A Gene-Free Formulation of Classical Quantitative Genetics Used to Examine Results and Interpretations Under Three Standard Assumptions

Abstract

Quantitative genetics (QG) analyses variation in traits of humans, other animals, or plants in ways that take account of the genealogical relatedness of the individuals whose traits are observed. “Classical” QG, where the analysis of variation does not involve data on measurable genetic or environmental entities or factors, is reformulated in this article using models that are free of hypothetical, idealized versions of such factors, while still allowing for defined degrees of relatedness among kinds of individuals or “varieties.” The gene-free formulation encompasses situations encountered in human QG as well as in agricultural QG. This formulation is used to describe three standard assumptions involved in classical QG and provide plausible alternatives. Several concerns about the partitioning of trait variation into components and its interpretation, most of which have a long history of debate, are discussed in light of the gene-free formulation and alternative assumptions. That discussion is at a theoretical level, not dependent on empirical data in any particular situation. Additional lines of work to put the gene-free formulation and alternative assumptions into practice and to assess their empirical consequences are noted, but lie beyond the scope of this article. The three standard QG assumptions examined are: (1) partitioning of trait variation into components requires models of hypothetical, idealized genes with simple Mendelian inheritance and direct contributions to the trait; (2) all other things being equal, similarity in traits for relatives is proportional to the fraction shared by the relatives of all the genes that vary in the population (e.g., fraternal or dizygotic twins share half of the variable genes that identical or monozygotic twins share); (3) in analyses of human data, genotype-environment interaction variance (in the classical QG sense) can be discounted. The concerns about the partitioning of trait variation discussed include: the distinction between traits and underlying measurable factors; the possible heterogeneity in factors underlying the development of a trait; the kinds of data needed to estimate key empirical parameters; and interpretations based on contributions of hypothetical genes; as well as, in human studies, the labeling of residual variance as a non-shared environmental effect; and the importance of estimating interaction variance.

Appendices

Appendices

1.1 Appendix 1: Gene-Free Formulations for Classes of Relatives Other Than Twins

Gene-free formulations can be derived and applied through five steps:

• (1) Define varieties in terms of the progenitors for individuals in the variety, e.g., a clone, a pair of parents, one mother and unrelated fathers, a pair of grandparents.

• (2) Specify the variant of Eq. 8 that encompasses the different kinds of relatives possible for such progenitors (e.g., sibs, first cousins).

• (3) Spell out the intraclass correlation equations for the different kinds of relatives in the different circumstances (e.g., raised together, raised apart).

• (4) Rearrange the intraclass correlation equations to produce estimators for the variance fractions and for any empirically determined parameter that had to be introduced to take degree of relatedness into account.

• (5) Collect data for the classes of relatives needed in order to estimate the variance fractions and parameters of interest using the equations from step 4.

1.2 Appendix 2: Numerical Illustrations of the Gene-Free Formulation of Sect. 2 and of the Contrasting Results under the Standard and Alternative Assumptions of Sect. 3

This Appendix provides results from applying the formulas of Sects. 2.2 and 3.2 to data sets generated for a range of values of broad-sense heritability and the other fractions of variance. First, data sets of 100 varieties and 100 locations were randomly generated with two MZ twins and two DZ twins for each variety–location combination. Each row of Table 4 corresponds to a pair of V/Y and γ values drawn from the ranges [.2–.8] and [.3–.8], respectively. L, VL, and E were set equal to (1–V)/3. V⁻ and VL⁻ were set to γ.V and γ.VL, respectively; T and TL were set to (1−γ)V and (1−γ)VL, respectively. Equations 3–7 were used to calculate from the full data set the actual fractions for V, L, VL, and E for MZ and for DZ twins. It is the average of the two values that is given in Table 4. (The actual fractions differ slightly from the pre-set values due to the random generation process). Equation 12 was used to calculate the actual values for γ. The values for broad-sense heritability and the fraction of variation due to differences among location averages were estimated first using the conventional formulas (Eqs. 18, 19) and then using the adjusted formulas that factor in γ (Eqs. 15, 16). The fraction due to noise is 1-the sum of these estimates for broad-sense heritability and the location fraction.

Table 4 Estimates of broad-sense heritability and L/Y derived from full simulated data set for a series of combinations of actual values

The results in Table 4 indicate that: (a) the adjusted estimates of broad-sense heritability are close to (V + VL)/Y and the adjusted values of L/Y are close to the actual values, a result that matches the theory in the body of the text; (b) the conventional formulas yield estimates of the two quantities that differ from the corresponding estimates from the adjusted formula by a close-to-equal and opposite amount. This amount is close to the value given by the difference between Eqs. 15 and 18, which can be shown to be (2γ−1) times the adjusted formula’s estimate of V⁺/Y.

To produce the results given in Table 5, 48 MZT pairs, 48 DZT pairs, and 48 UVT pairs were sampled from the full data sets of 10,000 MZ twin pairs and 10,000 DZ twin pairs. (Each row of the table corresponds to a row of Table 4, that is, to a particular pair of V/Y and γ values). The UVT pairs were sampled from the MZ data by choosing one member of the pairs for two different varieties at a given location. (One sample of pairs for pre-set values of V from .2 to .8 and γ = .5 is downloadable from http://bit.ly/TwinPairs). The values for broad-sense heritability and the fraction of variation due to differences among location averages were again estimated using both the conventional and adjusted formulas. After the sampling process was repeated 100 times for each data set, the mean and standard deviation of the various values were calculated.

Table 5 Estimates of broad-sense heritability and L/Y derived from repeated samples from full data sets in Table 4

The results summarized in Table 5 have the same features as those in Table 4 except that: (a) the means of the estimates of broad-sense heritability using the adjusted formula tend to overestimate slightly the actual values for (V + VL)/Y and the means of the corresponding estimates of L/Y tend to underestimate the actual values; and (b) the Standard Deviations of the estimates are substantial, but, in general, less for the adjusted formulas than for the conventional.

1.3 Appendix 3: Contrasting Formulations in a Selection of Articles That Point to the Potential Importance of Interaction Variance

This Appendix translates the main arguments about the potential importance of interaction variance made by a range of authors into the terms and notation of this article. This translation makes apparent the differences among the formulations and their differences from the account in this article. Two adjustments should be noted: (a) e is used below for environmental factors, not for noise or error; and (b) the equations are centered on the average value of the trait over all varieties, locations, and replicates. In other words, in Eq. 1, subtract m from both sides to produce the deviation of the trait value from the overall mean, then average over the replicates for each variety–location combination and call this average y′_ij.

$$ {\text{y}}^{\prime }_{{{\text{ij}}.}} = {\text{v}}_{\text{i}} + {\text{l}}_{\text{j}} + {\text{vl}}_{\text{ij}} $$

(20)

1.4 Layzer (1974)

Main argument: The contribution of variety–location (“genotype–environment”) interaction to trait variation consists of a direct interaction contribution and a covariance that is “negligible if, and only if, the genotypic and environmental variables are statistically independent to a high degree of approximation.” (Layzer’s discussion focuses on variety–location covariance or correlation for human traits.) In mathematical terms:

$$ {\text{y}}^{\prime }_{{{\text{ii}} .}} = {\text{v}}_{\text{i}} + {\text{l}}_{\text{i}} + {\text{vl}}_{\text{ii}} $$

(21)

where

$$ {\text{v}}_{\text{i}} = \int { \ldots \int {{\text{y }}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\text{g}}_{\text{i}} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\text{e}}){\text{ p}}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\text{g}}_{\text{i}} |\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\text{e}})de_{1} \ldots de_{s} } } $$

(22)

g _i is the set of genetic factors present in variety i; e is a set of possible values of environmental factors 1,….s; y (g _i,e) is the value of the trait for the set e; p(g _i|e) is the conditional probability of the occurrence of g _i for the set e; ∫…∫ … de ₁ …de _s is the integration over all of possible sets of environmental factors 1,….s

Similarly l_i is an integration over all of possible sets of genetic factors. The equation for variances corresponding to 21 is

$$ {\text{Y}} = {\text{V}} + {\text{L}} + 2{\text{ Covariance }}\left( {{\text{V}},{\text{ L}}} \right) + 2{\text{ Covariance }}\left( {{\text{V}} + {\text{L}},{\text{ VL}}} \right) + {\text{VL}} $$

(23)

which simplifies if covariances are zero to

$$ {\text{Y}} = {\text{V}} + {\text{L}} + {\text{VL}} $$

(24)

but covariances are zero if and only if p(g _i|e) = p(g _i) and p(e _i|g) = p(e _i).

1.5 Lewontin (1974)

Main argument: When the norm of reaction, that is, the response of a variety (or genetic type) to changes in some measurable environmental factor, varies for different varieties in slope or position, this can confound any attempt to extrapolate the relative ranking of varieties observed over part of the range of the environmental factor to the full range. This situation is equivalent to the vl_ij values being non-negligible in Eq. 25

$$ {\text{y}}^{\prime }_{{{\text{ij}} .}} = {\text{v}}_{\text{i}} + {\text{c}} . {\text{e}}_{\text{j}} + {\text{vl}}_{\text{ij}} $$

(25)

where e_j is the average value for the environmental factor in the jth location; c is a scaling constant; and vl_ij is the additional contribution from the i, jth variety–location combination.

1.6 Plomin et al. (1977)

Main argument: A proxy for variety–location interaction is constructed, namely, the interaction of two variants of a given variable, e.g., average educational attainment, first averaged for the individual’s biological parents (standing in for the variety contribution) and then for the individual’s adoptive parents (standing in for the location contribution). Low values of the interaction between the two variants are found for human behavioral traits. In other words, the v^rl_ij values are low in the following equation:

$$ {\text{y}}^{\prime }_{{{\text{ij}}.}} = {\text{v}}^{\text{b}}_{\text{i}} + {\text{l}}^{\text{a}}_{\text{j}} + {\text{vl}}^{\text{r}}_{\text{ij}} $$

(26)

where ${\text{v}}^{\text{b}}_{\text{i}}$ is the average value of y for the biological parents of person i; ${\text{l}}^{\text{a}}_{\text{j}}$ is the average value of y for the adoptive parents of person I raised in family j; v^rl_ij is the residual after allowing for the previous two contributions.

1.7 Jacquard (1983)

Main argument: In reality the condition is hardly ever fulfilled that the interaction contributions (estimated by vl_ij in Eq. 20) are all zero. Thus heritability in the broad sense (V/Y) has no meaning.

1.8 Wahlsten (2000)

Main argument: multiplicative functional relationships can result in a significant interaction variance even if there is no functional interaction term in the functional relationship. Consider the two functional relationships:

$$ {\text{y}}^{\prime }_{{{\text{ij}} .}} = {\text{g}}_{\text{i}} + {\text{e}}_{\text{j}} + {\text{ge}}_{\text{ij}} $$

(27)

$$ {\text{y}}^{\prime }_{{{\text{ij}} .}} = {\text{g}}_{\text{i}} {\text{e}}_{\text{j}} $$

(28)

where g_i is a functional contribution of the ith variety; e_j is the functional contribution of the jth location; ge_ij is the functional contribution from the i, jth variety–location combination.

If the ge_ij values in Eq. 27 are negligible, the values of vl_ij in Eq. 20 will also be negligible. However, even without any ge_ij values in Eq. 22, statistically significant values of vl_ij can arise.

1.9 Summary

In this article, the linear model underlying the partitioning of variation in the trait connects values of the trait for an individual to the summation of variety, location, variety–location interaction, and noise contributions (e.g., Eq. 1). It does so without reference to measurable genetic or environmental factors (unlike Layzer, Wahlsten), their functional relationships (unlike Wahlsten), probabilities of their occurrence (unlike Layzer), or a gradient underlying the differences in the variety or location contributions (unlike Lewontin). Interaction variance can be estimated (i.e., no proxy is needed; unlike Plomin) provided data are available from the appropriate classes of defined degrees of relatedness. In that case, broad-sense heritability can be estimated (unlike Jacquard).

Several of the authors also make arguments about the potential importance of variety–location (“genotype–environment”) correlation (or covariance). In this article lack of such correlations is assumed only to keep the focus on partitioning of variance under three standard assumptions and their alternatives. As noted in Sect. 2.4, whether these conditions can be met in human situations is a matter of ongoing controversy.