Chapter 4: Quantitative genetics I: Genetic variation …...Chapter 4: Quantitative genetics I: Genetic variation 4.1 Mendelian basis of continuous traits The previous chapters have

Conner and Hartl – p. 4-1

From: Conner, J. and D. Hartl, A Primer of Ecological Genetics. In prep. for Sinauer

Chapter 4: Quantitative genetics I: Genetic variation

4.1 Mendelian basis of continuous traits

The previous chapters have focused on the population genetics of single loci with only twoalleles. If a large majority of variation in a phenotypic trait is determined by one locus, the resultis a visible polymorphism like Mendel’s pea traits or flower color in Delphinium (Chapter 2).However, as noted in Chapter 2, most phenotypic traits do not fall into distinct categories, butrather are continuously distributed. Examples of continuously distributed traits are shown inFigure 4.1. These traits are called quantitative or metric traits because they need to bemeasured, not just scored as round or wrinkled or blue or white. In classical or statisticalquantitative genetics, the phenotypes of individuals of known genetic relationship (usuallyparents and offspring or siblings) are measured, and the genetic and environmental sources ofphenotypic variation are determined statistically. In the newer technique of quantitative traitlocus (QTL) mapping, variation in genetic markers that are scattered throughout the genome isstatistically related to phenotypic variation (Chapter 5). Both techniques have strengths andweaknesses, and both are valuable for the study of natural populations. The key similarity is thatthey focus on continuously distributed phenotypic traits, whereas population genetics is muchmore concerned with discrete genotypes. Population genetic techniques are applied directly tophenotypes only in the rare cases where phenotypic variation is discrete, i.e., visiblepolymorphisms. So for this chapter and the next two, the emphasis is shifting from genotypes tophenotypic traits.

Figure 4.1. Continuously distributed traits. Shown are frequency distributions, where the height of each bar givesthe number of individuals with the trait value on the X-axis. Note that these traits are approximate a normaldistribution, which is common for many traits (sometimes statistical transformation is necessary). Data are from afield study of wild radish (Conner et al. in press).

In population genetics the population is characterized with allele and genotype frequencies(Chapter 2). Because these are discrete, it is a simple matter to count individuals with each

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genotype and calculate the frequencies (proportions). We used F-statistics to understand howthis discrete variation is distributed within and among populations (Chapter 3). With continuousphenotypic traits, in most cases the alleles that are present in the population or even the lociaffect the trait are unknown, so we need to use the statistical measures of mean and variance(and later covariance, chapter 5) to characterize populations (Figure 4.2). This chapter isdevoted to relating these two methods by showing how the means and variances are determinedby allele frequencies and the environment. In this way we will show how statistical quantitativegenetics is derived from Mendelian genetics, and set the stage for a discussion of QTL mappingin the next chapter.

A

B

C

Phenotypic Trait

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Figure 4.2. Three normal distributions (idealized versions of real data such as that in Fig. 4.1) illustratingmean and variance. The mean (single-headed arrows) is just the average phenotype in the population, and thevariance (double-headed arrows) is a measure of how variable the population is; in other words, the width of thedistribution. Populations A and B have the same mean but different variances, while A and C have different meansbut the same variances. See Appendix 1 for the formulae for mean and variance.

Most phenotypic traits have this continuous distribution in spite of the fact that all geneticvariation is discrete, not continuous; for example, there are three distinct genotypes at a locuswith two alleles. This continuous distribution of most traits occurs for two reasons -- most traitshave more than one gene locus affecting them, and they are also affected, sometimes to a largedegree, by the environment. Remember that means, variances, and allele frequencies are allproperties of the population, but the first step in relating these is to discuss what determinesphenotypic value (P), which is simply the measurement of a given trait for a given individual.For example, the phenotypic value for the petal length of a plant in Fig. 4.1 might be 7.5millimeters, and the same plant’s for P for flowering time might be 44 days. The phenotypicvalue of an individual is determined by the individual’s genotype and the environment, which wewill define as all non-genetic effects on the phenotype:

P = G + E (4.1)

where G is the genotypic value and E is the environmental deviation. The genotypic value is thephenotype produced by a given genotype in the average environment, which means it can onlybe measured by replicating clones or highly inbred lines across environments, but it is a usefulconceptual tool for sexual species as well. The graphs in Figure 4.3 show the values of G foreach genotype with 2, 3, or six loci; this shows how adding loci makes the trait distribution morecontinuous even in the absence of any environmental effects.


Figure 4.3. A hypothetical example (based on the real petal length data in Fig. 4.1) showing genotypic values(along the X-axes). The three graphs show how increasing numbers of loci affecting a trait makes the traitdistribution more continuous in the absence of environmental deviations. In A, there are two loci with two alleleseach, which is the simplest case for a trait affected by more than one locus. The loci act additively (no dominance orepistasis), so each capital letter allele adds 1.5 mm of petal length over the aabb genotype, which has 5 mm petals.The table shows the frequency of each genotype under HWE and with p=q=0.5 for both loci, and the graph showsthe phenotypic distribution that results. B and C show the phenotypic distribution with 3 and 6 loci; in these cases

each capital letter allele adds 1 and 0.5 mm respectively, keeping all other conditions the same as in A.

Environmental factors usually vary continuously themselves – think of temperature, rainfall,sunlight, prey availability, etc. Therefore, the environment causes the phenotypic valuesproduced by different individuals with the same genotype to deviate continuously from G; thesedeviations are the E term in eq. 6.1. In other words, the environmental deviation is thedifference between the phenotypic and genotypic values caused by the environment. Thesepoints illustrated in Figure 4.4, which depicts a set of hypothetical results from raising 12individuals with the AABB genotype from Figure 4.3A each in a different environment. The

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resulting distribution of environmental deviations reflects the most common assumptions inmodeling environmental effects, that is that E is normally distributed with mean = zero. Thus,the most common environments cause little or no deviation from the genotypic value of four forAABB, and environments causing larger deviations being progressively rarer. Because the meanenvironmental deviation (E) is zero, when the means are plugged into equation 4.1, we find thatthe mean phenotypic value equals the mean genotypic value, which is a different way of statingour definition of G as the phenotypic value produced in the average environment.

Environment EPhenotypic

Value ofAABB

1 -0.9 3.12 -0.6 3.43 -0.5 3.54 -0.4 3.65 -0.1 3.96 0 4.07 0.1 4.18 0.2 4.29 0.4 4.410 0.6 4.611 0.7 4.712 1.0 5.0

Figure 4.4. An illustration of the effects of hypothetical environmental deviations on the phenotypes produced bythe AABB genotype in Fig. 4.3A. The deviations are continuously and roughly normally distributed with mean=0,which give rise to the same distribution of phenotypic values in the graph except that the mean=4.

We can combine Figures 4.3A and 4.4 by adding a normal curve representing theenvironmental deviations in Fig. 4.4 over the bar for each genotypic value in Fig 4.3A – theresult is shown in Fig. 4.5. You can see that these curves overlap, so for example someindividuals with the AaBb are actually taller than some AABb individuals due to theenvironmental deviations. This overlap further smoothes out the differences between genotypes,producing the smooth normal curve drawn over the entire histogram. Therefore, a continuousphenotype can result even if the trait is oligogenic, that is, affected by only a few loci (two in ourhypothetical case). Traits that are affected by many gene loci are often called polygenic traits.Note that by drawing the same curves over each bar, we are assuming that the environment hasthe same effect (i.e. the same E’s as in Fig. 4.4) on each genotype. This is rarely the case, so wewill relax this assumption when we discuss genotype-environment interaction in the nextchapter.

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Figure 4.5. Combining G and E to show how a continuous phenotypic distribution can result from only two lociwith two alleles each. The value on the x-axis for each bar is G as in Fig. 4.3A, and the small normal curves overeach bar represent the distribution of phenotypes produced by each genotype when the environmental deviation (E,from Fig. 4.4) is included. Since the small distributions overlap, the entire distribution approximates a smoothnormal curve (dashed).

The distinction between discretely polymorphic (Mendelian) traits and quantitative traits isthe magnitude of effect of a single locus on the trait relative to other sources of variation, i.e.,other loci and the environment. A polymorphism results when a single major gene isresponsible for the majority of phenotypic variation in the trait, and a quantitative trait resultswhen the variation is divided among several to many gene loci and the environment. Genes thatcontribute a small amount of variation are called minor genes, but in reality there is a continuumof effects, from very small to large. Only with the advent of QTL mapping are we beginning toget an idea of the number of gene loci affecting quantitative traits, and the distribution of effectsof those loci on phenotypic variation. Statistical quantitative genetics treats the genome as a"black box", providing no information on individual loci, but rather abstracting the effects ofgenes as statistical parameters such as variance and covariance (see below). Even so, statisticalquantitative genetics is very useful for answering a variety of evolutionary questions, and will bethe method of choice for many organisms until QTL mapping becomes simpler and cheaper.

4.2 Different types of gene action

Because quantitative genetics is focused on the phenotype, we need to be more explicitabout different modes of gene action, which we defined as how genotypes affect phenotypes inchapter 3. Additive gene action is called this because you can just add up the effects of eachallele. Hypothetical examples of additive gene action were given in Figure 4.3. Note that petallength in those examples is determined simply by the number of capital letter alleles present inthe two-locus genotypes. The affect of each allele is not affected by what other allele is presentat the same locus, nor by what alleles are present at the other loci. Note that additivity does notimply equal effects of all alleles at a locus or all loci affecting a trait; the continuum of effects ofloci can occur with any mode of gene action. Two other modes of gene action, dominance andepistasis, are characterized by interactions between alleles at the same locus and different loci,respectively (Table 4.1).


Interaction among alleles?

No interaction Interaction

Within a locus Additive Dominance

Between loci Additive Epistasis

Table 4.1. Summary of how interactions among alleles at different levels (within or between loci) causesdifferent types of gene action.

When genes act in a dominant fashion, the interaction between alleles at one locus meansthat the diploid genotype at each locus needs to be considered as a whole to determine thephenotypic effect (see Fig 3.8). An important general point for gene action is that it is specificfor a given locus and also for a given phenotypic trait, so that the degree of dominance orepistasis for a given locus can vary across traits at different levels of the phenotypic hierarchy(Chapter 1). As a hypothetical example, suppose that predation on a shrimp by fish depends onshrimp size, because above a certain threshold size they are too large for the fish to handle. If alocus has additive effects on body size, then this same locus could show dominance for predationresistance if heterozygotes at this locus are above the threshold size, because they would have thesame low predation risk as the large allele homozygote. Therefore, our hypothetical locus wouldbe additive for body size but dominant for predation resistance.

Epistasis is an interaction between alleles at different loci, in which the phenotypeassociated with a particular genotype depends on what alleles are present at another locus. Anexample comes from butterfly wing genes (Table 4.2; Smith 1980) which show both dominanceand epistasis. Note that with the BB and Bb genotypes at the B locus, there is underdominanceat the C locus (i.e., the heterozygote has the smallest wings), whereas with the bb genotype thereis overdominance at the C locus (Cc has the biggest wings). While epistasis is likely to be a verycommon feature of genomes, the importance of epistasis in evolution remains controversial.

BB and Bb bbCC 41.91 (46) 40.96 (119)Cc 40.81 (113) 42.13 (32)cc 40.94 (150) 41.62 (21)

Table 4.2. Mean forewing lengths (mm) of female Danaus butterflies of differing genotypes at two loci (samplesizes in parentheses). The B locus exhibits complete dominance. Note that these are estimates of genotypic values,because they are the averages of a number of individuals of the same genotype.

4.3 How can we quantify gene action?

Before we can show how population means and variances are determined by Mendeliangenetics, we need measures that quantify gene action. We will ignore epistasis for simplicity.Above we discussed the continuum of magnitudes of gene effects; the magnitude of the additiveand dominant action at a locus can be quantified as a and d respectively (figure 4.6). Thesevalues are of only conceptual interest in statistical quantitative genetics, because they cannot be


measured using those techniques, but they can be measured using QTL mapping so are ofincreasing practical interest (see next chapter). Figure 4.6 is similar to figure 3.8, except thehorizontal axis in figure 3.8 was fitness, whereas in figure 4.6 the horizontal is G (or mean P) forany phenotypic trait.

In the scheme shown in figure 4.6, the midpoint between the two homozygotes is set to zero,G for the two homozygotes are +a and –a, and for the heterozygote is d (d has the same purposeas h in Fig 3.8, but they are not the same). Figure 4.6A shows the additive case, 4.6B completedominance, and 4.6C partial dominance. From this we can see that the degree of dominance canbe expressed as d/a, which equals 0, 1, and 1/4 in these three cases respectively. Figure 4.6Ddepicts a locus with smaller a, that is, a smaller effect on G and thus P than the loci shown in A –C. Therefore, the magnitude of a gives a measure of the magnitude of the effect of that locus onthe phenotype, i.e., whether the locus is a major or minor gene (see above). Note that theabsolute value of d is the same in C and D, but since a is smaller in D the degree of dominanced/a is greater in D (1/2) due to the smaller overall effect of the locus in D. Finally, fig. 4.6Erepresents a locus with overdominance where d/a >1.

A A1A1A1A2A2A2

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Gene action

Additive

Completedominance

Partialdominance

Partialdominance

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GFigure 4.6. Diagram showing gene action quantified using a and d. The horizontal scale represents genotypicvalues.

We can illustrate the use of a and d with the data from the Danaus butterfly wing lengths inTable 4.2 (Figure 4.7). Since there is epistasis, the values of a and d for the C locus differdepending on the genotype at the B locus. The C locus has a larger effect on wing length with


the BB or Bb genotypes (a=0.48) than with the bb genotype (a= 0.33). Note that d is negativewith underdominance.

Figure 4.7. The effects of the genotypes at the C locus on wing length in Danaus butterflies; data from Table 4.2.For each line graph, the values of G are above the graph and a and d are below.

4.4 Population meanWe now are ready to show how the population mean for a given phenotypic trait is

determined by Mendelian genetics, i.e., allele frequencies and gene action. It is beyond thescope of this book to attempt this for multiple loci, so we will go back to the simplest case of onelocus with two alleles. This is adequate to demonstrate the main concepts, and in the absence ofepistasis the results from each locus can simply be summed to give the effects of all loci on themean for a phenotypic trait.

Genotype Frequency Genotypic value ProductAA p2 +a p2aAa 2pq d 2pqdaa q2 -a - q2a

Sum of products = a(p-q)+2pqd

Table 4.3 Derivation of the equation for genotypic mean. To simplify the sum of the products, note that p2-q2=(p+q)(p-q)=p-q because p+q=1

The derivation of the mean is shown in Table 4.3. The usual way of calculating a mean, thatis, summing values and dividing by the number of individuals summed, is equivalent to themethod shown in the table when you have multiple individuals with the same value. In thismethod, the value for each class (the three genotypes here) is multiplied by its frequency, andthen these products are summed to give the mean. The frequencies are the Hardy-Weinbergvalues, while the values are expressed in terms of a and d. The summation gives the equation forthe mean:

CC Cccc

0-a +a d

d/a

-1.27

2.54

BB and Bb

bb

-0.33 0.33 0.84

40.96 41.62 42.13

Cc CCcc

0d -a +a

-0.615 -0.485 0.485

40.81 40.94 41.91


G = P = a(p − q) + d2pq (4.2)

(A bar over a symbol denotes the mean of that variable). This equation is important because itshows how population means are determined by the allele frequencies, the magnitude of theadditive effect, and the degree of dominance. The first term represents the effects of thehomozygotes, and shows that as a increases the mean increases if p>q and decreases if p<q(recall that G for the aa homozygote is –a). The second term is the effect of heterozygotes.Again, in the absence of epistasis these terms can just be summed over all loci affecting the trait.Equation 4.2 is also important because it shows how changes in allele frequency affect the meanphenotype, which is how phenotypic evolution occurs.

4.5 Phenotypic variance

The next quantity we need to characterize continuous phenotypic traits in a population is thevariance. Variance is absolutely critical, because as we noted in earlier chapters variation is theraw material for evolutionary change. Variance is also the fundamental measure of variation instatistics, upon which many other statistical measures and tests are based (see also Fig. 4.2,Chapter 5, and Appendix 1):

(4.3)

The numerator of this formula is the “sum of squares” (SS), or the sum over all individuals of thesquared deviations from the mean. Therefore, if there are lots of individuals with values far fromthe mean in a population (i.e. curves A and C in Fig. 4.2) then the sum of the deviations will belarge and so will the variance. If most individuals have values close to the mean (e.g. curve B inFig. 4.2) then the deviations and thus the variance will be small. The denominator is the numberof individuals in the population minus one (the degrees of freedom; chapter 2), which makes thevariance an average squared deviation from the mean. This is why variance is sometimes calleda Mean Square (MS; see below).

If the phenotypic values in a population are used in eq. 4.3, then this is the phenotypicvariance (VP) for that trait. We can partition this total variance in the population into additivecomponents representing various genetic and environmental causes, assuming that there is nocorrelation or interaction between the genotypes and the environment. These assumptions canusually be met with proper experimental design and analysis, and the study of genotype byenvironment interactions is a very important endeavor in its own right (Chapter 5). The simplestpartition is:

VP = VG +VE (4.4)

where VG is the genotypic variance and VE is the environmental variance. This equationclosely parallels eq. 4.1, because these are the variances across the population of the individualphenotypic values, genotypic values, and environmental deviations respectively. Thispartitioning is most useful for clonal or highly selfing organisms, because they pass their diploidgenotypes on to their offspring intact, but it is less useful for sexually outcrossing species where

Vx =

(Xi − X )2i =1

n

∑n −1


the genotypes are created anew in each offspring by a random combination of an allele from eachparent at each locus. Therefore, for these species we need to further partition VG:

VG = VA +VD +VI (4.5)

where VA is the additive genetic variance, VD is the dominance variance, and VI is theinteraction or epistatic variance (the latter two are collectively referred to as non-additivegenetic variance). As the names imply, these are caused by the three modes of gene action.The additive genetic variance is the most important for sexually reproducing species, becauseonly the additive effects of genes are passed on directly from parents to offspring. Thedominance and epistatic effects are not passed on, because only one allele at each locus istransferred from each parent to create new dominance relationships in the offspring, andsimilarly independent assortment of alleles at different loci creates new epistatic effects (exceptwith tight linkage).

For these reasons VA is most important in determining changes in mean phenotypic valueacross generations, which is the definition of phenotypic evolution. It also means that VA is theeasiest of the genetic components to measure using the resemblance between relatives, becausethis resemblance is caused primarily by additive variation. This is the basis of the statisticalquantitative genetic methods that we will cover later in this chapter and the next. The effects ofdominance and epistasis cause offspring to not look like the average of their parents, which theywould under complete additive effects of genes. As an example, consider a cross between twoDanaus butterflies from Table 4.2, one with the BBCC genotype and the other with the bbccgenotype. Under complete additivity, the offspring (all BbCc) would have genotypic valuesequal to the average of the parents, i.e. (41.91 + 41.62)/2 = 41.76. However, due to bothunderdominance and epistasis in this case, the double heterozygote offspring have G = 40.81,less than either parent.

The equation for additive variance in terms of allele frequencies and gene action is:

VA = 2pq[a + d(q − p)]2 (4.6)

As for the equation for the population mean (eq. 4.2), this is for one locus only. Recall fromfigure 2.10 that the first term, 2pq, is maximum at what p=q= 0.5. This means that geneticvariance is high at intermediate allele frequencies. This makes perfect sense, because if oneallele is rare, most individuals are homozygous for the other allele, and therefore there is littlevariance in the population. If there is no dominance, d=0, and the equation reduces to:

VA = 2pqa2 (4.7)

which means that additive variance is maximized at p=q= 0.5 (Figure 4.8A). When there isdominance, the maximum variance occurs when the recessive allele is more common (q=0.75),making the d(q-p) term large and positive (Figure 4.8B). Intuitively, this is because withdominance and equal allele frequencies, 75% of the individuals in the population have thedominant phenotype. As q becomes larger than 0.75, additive variance drops because the first2pq term drops faster then the d(q-p) term increases. Note that dominance variance does peak at


p=q= 0.5 in Figure 4.8B; this is because the equation for dominance variance is similar toequation 4.7 in that the allele frequencies are only in the 2pq term:

VD = (2pqd)2 (4.8)

Finally note that all these equations for variance have a squared term because variance is definedas the squared deviation from the mean (eq. 4.3). This also means that they can not be negative,because negative variability is meaningless. In practice, however, estimates of variances can benegative due to experimental error.

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Figure 4.8. Genotypic variance VG, additive genetic variance VA, and dominance variance VD for a single locus withtwo alleles in a population in HWE. Note that the X-axis is the frequency of the a allele, which is recessive in panelB. Because this is a single locus there is no epistatic variance. A. A completely additive locus, a = 0.1, d = 0. B.Complete dominance, a = d = 0.0707. From equations 4.6-4.8.

4.6 Breeding value

The genotypic value is a property of the genotype, but if we want to understand theevolution of phenotypic traits, we need to understand how the trait is inherited, that is, how it ispassed from parents to offspring. In a sexually reproducing organism, genotypes are not passedon from parents to offspring, but rather are created anew in each offspring from combining anallele from each parent at each locus. The breeding value can be defined as the effect of anindividuals genes on the value of the trait in its offspring; this effect is caused by the additiveeffect of genes. The breeding value is sometimes called the ‘additive genotype’, and thevariance of these breeding values is VA. Thus, the breeding values are more important than G insexual species. Breeding values are becoming increasingly common in ecological genetics,because there are improved methods of estimating them, they provide a useful method ofestimating genetic correlations (below), and because they may reduce bias in measuring selection(next chapter). The improved method of estimation is called Best Linear Unbiased Prediction(BLUP; for details see Littell et al. 1996 chapter six).

The name breeding value comes from its roots in animal breeding. For example, if a givenbull had genes that tended to increase meat production in its offspring, then it has a high valuefor use in breeding to increase meat production in the next generation. The breeding value for agiven sire is usually estimated as two times the deviation of the mean of his offspring from thepopulation mean in a paternal half-sibling mating design. Therefore, if a bull’s offspring tend to


have higher meat production than average, then this bull has a positive breeding value for thistrait.

The breeding value is two times the deviation from the mean because the sire onlycontributed half of the offspring’s alleles. Because the sire was mated to several randomlychosen dams it represents the effect of his alleles in combination with a random sample of otheralleles in the population. This emphasizes the point that breeding value, like all quantitativegenetic parameters, depends on population allele frequencies. If the hypothetical bull mentionedabove was moved to a population of higher meat producers (i.e., higher frequencies of high meatproduction alleles), he could have negative breeding value, that is, his offspring could be belowthe population mean. One can also measure a breeding value for traits that the sire does notpossess, for example, milk production in a bull, because it is based on offspring measurementsonly.

4.7 Heritability

The heritability is the most measured and discussed quantity in quantitative genetics, bothin plant and animal breeding and also when applied to natural populations. It is so importantbecause it partly determines how rapidly the mean phenotype evolves in response to artificial ornatural selection. Heritability is the proportion of the total phenotypic variance that is due togenetic causes; in other words, the relative importance of heredity in determining phenotypicvariance. There are two definitions of heritability. Broad-sense heritability is based ongenotypic variance:

hB2 = VG

VP(4.9)

Therefore, broad sense heritability measures the extent to which phenotypic variation isdetermined by genotypic variation. Broad-sense heritability includes the effects of dominanceand epistatic variance and therefore is most useful in clonal or highly selfing species in whichgenotypes are passed from parents to offspring more or less intact; in these species it is the broadsense heritability that affects evolutionary rates. In outbreeding species evolutionary rates areaffected by narrow-sense heritability, the proportion of total phenotypic variance that isdetermined by additive variance:

hN2 = VA

VP(4.10)

Since we focus on sexual species in this book, from now on when the term heritability isused alone it will mean narrow-sense. The symbol h2 derives from Sewall Wright’s definition ofh as the ratio of standard deviations (the square root of a variance), which is rarely used. Thesymbol can serve as a reminder that heritability is a ratio of variances, which as we have notedare based on squared deviations from the mean.

Heritability is often interpreted as the degree of genetic determination, or the extent towhich the phenotype is determined by the genotype, or the extent to which the phenotype isdetermined by genes from parents. These are all incorrect, because there can be lots of loci that


affect the trait but are fixed and thus do not contribute to variation. Note that in equations foradditive and dominance variance above (eq. 4.6-4.8), when one allele is fixed then the frequencyof the other is zero and the entire equation equals zero. However, the equation for populationmean (eq. 4.2) does not equal zero when one allele is fixed. Thus, a locus can still affect a traitmean even when it is not variable. For example, the fact that humans have two eyes isundoubtedly genetically determined, but the heritability of eye number is essentially zero. It isextremely important to remember that heritability refers to variation; note that all of thedescriptions and definitions above are based entirely on variation.

Heritability measures the extent to which resemblance among relatives, in comparison tounrelated individuals of the same species, is due to heredity. Obviously all members of a speciesresemble each other more than they resemble another species; this is due to loci that are fixedthroughout the species, or loci for which few or no alleles are shared with related species. Thecloser resemblance of relatives within the species or population is determined by additivevariation (and often the environment).

There are a large number of misconceptions relating to heritability, even among biologists,so it is important to stress the following points. First, there is not one heritability for a given traitin a given species, because heritability can and often does differ among populations and amongenvironments. It can differ among populations because VA depends on allele frequencies (eq.4.6), so if populations are differentiated for loci affecting the trait of interest, heritability maydiffer. For example, Mitchell-Olds (1986) found significantly different heritabilities forgermination time and two size measures between two Wisconsin populations of jewelweed(Impatiens capensis) grown in a common environment. However, clear examples like this ofdifferences among natural populations in heritability in a common environment are rare.Heritability can differ among environments because VE is part of VP (eq. 4.4). This can be seenmore clearly when the equation for heritability is rewritten with the components of VP writtenout:

h 2 = VAVA +VD +VI +VE

(4.11)

Therefore, as VE increases, heritability decreases, because less of the phenotypic variance isadditive genetic. For example, the heritability of wing width in male Drosophila melanogasterwas much greater under control as compared to stressful conditions (h2 = 0.69 vs. 0.09respectively) even though VA was slightly greater under stress (Hoffmann and Schiffer 1998).This lower heritability was caused by a much greater environmental variance under stress (VE =9.2 and 0.9 under stress and control conditions respectively). The numerator of heritability canalso be affected by the environment, because expression of genetic variance can be affected bythe environment; this is called genotype by environment interaction, and is discussed further inthe next chapter.

Since heritability is a proportion of variances, and the numerator is contained in thedenominator, it is dimensionless and theoretically varies from zero to one, although withexperimental error estimates of heritability can be outside these bounds. A second commonmisconception is that a heritability of zero means that the trait is not affected by genes. Thereason this is not true is due to the effects of fixed loci mentioned above; in fact, all traits have


some genetic component, because organisms are built according to the instructions in genes. Theconfusion disappears if one focuses on variation. A heritability of zero does mean that there isno additive variance in that population and environment, so that the phenotypic variance isentirely environmental and non-additive genetic.

A final misconception is that a heritability of one means that the environment cannot affecta trait. A heritability of one means only that the environmental variation that was present in thatparticular population did not affect the phenotypic variance. If the environment changes, thiscould still change the mean and variance for the trait.

Because heritability is dimensionless, it is useful for comparing across species and traits thatare measured on very different scales, and it is therefore very widely used and reported (Table4.4). However, the fact that heritability can change with changes in either additive variance orenvironmental and non-additive variance can be confusing and misleading if the focus of interestis on additive variance. It is really the amount of additive variance that determines the rate ofevolutionary change, but directly comparing variances is difficult because they are notdimensionless and therefore vary with the scale of the trait or organism being measured.

Table 4.4. Examples of heritability in natural populations; many of these were estimated in the field.

Organism Trait h2 Reference

Guppy Orange spot size 1.08 Houde 1992

Garter snake Chemoreceptiveresponse

0.32 Arnold 1981

Red deer Female fecundity 0.46 Kruuk et al. 2000

Meadow voles Growth rate 0.54 Boonstra and Boag 1987

Darwin’s finch Bill length 0.65 Boag 1983

Collared flycatcher Male lifespan 0.15 Merila and Sheldon2000

Seaweed fly Male body size 0.70 Wilcockson et al. 1995

Cricket Development time 0.32 Simons and Roff 1994

Milkweed bug Flight duration 0.20 Caldwell and Hegmann1969

Scarlet gilia Corolla width 0.29 Campbell 1996

Jewelweed Height 0.08 Bennington andMcGraw 1996


4.8 Estimating Additive Variance and Heritability

Recall that additive genetic variance creates resemblance among relatives over and abovethe resemblance among unrelated members of the population. Statistical quantitative geneticsuses this fact to separate VA from non-additive variance and VE. A common misconceptionabout measuring heritability is that one can separate VG from VE by eliminating VE throughraising the organisms in a controlled laboratory environment. However, no matter how excellentthe lab facilities and protocols are, there will inevitably be at least small differences amongindividuals in temperature, light, food, water, etc. Environmental variance can certainly bereduced, but even then the estimate of heritability is not relevant to the field where evolution isoccurring. This is because VE is in the denominator of heritability (eq. 4.11 above), so byreducing VE in the lab one overestimates heritability in the field. In addition, if there is genotypeby environment interaction (next chapter), then the lab estimate of VA does not accurately reflectVA in the field either. For example, Conner et al. (in press) reported greatly decreasedheritability estimates for six floral traits in the field compared to the greenhouse for the samenatural population. These reduced heritabilities were due both to increased VE and decreased VA

in the field relative to the greenhouse, a pattern matched by most other similar studies.

In addition to these problems separating VA from VE, additive and non-additive variancestill need to be separated to understand phenotypic evolution in outbreeding species. Thus, inquantitative genetics, estimates of variances and heritabilities are made using phenotypicmeasurements on individuals of known genetic relationship, typically either parents andoffspring or siblings. We will cover each of these in turn.

Offspring – Parent Regression

To estimate heritabilities using parent-offspring regression, you first need an adequatenumber of monogamously mated pairs for the parental generation. Fifteen pairs is a minimum,30 to 50 pairs are usually necessary for reasonably precise estimates. The procedure is tomeasure the trait(s) of interest on one or typically both of the parents and raise their offspring.When the offspring reach the same developmental stage at which the parents were measured, thesame trait(s) are measured on the offspring and the average is taken over all the offspringmeasured in each family. This offspring average is then regressed on the measurements of thefathers, mothers, and/or the average of the two parents (called the mid-parent; Fig 4.9).

Offs

prin

g m

ean

tars

us L

.

Midparent tarsus L. (mm)


Figure 4.9. Offspring-parent regression of tarsus length in a natural population of tree swallows. Reprinted, withpermission, from Wiggins 1989.

In an offspring-parent regression such as that in Fig. 4.9, each family is represented onepoint. Therefore, each of the 13 points in Fig. 4.9 represents the average tarsus length of the twoparents on the X-axis and the average tarsus length of all the offspring of those two parents onthe Y-axis; two to four offspring per family is typical. Linear regression is a statisticaltechnique of drawing the best fitting straight line through these points, which produces anequation for the line in the familiar form:

Y = a + bX (4.12)

In this equation, b represents the slope, or the change in Y per unit change in X (∆Y/∆X).The slope of offspring on mid-parent is the estimate of heritability. Therefore, the steeper theslope, the more directly the offspring resemble their parents, and the higher the proportion ofphenotypic variance that is additive genetic, that is, passed on from parents to offspring. Forexample, if the slope is one, then for an increase of one unit of phenotypic value of the parents,you get an increase of one unit in the offspring, that is, the average offspring has exactly thesame phenotypic value as the average parent. A slope of one also means that the spread of pointsalong the X axis is the same as the spread of points along the Y-axis; in other words, all of thephenotypic variance in the parents is passed on to the offspring because it is all additive genetic.

The slope of 0.5 in Fig. 4.9 means that a one unit increase in the parents produces only halfa unit increase in the offspring, that is, only half of the phenotypic variance is additive geneticand therefore passed on. What happened to the other half of the phenotypic variance? It iscaused by non-additive genetic and environmental variance, so does not contribute to theresemblance between parents and offspring. Does this mean that the total phenotypic variance inthe offspring is half that of the parental generation? No, because the environmental and non-additive variance is expressed in the variance among the full-siblings within each family (seebelow), and this variance is removed from the analysis by using a single average for theoffspring from each family rather than the individual offspring values.

Box: regression details

In any regression analysis, there is a standard error (SE) for the slope, a P-value giving theprobability that the slope is zero, and an R2 value that is the proportion of the variance in thedependent (Y) variable that is explained by the variance in the independent (X) variable. As thescatter of points around the regression line increases, the R2 decreases (see below). In a simplelinear regression with only one X variable, these three are very closely related to each other, andquantify how confident we are in the value of the slope.

In an offspring-parent regression, increasing the number of offspring included in eachfamily will increase the precision of the estimate of the average trait value for that family.Because this increase in family size decreases random error in the Y-variable, it will tend todecrease the standard error and the P-value for the slope and increase the R2. This means that wehave more confidence in our estimate of the slope, and therefore our estimate of heritability, butthe increased R2 does not mean an increased heritability. The increased number of offspring perfamily will not have a systematic effect on the estimate of the slope (h2), only our confidence inthe estimate. Increasing the number of points in the regression, that is, the number of families,


has a similar effect; it will decrease the standard error and the P-value, but will not cause apredictable change in the slope. Increasing the number of families should not cause a predictablechange in the R2 value either, because it does not affect the accuracy of estimation of each familymean, and therefore does not affect the scatter of points around the regression line.

For example, the two offspring-parent regressions in figure 4.10 were calculated from thesame data, with the average of only two offspring per family used in fig. 4.10A and the mean ofall four offspring that were measured in each family in fig. 4.10B. The two regressions havevery similar slopes (not statistically different as illustrated by the overlapping standard errors)and therefore make very similar estimates of heritability. The difference is that the scatteraround the regression line is greater in fig. 4.10A (reflected in the higher SE and lower R2

compared to Fig. 4.10B), and so we have less confidence in this estimate. In figure 4.10B wehave greater confidence in our heritability estimate (lower SE) due to less scatter around theregression line, which in turn leads to higher R2 because each family mean was estimated withless random error.

Figure 4.10. Offspring parent regressions estimating heritability for pistil length in wild radish flowers. Theslope of each fitted regression line is given with the standard error of the slope in parentheses. Four offspring weremeasured in each of 50 families; in A only two offspring per family were used to calculate the offspring mean,whereas all four were used in B. Note that the increased number of offspring per family does not increase theheritability estimate, but does decrease the scatter around the regression line, which is reflected in the lower standarderror and the higher R2. Data from Conner and Via 1993.

End box.

A few final notes on offspring parent regression. The first is that the statistical technique ofregression, now widely used throughout science, business, etc, was developed by Francis Galtonin the late 1800’s for the purpose of examining inheritance (Provine 1971). The name regressioncomes from Pearson’s observation that the offspring-parent regression was always less than onebecause heritabilities are rarely, if ever, one. This means that the average offspring “regress”toward the parental mean, that is, there is less variance among the offspring means compared tothe parental means as we noted above. Second, if the phenotypic values for only the mother or

111213141516171819

11 12 13 14 15 16 17 18Midparent pistil length

Two

offs

prin

g m

ean

pist

il len

gthA Slope = h2 = 0.74 (0.09)

R2 = 0.59Slope = h2 = 0.79 (0.12)R2 = 0.47

B

111213141516171819

11 12 13 14 15 16 17 18Midparent pistil length

Four

offs

prin

g m

ean

pist

il len

gth


father is used on the X-axis, then the slope is half the heritability, because each parent onlycontributes half of the nuclear genes in the offspring. Third, the additive variance is 1/2 theoffspring-parent covariance (not regression slope; see the next chapter), regardless of whetherone or the mean of both parents is used.

Maternal and paternal effects

The discussion of offspring-parent regression above included the unstated assumption thatall of the resemblance between offspring and parents was due to genetics. This is not a goodassumption in species or experiments in which there is parental care or provisioning of theoffspring. If the amount of parental care depends on the value of the phenotypic trait of interestin the parents, and in turn the amount of care partly determines the phenotypic value of the sametrait in the offspring, then this gives rise to a non-genetic resemblance between parents andoffspring.

The most important cause of non-genetic resemblance between parents and offspring is amaternal effect, because mothers often have profound effects on their offspring throughprovisioning of the seed or egg in most plants and animals, gestation and lactation in mammals,and direct care of the young in many animal species (Mousseau and Fox 1998). There are alsomaternal genetic effects that does not contribute to additive variation because it is the mother thattransmits most or all of the non-nuclear genes in the mitochondria and chloroplasts (Kirkpatrickand Lande 1989). For example, larger male and female dung beetles (Onthophagus taurus)provide larger amounts of dung to their offspring than do smaller beetles, and this increasedparental provisioning produces larger offspring (Hunt and Simmons 2000).

The best way to obtain estimates of additive variance and heritability that are notcontaminated by maternal effects is to use data on fathers only, e.g. do an offspring - fatherregression. This works extremely well in the vast majority of plants, arthropods, and reptiles inwhich the fathers only contribute sperm to the offspring. If the offspring-mother regressionslope is greater than the offspring-father slope, then this suggests the presence of a maternaleffect. In other animals, paternal care (care by fathers) is common so paternal effects arepossible. Paternal care is particularly common in birds, so the technique of cross-fostering isoften used. In a cross-fostering experiment, eggs are switched between nests as soon as possibleafter laying. After the offspring mature and are measured, their phenotypic values can beregressed on those of their genetic parents, which gives a good estimate of heritability becausethe genetic parents did not care for them. The offspring values can also be regressed on thefoster parents, which gives a good estimate of the parental effects. Cross-fostering does notwholly eliminate parental effects, because there could be effects of egg provisioning; since theseare through the mother, comparison of the offspring-mother and offspring father regressionslopes provides an estimate of the strength of egg maternal effects. See the discussion paper byWiggins for an example of a cross-fostering experiment.

Sibling analysesBox: ANOVA

Just as we needed to understand the statistical technique of regression to interpret the resultsof an offspring-parent regression, we now need to understand the basic concepts of analysis of


variance (ANOVA), which is the technique that Sir Ronald Fisher devised to do quantitativegenetics on groups of siblings. ANOVA is a way of testing whether the means of a number ofdifferent groups differ in some measurement, or similarly whether there is significant varianceamong the means. The simplest case is when there are only two groups, and then the ANOVA isthe same thing as a t-test. For example, Figure 4.11 shows the distribution of adult dry weightsfor male and female flour beetles (Tribolium castaneum). Note that the mean for females isgreater than that for males, but that the distributions overlap broadly. The ANOVA tests whetherthis difference in means is due only to random sampling error, or whether the difference is real.The test statistic for ANOVA has an F distribution, and the P value gives the probability that thedifference in means is due to chance alone; because that chance is less than one in 10,000 here,we can safely conclude that females really are heavier than males in this population even withthe large overlap in the distributions..

Figure 4.11 Adult dry weights in male vs. female flour beetles (Tribolium castaneum). Arrows show themeans for males and females, and the F and P values are from the ANOVA testing whether these means aredifferent from each other. Data from Conner and Via 1992.

We will be more confident of a difference between means, and therefore the P-value will belower, the less overlap there is between the two distributions. Figure 4.12B and C show the twoways that overlap can be reduced relative to the largely overlapping distributions in fig. 4.12A.In fig. 4.12B the variances are the same as in A, but the means are farther apart, and in fig. 4.12Cthe difference in means is the same as in A but the variances within each group are smaller. Thekey to our confidence that there is a real difference, then, depends on the magnitude of thedifference between the group means relative to the variance within the groups.

0

2

4

6

8

10

12

14

16

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6Adult dry weight (mg)

Freq

uenc

y

FemalesMales

Males Females

F = 21.5P < 0.0001


A

C

Phenotypic Trait

Fre

quen

cy

B

Figure 4.12 Trait distributions from three hypothetical pairs of populations. In A we have little confidence that thegroups differ because the distributions overlap greatly (shaded area), whereas in B and C the overlap is less due togreater differences between the means and smaller variances respectively.

ANOVA is usually applied when there are more than two groups being compared; inquantitative genetics these groups are typically families (fig. 4.13). How can we express thedifferences among all these group means? We could follow the approach depicted in Fig. 4.12and calculate the differences in all possible pairs of families, but with more than a few groupsthis becomes extremely inefficient. It is much simpler to calculate an average deviation of eachfamily mean from a single standard, and for this statisticians use the grand or overall mean:

(4.13)

Here Y i represents the mean value of group i, n is the number of groups, and Y is the grandmean, or the mean of all the group means Y i . This should remind you of equation 4.3 for avariance, because that is exactly what it is, the variance among group means instead of thevariance among individuals. So instead of testing how different the means of two groups arerelative to the variance within the groups, ANOVA with many groups tests whether the varianceamong groups is significantly greater than the variance within groups. This is why it is calledanalysis of variance. We use Y instead of X to remind us that the measurements are thedependent variable in the ANOVA. The numerator of this is usually called the Sums of Squares

Vmeans = MS =

(Y i −Y )2

i=1

n∑

n −1=

SSd. f .


(SS), short for summed squared deviations from the mean, and the entire variance is usuallycalled the Mean Squares (MS) in ANOVA.

Phenotypic Trait

Fre

quen

cy

YY1 Y2 Y3 Y4 Y5 Y6

Y1 Y_ YY6_

Figure 4.13 A hypothetical plot showing the phenotypic distributions of six families with their respective means, as

well as the grand mean of all individuals (Y ). The deviations of all six family means from the grand mean aresquared and summed to calculate the Sum of Squares (SS); the deviations of two families (1 and 6) are shown asexamples. Note that the deviations for families 1 – 3 are negative, but that the squaring is done to make all of thempositive.

The results of analysis of variance are usually presented in an ANOVA table (Table 4.5).The source of variance is rarely spelled out as it is here, so it is important to keep in mind whatexactly each factor (a row in the table) represents. It is also important that the degrees offreedom and either the SS or the MS be presented (or both). The ratio of the MS is a measure ofwhether there is more variance among groups (families here) than within groups. The MS ratiois the test statistic, and it has an F-distribution. The P-value tells the probability that the greatervariance among groups compared to within groups is due to chance alone, so the small P inTable 4.5 gives us a high degree of confidence that the families differ.

Factor Source ofvariance

SS d.f. MS F P

Family Amongfamilies

500 5 100 10 <0.005

Error Withinfamilies

90 9 10


Table 4.5 Hypothetical ANOVA table for the data shown in Figure 4.13. There are six families with 10 offspringin each family, so the degrees of freedom (d.f.) at each level are one less than these values. The MS isSS/d.f.(equation 4.13), and the ratio of the mean squares has an F-distribution.

ANOVA and regression are closely related techniques, and both have continuous dependentvariables. The difference is that the independent variable is continuous in regression (e.g., aquantitative trait) and it is categorical or discrete in ANOVA (e.g., different treatment groups orfamilies).End boxHow is ANOVA used to estimate genetic variance?

Just as we did in offspring-parent regression, with sibling analysis using ANOVA we needto separate additive genetic variance from environmental and non-additive genetic variance. Todo this, a large number of families of known parentage need to be raised, and a reasonablenumber of siblings from each family need to be placed in randomized positions in the lab, field,or greenhouse so that there are no environmental differences among these families on average. Ifthis is done, then the environmental variance is among siblings within families, and differencesamong family means are genetic. Therefore, the ANOVA test of whether there is significantlygreater variance among family means than within families for the phenotypic trait of interesttests for significant genetic variation.

The simplest sibling analysis is based on full-sibling families similar to those used inoffspring parent regression. The difference here is that data on the parental phenotypes are notused, but rather a simple one-way ANOVA is used to measure among-family variance as inFigure 4.13. However, as we will see below, this measure of variance is not good estimate of VA

because it also includes dominance variance and a different kind of maternal effect.

A better way to estimate additive variance and heritability with siblings is with a nestedpaternal half-sibling design (Figure 4.14). In this design, a random sample of males (oftencalled sires) are each mated to a unique set of randomly chosen females (called dams). Asample of offspring from each dam (three in Fig. 4.14) are raised and the phenotypic traits ofinterest measured. Thus, the offspring of each dam represent full-sibling families (because theyhave the same mother and father), and these are nested within half-sibling families, that is, all theoffspring of the same father. The latter are called half-sibling families because the offspring ofdifferent mothers share the same father. Nesting means that there are different dams mated toeach sire, and this creates the hierarchical structure shown in Fig. 4.14. As we will see below,the variance among half-sibling families is used to estimate VA; significant variance among siresin the ANOVA is good evidence for additive variance in the population. The reason paternalhalf-sibling families are used is to eliminate maternal effect from the estimate, but just as inoffspring parent regression, if there is paternal care then this estimate can be contaminated bypaternal effects. It is important that offspring are randomized across the environment they areraised in, so that the environments experienced by the offspring of each sire do not differ onaverage. This is how the environmental variance is removed from the variance among half-sibfamilies.


A B CSires

Dams 1 2 3 4 5 6 7 8 9

OffspringAdult

Weights

Fac

tors

1.1 0.8 0.9 0.8 0.8 1.2 1.0 0.9 1.3

Half-sib family Full-sib family

(offspring distributed randomly in the rearing environment)D

ata

Figure 4.14 Diagram of a nested paternal half-sibling design. Only three sires are shown for clarity, although50 or more sires are necessary for precise estimates of additive variance. Each sire is mated to three dams in thisexample, and three offspring from each dam are raised and measured (only the offspring of one sire are shown forclarity). In the ANOVA, Sire and Dam nested within Sire are the independent variables or factors, and thephenotypic measurements of the offspring are the data used for the dependent variable.

The key parts of a nested ANOVA table on half-sibling data are shown in Table 4.6. Thevariance or Mean Squares (MS) is partitioned into three sources. The variance among sires isfound by taking the mean of all offspring of each sire and calculating the variance of these siremeans as in equation 4.13. Similarly, the variance among dams is calculated by taking theaverage value for each full-sib family (the offspring of each dam) and averaging these means,after correcting for differences among sire means. This is referred to as the variance among damswithin sires, because each group of dams was mated to a different sire. The average varianceamong the offspring within each full-sib family is referred to as the error variance. The expectedMean Squares shows the theoretical composition of each MS (σ2 is the symbol used for thetheoretical expectation of a variance). Note that each MS contains the variance from all levelsbelow it in the nested hierarchy – the error variance includes only the average variance withinfull-sib families, the dam within sire variance includes the error variance plus the varianceamong full sib families times the number of offspring/dam (k), and the sire MS includes theseplus the variance among half-sib families times k and the number of dams/sire (d).

Table 4.6 Nested analysis of variance for half-sibling data. The expected mean squares (MS) show the theoreticalcomposition of the variance at each level.

Title Source of variance d.f. Expected MSSire Among sires s - 1 σ2

W+kσ2D+dkσ2

S

Dam (Sire) Among dams (within sires) s(d – 1) σ2W+kσ2

D

Error Within full-sib families sd(k – 1) σ2W

s = # of siresd = # of dams per sirek = # of offspring per dam

The ratio of two variances (MS) has an F-distribution, and this is how the statistical tests aredone in a standard ANOVA. For example, the F value testing for significant variance amongsires (and thus for significant VA) is the Sire MS divided by the Dam MS. Because the MS for


each level in the hierarchy includes variance from all lower levels, the MS do not give anestimate of the variance at each level alone except in the case of the error variance for whichthere are no lower levels. The expected MS are used to find the variance components at eachlevel from the MS by subtracting the lower level MS and dividing by the coefficients. Forexample, the sire variance component is estimated as:

σ S2 =

MSS − MSDdk

(4.14)

where MSS and MSD refer to the sire and dam mean squares, respectively.

Exercise: Derive the formula for the Dam variance components from the expected MS in Table4.6.

Answer:

σ D2 =

MSD − MSWk

(4.15)

These sire, dam, and error variance components are observational components, i.e., variancecomponents due to the different factors in the mating design. These variance components sum tothe total phenotypic variance, VP. From these we can obtain the causal variance components,that is, those due to the underlying genetic and environmental sources of variance:

σ S2 =

14

VA

σ D2 =

14

VA +14

VD + VEc

Thus, four times the sire variance component is the estimate of VA from a nested paternalhalf-sibling design. Note that both also include some epistatic variance, VI, but thesecomponents are small, especially in the case of the sire variance, and are generally ignored. Sowith traditional ANOVA analyses, there are three steps – first the MS is calculated, then theobservational (sire, dam) variance components are calculated from the MS, and then the causal(additive, dominance, environmental) variances are calculated from the observed variancecomponents. This was the process for almost all half-sibling analyses done before 2000. Nowthese analyses are usually done using the newer statistical techniques of maximum likelihood(ML) or more often restricted maximum likelihood (REML), which generally produce theobservational components directly. This saves the first step, only requiring the calculation ofcausal components from these. The major advantage of maximum likelihood is that it is lesssensitive to unbalanced data, that is, where there are unequal numbers of dams mated to eachsire or unequal numbers of offspring per dam. Unbalance is common in experiments due touneven mortality of individuals.

Futuyma et al. (1993) used nested half-sibling designs to test whether populations of abeetle, Ophraella communa, that feeds on the leaves of only one plant species possess geneticvariation for feeding ability on the hosts of other Ophraella species. There were 55 sires eachmated to two dams, and three offspring from each dam were tested on each plant species. The

(4.16)

(4.17)


final data were slightly unbalanced, because nine dams produced no offspring and for somefewer than three larvae per plant were successfully tested. The results of larval feeding trials forone of the alternative host plants, marsh elder (Iva frutescens), are shown in Table 4.7. TheANOVA shows significant variance among the offspring of different sires, which is goodevidence for significant additive genetic variance, although the heritability is fairly low. Thismeans that these beetles have the genetic variation necessary for the evolution of increasedfeeding ability on the alternate host, perhaps facilitating a shift to that host in the future orindicative of a recent ancestor that fed on marsh elder.

Table 4.7. ANOVA table from a half-sibling design testing for genetic variation in the amountof marsh elder (Iva frutescens) leaves consumed by Ophraella larvae. From Table 3 of Futuymaet al. 1993.

d.f. MS F PSire 54 1.42 1.7 <0.05Dam (Sire) 44 0.83 2.1 0.0003Error 193 0.39

s = 55d = 101/55 = 1.84k = 2.91

Sire variance component (eq. 4.14): σS2 =

MSS − MSDdk

=1.42 − 0.831.84 × 2.91

= 0.11

Therefore, VA is 0.44, because it is four times the sire variance component. The sum of the threeMean Squares is an estimate of the total phenotypic variance, VP, which here equals 2.64. Theheritability is then 0.44/2.64 = 0.17.

Dam variance component (eq. 4.15): σ D2 =

MSD − MSWk

= 0.83− 0.392.91 = 0.15

The Dam variance component is slightly larger than the sire variance component, suggesting thepresence of dominance variance or common environment effects (compare eq. 4.16 to 4.17).

Exercise: Table 4.8 shows the ANOVA table for the amount of leaf consumed by Ophraellalarvae feeding on a different plant, boneset (Eupatorium perfoliatum). Calculate VA and h2

for these data. Is there evidence for dominance variance or common environmental effects?Are the degrees of freedom correct based on the formulae in Table 4.7?

d.f. MS F PSire 54 1.04 2.0 <0.025Dam (Sire) 44 0.52 1.7 0.007Error 198 0.30

s = 55, d = 1.84, k = 2.96

Answer: VA = 0.38; h2 = 0.20. No evidence for dominance or common environmentaleffects because the sire variance component (0.10) is actually greater than the dam variance


component (0.07). This is theoretically impossible, because variance components cannot benegative and the dam variance component contains more causal components than the sirevariance component (eqs. 4.16 and 4.17), but is due to estimation error, which is alwayspresent. The degrees of freedom seem correct, except that the dam d.f. should be 46 basedon 55 sires and 101 total dams. It is important to pay attention to the degrees of freedom inan ANOVA table, because a lack of a match to the values expected from the formulae inTable 4.6 is a good clue that there is a mistake or problem with the analysis. In this case thedifference is small and therefore likely to be inconsequential to the interpretation of theresults.

The sire observational component includes only VA, because you are estimating the effectsof each male’s genes, averaged over several mothers so maternal effects cancel out, as does VE

because the offspring were raised in random positions across the environment. There is nodominance variance, VD, in the sire variance because only the effects of one of the parents (thesire) is included, whereas dominance depends on the alleles contributed by both parents at eachlocus. The dam observed variance component includes three causal components: VA, VD, andVEc. VD is included because these are full-sib families, and thus the effects of both parents areincluded. VEc is variance among dams due to differences among the full-sibling families in thecommon rearing environments shared within these families. This is a different kind of maternaleffect than we discussed above. With offspring-parent regression, the maternal effects are thosethat create similarities between mothers and offspring for the trait of interest. With sib analysisthe parental measurements are not included, so this is a maternal or other environmental effectthat causes similarities among the offspring within full-sib families (and thus differences amongthe families) over and above the genetic similarities. In fact, it often is not the same trait; forexample, differences in milk production among mothers might lead to similarities within, anddifferences among, full-sib families in offspring body size.

If fathers contribute more than sperm to their offspring, then differences among fathers inpaternal care can cause VEc to be present in the sire variance component. This can be reduced oreliminated by splitting up the offspring and having them be raised by a number of differentfathers and mothers, similar to cross-fostering in offspring-parent regression. However, alsosimilar to cross-fostering, this does not remove any effects that occur before the offspring can betransferred. For example, in some insects males provide a nuptial gift of food or transfernutrients and other substances during mating. If the female uses these to provision the eggs, thenthere could be VEc in the sire variance component that would be very difficult to control for. It isnot known how important these effects are in nature.

Comparison of methods

There are a number of factors to consider when deciding whether to use offspring-parentregression, full-sib analysis, or half-sibling analysis to determine genetic variance andheritability. In natural populations, often the overriding factor is feasibility – there are manyanimals and some plants for which the nested half-sibling mating design is extremely difficult orimpossible to implement. This is one reason why offspring-parent regression is very common inbirds. The other is an advantage of offspring-parent regression, that is, it can be done inunmanipulated natural populations of monogamously-breeding animals. Ideally molecularmarkers should be used to determine paternity of the offspring, because in behaviorally


monogamous birds mating outside the pair can be common (‘extra-pair copulations’ or EPCs;Westneat et al. 1990). Even maternity needs to be checked in some bird species in whichfemales lay eggs in other female’s nests (‘egg-dumping’). If there are offspring included in theregression that are not genetically related to the parents they are being regressed on, then theheritability estimate will be biased downward, that is, lower than the actual heritability, becausethe genetic relatedness is lower than is assumed by the analysis.

Another advantage of offspring-parent regression is that it is simpler statistically, but thissimplicity comes at a price in flexibility and sometimes bias. The bias can occur if there isgenotype by environment interaction (GxE; see next chapter), that is, if genotypes differ in theirphenotypic response to the environment. This can bias offspring-parent regressions becausesince offspring and parents develop at different times there is no way to make their environmentsin exactly the same. The price in flexibility comes if genotype by environment interaction is theobject of study. The usual technique (next chapter) is to split full-sibling families and raise theoffspring in the different environments. This cannot be done with offspring-parent regressionbecause there are only two parents, and even they cannot be raised in different environments.

Other issues in choosing a technique are precision and bias. The closer the geneticrelationship between the relatives being measured, the higher the precision. Therefore, full-sibling and offspring-parent regression have better precision than half-sibling, because theformer are based on a relatedness of 0.5 versus 0.25 for the latter.

Table 4.7. Summary of advantages and disadvantages of different methods of estimating additive genetic variancein outbreeding species.

Method Advantages Disadvantages

Offspring-parent SimplePreciseCan use unmanipulatedpopulations

Environmental differences betweenparents and offspring

Biased by maternal effects (exceptfather-offspring)

Cannot test for GxE

Full-sibling Precise High bias from maternal effects anddominance

Half-sibling Low biasCan include other factors, e.g.,

different environments

ImpreciseComplexLarge sample sizes needed

Bias is usually a more important issue than precision, and comes mainly from maternaleffects (including common environment) and dominance. The former affects both offspring-mother and offspring-midparent regressions, and both can bias full-sibling analyses. Therefore,offspring-father regression and paternal half-sibling analysis are usually the least biased, but canbe biased if there is paternal care. Typically, however, half-sibling designs are used in organismslike plants and insects in which paternal investment is rare or nonexistent. Note that half-sibling


mating designs can be difficult to accomplish with vertebrates, so this design is usually used forplants and insects.

Because full-sibling designs have two sources of bias, they are definitely less useful forsexually-reproducing organisms. They are useful for highly selfing plants, however, becauseselfers pass their diploid genotype on to their offspring so that VD does contribute directly toevolutionary change. Similarly, for clonal organisms the variance among clones is the relevantdeterminant of the rate of phenotypic evolution.

Full-sibling designs have been commonly used for natural populations of outbreedingspecies when the other designs were too difficult to implement. The variance among full-sibfamilies provides an upper bound to VA, so perhaps this is better than nothing, but because ofdominance and common environment often there will be significant full-sib variance when thereis not significant VA. For example, Agrawal et al. (2002) found highly significant varianceamong full sibling families for leaf trichome density in wild radish, but no evidence forheritability or additive variance.

In quantitative genetic experiments of any design, the number of families that the estimateof VA is based upon is of prime importance to the power of the analysis. Twenty families shouldbe considered an absolute minimum, and 50 or more is necessary for reasonable statisticalpower. In particular, a finding of no significant additive variance with less than 50 familiesshould be interpreted with caution, because the lack of significance could easily be due to a lackof statistical power rather than a real lack of variance.

Readings questionsWiggins 1989 :

1. Describe how he performed his experiment. How were the experimental and control familieseach handled? Was his control a good one? If not, how would you improve it?

2. Describe what animals are involved in the two regressions in Fig. 1. What does each pointrepresent in each? Who is being regressed on who in each?

3. Explain the biological significance of the differences between the slopes of the two regressionplots in figure 1. What is the relationship between the data in these plots and the data in Table 1?Was there any evidence for maternal or paternal effects?

4. Why are measurements of heritabilities of these traits important and interesting?

Janssen et al. 1988:

1. What is a negative maternal effect?2. What is their evidence for a negative maternal effect?3. Why is this data interpreted as evidence for a negative maternal effect?


Mazer and Schick 1991:1. The mean phenotypic values of some of the traits differed significantly across densitytreatments. Which traits were these, and which figure and/or table presents these data? Are thedifferences in the direction you expect?

2. For which traits is there evidence for significant narrow-sense heritability (additive geneticvariance) at each of the three densities individually? What specifically is this evidence, and inwhich table and/or figure is it presented?

References and further reading:

For general reference see Falconer and Mackay 1996; Lynch and Walsh 1998; Roff 1997.

Agrawal, A. A., J. K. Conner, M. T. J. Johnson, and R. Wallsgrove. 2002. Ecological genetics ofan induced plant defense against herbivores: additive genetic variance and costs ofphenotypic plasticity. Evolution 56:2206-2213.

Arnold, S. J. 1981. Behavioral variation in natural populations. I. Phenotypic, genetic andenvironmental correlations between responses to prey in the garter snake, Thamnophiselegans. Evolution 35:489-509.

Bennington, C. C., and J. B. McGraw. 1996. Environment-dependence of quantitative geneticparameters in Impatiens pallida. Evolution 50:1083-1097.

Boag, P. T. 1983. The heritability of external morphology in Darwin's ground finches (Geospiza)on Isla Daphne Major, Galapagos. Evolution 37:877-894.

Boonstra, R., and P. T. Boag. 1987. A test of the Chitty hypothesis: Inheritance of life-historytraits in meadow voles Microtus pennsylvanicus. Evolution 41:929-947.

Caldwell, R. L., and J. P. Hegmann. 1969. Heritability of flight duration in the milkweed bugLygaeus kalmii. Nature 223:91-92.

Campbell, D. R. 1996. Evolution of floral traits in a hermaphroditic plant: field measurements ofheritabilities and genetic correlations. Evolution 50:1442-1453.

Conner, J., and S. Via. 1992. Natural selection on body size in Tribolium: possible geneticconstraints on adaptive evolution. Heredity 69:73-83.

Conner, J. K., and S. Via. 1993. Patterns of phenotypic and genetic correlations amongmorphological and life-history traits in wild radish, Raphanus raphanistrum. Evolution47:704-711.

Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to Quantitative Genetics. Longman,Harlow, UK.

Futuyma, D. J., M. C. Keese, and S. J. Scheffer. 1993. Genetic constraints and the phylogeny ofinsect-plant associations: Responses of Ophraella communa (Coleoptera:Chrysomelidae) to host plants of its congeners. Evolution 47:888-905.

Hoffmann, A. A., and M. Schiffer. 1998. Changes in the heritability of five morphological traitsunder combined environmental stresses in Drosophila melanogaster. Evolution 52:1207-1212.

Houde, A. E. 1992. Sex-linked heritability of a sexually selected character in a natural populationof Poecilia reticulata (Pisces: Poeciliidae) (guppies). Heredity 69:229-235.


Hunt, J., and L. W. Simmons. 2000. Maternal and paternal effects on offspring phenotype in theDung Beetle Onthophagus taurus. Evolution 54:936-941.

Janssen, G. M., G. DeJong, E. N. G. Joosse, and W. Scharloo. 1988. A negative maternal effectin springtails. Evolution 42:828-834.

Kirkpatrick, M., and R. Lande. 1989. The evolution of maternal characters. Evolution 43:485-503.

Kruuk, L. E. B., T. H. Clutton-Brock, J. Slate, J. M. Pemberton, S. Brotherstone, and F. E.Guinness. 2000. Heritability of fitness in a wild mammal population. Proceedings of theNational Academy of Sciences of the United States of America 97:698-703.

Littell, R. C., G. A. Milliken, W. W. Stroup, and R. D. Wolfinger. 1996. SAS System for MixedModels. SAS Institute, Inc., Cary, NC.

Lynch, M., and B. Walsh. 1998. Genetics and Analysis of Quantitative Traits. SinauerAssociates, Sunderland, MA.

Mazer, S. J., and C. T. Schick. 1991. Constancy of population parameters for life-history andfloral traits in Raphanus sativus L. II. Effects of planting density on phenotype andheritability estimates. Evolution 45:1888-1907.

Merila, J., and B. C. Sheldon. 2000. Lifetime reproductive success and heritability in nature. TheAmerican Naturalist 155:301-310.

Mitchell-Olds, T. 1986. Quantitative genetics of survival and growth in Impatiens capensis.Evolution 40:107-116.

Mousseau, T., and C. Fox. 1998. Maternal effects as adaptations. Oxford Univ Press, NY.Provine, W. B. 1971. The Origins of Theoretical Population Genetics. University of Chicago

Press, Chicago.Roff, D. A. 1997. Evolutionary Quantitative Genetics. Chapman & Hall, New York.Simons, A. M., and D. A. Roff. 1994. The effect of environmental variability on the heritabilities

of traits of a field cricket. Evolution 48:1637-1649.Smith, D. A. S. 1980. Heterosis, epistasis and linkage disequilibrium in a wild population of the

polymorphic butterfly Danaus chrysippus (L). Zool. J. Linn. Soc. 69:87-109.Westneat, D. F., P. W. Sherman, and M. L. Morton. 1990. The ecology and evolution of extra-

pair copulations in birds. Current Ornithology 7:331-369.Wiggins, D. A. 1989. Heritability of body size in cross-fostered tree swallow broods. Evolution

43:1808-1811.Wilcockson, R. W., C. S. Crean, and T. H. Day. 1995. Heritability of a sexually selected

character expressed in both sexes. Nature 374:158-159.


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