Linearity Versus Nonlinearity of Offspring-Parent Regression: an Experimental

Linearity Versus Nonlinearity of Offspring-Parent Regression:An Experimental Study of Drosophila melanogaster

A. Gimelfarb and J. H. Willis

Department of Biology, University of Oregon, Eugene, Oregon 97403 Manuscript received November 20, 1993 Accepted for publication June 9, 1994

ABSTRACT An experiment was conducted to investigate the offspring-parent regressionfor three quantitative traits (weight, abdominal bristles and wing length) in Drosophila melanogaster. Linear and polynomial models were fitted for the regressions of a character in offspringon both parents. It is demonstrated that responses by the characters to selection predicted by the nonlinear regressions may differ substantiallyfrom those predicted by the linear regressions. This is true even, and especially, if selection is weak. The realized heritability for a character under selection is shown to be determined not only by the offspring-parent regression but also by the distribution of the character and by the form and strength of selection.

HE following well known formula of quantitative with the wing tip and the anteriorcrossvein. KELLER and T genetics is customarily used to predict theresponse MITCHELL(1962) found no evidence of either domi- to selection by a quantitative trait: nance or epistasis for the wing length (they measured the distance from the humeral plate at the base of the = h2S, (1) R wing to its tip). where R is the response, S is the selection differential The following notations for the traits will be used in and h2 is the narrow sense heritability. Implicit in this the paper: WTD and WTS for weight of daughters and formula is the assumption that the offspring-parent re- sons, BD and BS for the numberof bristleson daughters gression is linear. Thisassumption is rarely questioned. and on sons, WGD and WGS for the wing length in Yet, FRANKHAM(1990) has compiled data illustrating the daughters and in sons. asymmetry ofresponses to directional selection on com- ponents of reproductive fitness indicating apossibility of MATERIALS AND METHODS nonlinearity in the offspring-parent regression for such The data on the weight and abdominal bristles were col- traits. It has also been demonstrated theoretically that lected from a base population originated from a stock pro- dominance of allelic effects (BULMER1985; GIMELFARB vided to us by BRIANCHARLESWORTH. The population was maintained in 10 bottles (approximately 150 flies per bottle) with 1986a) or skewness in the distribution of the environ- a mixing every two weeks of adult flies between the bottles as mental component (NISHIDAand ABE 1974) can cause a described by ROSE and CHARLESWORTH(1981). A batch of 10 nonlinearity in the regression. There are,undoubtedly, families was initiated by collecting 20 virgin females and 20 other hereditary and developmental mechanisms that males (grandparents) from the base population and mating can render the offspring-parent regression nonlinear. them randomly (one pair in a vial). On the 14th day after mating, 10 virgin females and 10 males (onlyone fly from each The purpose of the current study was to investigate vial) were collectedand kept separately. At the age of two days directly the linearity (or nonlinearity) of the offspring- these twenty flies (parents) were weighed, their bristles were parent regression for a number of quantitative traits in scored, and they were randomly mated(one pair in a vial). On Drosophila melanogaster, and to see how nonlinearities, the 14th day after mating, two virgin offspring of each sex were if found, affect our inferences about responses by the collected from each vial. Like their parents, they were kept separately, and their weight and bristle scorewere obtained on characters to selection. The following three traits were the second day of their life. The total of 22 batches were es- involved in the study: tablished, out of which 176 families with complete data have Weight (in milligrams). For this trait, KF,ARSEY and been recovered. The complete data for a family included KOJIMA(1967) have presented evidence ofweak epistasis. weights and bristle scoresof the mother, the father, two sisters and two brothers. No special effort was made tocontrol for the Abdominal bristles (on segments 4 and 5 in females density other than always removing parents from a vial on the and onsegments 3 and 4 in males). KELLER and MITCHELL 8th day and collecting offspring on the 14th day. All batches

(1962) found evidence of weakdominance andepistasis were maintained at 24( ?l)O. Collections were made using for this trait. CLAWONet al. (1957) attributed 9% of the CO,, whereas for weighing and bristle scoring the flies were total phenotypic variance by the abdominal bristles in a etherized. laboratory population to “genetic complexities.” The data on the wing length of parents and their offspring (two sisters and two brothers) in 159 families were given to us Wing length(in millimeters) measured as the distance by JERRY COYNE.They come from the experiment 3 (parents between the intersections of the third longitudina1 vein and offspring reared in the laboratory) reported by COWEand

Genetics 138: 343-352 (October, 1994) 344 andA. Gimelfarb J. H. Willis

TABLE 1 Parameters of parental and offspring distributions

~~~ Weight (mg) Bristles Wing (mm)

Mean SD Mean SD Mean SD

Mothers0.112 1.202 3.92 45.18 0.058 1.540 Daughters0.127 1.075 2.87 45.21 0.056 1.514 Fathers0.075 0.862 3.35 38.06 1.354 0.046 Sons 0.813 0.068 37.83 0.0442.74 1.355

BEECHAM(1987), and these authorscan be consulted for the is not significant (P = 0.3). There is a significant dif- description of the data and experimental procedures. ference (P<0.05) between the regressions of daughters Table 1 shows the mean and standard deviation for the characters under consideration. Thereis a significant difference in and of sons for all traits. the mean of each character between sexes. A significant dif- As has been mentioned earlier, thecoefficient of lin- ference existsalso between parents and offspring in their ear regression of offspring on the mid-parental value is mean weight: the offspring weigh less, on the average, than used as an estimate of the heritability defined as the ratio parents of the same sex. This can beattributed to the fact that of the additive component of variance to the total phe- theparental parents (grandparents) were notexposed to ether, whereas the offspring’s parents were etherized twicein notypic variance. If a character is standardized, this co- order to weigh themand score their bristles.It has been dem- efficient is equal to the sum of the coefficients in (3). onstrated by GIMELFAREIand WILLIS(1988) that exposing flies Hence, for thestandardized characters discussed in the to ether reduces theweight of their offspring.Also, the mean paper theheritability estimated by the regression on the wing length in daughters and mothers differ significantly,but mid-parental value is an explanation for this is not known. The offspring-parent regressions were fitted separately for h2 = b, + b/. (4) daughters and for sons. The charactersof the two sibs of the same sex were averaged so that an “offspring character” ina The estimated heritabilities are shown in Table 2. Also regression is in fact the arithmetic average of the character shown in the table are thecoefficients of determination, among the two sibs. Also, all characters (in parents as well as R2.This parameter measures the proportion of the vari- in offspring) were standardized prior to fittinga regression by subtracting the mean and dividing over the standard deviation.ance among the offspring explainable by thecorre- Hence, each character ina regression has zero mean andunit sponding regression on parents. The higher R2is, the variance. Statistical analyses were performed using the soft- more accurate prediction about the character in off- ware SYSTAT (WILKINSON1990) on a PC 486 computer. spring can be made based on the charactersof parents. Nonlinear offspring-parent regressions RESULTS AND DISCUSSION Linear offspring-parent regressions Before fitting nonlinear regressions we have excluded observations that were outliers in the linearregressions The coefficient of linear regression of the offspring’s (the coefficients reported in Table 3 are, in fact, with the character on the characterof onlyone of the parentsor outliers left out). An observation was considered anout- on the mid-parental value is often used to estimate the lier if the actual value of the characterdiffered by more narrow sense heritability and, consequently, to predict than 2.5 SD from the value predicted by the regression. the response by the character toselection based on for- It is usually recommended in text books on linear re- mula (1).In reality, though, theresponse is determined gression to check for outliers as a possible indication of by the regression on the charactersin both parents.For the nonlinearity in the regression. We have decided, this reason, we fitted the regression however, to be conservative and tomake sure that a nonlinearity, if found, is not due to just few outliers but is, f(x,y) = x, yl (2) indeed, a “solid” nonlinearity. Therefore, the nonlin- of the character in offspring, z, on the character of the earities that we have found probably underestimate the mother, x, and of the father, y. Table 2 shows the CO- nonlinearities in the actual offspring-parent regressions efficients of linear regression, for the characters under consideration. In the three-dimensional space, the linear regression f(x, y) = b,x + by, (3) function f(x,y) in (3) represents a plane. There are fitted to the family data for all of the characters. There many different biological mechanisms that can make is no constant term in the regression function since the the offspring-parent regression nonlinear so that the characters in parents and in offspring arestandardized. surface f(x, y) becomes curved rather than plane. For All of the regression coefficients in the table are signifi- the characters inour study we do notknow whatmecha- cant at P = 0.001 level, except for bfin WTD and WTS nisms can be atwork making the regressions nonlinear. which are significant at P = 0.01, and b, for WTS which Moreover, it is almost certain that different mechanisms O ffspring-Parent Regression Offspring-Parent 345

TABLE 2

Linear offspring-parent regressions

Trait

WTD WTS BD WGS BS WGD

0.35 0.29 0.23 0.45 bm 0.29 0.08 0.27 0.42 0.37 0.28 bf 0.18 0.18 0.47 0.26 0.62 0.71 0.60 0.73 0.14 0.04 0.18 0.25 0.24 0.36 b,, by coefficients of regression on mother and on father, respectively; 2,coefficient of determination of linear regression. All regression coefficients are significant at P = 0.001 level, except for bf in WTD and WTS which are significant at P = 0.01, and b, in WTS which is not significant (P = 0.3). TABLE 3

Polynomial offspring-parent regressions

~~ Trait Monomials in the regression and their coefficients RX PNL 3 WTD X4 Y5 XY XY3 2y3 0.21 0.019 0.14 0.02 0.03 0.28 -0.14 -0.04

WTS Y XY‘ XY3 x2y3 0.10 0.015 0.24 0.21 -0.07 -0.06

BD X Y XY3 X3Y 0.21 0.068 0.37 0.27 -0.08 0.04 4 BS X Y XY2 X XY X2Y X3Y 0.32 0.010 0.41 0.33 -0.02 -0.13 0.32 0.12 -0.12 WGD X Y X2 x2y2 0.29 0.017 0.22 0.36 0.17 -0.09

WGS X Y XY x*y2 0.39 0.053 0.43 0.28 0.23 -0.11 R;N, coefficients of determination of polynomial regression; PNL,significance level for nonlinearity of regression.

can be responsible for the nonlinearities in different model does not contain a linear term (WTD does not traits. Our purpose, however, is not to model aspecific have either x or y, and WTS does not have x), the biological mechanism but only to seewhether theac- absent term was added to the model for making com- tual (but unknown to us) regression surface for a par- parisons. Tests based on the expanded models could ticular trait can be approximated better by a curved only underestimate the significanceof the nonlinear- surface than by a plane. To do thatwe have fitted for ity of the original polynomial models since changesin eachcharacter afifth-order polynomial regression due tosuch expansions were negligibly small, function whereas the number of degrees of freedom in the model increased. f(x, y) = bpy (0 j 5) (5) x 2 5 i + 5 The polynomial model provides a significantly better ~j fit ( PNLI 0.05) for almost all characters, except BD, for in the standardized variables. Stepwise “forward” and which the polynomial model is only marginally signifi- “backward” multivariate regression procedures in SYS cant (PNL= 0.068). The improvements in the fit by the TAT (WILKINSON1990) were employed to fit the regres- nonlinear regressions are not very substantial, however. sion model. The model with the highest coefficient of The coefficients of determination for nonlinearregres- determination, R2, and with the significance level of sions are not much higher thanthose for linear regres- each term in the polynomial lower than P = 0.05 was sions. Even for WTS, inwhich case R;N morethan chosen as the “best.” The monomial terms included in doubles R2 for the linear regression (0.1 vs. 0.04),its the polynomial regression functions for thetraits under value is still quite low (only 10% of the variation in consideration are shown in Table 3. Also shown are the weight among sons is explained by the nonlinear trans- coefficients of determination for thepolynomial regres- mission of the weights by parents). sions, R:N, as well as the significance levels for the non- Figures 1, 2 and 3 show the offspring-parent poly- linearity of a regression, PNL.These significance levels nomial regression surfaces in the three-dimensional were obtained by comparing R:N for the “full” polyno- space for the weight, bristle number and wing length. mial model to R2 for the “partial” linear model (Table The regression for daughtersand for sons are on theleft 2) (SOW and ROHLF1981). In cases in which the full and right side of each graph, respectively. Also shown 346 A. Gimelfarb and J. H. Willis

FIGURE1.-Linear (plane) and polynomial (curved surface) regressions of weight of offspring on the weights of mothers and fathers (scales on all axes are the distances from the meanin standard deviation units).

FIGURE2.-Linear (plane) and polynomial (curved surface) regressions of number of abdominal bristles in offspring on the number of abdominal bristles in mothers and fathers (scales on all axes are the distances from the mean in standard deviation units).

are the planes corresponding to the linear regressions a nonlinear regression of a trait in offspring of parents (Table 2). The scales on all axes are in standard devia- whose traits deviate substantially from the mean. Let us tion units. A dot on the bottomof a graph represents not forget, however, that the same is true for linear re- a family (the value of thetrait in mother and in gressions as well. The error of a linearregression is also father). Thesurfaces appear to besufficiently smooth, greater for parentswhose traits are fartheraway from the withoutlocal “wobbling” which means thatthe sample mean and, hence, predictions for the offspring polynomial models are not overfitting. The general of such parents based on a linearregression are also not shapes of the regression surfaces in daughters and reliable. It is not certain whether the erroris higher for sons appear to be similar, even though differences linear or for nonlinear regression. between regressions in two sexes are statistically sig- Recall that the coefficient of linear regression on nificant for all traits. mothers was not significant for WTS (Table 2). Thus, There arenoticeable nonlinearities in the regression judging by the linearoffspring-parent regression, moth- surfaces for all ofthe traits. The nonlinearities are more ers do not appear tocontribute significantly tothe pronounced in the corners of a graph, i.e., if the traits weight of their sons. This has been confirmed by the in parents deviate substantially from the mean. Since univariate linear regression of the weight in sons on the parents with such traits areunderrepresented in a weight in mothers which turned out to be also nonsig- sample, the errorof the regression is higher for families nificant. Notice, however, that this conclusion is con- with such parents than forfamilies in which parents have tradicted by the polynomial regression. Indeed,the traits that are closer to the mean. Consequently, not value of R:N is twofold higher if the monomials with much credence should givenbe to a predictionbased on variable x are presentin the regression function (Table Offspring-Parent Regression 347

IA I4 A

FIGURE3.-Linear (plane) and polynomial (curved surface) regressions of wing length of offspring on the wing length of mothers and fathers (scales on all axes are the distancesfrom the mean in standard deviation units).

’L. ?.

3) than if they are excluded. Thus, mothersdo contrib- (1986b) has demonstrated that genotype-environment ute to theweight of their sons, but their contribution is interaction, forexample, may cause a reversed response not linear. under strong selection. For a linearoffspring-parent regression, the expected realized heritability is the same Nonlinear regression and realized heritability under any selection as the heritability estimated by the We have established that there is a statistically signifi- regression itself. cant (P5 0.05) nonlinearity in the offspring-parent re- Truncation selection:Table 4 shows the realized heri- gression for all considered traits, except the bristles in tabilities expected under truncation selection for the daughters, BD, for which the nonlinearity is significant characters considered in the paper assuming that the at P = 0.068. Such a finding is not very interesting in actual offspring-parent regressions are polynomial itself since no one expects the regression to be exactly (Table 3). An infinitely large population size and ran- linear forany real trait. It is important to know, however, dom mating were assumed, and the distribution of a whether the detected nonlinearities may affect our in- character among females and among males before se- ferences about the dynamics of the traits. Do, for ex- lection was assumed as a standardized normal (with zero ample, linear and nonlinear regressions predict similar mean and unit variance). responses by the traits to selection, and, hence,as far as The left column in Table 4 shows the strength of se- selection is concerned the nonlinearities can be disre- lection expressed as the percentageof individuals being garded? selected. In parentheses are selection differentials (in One of the measures of a response to selection by standard deviation units) produced by truncation selec- different traits is the realized heritability (FALCONER tion of the corresponding strength on a normally dis- 1983): tributed trait. Positive and negative values of selection R differential correspond to positive and negative selec- h:=Z, (6) tion. Only individuals with the character above a speci- fied “threshold” are selected under positive selection, where S is the selection differential and R is the response whereas only individuals with the character below the to selection. It is quite unfortunate that the term heri- threshold areselected under negative selection. The cal- tability and thenotation h2 are used customarily for both culation of the threshold value for selection of a par- the ratio (6) and theratio of the additive component of ticular strength is discussed in APPENDIX A. variance to thetotal phenotypic variance called the nar- It is seen that forall the traits in Table 4 their realized row sense heritability. It should be kept in mind, how- heritability depends on the strength of selection and ever, that the two heritabilities are the same only if the it candiffer substantially from the heritability estimated offspring-parent regression is linear, otherwise they may by linear offspring-parent regression (shown under differ. While the narrow sense heritability cannot, ob- the corresponding traits in the table). Also, a linear viously, be either greater than one or negative, the re- offspring-parent regression cannot, obviously, predict alized heritability can. A value of h: greater than one an asymmetry of responses to positive and negative se- means only that the response to selection exceeds the lection which is not uncommon in selection experi- selection differential. A negative value means that ments. On the other hand, atleast some degree of asym- selection producesa reversed response. GIMELFARB metry in responses to selection (realized heritabilities) 348 A. Gimelfarb and J. H. Willis

TABLE 4 responses is also present even if selection is weak,and for Realized heritabilities predictedby nonlinear offspring-parent some traits it is more pronounced under weaker selec- regressions under truncation selection tion. These results may appear so bizarre as to suggest Percent Trait and its estimated heritability a computational error.Yet, there is no error. Indeed, it selected follows from expressions (A14) and (A15) in APPENDIX A (selection WTD WTS BD BS WGD WGS that if a character is distributed with density p(x) over differential) 0.47 0.26 0.62 0.71 0.60 0.73 an interval -B 5 x 5 B, and the offspring-parent re- 15 ( 1.55) 0.31 0.04 0.49 0.29 0.42 0.75 gression for the character is f(x, y), 15 (-1.55) 0.52 0.62 0.79 0.98 0.72 0.78 lB 30 ( 1.16) 0.39 0.20 0.56 0.56 0.54 0.90 = [fh-B) + f(-B, x)lp(x) dx 0.37 0.52 0.73 0.65 0.60 0.63 x LE 30 (-1.16) (7a) 50 ( 0.80) 0.39 0.23 0.59 0.68 0.57 0.96 50 (-0.80)0.37 50 0.430.57 0.59 0.69 0.58 (positive selection), 70 ( 0.50) 0.600.23 0.41 0.74 0.980.56 70 (-0.50)0.550.58 0.59 0.68 0.36 0.51 lB h:, = - I_,[f(x, B) + f(& X)lP(X) dx 90 ( 0.20)1.07 0.550.91 0.62 0.20 0.62 90 (-0.20) 0.97 0.28 0.66 0.52 0.470.59 (7b) 95 ( 0.10) 0.83 0.17 0.63 1.10 0.58 1.14 (negative selection), 95 (-0.10) 1.36 0.22 0.66 0.41 0.58 0.39 where hi denotes the limit of the realized heritability under truncation selection when the strength of selec- is predicted by polynomial regressions for all of the tion approaches zero. The first thing to notice is that the characters. absolute values of the integrals in (7a) and in (7b) are For most of the characters therealized heritability pre- not necessarily the same. This means that there can be dicted under moderate selection (70% selected) ap- asymmetry in responses to very weak positive and nega- pears to be closer to theestimated heritability than that tive truncation selection. Notice also that if the regres- predicted under stronger selection, and thediscrepancy sion f (x, y) is a polynomial in x and y, the integrals in between the realized and estimated heritabilities can be- (7a) and in (7b) are polynomials in B. Consequently, come quite large when selection is strong(15% se- hi is also a polynomial in B with coefficients that depend lected). The asymmetry of responses seems also to in- on p(x). Thus, the limit of the realized heritability of a crease with stronger selection, and for most of the character under truncation selection is determined not characters it becomes quite pronounced under strong only by the offspring-parent regression, but also by the selection. The values of the realized heritability pre- width of the interval over which the character is distrib- dicted for strong selection should be taken, however, uted and by the shape of its distribution. The last two with skepticism. Indeed, traits among parents selected rows in Table 5show the limits ofthe realized heritability under strong selection deviate substantially (more than predicted by the polynomial regressions under trunca- a standarddeviation) from the population mean.At the tion selection assuming P( x) as truncated normalin the same time, as has been discussed earlier, a regression interval -3 5 x 5 3. Since hi is a polynomial in B, the function is not a very good predictor of a trait in off- limits of the realized heritability under truncation se- spring of parents whose traits deviate much from the lection will be higher thanthose shown in Table 5 if the mean. characters aredistributed in an interval wider than -3 5 The least squares method of fitting a regression func- x 5 -3. No limit exists if the distribution interval is tion to sample data ensures that in the absence of se- infinite. lection linear and polynomial regressions predict the It should be noted that thefact that therealized heri- same mean value of a character among offspring (zero tability may become higher under weaker selection does if the character is standardized). It would seem, there- not mean,of course, thatweaker selection yields a stron- fore, thatif the meanis not changed muchby selection, ger response in such instances. In fact, the response to i. e., selection is sufficiently weak,linear and polynomial truncation selection approaches zero when the strength regressions should predictsimilar responses. Yet data in of selection goes to zero, but so does also the selection Table 4for selection that is relatively weak (between 90 differential. and 95% selected) do not supportthis conclusion. It is Exponential and Gaussian selection: Truncation fit- seen that not only the realized heritability predicted un- ness function which is common in artificial selection der weak selection by nonlinear regressions may differ may not be adequate for describing selection in nature. substantiallyfrom theheritability estimated by linear re- Such selection may be better approximated eitherby an gression, but the discrepancy between the two herita- exponential fitness function, bilities actually increases with decreasing strength of selection for most of the characters. The asymmetry of w(x, Q, = exp(Q4, (8) O ffspring-Parent Regression Offspring-Parent 349

TABLE 5

Limit values of realized heritability when strength of selection approaches zero

Traits and their estimated heritabilities

Form of WTD WTS BD WGSBS WGD selection 0.47 0.26 0.62 0.71 0.60 0.73

Exponential0.27 0.52 0.69 0.64 0.72 0.57 Gaussian (0 = 3) 0.68 0.27 0.64 0.76 0.58 0.79 Gaussian (0 0.27= -3) 0.76 0.64 0.65 0.62 0.57 Gaussian (00.27 = 1) 0.60 0.86 0.64 0.58 0.93 Gaussian (0 0.27= -1) 0.84 0.64 0.61 0.56 0.51 Gaussian (00.27 = 0.5) 0.48 0.99 0.64 1.14 0.59 Gaussian (0 = -0.5) 0.96 0.490.27 0.64 0.30 0.55 Gaussian (0 0.27= 0.25) 0.24 0.64 1.24 0.62 1.56 Gaussian (6 = -0.25) 1.20 0.230.27 0.64 0.53 -0.12 Truncation (positive) 2.66 -0.07 0.64 1 .a7 0.59 1.31 Truncation (negative) 3.69 -0.07 0.64 0.22 0.11 0.55 0, "optimum"phenotype in standard deviation units.

or by a Gaussian fitness function, sions assuming infinite population size and parental distributions as standardized normal (seeAPPENDIX B). 4x9 €J = exp[- Q(. - e)'] (Q2 01, (9) The expected heritabilities differ slightly from those with the optimum, 8, away from the mean of the char- estimated in our samples (shown in Tables 4 and 5 acter. The direction of exponential selection is indi- underthe corresponding traits). The difference is cated by the sign ofQ, whereas the directionof Gaussian not surprisinggiven that thesample size is not infinite selection is indicated by the sign of 8. The strength of and the actual parental distributions are not exactly selection is determined in both cases by the absolute normal. value of Q. No general conclusion regarding the limit of the re- Regarding the limit of the realized heritability under alized heritability can be drawn from Table 5.While for exponential and Gaussian selection when the strength one trait (BD) it is similar to the estimated heritability of selection approaches zero, let under any form of selection, for another trait (WTS) it is similar to the estimated heritability under any Gaus- a = [1; y)P(x)P(r) dx dY, (104 sian selection but is quite different from it under truncation selection. For the majority of the traits, the limit of the realized heritability depends very much on the b = [ Yf(X, y)P(x)P(r)dx dY. (lob) form of selection and may differ remarkably from the estimated heritability. It follows from equations (A16), (A17) and (A18) (A19) The only general conclusion that can be drawn from in APPENDIX A that, given the distribution of the character Tables 4 and 5 is that predictions of the response by the is standardized normal, mean of a character to selection, and to weak selection in particular, based on the estimated heritability of the h; = a + b, (11) character may be quite unreliable. The reason for this if selection is exponential, and is simple. If the offspring-parent regression is linear, the mean of a character in offspring population depends h2=a+b" [(2 + r')f(m Y>Pcap(y)dx dy, only on the mean among parents. Consequently, a response by the mean to selection is completely deter- (12) mined in such a case by the change in the mean among if selection is Gaussian. Table 5 shows the limits of the parents (selection differential), and formula (1) is a realized heritability predicted by the polynomial succinctformulation of this. If, however, the offspring-parent regressions under exponential and offspring-parent regression is not linear, the mean Gaussian selection assuming that the parental distribu- among offspring depends not only on the parental tions are standardized normal.The optima forGaussian means but also on higher moments of parental dis- selection are in standard deviation units. If selection is tributions. Consequently, a response by the mean to exponential, the limits of the realized heritability are selection is determined in such a case not only by exactly equal to theheritabilities expected from theleast the selection differential, but also by changes caused squares linear approximationsof the polynomial regre+ by selection in highermoments of theparental 350 andA. Gimelfarb J. H. Willis distributions. Hence, the realizedheritability for a The discrepancy between the responses by a character character with nonlinear offspring-parent regression to selection predicted by nonlinear and linearoffspring- depends on the parental distribution and on the formparent regressions can be substantial not only if selec- and strength of selection. tion is strong, but also, and even more so, if selection is The effect of the form and strength of selection on the weak. realized heritability can be seen in Tables 4 and 5. Also, Predicting responses by a quantitative character to the assumption of normality of parental distributions weak selection can be a very uncertain exercise. made in computing these tables implies that all odd mo- We wish to thank BRIANCHARLESWORTH for supplying to us flies and ments of the distributions are zero and any even mo- JERRY COYNEfor sharing his data. We are very grateful to STEVENORZACK ment is expressed in a particular way through thesecond for hisencouragement andto THOMASNACW for useful discussions. moment. This, however, may not be true for distribu- This work was supported by the U.S. Public Health grant GM27120. tions otherthan normal. Consequently, the realized heritabilities for the same traits under the same form LITERATURE CITED and strength of selection may differ from those shown BULMER,M. G., 1985 TheMathematical Theory of Quantitative in Tables 4 and 5, if parental distributions arenot Genetics. Claredon Press, Oxford. normal. CLAYION, G. A,,J. A. MORRIS,and A. ROBERTSON,1957 An experimental check on quantitative genetical theory.I. Short-term responses The effect on the mean amongoffspring of changes to selection. J. Genet. 55: 131-151. in higher moments of parental distributions caused by COYNE,J. A., and E. BEECHAM,1987 Heritability of two morpho- selection may become particularly noticeable if the se- logical charactersin Drosophila melanogaster. Genetics 117: 727-737. lection differential is small, i.e., selection on the mean FALCONER,D. S., 1983 Introduction to Quantitative Genetics. Long- among parentsis weak.There can even be a “response” man, New York. by the mean in offspring to selection that does notaffect FRANKHAM,R., 1990 Are responses to artificial selectionfor reproductive fitness characters consistently asymmetrical? Genet. Res. the parental mean at all. For example, Gaussian selec- 56 35-42. tion with 8 = 0 does not change the mean in a parental GIMELFARB,A., 1986a Offspring-parent genotypic regression: how lin- population with normally distributedcharacter, but ear is it? Biometrics 42: 67-71. GIMELFARB,A,, 1986b Multiplicative genotype-environmentinterac- doeschange the variance. Given the polynomial tion as acause of reversed response to directionalselection. offspring-parent regressions (Table 3), it is not difflcult Genetics 114: 333-343. to see that the mean among the offspring for WTD, BS, GIMELFARB,A,, and J. WILLIS,1988 Etherizing parents reduces the weight of their offspring. Drosophila Inf. Sew. 67: 43. WGD and WGS will be affected by a change in the pa- KFARSEY, M. J., and K. KOJIMA,1967 The genetic architecture of body rental variance, and, hence, the meanfor these traits will weight and egg hatchability in Drosophila melanogaster. Genetics “respond” toselection even though the selection differ- 56: 23-37. KELLER,E. C., and D.F. MITCHELL,1962 Interchromosomal genotypic ential is zero. It is clear that the realized heritability will interactions. I. Analysis of morphological characters. Genetics47: be infinity in such a case. 1557-1571. NISHIDA,A,, and T. ABE,1974 The distribution of genetic and envi- ronmental effects and the linearity of heritability. Can. J.Genet. Cytol. 16: 3-10. CONCLUSIONS ROSE,M. C., and B. CHARLESWORTH,1981 Genetics of life history in Drosophila melanogaster. I. Sib analysis of adult females. Genetics The offspring-parent regression is significantly non- 97: 173-186. linear for all of the characters considered in the paper, Sow, R. R., and F. J. ROHLF,1981 Biometry, Ed. 2. W. H. Freeman except the bristles in daughters for which the nonlin- & Co., San Francisco. WILKINSON,L., 1990 SYSTAT: The Systemfor Statistics.SYSTAT Inc., earity is marginally significant (P = 0.068). Evanston, Ill. Nonlinear contributions by parents to a character in Communicating editor: A. G. CLARK their offspring may not berevealed by a linearoffspring- parent regression, as the weight in sons (WTS) demon- APPENDIX A strates. The response by a character to selection predicted Consider a quantitative character having zero mean by a nonlinear offspring-parent regression fitted to and unit variance distributed in an interval -B 5 x IB family data can be quite different from the response (B5 w) with a density function p(x) for both sexes. Let predicted by the linear regression fitted to the same the character be under selection with the phenotypic family data. fitness function w( x, (3). Besides the individual’s phe- The response by a trait to selection is a result of an notype, the fitness function also includes a parameter(3 intricate interplay between the offspring-parent regres- characterizing the strength of selection: smaller Q im- sion function, the distribution of the character,and the plies weaker selection. There is no selection if Q = 0, form and strength of selection. i.e., w(x, 0) = 1. The mean fitness of a populationis, by The nonlinearity of the offspring-parent regression definition, can be responsible for the asymmetryof responses to positive and negative selection observed in many experiments. Offspring-Parent Regression Offspring-Parent 35 1

In the case of truncation selection, the fitness function The following conditions are straightforward when can be expressed as the strength of selection approaches zero:

1 if xr-B+Q lim S = 0, 4x9 @ = 0 otherwise Q.0 (positive selection), lim R = 0, e-0 1 if xlB- Q lim w(x, Q, = W= 1. 0 otherwise Q.0

(negative selection). Given (A7) and (AB), the limit of h: = R/S when Q approaches zerois ofthe indeterminatetype O/O. It can, Substituting these expressions into (Al) yields for the however, be evaluated by applying L’H8pital’s rule: mean fitness

E W= 1p(x)dx, T= B- Q (Ma) It can be shown, taking into account (A7), (A8) and (positive selection), (A9), that

W= [lp(~)d~,T= -B+ Q (Mb)

(negative selection).

The meanfitness under truncationselection is equiva- lent to the proportionof individuals that are selected. Consequently, the threshold corresponding to a particular proportion of selected individuals can be ob- where tained by solving with respect to T equations (A3a) and (A3b). Given that themean of the characterbefore selection is zero, selection differential, S, and response, R, are For the fitnessfunction of truncationselection, obtained as +(x) = 6(x + B) if selection is positive (A2a), and +(x) = -6( x - B) if it is negative (A2b), where ti(.) denotes delta function. Substituting such +(x) into (All) and (A12) we obtain

dS lim - = Bp( -B), (A14a) Q.0 aQ where f (x, y) = E[z I x, y] is the offspring-parent regres- aR sion function. It is easy to see that if f(x, y) = ax + by, lim - (A15a) ie., the offspring-parent regression is linear, R = US+ Q.0 aQ bS, and, hence,

h:=a+b for any form of selection and character distribution.The under positive selection, and realized heritabilities under truncation selection in Table 4were obtained by first evaluating the integrals (A5) for the regression function corresponding to a (A14b) particulartrait andfor the fitness functioneither dR (A2a) or (A2b), and then dividing the result by (A4) 1‘ also computed for the fitness function either (A2a) Ea (A15b) or (A2b).A package of mathematicalprograms MATHCAD was used for computing the integrals on a 486 PC computer. 352 A. Gimelfarb and J. H. Willis under negative selection. Dividing(A15a) by (A14a) Dividing (A17) by (A16) and (A19) by (A18) results in and (A15b) by (A14b) yields expressions (7a) and (7b) expressions (11) and (12) in the text. for hi in the text. If selection is exponential (8), the derivative (A13) is Ql( x) = x. Given that p(x) is standardized normal, APPENDIX B the substitution of such +(x) into (All) and (A12) The least squares linear approximationof a nonlinear results in offspring-parent regression is a function ax + Py that as delivers the minimum of the integral

1. l[f(x, y) - (ax + Py)]*p(x)p(y)dxdy, (Bl) where p(x) and p( y) are the distributions of the character amongmothers and fathers, respectively. The

If selection is Gaussian (9), +(x) = -(x - 0)', and the minimum of (Bl) is delivered by substitution of such +(x) into (All) and (A12) yields as a! = 1.1Xfb, y>P(x)P(y)dx dy, (B24 lim - = - (x - 0)'xp(x) dx = 20,(A18) Q.0 aQ P = J$!f(x. r)P(x)P(y) dxdr. (B2b) aR lim - eoaQ Given that the characters in parents are standardized, the heritability expected from such linear approxima- tion is Q + P (see (4)). Itis seen that (B2a) is the same as (1la) and (B2b) is the same as (11 b) .Hence, h: under exponential selection is equal to the heritability expected from the linear approximationof the offspring- parent regression.