONER1989). In the absence of perturbing phological traits are generally positively correlated, forces, Equation 8 does predict that VA,and therefore CVs of volumes will be larger than those of areas, 198 D. Houle which will be larger than those of linear dimensions. to be far fromgenetic equilibrium if they had recently Trends based on dimensionality arise for non-biolog- been founded from inbredlines or very smallnumbers ical reasons, and so ought to be factored out before of individuals, had been subjected to artificial selec- any analyses of variability. On the other hand, a life tion, or were formed by crossing or mixing individuals history trait, such as lifetime fecundity is in part due from more than one population. However, popula- to the products of age specific mortalities, thus we can tions many generations beyond such events were in- also predict that the CV of lifetime fecundity might be cluded. Some Drosophila and Tribolium laboratory higher than that of fecundity over shorter periods, if populations were included as a result of this exception. the mortalities are generally positively correlated. To meet criterion (iv), only studies which partitioned Such relationships are biologically more interesting. phenotypic variance using half-sib correlations, par- If the parts combine additively to make up thewhole, ent-offspring regression, or response to initial gener- then the CV of the whole equals that of the parts only ations of selection were used. Studies using inappro- if the parts are perfectly correlated, and less than that priate nonrandom samples of individuals were also of the parts otherwise (LANDE1977; BRYANT1986). excluded. Maternal-offspring regressions were in- BRYANTuses this to explain decreasing CVs through cludeddespite the possibilityof some bias due to parts of . maternal effects. Estimates of the same parameters by different methods were combined before analysis, as VARIATIONDATA BASE described below, to help mitigate the effects of differ- ent estimation methods. To reexamine evolvabilities and variabilities on the Criterion (v) is necessary because thearguments basisof mean-standardized measures, I reviewed advanced above forthe use of coefficients of variation quantitative genetic studies of animal populations. do not apply on alternative scales of measurement. The data set was restricted to those estimates which For example, any transformation of fitness invalidates met five criteria: (i) the characters may be assumed to the simple prediction of Equation 8, since the fitness be under either strong directionalor weak stabilizing function would also need a corresponding transfor- selection; (ii) the population studied may be assumed mation. In addition, calculating coefficients of varia- to be near genetic equilibrium; (iii) the population tion on scales which potentially yield negative as well studied was outbred; (iv) the additive genetic variance as positive means, such as log scales, renders coeffi- was estimated using relatively unbiased methods; and cients of variation meaningless. (v) the variances are reported on an untransformed From 2 13 quantitative genetics papers reviewed, scale. Some of these criteria require further clarifica- 153 included narrow-sense h2 estimates for characters tion. which met these criteria. Of these, 61 studies did not First, I chose characterswhere there is a clear contain either charactermeans or variances, or lacked expectation of the kind of selection pressure the char- both, precluding calculation of the appropriate coef- acter is subjected to.Traits assumed tobe under ficients of variation. From the 92 remaining papers, directional selection, which I term fitness traits, in- 842 acceptable sets of estimates of means and vari- clude many life history traits, and traits highly corre- ances were extracted. A substantial number of esti- lated withsize during growth, such as weight at a mates from within these papers were also excluded particular age. Some life-history characters, such as for the set of reasons detailed above. The data, and development times of univoltine insects, were ex- references to acceptable and excludedpapers, are cluded from theanalyses because they are morelikely available from the author in printed form, or on disk to be subject to optimizing selection. The traits which as an ASCII, Quattro Pro orLotus file. were assumed to suffer weak optimizing selection are Of the 201 studies cited by ROFF and MOUSSEAU primarily morphological traits which are not directly (1987) and MOUSSEAU and ROFF (1987) which in- a function of growth rate. These are principally sizes cluded “life history” or “morphological” characters, I of adult body parts, or meristic traits such as bristle was able to examine 147. Of these, 108 met criteria numbers. Many of the studies included in this data set i-v, and 65 included the necessary means and vari- were also reviewed by ROFF and MOUSSEAU(1987) ances to calculate coefficients of variation. The 27 and MOUSSEAUand ROFF (1987). My definition of otherpapers I include consist primarily of papers directionally selected and weak optimally selected published between 1985 and 1990, and laboratory traits roughly corresponds to their definitions of “life studies of Tribolium. history” and “morphological” traits, with the excep- tion of the sorts of traits noted above. Behavioral and ANALYSES physiological traits were excluded from this analysis because there is rarely a clear expectation about their Three potential measures of variability were calcu- relationship to fitness. lated from these data: narrow sense h2, CVA,and the To meet criterion (ii), populations were presumed residual coefficient of variation Com paring Genetic Variabilities Genetic Comparing 199 high evolvabilities, as well as high h2,but wing length, CvR = 100 -/x. (1 3) which has similar h2, has far lower evolvability in each When the VAestimate was negative, h2 and CVA were case. Comparison of CVA and CVR show that this is due both set equal to zero. Analyses which included the to approximately fourfold lower mean-standardized negative estimates (with CVA replaced by ZA) yield variability. Similarly, fecundity generally has a high similar conclusions to those reported below. All three evolvability, because of its high additive genetic vari- measures are far from normally distributed. ance, in spite of relatively high residual variance. For analysis traits were classified on the basisof Corrections for dimensionality: As noted above, their relationship to body size as well as the sort of the variances of traits are expected to be a function selection they are assumed to experience. Direction- of their dimensionality. Only S and G classes contain ally selected traits which represent sizes of whole substantial number of traits with differentdimen- organisms or body parts duringgrowth are designated sionalities in this data set. In these classes, analysis of “G” traits, while the remaining traits, primarily life covariance of means and CVs on a log-log scale shows history traits, are designated “L.” Similarly, weakly that the slopes are homogeneous, but the intercepts selected traits which represent sizes of adult body parts are significantly differentamong dimensionality are designated “S” traits. The remainingtraits are classes for both CVA and CVR (P < 0.001). In each primarily meristic in nature, and are designated “M” case, the differences among intercepts are not signif- traits, although a few non-meristic traits which were icantly differentfrom the 1:2:3 relationship (lin- demonstrablyuncorrelated with adult size are also ear:area:volume) expectedif the linear dimensions are included in this category. perfectly correlated (YABLOKOV1974; LANDE1977). Many of the 842 sets are repeated estimates of the Therefore, both CVs were divided by their dimen- sameparameters by differentmethods. Therefore sionalities before further analysis. median heritabilities and coefficients of variation for Variability comparisons: The possibility of nonin- the same trait in the same population, sex, and envi- dependence in this data set exists because alarge ronment were calculated before further analysis. This proportion of the estimates come from a few species reduced the number of estimates to 400. (44% from D.melanogaster) and a few studies (the five To correct for scale effects I use a “robust” regres- largest studies contribute 31 % of the estimates). If sion algorithm which minimizes the absolute devia- species differ in the parameters being estimated, or tions from the regression line (PRESSet al. 1989, pp. individual studies estimate parameters with bias, the 595-597), rather than the conventional least squares overall tests may be misleading. Therefore, analyses or major axis methods. The lack of normality of the were carried out on medians of variables calculated data results in undue weight being given to outlier within four different groupings of the data. In order points under least squares, and renders significance of decreasing sample size these are medians of vari- tests invalid. Instead, significance tests of the robust ables within (1) species, phenotype and study (the full regression were constructed from the proportion of 400 estimates); (2) species and phenotype; (3) study positive slopes underbootstrap resampling, with a and class of trait; and (4) species and class of trait. sample size of200. Differences in slopes and intercepts For grouping (l),there were significant regressions among groupswere assessed using randomization tests of h2, CVR, and CVA on trait means. For the other with a sample size of 1000. The probability values three groupings, thereare significant relationships represent the proportionof randomizations wherethe between mean, and CVR and CVA. In each case, the absolute value of the parameter difference exceeds estimated slopes are positive for h2 and negative for that based on theoriginal data. Comparisonwise error CVR.This may suggest that VRmay be disproportion- rates were adjusted for multiple comparisons using ately inflated at small scales due to measurement or the Sidik method (SOKALand ROHLF198 1, p. 242). scoring errors. CVA has a negative relationship with mean which is most easily explained by the part to COMPARISONS OF VARIATION whole relationships described under VARIABILITY Evolvability comparisons: As an example of the above. usefulness of mean-standardized measures of evolva- These significant regressions require that analyses bility, Table 1 presents variability measures for five of differences among trait classes should proceed in a commonly studiedcharacters in Drosophilamelano- manner analogous to analysis of covariance. To com- gaster. The last three columns present standardized pare thevariability of two classes of traits, Ifirst tested evolvabilities assuming threedifferent selection re- whether their slopes were homogeneous. I thentested gimes: truncation, linear directional, andgaussian op- whether the residuals from a jointregression differed timizing wherethe population mean is not atthe in location. This approach yields ambiguous results if optimum. In nocase is heritability a good indicator of the slopes differ and the regression lines cross within evolvability. Sternopleural bristles do generally have the range of the data, or athigher means. For h2 and 200 D. Houle

TABLE 1 Evolvabilities of representative traits in D. melanogaster

Evolvability

Shift Truncationoptimum Linear Trait N" h' CVR CV, VAAZfi) I, x 10' V*IZ Sternopleural bristles 7.9721 0.44 8.39 0.061 0.70 0.13 Wing length 31 0.36 2.09 1.56 0.009 0.02 0.05 Fecundity 12 0.06 39.02 1 1.90 0.035 1.42 5.38 Longevity 7 0.1 1 27.73 9.89 0.033 0.98 0.19 Development time0.28 2 4.48 2.47 0.012 0.06 0.12

a Number of studies median estimates calculated from.

CV, there is a strong expectation that they will be influenced by measurement error atsmall scales, so it might be that the class with the more positive slope would have larger parameter values in the absence of error. However, since these data include traits meas- ured on a wide variety of scales, and measurement 0.5 error is not the only reason for a relationshipbetween means and the other parameters, I prefer to regard such results as inconclusive. As an example, the data for grouping 2, medians 0.0 within phenotype within species, are graphed in Fig- A ure 1. Since thetraits measured spanned so many Ib I orders of magnitude in their means and coefficients of variation, I log transformed these variables in all groupings of the data before analysis to increase the influence of points near the centerof the distribution and aid the presentation. Similar results were obtained from analyses of untransformed data. The analysisof grouping 2 is presented in some detail in Tables 2-4. In this grouping, for all three parameters, none of the individual trait class regres- 2.0 A sions are significantly differentfrom 0 (resultsnot C shown). The analyses of h2 in Table 2 show that none of the tests for homogeneity of slopes are significant when adjusted for the fact that six comparisons are made. Comparisons of the residuals from common regressions confirms results previously found for h2 data (ROFFand MOUSSEAU1987; MOUSSEAUand ROFF 1987). S traits have significantly higher h2 than G or L traits, and M traits higher h2 than L traits. I Table 3 presents analyses of CVR,and hereopposite -2-1 0 123 4 results are obtained. As may also be seen in Figure loglo mean 1b, L traits have the highest CVR, while S traits per- haps the lowest. Comparison of S and M traits yields FIGURE1.-Plot of h', CVR and CV, us. trait means for each of an ambiguous result. The results for CV,, presented the four trait classes. Points are medians forphenotypes within in Figure ICand Table 4 are similar to those for CVR. species. The lines are the robust regressions within trait classes. S traits have significantly lower CV, than each of the other three classes. ties for discrimination among classes. In almost all A summary of the results obtained from data of all cases the above conclusions that trait classes with high groupings is presented in Table 5. These results are h2 are the trait classes with the lowest CVs are con- similar to those for grouping 2, although the group- firmed. When using CVs as an indicator of variability, ings with smaller sample sizes allow fewer opportuni- there is certainly no evidence that traits which are ComparingVariabilities Genetic 20 1

TABLE 2 TABLE 5 Comparisons of h4among trait classes for data grouping 2 Summary of trait class comparisons for data groupings 1-4

Class of trait M G L Parameter S Grouping N h' CV, CVA Slopes differ? NS NS t 1 400 S residual 0.073 0.095** 0.159** 1: G M?S S?M G 4 EM-G L Column residual 0.010 -0.098 -0.125 2 148 EMS s?#GL SMGL M -- - Slopes differ? NS NS 3 102SMGL LGMS M residual 0.0997 0.165** Column residual -0.020 -0.056 4 50 LGMSgGL SMG G - Slopes differ? NS G residual 0.0807 Trait classes are listed in order from smallest to largest. Hori- Column residual -0.009 zontal lines connect classes which are not significantly different at P < 0.05 (adjusted for multiple comparisons) by the analysis of The first row in each cell reflects the probability that slopes do covariance-randomization tests described in the text. Order of not differ between groups. The numbers in the second and third classes within homogeneous sets does not necessarily reflect relative rows are means of residuals from a common regression on the row ranking. A question mark between classes represents an ambiguous and column class data. The upper value is for the row class, the result where classes have significantly different slopes, andthe lower for the column class. Significant differences among residuals regression lines cross within or above the range of the data. N is are indicated in row 2. the sample size of each data grouping. tP< 0.05, uncorrected for simultaneous tests; * P < 0.05, corrected for simultaneous tests; ** P < 0.01, corrected for simul- TABLE 6 taneous tests; NS, not significant. Spearman ranktorrelations among variability parameters

TABLE 3 Param- eter CV, CVA 1, V*b Comparisons of CV, among trait classes for data grouping 2 h2 -0.5266****0.0637 0.0650 0.1879*** Class of trait M G L CVH 0.3030****0.7167****0.7217**** CV* 0.5754**** 0.9980**** S IA 0.5793**** Slopes differ? * NS NS ***P

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