The analysis of variance and the analysis of causes

This issue of the American Journal of Human Genetics contains two articles by Newton Morton and his colleagues1,2 that provide a detailed analytic critique of various estimates of heritability and components of variance for human phenotypes. They make especially illuminating remarks on the problems of partitioning variances and covariances among groups such as social classes and races. The most important point of all, at least from the standpoint of the practical, social, and political applications of human population genetics, occurs at the conclusion of the first paper1 in which Morton points out explicitly the chief programmatic fallacy committed by those who argue so strongly for the importance of heritability measures for human traits. The fallacy is that a knowledge of the heritability of some trait in a population provides an index of the efficacy of environmental or clinical intervention in altering the trait either in individuals or in the population as a whole. This fallacy, sometimes propagated even by geneticists, who should know better, arises from the confusion between the technical meaning of heritablility and the everyday meaning of the word. A trait can have a heritability of 1.0 in a population at some time, yet this could be completely altered in the future by a simple environmental change. If this were not the case, ‘inborn errors of metabolism’ would be forever incurable, which is patently untrue. But the misunderstanding about the relationship between heritability and phenotypic plasticity is not simply the result of an ignorance of genetics on the part of psychologists and electronic engineers. It arises from the entire system of analysis of causes through linear models, embodied in the analysis of variance and covariance and in path analysis. It is indeed ironic that while Morton and his colleagues dispute the erroneous programmatic conclusions that are drawn from the analysis of human phenotypic variation, they nevertheless rely heavily for their analytic techniques on the very linear models that are responsible for the confusion. I would like to look rather closely at the problem of the analysis of causes in human genetics and to try to understand how the underlying model of this analysis moulds our view of the real world. I will begin by saying some very obvious and elementary things about causes, but I will come thereby to some very annoying conclusions. We must first separate two quite distinct problems about causation that are discussed by Morton. One is to discriminate which of two alternative and mutually exclusive causes lies at the basis of some observed phenotype. In particular, it is the purpose of segregation analysis to attempt to distinguish those individuals who owe their phenotypic deviation to their homozygosity for rare deleterious gene alleles from those whose phenotypic peculiarity arises from the interaction of environment with genotypes that are drawn from the normal array of segregating genes of minor effect. This is the old problem of distinguishing major gene effects from ‘polygenic’ effects. I do not want to take up here the question of whether such a clear distinction can be made or whether the spectrum of gene effects and gene frequencies is such that we cannot find a clear dividing line between the two cases. The evidence at present is ambiguous, but at least in principle it may be possible to discriminate two etiological groups, and whether such groups exist for any particular human disorder is a matter for empirical research. It is possible, although not necessary, that the form of clinical or environmental intervention required to correct a disorder arising from homozygosity for a single rare recessive allele (the classical ‘inborn error of metabolism’) may be different from that required for the ‘polygenic’ class. Moreover, for the purposes of genetic counselling, the risk of future offspring being affected, will be different if a family is segregating for a rare recessive than if it is not. Thus, the discrimination between two alternative causes of a human disorder is worth making if it can be done. The second problem of causation is quite different. It is the problem of the analysis into separate elements of a number of causes that are interacting to produce a single result. In particular, it is the problem of analyzing into separate components the interaction between environment and genotype in the determination of phenotype. Here, far from trying to discriminate individuals into two distinct and mutually exclusive etiological groups, we recognize that all individuals owe their phenotype to the biochemical activity of their genes in a unique sequence of environments and to developmental events that may occur subsequent to, although dependent upon, the initial action of the genes. The analysis of interacting causes is fundamentally a different concept from the discrimination of alternative causes. The difficulties in the early history of genetics embodied in the pseudo-question of ‘nature versus nurture’ arose precisely because of the confusion between these two problems in causation. It was supposed that the phenotype of an individual could be the result of either environment or genotype, whereas we understand the phenotype to be the result of both. This confusion has persisted into modern genetics with the concept of the phenocopy, which is supposed to be an environmentally caused phenotypic deviation, as opposed to a mutant which is genetically caused. But, of course, both ‘mutant’ and ‘phenocopy’ result from a unique interaction of gene and environment. If they are etiologically separable, it is not by a line that separates environmental from genetic causation but by a line that separates two kinds of genetic basis: a single gene with major effect or many genes each with small effect. That is the message of the work by Waddington3 and Rendel4 on canalization. Second, equation (1) contains population means at two levels. One level is the grand mean phenotype μY and the other is the set of so-called ‘marginal’ genotypic and environmental means, E and G. These, it must be remembered, are the mean for a given environment averaged over all genotypes in the population and the mean for a given genotype averaged over all environments. But since the analysis is a function of these phenotypic means, it will, in general, give a different result if the means are different. That is, the linear model is a local analysis. It gives a result that depends upon the actual distribution of genotypes and environments in the particular population sampled. Therefore, the result of the analysis has a historical (i.e., spatiotemporal) limitation and is not in general a statement about functional relations. So, the genetic variance for a character in a population may be very small because the functional relationship between gene action and the character is weak for any conceivable genotype or it may be small simply because the population is homozygous for those loci that are of strong functional significance for the trait. The analysis of variation cannot distinguish between these alternatives even though for most purposes in human genetics we wish to do so. What has happened in attempting to solve the problem of the analysis of causes by using the analysis of variation is that a totally different object has been substituted as the object of investigation, almost without noticing it. The new object of study, the deviation of phenotypic value from the mean, is not the same as the phenotypic value itself; and the tautological analysis of that deviation is not the same as the analysis of causes. In fact, the analysis of variation throws out the baby with the bath water. It is both too specific in that it is spatiotemporally restricted in its outcome and too general in that it confounds different causative schemes in the same outcome. Only in a very special case, to which I shall refer below, can the analysis of variation be placed in a one-to-one correspondence to the analysis of causes. The real object of study, both for programmatic and theoretical purposes, is the relation between genotype, environment, and phenotype. This is expressed in the norm of reaction, which is a table of correspondence between phenotype, on the one hand, and genotype-environment combinations on the other. The relations between phenotype and genotype and between phenotype and environment are many-many-relations, no single phenotype corresponding to a unique genotype and vice versa. In order to clarify the relation between the two objects of study (i.e. the norm of reaction and the analysis of variance, which analyses something quite different), let us consider the simplified norms of reaction shown in figures 1a–h. We assume that there is a single well-ordered environmental variable E, say temperature, and a scale of phenotypic measurement P. Each line is the norm of reaction, the relationship of phenotype to environment, for a particular hypothetical genotype (G1 or G2). a–h, Examples of different forms of reaction norms. In each case the phenotype (P) is plotted as a function of environment (E) for different genotypes (G1, G2). The first thing to observe is that in every case the phenotype is sensitive to differences in both environment and genotype. That is, each genotype reacts to changing environment, and in no case are the two genotypes identical in their reactions. Thus in any usual sense of the word, both genotypes and environment are causes of phenotypic differences and are necessary objects of our study. Figure 1a is in one sense the most general, for if environment extends uniformly over the entire range and if the two genotypes are equally frequent, there is an overall effect of genotype (G1 being on the average superior to G2) and an overall effect of environment (phenotype gets smaller on the average with increasing temperature). Nevertheless, the genotypes cross so that neither is always superior. Figure 1b shows an overall effect of environment, since both genotypes have a positive slope; but there is no overall effect of genotype, since the two genotypes would have exactly the same mean phenotype if all environments wereconsidered equally. There is no a priori way from Figure 1b of ranking the two genotypes. However, if because of particular circumstances the distribution of environments were heavily weighted toward the lower temperatures, then G1 would be consistently superior to G2; an analysis of variance would show a strong effect of genotype as well as of environment, but very little genotype-environment interaction. Thus the analysis of variance would reflect the particular environmental circumstances and give a completely incorrect picture of the general relationship between cause and effect here, where there is overall no effect of genotype but a strong genotype-environment interaction. Figure 1c is the complementary case to that shown in Figure 1b. In figure 1c there is no overall effect of environment, but G1 is clearly superior to G2 overall. In this case a strong environmental component of variance will appear, however, if either one of the genotypes should predominate in the population. So the historical events that mould the genotypic distribution of a population will have an effect on the judgment, from the analysis of variance, of the importance of environment. The overall lack of genetic effect in Figure 1b and of environmental effect in Figure 1c can both appear in a trait like that shown in Figure 1a, which overall has both effects if the distribution of environments or of genotypes is asymmetric. Thus if environments are distributed around the middle in Figure 1a, there will appear to be no average effect of genotype, while if the population is appropriately weighted toward an excess of G1, the average phenotype across environments will be constant as shown by the dashed line. Here real overall effects are obscured because of spatiotemporal events, and the analysis of variance fails to reveal significant overall differences. These last considerations lead to two extremely important points about the analysis of variance. First, although equation (1) appears to isolate distinct causes of variation into separate elements, it does not do so because the amount of environmental variance that appears depends upon the genotypic distribution, while the amount of genetic variance depends upon the environmental distribution. Thus the appearance of the separation of causes is a pure illusion. Second, because the linear model appears as a sum of variation from different causes, it is sometimes erroneously supposed that removing one of the sources of variation will reduce the total variance. So, the meaning of the genetic variance is sometimes given as “the amount of variation that would be left if the environment were held constant,’ and the environmental variance is described as ‘the amount of variance that would remain if all the genetic variation were removed,’ an erroneous explanation offered by Jensen5, for example. Suppose that the norms of reaction were as in figure 1a and a unimodal distribution of environments were centered near the middle, with a roughly equal mixture of the two genotypes. Now suppose we fix the environment. What will happen to the total variance? That depends on which environment we fix upon. If we choose an environment about 1 sd or more to the right of the mean, there would actually be an increase in the total variance because the difference between genotypes is much greater in that environment than on the average over the original distribution. Conversely, suppose we fix the genotype. If we chose G2 to be our pure strain, then, again, we would increase the total variance because we had chosen the more environmentally plastic genotype. The apparent absurdity that removing a source of variance actually increases the total variance is a consequence of the fact that the linear model does not really effect a separation of causes of variation and that it is a purely local description with no predictive reliability. Without knowing the norms of reaction, the present distribution of environments, and the present distribution of genotypes, and without then specifying which environments and which genotypes are to be eliminated or fixed, it is impossible to predict whether the total variation would be increased, decreased, or remain unchanged by environmental or genetic changes. In Figure 1d there is no overall effect of either genotype or environment, but both can obviously appear in a particular population in a particular environmental range as discussed above. The case shown in 1e has been chosen to illustrate a common situation for enzyme activity, a parabolic relation between phenotype and environment. Here genotypes are displaced horizontally (have different temperature optima). Neither geontype is superior overall, nor is there any general monotone environmental trend for either genotype. But for any distribution of environments except a perfectly symmetrical one, there will appear a component of variance for genotypic effect. Moreover, if the temperature distribution is largely to either side of the crossover point between these two genotypes, there will be very large components of variance for both genotype and environment and a vanishingly small interaction component; yet over the total range of environments exactly the opposite is true! Figure 1e also shows a second important phenomenon, that of differential phenotypic sensitivity in different environmental ranges. At intermediate temperatures there is less difference between genotypes and less difference between the effect of environments than at more extreme temperatures. This phenomenon of canalisation, is more generally visualized in Figure 1f. Over a range of intermediate phenotypes there is little effect of either genotype or environment, while outside this zone of canalisation, phenotype is sensitive to both4. The zone of canalisation corresponds to the range of environments that have been historically the most common in the species, but in new environments much greater variance appears. Figure 1f bears directly on the characteristic of the analysis of variance that all effects are measured in phenotypic units. The transformations, equations (2) and (3), that express the relationship between the phenotypic deviations ascribable to genotype or environment and the actual values of the genotypes or environmental variables are not simple linear proportionalities. The sensitivity of phenotype to both environment and genotype is a function of the particular range of environments and genotypes. For the programmatic purposes of human genetics, one needs to know more than the components of variation in the historical range of environments. Figures 1a-f are meant to illustrate how the analysis of variance will give a completely erroneous picture of the causative relations among genotype, environment, and phenotype because of the particular distribution of genotypes and environments in a given population at a given time picks out relations from the array of reaction norms that are necessarily atypical of the entire spectrum of causative relations. 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A may from to to to discrimination to lower but equation (1) would simple heritability for The effects of historical and environment are up in the outcome of variance analysis which is not a for the of functional relations. 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However, most genotypes are variable in their and there is a of genotype-environment interaction with each other and quite different environmental reaction norms for of of from and as the objects of analysis are different we causes and we variance, so the purposes of these analyses are different. The analysis of causes in human genetics is meant to provide us with the knowledge we for correct schemes of environmental and with a knowledge of the frequencies of different human genotypes, a knowledge of norms of reaction can also predict the and of environmental changes. of variance can do neither of these because its are a unique function of the present distribution of environment and genotypes. The purposes of the analysis of variance in human genetics are to predict the at which may the genotypic of human and to in some the history of the Neither of these to be a problem since both are in the genotypic of the take so as to the of human social and that human activity and are to upon such genetic change. The of genetic while is an activity of rather than of At any both these not simply the analysis into genetic and environmental components of variation, but a analysis of genetic variance into its and The simple analysis of variance is for these purposes and indeed it has no at In view of the that has been by the spatiotemporally local analysis of variance with the analysis of causes, I that we the for of There are of real