Statistical Genetics of an Annual Plant, Impatiens capensis. I

Copyright 0 1990 by the Genetics Society of America

Statistical Genetics of an Annual Plant, Impatiens capensis. I. Genetic Basis of Quantitative Variation

Thomas Mitchell-Olds* and Joy Bergelson+ *Division $Biological Sciences, University $Montana, Missoula, Montana 59812, and ?Department of Zoology, University of

Washington, Seattle, Washington 981 95 Manuscript received May 2, 1989

Accepted for publication October 10, 1989

ABSTRACT Analysis of quantitative genetics in natural populations has been hindered by computational and

methodological problems in statistical analysis. We developed and validated a jackknife procedure to test for existence of broad sense heritabilities and dominance or maternal effects influencing quantitative characters in Impatiens capensis. Early life cycle characters showed evidence of dominance and/ or maternal effects, while later characters exhibited predominantly environmental variation. Monte Carlo simulations demonstrate that these jackknife tests of variance components are extremely robust to heterogeneous error variances. Statistical methods from human genetics provide evidence for either a major locus influencing germination date, or genes that affect phenotypic variability per se. We urge explicit consideration of statistical behavior of estimation and testing procedures for proper biological interpretation of statistical results.

U NDERSTANDING the potential for life history evolution in natural populations requires infor-

mation on genetic and environmental determinants of life history variation, as well as knowledge of natural selection operating in the field (LANDE 1982). Applications of quantitative genetic methods have provided estimates of genetic variances and covari- ances in natural plant populations (e .g . , VENABLE 1984; MITCHELL-OLDS 1986), which quantify constraints on rate and direction of evolutionary change. Despite recent extensions of quantitative genetic theory, analysis of empirical results often poses severe statistical problems (MITCHELL-OLDS and RUTLEDGE 1986; MITCHELL-OLDS and SHAW 1987). Choice of optimal statistical procedures may be difficult (e.g. ,SEARLE 197 1; SWALLOW and MONAHAN 1984), and analysis of some data sets may be computationally infeasible (SHAW 1987). Nevertheless, understanding of underlying biological patterns and principles may be strongly influenced by choice of genetic models and analytical procedures. This paper illustrates the importance of considering statistical behavior of testing procedures in order to provide proper biological interpretation of genetic variation and natural selection operating in the wild.

Basic models of polygenic, additive genetic variance (BULMER 1980; FALCONER 198 1) have been extended to consider nonadditive genetic variance, maternal effects, and major gene effects (COCKERHAM 1963; WILLHAM 1963; MORTON 1982). Nonadditive genetic

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Genetics 124: 407-415 (February, 1990)

variance is typically partitioned into dominance and epistatic effects. Dominance genetic variance is frequently found to be important in plant populations (THOMAS and GALE 1977; MATHER and JINKS 1982; SPRAGUE 1983; LANDE and SCHEMSKE 1985). Since directional dominance is a necessary condition for occurrence of inbreeding depression under many genetic models (CROW and KIMURA 1970), the wide- spread observation of inbreeding depression provides further evidence for the importance of dominance genetic variance. Epistatic variance is difficult to estimate with precision due to large statistical sampling error (HALLAUER and MIRANDA 1981), especially in field situations. Maternal effects, due to either cytoplasmic inheritance, maternal nuclear genes, or long term maternal environmental effects (FALCONER 198 1) are well known in plants and animals (WILLHAM 1972; ROACH and WULFF 1987). Last, the number of loci influencing continuous characters in natural populations and the magnitude of their effects has been the topic of considerable discussion for many years, since the genetic architecture of quantitative variation affects the response to selection, and differs among competing models of evolutionary change (WRIGHT 1968; MAYO 1980; LANDE 198 1, 1983; MATHER and

JINKS 1982; GOTTLIEB 1984). Despite the importance of this question, the genetic basis of quantitative variation is poorly understood, since most statistical methods for examining possible major genes influencing continuous characters have statistical or empirical lim- itations that hinder their application to natural populations (MITCHELL-OLDS and RUTLEGE 1986). In this paper, we show that most quantitative genetics exper-

408 T. Mitchell-Olds and J. Bergelson

iments involving sibling families contain statistical information that may provide weak evidence regarding patterns of genetic control of continuous characters.

Previously MITCHELL-OLDS (1 986) estimated levels of additive genetic variance in field populations of Impatiens capensis, using the sires component of variance from a nested design (North Carolina type I). Although additional information regarding determinants of phenotypic variation can be extracted from such experiments, more complete analysis of such data has been hindered by lack of statistically valid and numerically feasible methods (MITCHELL-OLDS and RUTLEDCE 1986; SHAW 1987). Here we develop and validate statistical methods for further analysis of such data, permitting significance testing of possible maternal and/or dominance genetic influences on phenotypic variation. We also show how methods from human statistical genetics may provide a crude test of whether continuous phenotypic variation is caused by many genes of small effect, or whether variation due to a locus with large differences between alleles (a “major gene” or “leading factor” (WRIGHT 1968) might be present in addition to polygenic variation. (Nothing in this paper should be construed as advo- cating a purely oligogenic model.)

Analysis of data from MITCHELL-OLDS (1986) will be used to address the following questions: 1) Is there evidence for maternal or dominance genetic variance? 2) Do alternative statistical estimators provide evidence for previously undetected genetic variation? 3) Is life history variation under polygenic control, or is there evidence for loci of large effect? 4) Are there genetic influences on levels of phenotypic variation?

MATERIALS AND METHODS

Impatiens natural history: Impatiens capensis Meerb., or jewelweed, is an erect summer annual native to northeastern North America, frequently found on streambanks and in floodplains that are partially disturbed. It is an obligate annual in that all viable seed germinate in the spring following their production (WALLER 1984). Their innate dor- mancy is broken by a period of cool moist stratification (LECK 1979). Given moist conditions, growth continues until late August, when their height may exceed 2 m. Jewelweed exhibits a mixed mating system as consequence of production of both cleistogamous (CL) and chasmogamous (CH) flowers. The CL flowers are obligately self-fertilized at an early stage in their development, while the CH flowers are predominantly outcrossed (MITCHELL-OLDS and WALLER 1985; KNIGHT and WALLER 1987). Plant size is the major component of maternal fitness in this strict annual: dry weight is highly correlated with total seed production (? = 0.92) (WALLER 1979), while juvenile mortality is negligible in this study (MITCHELL-OLDS 1986).

Experimental design: Experimental design has been described in detail in MITCHELL-OLDS (1 986). Briefly, pa- rental plants were collected from near Madison and Milwau- kee, Wisconsin, and controlled pollinations were conducted within each population in a nested sires/dams/progeny design (North Carolina design I ) (COMSTOCK and ROBINSON

1948). Progeny were planted into a naturally seeded stand of I. capensis, and morphological and phenological characters were measured. Here we report on seed weight, germination date, plant size at monthly intervals in June, July, and August (estimated as height X a X radius’), and final dry weight, which is highly correlated with fecundity and will be taken as an indicator of fitness. We calculated growth rate between monthly size estimates as (final volume - initial volume)/time interval. Since quantitative genetic interpretation of selection gradients requires multivariate normality of predictor variables, all predictor variables were subject to a Box-Cox transformation (SOKAL and ROHLF 1981). Dry weight was divided by mean dry weight to provide an index of relative fitness (LANDE and ARNOLD 1983).

Quantitative genetic analysis and assumptions are reviewed in MITCHELL-OLDS (1 986) and MITCHELL-OLDS and RUTLEDCE (1986). Proper randomization of crosses and planting location precluded most sources of bias. F,,y was estimated by isozyme analysis to be 0.47 in each population, permitting estimation of additive genetic variance from the covariance of siblings (COCKERHAM 1963; MITCHELL-OLDS 1986).

Variance component analysis: Alternative statistical procedures may differ in bias of estimates, ability to test particular hypotheses, statistical power of significance tests, robustness to violation of assumptions, and computational feasibility (e.g., SOKAL and ROHLF 1981; SWALLOW and MONAHAN 1984; SHAW 1987). Our inquiry required a test for existence of dominance and/or maternal effects, as well as the ability to pool sires and dams information regarding broad sense heritabilities. Although least squares ANOVA permits estimation of a variance component attributable to maternal and dominance effects using a linear combination of the observational components of variance (below), standard least squares methods do not provide a significance test for these effects when data are unbalanced. Maximum likelihood (SHAW 1987) proved computationally infeasible, and has unknown statistical performance with heterogeneous variances. Randomization tests of significance could not be applied since the appropriate resampling design for compar- ing components of variance due to sires and dams (at two different hierarchical levels) is unknown. [SOKAL and ROHLF (1981) discuss how deletion of groups (e.g. , sire families) is appropriate for testing of effects at lower levels (e.g. , dams). However, methods for testing equality of sires and dams components of variance using randomization tests are un- clear, since significance tests for a particular component require randomization at that level.] Jackknife procedures have demonstrated good statistical behavior for estimation and testing of variance components in quantitative genetic applications (ARVESEN and SCHMITZ 1970; KNAPP, BRIDGES and YANC 1989), and are less computationally burdensome than bootstrap methods (EFRON 1982). For these reasons, we employed a nested ANOVA for unequal sample sizes (SOKAL and ROHLF 198 1, pp. 293-300) using unweighted, delete-one-sire-familyjackknife estimates (SOKAL and ROHLF 1981, p. 796) and approximate tests of significance (WU 1986). O u r initial simulations found that the log transformation of ARVESEN and SCHMITZ (1 970) was unnecessary, and that truncation of negative estimates of variance components caused rejection of the null hypothesis at undesira- bly high frequencies, so we report here jackknife results from an untransformed, untruncated variance component estimator (see also KNAPP, BRIDGES and YANG 1989).

Let the observational components of variance, u: = u,? + ui + uf, where u; = total phenotypic variance, a,: = the sires component of variance, ui = the dams component of variance, and u:v = the variance within full-sibships. In

Statistical Genetics of Impatiens 409

random mating populations the causal components of variance are obtained from uz = 1/4VA, and uz = 1/4VA + 1/4VD + V.+, = 1/4VA + u&,,, where VA = the additive genetic variance, VD = the dominance genetic variance, and VM = the maternal variance (which may be due to cytoplasmic, indirect genetic effects, and permanent maternal environmental effects) (FALCONER 1981). Because Vo and VM are confounded in this design, we use u z ~ to indicate a combination of dominance and maternal effects.

Least squares estimates of the observational and causal components of variance were obtained by equating observed mean squares to their expectations (SEARLE 197 1 ; FAL%ONER 1281; SOKAL and ROHLF 1981). We estimated aiM as ai - u,< (“caret notation” denotes “an estimator of”), and a pooled, reduced variance estim%te of the additive gefetic variaye can be obtained as 4&(1 + F,.& where aiD) = 1/2(a,? + uz), provided that u?>M = 0 (COCKERHAM 1963; FALCONER 1981, p. 156). Finally, jackknife estimates and standard errors were obtained by an unweighted, delete- one-sire-family jackknife (Wu 1986) and compared to a t- distribution. ,We also examined an alternative weighting scheme for a&, weighting ui and us In inverse proportion to their estimated variances (SEARLE 197 1, p. 476), but this proved infeasible because negative estimates of var (u;) and var(u,:) were obtained due to sampling error.

Since jackknife tests of significance are only approximate, we conducted Monte Carlo simulations to determine the statistical behavior of this test. Ideally, a statistical test will reject a true null hypothesis only lOOa percent of the time. However, a statistical test may be too conservative (reject true Ho fewer times than expected) or too liberal (reject true Ho too many times) (SOKAL and ROHLF 1981, p. 301). This can be tested by generating large numbers of pseudo-random data sets where Ho is true, and determining whether the putative level of significance is actually provided by the test. In addition, this procedure can be used to examine the power of a test (how often Ho is rejected when it is false) by generating simulated data where variance components have true values greater than zero. Biologically, such a power curve (see Figure 1) enables us to ask how frequently we attain significance for a particular heritability in an experi- ment. Finally, Monte Carlo simulations enable us to assess whether jackknife tests of significance are robust to possible heterogeneity of within-family variance.

For each assumed parametric value of VA and aiM we generated 3000 pseudo-random data sets (detailed in Figure l ) , then calculated jackknife estimates of variance components and approximate t-tests. We compared t-values with the appropriate critical t-value at a = 0.05, and we computed the frequency of rejection of Ho. T o examine the effect of heterogeneous within-family variances we randomly as- signed 50% of simulated full-sibships to have within-family variance increased by a factor of two.

Major gene analysis: Little information exists on the genetic basis of quantitative variation (FALCONER 198 1 ; BAR- TON and T’URELLI 1989). Although it is often assumed that there is a continuous distribution of allelic effects, with many genes of small effect and few genes of large effect, it is difficult to fit phenotypic data to these complex models. Instead, most segregation analyses of phenotypic data have considered simplified models. Purely polygenic models and oligogenic models represent opposite ends of a presumably continuous spectrum of gene effects. An intermediate (although equally arbitrary) approach considers a “mixed model” with a major locus and a segregating polygenic background (e.g. , MORTON 1982). Existing statistical methods for distinguishing between polygenic inheritance, oli-

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gogenic segregation, and mixed models are complex and have limited statistical power (reviewed in MITCHELL-OLDS and RUTLEDCE 1986). One crude but simple method for detection of a major locus was suggested by PEARSON (1904) and FAIN (1 978). They proposed that segregation of a major gene might be detected by graphical plots of family variance against family mean, since families homozygous at the major locus would have low variance and extreme mean values, while families that were segregating at the major locus would have large within family variance and an intermediate mean phenotype. Consequently, as a test for the presence of major genes, FAIN suggested a test for heterogeneity of within- family variances. The statistical power of this test is known to be limited (FAIN 1978; MAYO, HANCOCK and BACHURST 1980), but it has the advantage of being usable in many sib analyses from quantitative genetic studies. Furthermore, since most statistical tests assume homogeneity of error variances, this test should always be performed in routine statistical explorations. FAIN’S approach has been criticized because loss of developmental homeostasis might also cause heterogeneity of family variances (MAYO, HANCOCK and BACHURST 1980). However, we show below that graphical analysis easily discriminates between these two hypotheses.

RESULTS

Variance component analysis: Components of variation for each character are presented in Table 1. The data differ in several respects from the analysis of MITCHELL-OLDS (1986). Slightly more plants are included in this broad sense analysis, since previous statistics were restricted to sires that were mated to more than one dam. The current approach also includes sires mated to only a single dam, which consti- tute full-sibships that confound additive genetic variance with dominance and maternal effects. Also, we analyzed a different set of predictor variables, each subject to Box-Cox transformation, while the previous analysis examined predictor variables on their original scale of measurement. Finally, these results utilize a different statistical estimation and testing procedure ljackknife estimates of ANOVA method variance components (SEARLE 1971) and jackknife tests of significance (WU 1986), us. ANOVA method variance components subject to randomization tests of significance]. Estimates of partitioning of variance (Table 1) are in general accord with MITCHELL-OLDS (1986). Seed weight and germination date show significant dominance and/or maternal effects in the Madison population, while analysis of the Milwaukee families is only suggestive of dominance or maternal effects. When families from the two populations are analyzed together, results generally follow this same pattern. There is one result in which new analyses are at odds with previous ones: estimates of heritability of fitness in the Milwaukee population are noticeably smaller and are no longer significantly different from zero.

Statistical simulations of variance components: The jackknife estimators of variance components were unbiased (data not shown), and provided useful


TABLE 1

Genetic and environmental components of variance

Madison: SW -0.251 0.769*** 0.259*** 1.019*** 0.482 GD -0.107 0.325* 0.109** 0.432* 0.782 JS -0.053 0.052 -0.001 0.105 1.002

EG 0.002 0.029 0.015 0.028 0.969 LG -0.068 0.052 -0.008 0.120 1.016 F T -0.017 -0.008 -0.013 0.009 1.025

Milwaukee: SW 0.098 0.361* 0.229*** 0.264 0.541 GD 0.147* -0.153 -0.003 -0.300 1.006 JS 0.070 -0.012 0.029 -0.082 0.942

EG -0.008 0.033 0.013 0.041 0.975 LG 0.040 -0.045 -0.003 -0.085 1.005 FT 0.041 -0.006 0.018 -0.047 0.965

Pooled: SW -0.109 0.607*** 0.249*** 0.716*** 0.502 GD -0.051 0.203* 0.076** 0.254 0.848 JS -0.016 0.052 0.018 0.069 0.964

EG -0.014 0.063 0.025 0.077 0.951 LG -0.07 1 0.13 1 0.030 0.202 0.939 F T -0.026 0.043 0.009 0.068 0.983

Variance component analysis of Madison and Milwaukee populations, and pooled data from both populations. SW = seed weight, GD = germination date, JS =June size, EG = early growth rate, I X = late growth rate, and F T = relative fitness. u?, uf, u,$),

U ~ M , and uf represent components of variance attributable to sires, dams, pooled sire plus dam effects, dominance and/or maternal effects, and within family variation, respectively. All variance components have been standardized so that total variance = 1 .O. Neg- ative estimates are due to sampling error, and may be considered to indicate zero variance attributable to a particular component. Estimates of variance components are from an unweighted, delete- one jackknife (see text). In Madison there are 38 sires, 73 dams, and 243 progeny. Milwaukee has 31 sires, 50 dams, and 212 progeny. Significance tests are from one-tailed jackknife t-tests. * indicates P < 0.05, ** indicates P < 0.01, and *** indicates P < 0.001. Narrow sense or brodisensLestimates of heritabilities may be estitnated by multiplying u: or u:,) by 4/(1 + F J 5 ) = 2.72.

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tests of stGist3al signikcance (Figure 1). Significance tests on u,:, &, and ailLI were slightly conservative, rejecting a true null hypothesis 3.5%, 2.1 % and 1.8% of the time at a nominal a = 5% level when n = 250. The jackknife test of significance was extremely robust to the presence of heterogeneous within-family variances: simulations with heteroscedastic within- family variances produced type I and I1 error rates virtually identical to those portrayed in Figure 1 (data not shown). In the absxnce of dominance and maternal variance, tests using a& were more likely to detep the existence of additive genetic variance than,,d; the power of jackknife t-tests increased from a: to UDM

to u , ~ ~ . 9 With 250 individuals, a heritability2f 40% "3s detectable only 33% of the time using a:, while U ~ D

was found to be .significant 90% of the time at this heritability. Hoxever, due to the low statistical power of tests on a u : . ~ (Figure l), caution musk be used when a strictly additive interpretation of a a& is being

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Per Cent of Variance FIGURE 1.-Power curves for variance components. Percent

probability of rejection of null hypothesis that a particular variance component equals zero plotted against percent of total variance attributable to that particular component of variance. From bottom to top, the first three curves show results with n = 250, representing US, uDh, and u,?~~ , respectively. The topmost curve indicates u:,) when + * A

n = 500. When percent of variance equals zero then the null hypothesis is true, so values at intersection with left vertical axis are type I error rates. When percent of variance attributable to a particular component is greater than zero then each curve indicates the power of the test ( i e . , the probability of rejecting the null hypothesis when it is false, or one minus the type I1 error rate). Simulatiogs involving 0% and u% assume uiM = 0, and simulations involving assume a,: = 0.

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considered. At low levels of genetic variation it is unlikely that her2abilities could be shown to be significant: when h' = 10% the null hypothesis was rejected in 11 % and 18% of the trials using u,: and u&, respectively. However, larger sample sizes (e .g . , n = 500, Figure 1) gave substantially greater power for detection of moderate heritabilities.

Major gene analysis: Most characters showed strongly heterogeneous within-family variances (Table 2). Graphical and regression analysis (not shown) indicated that within-family variance increased with increasing trait values in both populations for all characters except germination date. This pattern of increasing variances presumably represents a scaling effect. However, in both populations germination date had maximum within-family variance in intermediate families. We tested this hypothesis against hypotheses that within-family variability increases or decreases monotonically, since maximum variability at extreme trait values might be an artifact of scale of measurement (DRAPER and SMITH 198 1). Using Equation 5 of MITCHELL-OLDS and SHAW (1987), the monotonic hypotheses are strongly rejected (Table 3, P 0.001). Thus FAIN'S (1978) predicted pattern of maximum

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Statistical Genetics of Impatiens 41 1

TABLE 2

Levene’s test for heterogeneity of family variances

Madison Milwaukee

Charactet- t’f F Vf F

TABLE 3

Test for maximum within-family variation at intermediate family mean

Seed Mright 21.3 2.035*** 9.3 1.469* Gernlination date 11.0 1.473* 34.9 3.396*** June s ix 10.5 1.447* 4.33 1.202

Late growth 15.0 1.674** 15.1 1.796** Fitness 33.0 2.884*** 20.5 2.158***

E:arl) grolvth 17.8 1.831** 17.2 1.927**

Levene’s test (LEVENE 1960) for heterogeneity of within full-sib family variances. The dependent variable is the absolute value of individual deviation from full-sibship mean, lztl - i , ] . One way analyses of variance of absolute deviations give V,, the percent of differences i n variability attributable to full-sib family, and F , the F-ratio from Levene’s test ANOVA. Madison df = 59, 70. Milwau- kee d.f. = 44, 157. Significance levels as in Table 1.

variability in intermediate families is observed for germination date in both populations.

DISCUSSION

Genetic and maternal variation: Quantitative ge- neticists use broad sense heritabilities as a rough estimate of the overall level of genetic variation for a trait. Broad sense estimates may be easier to obtain than narrow sense estimates, but they are likely to be upwardly biased by nonadditive genetic variance (FAL- CONER 1981). Comparison between broad sense and narrow sense estimators typically involve a tradeoff bEtween unbiased estimators of low precision (e .g . , af and biased estimators with greater statistical power (asn, 3 Figure 1). In partially inbred populations, the regression of offspring on parent includes part of the dominance variance, so choice of optimal experimental design procedures can be especially difficult in natural plant populations (COCKERHAM 1963; MITCH- ELL-OLDS and RUTLEDGE 1986). We chose to investi- gate broad sense estimators in order to take advantage of their generally greater statistical precision (lower mean square error) and hopefully detect family effects on life history variation that were not detectable in the narrow sense heritability analyses of MITCHELL- OLDS (1986). Yet, we found that most estimates of broad sense heritabilities do not differ significandy from zero in either the Madison or Milwaukee populations, or when data from both populations are pooled.

These analyses also permit examination of evidence for dominance genetic and maternal effects for life history traits. In the Madison population both seed weight and germination date show significant dominance and/or maternal variance. This agrees with many studies on the genetics of plant and animal populations (FALCONER 198 1 ; ATCHLEY 1984; ROACH and WULFF 1987), in which maternal effects are often

Population Variance us.

mean 1 P

Madison Increasing 5.510 0.001 Madison Decreasing 5.096 0.001 Milwaukee Increasing 6.737 0.001 Milwaukee Decreasing 4.302 0.001

Hypothesis test of whether within full-sib family variance is a monotonic increasing or decreasing function of family mean, employing the jackknife test of significance developed by MITCHELL- OLDS and SHAW (1 987), Equation 5. The Statistical test is similar to traditional tests of a general linear hypothesis, employing a second order polynomial regression, and asking whether the maximum occurs at an intermediate value, as indicated by least squares regression, or whether the maximum can be forced to occur at the most extreme (minimum or maximum) observed value of family mean, corresponding to monotonic decreasing or increasing functions, respectively.

found to be most important in early stages of the life cycle. The Milwaukee population shows additive genetic variance for germination date, and a significant broad sense heritability for seed weight. This broad sense variation apparently is not due to dominance or maternal effects, although the low statistical power of the test on must be considered.

Although we had anticipated that broad sense estimates would reveal statistically significant heritabilities for more characters than the narrow sense estimators calculated previously (MITCHELL-OLDS 1986), none of the mid- or late-life cycle characters had detectable genetic or maternal variances. This indicates that virtually all observed phenotypic variation is due to environmental differences. In contrast to previous results from the Milwaukee population (MITCHELL-OLDS 1986), plant dry weight, a major indicator of individual fitness, was not significant in these new analyses (Table 1). Although the jackknife tests of significance were somewhat conservative (Fig- ure l) , this does not explain the failure to attain statistical significance with these estimators. Thejack- knife estimator of broad sense heritability, Hf = 0.05, was substantially lower than the previous least squares estimate of narrow sense heritability, eS = 0.29. This difference is due primarily to use of a larger set of families than before (including families where sire and dam effects are confounded), as well as the negative correlation expected between estimates of the sires and dams components of variance (Figure 2).

What biological conclusion can be drawn from con- tradictory results from two statistical tests for heritability of fitness? Although each estimation and testing procedure is statistically valid (SEARLE 1971; SOKAL and ROHLF 1981; WU 1986), our Monte Carlo simulations provide information on the weight that may

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FIGURE 2."Scatter plot of u? vs. u;, showing that jackknife estimates of variance components are negatively correlated due to statistical properties of the estimation procedure. Data are from Monte Carlo simulation of 25 sires, 2 dams per sire, and 5 progeny per dam, with parametric (ie., true) values u.: = u;) = 0.1 and = O.X, replicated 500 times. The correlation between estimates of u: and UX equals -0.55. Such correlated estimates are comnlon to many estitnation procedures, e .g . , SHAW (1987) Figure 1.

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be given tohhe alternative conclusions. The pooled estimator, a,&, provides greater statistical precision (48% smaller standard error of the estimate in Monte Carlo simulations, data not shown) and Fneral ly greater statistical power (Figure 1) than a;. Conse- quently, we are more inclined to believe the results from the present analysis, indicating that there is little or no additive genetic variation for plant size (and therefore level of seed production) in these populations of I. capensis. Nevertheless, other studies on this species (WALLER 1984; MITCHELL-OLDS and WALLER 1985) have found inbreeding depression for major components of fitness, which implies the existence of dominance genetic variance in these populations (CROW and KIMURA 1970). Failure to detect significant broad sense heritabilities for several characters in this study is likely due to the low statistical power of variance component analysis (Figure 1).

Major genes and phenotypic variance: We found heterogeneity of within-family variances for nearly all characters in both populations. In most cases, variance increased monotonically with family mean, indicating a simple effect of scale of measurement. The one exception to this pattern was germination date, which had maximum within-family variance at intermediate trait values in both populations. There are several possible explanations for heterogeneous patterns of within-family variation. Heterogeneity of family variances might be due to 1 ) artifactual effects of scale of

measurement, 2) differences in developmental homeostasis, 3) mortality selection against extreme phe- notypes, 4) variation in levels of inbreeding between sibships, 5) segregation of major genes, or 6) genes for variability per se.

Several of these hypotheses can be eliminated as explanations of our results. Each will be considered in turn: 1) If heterogeneity of variances is an effect of scale of measurement then we would expect greater variability at larger values of the trait (SOKAL and ROHLF 198 1). This pattern is observed for most characters in both populations, but not for germination date. 2) Differences in developmental homeostasis are expected to cause high variability in families that deviate from the mean or optimum, but lower variance near the center of the distribution (LERNER 1954; MITTON and GRANT 1984). This is reverse to the observed pattern (Figure 3). In related work, MITCHELL-OLDS and WALLER (1 985) tested for differences in developmental homeostasis in several leaf characters, but found no evidence for such canaliza- tion in those traits. 3) Mortality selection against extreme individuals might reduce variance in extreme families. However, levels of mortality in this experi- ment were extremely low (MITCHELL-OLDS 1986), so this factor cannot explain these observations. 4) Var- iability in level of inbreeding among families might cause heterogeneity of family variances (COCKERHAM 1963; MCCALL, MITCHELL-OLDS and WALLER 1989). We avoided this problem by performing controlled crosses among unrelated individuals to create outbred progenies. Note that this factor could not be eliminated had open pollinated maternal sibships simply been collected in the field.

The last two hypotheses cannot be distinguished with our data: 5) FAIN (1978) suggested that larger variance is expected in phenotypically intermediate families if they are segregating at a locus of large effect, while families at both extremes would be homozygous, and consequently have less segregating genetic variance. In the Madison population there is a significant dominance and/or maternal influence on germination date, so that differences in full-sibship variability might not be of genetic origin. However, in the Milwaukee population there is no trace of maternal influence on germination date. Conse- quently, different levels of variability among full-sib families must be due to genetic differences. There- fore, we may be seeing effects of major genes for quantitative characters, or 6) effects of genes that directly influence variability of phenotypic expression, per se. While these data cannot distinguish between these possibilities, they are both of considerable evolutionary interest.

The relative importance of polygenic variation us. loci of large effect (major genes or "leading factors")


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Family Mean Family Mean

FIGURE 3,”Scatter plot of absolute deviation of each individual from its full-sib family mean, IzY - f,l, against full-sib family mean value in Madison (A) and Milwaukee (B). Data have been Box-Cox transformed from their original germination cohort values. Only families with two or more progeny are shown. Families have significantly heterogeneous within family variances by Levene’s test (Table Z) , and maximum variability occurs at intermediate germination dates (Table 3). This is exactly the pattern predicted by FAIN (1978) as an indicator of segregation of major genes. This pattern is expected if intermediate families are segregating at a locus of large effect, while families at both extremes are homozygous, and consequently have less segregating genetic variance.

(WRIGHT 1968)) has been the subject of continuing debate for many years (WRIGHT 1968; LANDE 1983; GOTTLIEB 1984, 1985; COYNE and LANDE 1985; SHRIMPTON and ROBERTSON 1988a, b; SING et al. 1988; LANDER and BOTSTEIN 1989) and may influence strongly the response to selection. Quantitative genet- icists have argued convincingly that most metrical characters are under polygenic control (FALCONER 1981; LANDE 1981; BARTON and TURELLI 1989), although WRIGHT (1968, p. 41 1) has suggested that “It is probable that a leading factor or factors could be isolated much more often than has been the case.” Although existing statistical approaches provide limited insight regarding this problem (MAYO, HANCOCK and BAGHURST 1980; COCKERHAM 1986), maximum likelihood analysis of segregation in pedigrees permits examination of discrete or metrical characters with flexible models incorporating major loci and polygenic variance (E. A. THOMPSON and MITCHELL-OLDS, unpublished results). With the advent of extensive genetic maps employing large numbers of codominant molecular markers, the potential exists for resolution of this question in the near future (MICHELMORE and HULBERT 1987; CHANG et al. 1988; PATERSON et al. 1988; MICHELMORE and SHAW 1988; LANDER and BOTSTEIN 1989).

Genetic influences on levels of phenotypic variabil- i ty have been analyzed in plants and animals (MITTON and GRANT 1984; MITCHELL-OLDS and WALLER

1985; SCHLICTING 1986; LEARY and ALLENDORF 1989). Genetic differences in fluctuating asymmetry have been detected in many studies (e .g . , VRIJENHOEK and LERMAN 1982; LEAMY 1984; LEARY et al. 1985), although MITCHELL-OLDS and WALLER (1985) found that selfed and outcrossed plants do not differ in their levels of phenotypic variance for leaf shape in I. capensis. The pattern of genetic changes in variability observed (Figure 3) are exactly opposite to the predic- tions of standard theories of developmental homeostasis. Better understanding of the evolutionary implications of these observations would require detailed genetic analysis of these populations. However, regardless of the genetic basis of these heterogeneous variances, they have important statistical consequences. Most parametric tests of statistical significance assume homogeneous error variances (SEARLE 197 1 ; SOKAL and ROHLF 198 l), and violation of this assumption may invalidate statistical and biological conclusions. Accordingly, examination of family variances is an essential aspect of quantitative genetic analyses which influences genetic and statistical interpretation.

Statistical methodology: Analysis of quantitative genetics in natural populations has been hindered by computational and methodological problems in statistical analysis (MITCHELL-OLDS and RUTLEDGE 1986; SHAW 1987). Maximum likelihood offers many statistical advantages but proved computationally infeasible


in this instance. We have shown that jackknife analysis of variance components provides a flexible method for estimation and testing of biologically important questions regarding influences on life history variation. For example, classical least square methods do not allow tests of dominance/maternal effects. Fur- thermore, these jackknife tests of significance proved to be extremely robust to heterogeneity of within family variances.

When statistical procedures, such as maximum likelihood or resampling methods, rely on asymptotic properties, then Monte Carlo simulations are essential to provide full confidence in estimates and tests. These simulations permit evaluation of rates of type I and type I1 errors, bias and standard error of an estimator, performance at finite sample sizes, and robustness to assumption violations. It is essential to understand the statistical behavior of one’s chosen method for proper biological interpretation of statistical results. Failure to detect significant levels of genetic variance has little meaning if statistical power is low, but can be taken as strong evidence for lack of genetic variation when statistical power is high. For example, in the pooled analysis of the Madison and Milwaukee populations together (n = 454), we can conclude with strong statistical confidence that heritabilities for most characters are less than 0.20 (Figure 1). We urge explicit consideration of statistical characteristics of estimators, and in particular of statistical power, in quantitative genetic analyses.

We are grateful to F. ALLENDORF, C. DENNISTON, R. NAKAMURA, E. THOMPSON, D. WIPRNASZ, S. VIA, D. WALLER, B. WEIR, and three anonymous reviewers for conments and discussion. T.M.O. was supported by National Science Foundation grants BSR-83 1 181 7, BSR-842 1272, and U.S. Department of Agriculture Com- petitiveGrant88-37151-3958.J.M.B.wassupportedbyaNational Science Foundation predoctoral fellowship and a National Science Foundation dissertation improvement grant. This is contribution number 102 from the University of Montana Herbarium and Center for Plant Diversity.

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Communicating editor: B. S. WEIR

Documents

Statistical Genetics of an Annual Plant, Impatiens capensis. I