John C. Loehlin and John C. DeFries
Received 22 Feb. 1986–Final 22 Jan. 1987
The estimation of various forms of genotype-environment (GE) correlation is considered. Two methods of estimating “passive” GE correlation from adoption studies are presented and illustrated with IQ data from five adoption studies. One method involves comparison of variances in adoptive and nonadoptive families, whereas the other compares parentchild correlations in such families. The second method yields more precise results for given sample sizes. Ignoring measurement errors leads to marked underestimation of GE correlations. Application of these methods to available adoption data on IQ suggests that typical passive GE correlations for IQ lie in the neighborhood of 0.30.
When the same parents supply both genes and environments to their children, it is likely that the children’s genes and environments will be correlated. This is a passive genotype-environment (GE) correlation, in the sense of Plomin et al. (1977). In addition, active and reactive GE correlations may arise between children’s genotypes and their environments, if genetically influenced traits of the children result in the selection of certain environments (active GE correlation) or provoke responses from others that in turn affect the traits (reactive GE correlation).
GE correlations have been the object of considerable interest, because they represent a source of trait variation that complicates heredity-environment interpretations. If some appreciable fraction of the variation of a trait is due to the covariation of genes and environments, that portion of the variance can be assigned neither to heredity nor to environment – it is attributable jointly to both. If one’s analytic method proceeds by estimating the genetic effect directly and the environmental effect by subtraction, the presence of GE correlation could lead to a substantial underestimation of the latter (Jencks and Brown, 1977).
The presence of GE correlation may also have implications vis-a-vis evolutionary theory (Eaves, 1976a). Suppose that a mother’s intelligence affects the efficiency with which she teaches her children and that intelligence is partially heritable and positively related to relative fitness. There will be a GE correlation present, since the children receiving favorable genes will also experience a more favorable environment. Although natural selection would occur directly for the IQ phenotype, indirect selection for IQ’s contribution to the mother’s teaching skills would also take place – a form of kin selection, since the genes of the mother are selected in part on the basis of their contribution to the fitness of her offspring. Thus, the presence of GE correlation may suggest the presence of evolutionary processes of theoretical interest.
GE correlation may also have implications for developmental behavioral genetics. For example, Scarr and McCartney (1983) have recently proposed a theory of child development that posits that the importance of the three types of GE correlations changes with development. According to this theory, the relative importance of the passive type wanes from infancy to adolescence, whereas that of the active and reactive types increases over the same developmental period. General discussions of GE covariation include those by Plomin et al. (1977) and Eaves et al. (1977).
There have been a number of efforts to estimate GE correlations for human behavioral traits. For example, (passive) GE correlation for IQ has been estimated in several studies using complex path models. Jencks et al. (1972, Appendix A) obtained a figure of about 0.24 for this correlation, based on IQ data from the literature. Loehlin et al. (1975, Appendix I) suggested that a corrected version of Jencks’ model would yield a figure closer to 0.18. Rao et al. (1976), using a data set similar to Jencks’ and a somewhat different path model, obtained estimates for passive GE correlation for IQ ranging from 0.0 to 0.36 under various assumptions. Rice et al. (1980) obtained estimates of 0.14 to 0.19 with another path model.
Horn et al. (1982), also using a path model, estimated passive GE correlation from their own adoption data. Their basic estimate of GE correlation for performance IQ was 0.19. The correlation did not differ notably by age or sex of child or family socioeconomic status but did vary considerably under simulated sampling fluctuation (0.05 to 0.49). It appeared to be somewhat higher in subsets of the data based only on Wechsler scales (0.36 to 0.47). In another independent data set, from the Colorado Adoption Project, Fulker and DeFries (1983), using a path model, estimated GE correlations of 0.03 and 0.04 for cognitive abilities in very young children (aged 1 and 2 years). Because of the quite modest relationships between cognitive measures at these ages and later IQs, this should probably not be taken as an estimate of the same parameter as that in the other studies cited, although it remains of interest in its own right.
While a typical estimate in the above studies (except the last) seems to be a GE correlation of about 0.20 for IQ, other approaches have been considered which have led to rather different estimates. Cattell incorporates GE correlations in his Multiple Abstract Variance Analysis (MAVA) equations (e.g., Cattell, 1960, 1982). For measures of fluid and crystallized intelligence he reports estimates of GE correlation that range from -0.81 to -0.52 for one method of solving the MAVA equations and from -0.41 to +1.00 for another (Cattell, 1982, pp. 305-312). The estimates are more often negative than not (in contrast to those reported in the studies mentioned above) and are often quite large; however, a statistical test of the goodness of fit of the overall model failed to reject the null hypothesis of zero GE correlations for both measures.
Jensen (1976) considered the estimation of GE correlation from data from monozygotic (MZ) and dizygotic (DZ) twins. His two equations in six unknowns were not determinate, but apparently following an earlier paper by Hogarth (1974), he solved them for some 60 combinations of assumed parameters and found only one combination that gave a “realistic” solution: this yielded a GE correlation of 0.08 for IQ.
Taubman (1976) explored various assumptions about the effects of GE correlation on data from MZ and DZ twins, and Thomas (1975) dealt more generally with kinship correlations. Goldberger (1977, 1978) sharply criticized these approaches.
Eaves has developed a model of passive GE correlation in his treatment of “cultural transmission” (1976a) and a model of one form of reactive GE correlation in his treatment of “sibling effects” (1976b). These models are derived assuming random mating and absence of shared environments, so they would not be directly applicable to traits such as IQ and academic achievement, for which one or both of these factors are likely to be substantial. However, such models might be appropriate for other traits, such as those in the personality domain, where assortative mating tends to be slight (e.g., Price and Vandenberg, 1980) and shared familial environment appears to have a minimal effect (Rowe and Plomin, 1981).
In the present paper, some simple path models are developed for estimating GE correlations from adoption data, models which do not require the assumption of random mating or the absence of shared environment among siblings. They have the advantage over more complex models of being focused specifically on the estimation of GE correlation and, thus, not confounding its estimation with the simultaneous estimation of other parameters of the model.
We have chosen to focus on two contrasting estimates, one based on parent-child correlations and one based on a comparison of children’s variances, rather than on a single joint estimate, in order to bring out more sharply the difference in power between the two, as well as to allow for circumstances where only one or the other might be available. We return briefly in the final section to a consideration of combining the two estimates.
ESTIMATING GE CORRELATION FROM ADOPTION DATA
Figure 1 shows path diagrams representing hypothetical causal relationships in ordinary biological families and adoptive families. As can be seen, Fig. 1b assumes that adopted children have not been selectively placed to resemble their adoptive parents on the trait in question: Gc is uncorrelated with Pp. (Mild departures from this condition, on the scale typically observed in practice, will not greatly distort the estimates to be discussed.) The path model implies a passive GE correlation in nonadoptive families, taking the value xkg. There is no corresponding correlation in adoptive families.
There are two different ways in which the passive GE correlation could be estimated from adoptive family data. One would be by estimating the path xkg connecting Ec to Gc. The other is to compare the variances of Pc, the observed trait of the child, iia nonadoptive and adoptive families. In nonadoptive families, the total variance of Pc contains components due to heredity, environment, and their covariation. In adoptive families no covariance term is present, since genotype and environment are uncorrelated.
Comparison of Variances
If we assume that the separate genetic and environmental variances are the same in adoptive and nonadoptive families, by virtue of the fact that the direct effects h and e of genotype and environment on a child’s development do not differ in the two kinds of families, we can take the difference between the variance of Pc in the adoptive and comparable nonadoptive families to be a direct measure of twice the GE covariance (Jinks and Fulker, 1970).
This approach is conceptually straightforward; its major disadvantage is that it requires large samples in order to estimate the GE correlation with reasonable accuracy. For example, in a study with 150 adopted and 150 nonadopted children from comparable families, the standard error of a difference in IQ standard deviations, assuming normality, is about 1.2 IQ points (SQRT(2*225/300)). As we shall see, a difference of an IQ point or two makes a considerable difference in the estimated GE correlation, so that estimates derived from samples of this size – which is typical – will be subject to large sampling fluctuations. Nevertheless, the central tendency observed in several such studies may still be informative.
Table I presents the standard deviations of adopted and nonadopted children from five adoption studies. The first three studies, the classical adoption studies of Burks (1928) and Leahy (1935) and the modern study by Scarr and Weinberg (1978), were each based on a sample of adoptive families and a sample of ordinary nonadoptive families matched to the adoptive family sample on various socioeconomic and other characteristics. The last two studies in Table I, by Scarr and Weinberg (1977) and Horn et al. (1979), involved both adopted children and biological children of adoptive parents reared in the same families. In this design, the question of the comparability of the two kinds of families does not arise, since they are the same families.
As can be seen from Table I, the difference in standard deviations in four of the five studies was in the expected direction of less IQ variability in the adopted children – there was a trivial difference in the opposite direction in the study by Horn et al. The range of differences, from about 0 to about 3 IQ points, is consistent with the standard errors of about 1.2 IQ points which would be expected with these sample sizes (see above).
The last column in Table I shows estimates of GE correlation derived from each study in the following way. The differences between variances, expressed as a proportion of the variance in ordinary families, gives an estimate of (relative) GE covariance. For the Burks study, this is (15.13²-15.00²)/15.13² or 0.0171. This is an estimate of 2hse, where s is the desired GE correlation. If we knew h and e, we could solve directly for s; but in fact we can obtain a reasonable approximation of s without knowing h and e separately, by taking advantage of the fact that over a considerable range of relative sizes of h² and e², the product h*e does not vary much. In the present case (neglecting measurement error) h² + e² + 2hse = 1.0, so h² + e² = 0.9829. For ratios of h 2 to e 2 up to 1 to 5 in either direction, the product h*e varies only in the range 0.366 to 0.491. This yields an estimate of a GE correlation of about 0.02 (the range is 0.017 to 0.023). At the opposite extreme, in Leahy’s study, the standardized covariance term is 0.341, h*e varies from 0.246 to 0.330, and the GE correlation is estimated as being between 0.52 and 0.69.
It is evident from Table I that estimates of GE correlation by this method from samples of these sizes vary widely, although the differences in standard deviation, as we have noted, vary only over a range of about 3 IQ points.
Comparison of Correlations
An alternative approach to estimating GE correlation from adoption study data starts with parent-child correlations rather than children’s variances.
From the path diagrams in Fig. 1 it can be seen that the correlation between the child and the average of his/her parents can be expressed in nonadoptive families as xe + kgh, whereas in adoptive families it is simply xe. The difference between these two correlations thus yields the value kgh. If we then multiply together xe and kgh, we obtain the product xekgh, which is just hse, since xkg is the GE correlation s. As before, from hse we can estimate a fairly narrow range of values for s, since the product h*e does not vary much over a considerable range of values of h and e.
There is one complication. The quantity xe can be expected to be the same across the two samples only if we first adjust the adoptive correlation for the effects upon it of the reduced variation of adopted children. This may be done by multiplying the adoptive correlation by SQRT(1-2hse), the theoretical ratio of the standard deviations of adopted to nonadopted children (see, e.g., Rao et al., 1976). This might seem to defeat the method, since it requires a prior knowledge of hse, the very quantity we seek to estimate. But in fact one can proceed iteratively, by initially assuming the absence of rGE and solving the equation to arrive at an initial estimate of hse, substituting this on the right-hand side of the equation, and repeating until a stable value is reached, usually in a very few iterative cycles.
To illustrate this approach to estimating rGE, we again use the Burks data. Here xe = 0.20, kgh = 0.32, and the initial estimate of hse = 0.064. After iterative adjustment, hse is 0.0623. This implies that h² + e² = 0.8754, h*e varies from 0.326 to 0.428 for ratios of h 2 to e 2 as extreme as 5 to 1 either way, and s, the GE correlation, is estimated as lying between 0.14 and 0.19. Estimates for all five adoption studies are given in Table II. In fact, the iterative adjustment for the reduced variance in adopted children made little difference to the eventual estimates of rGE – the effect was typically of the order of 1 in the second decimal place. In most practical situations one could probably dispense with the adjustment, unless there were reason to suspect very high GE correlations.
Comparison of Tables I and II reveals that the use of parent-child correlations yields estimates of the GE correlation that vary over a much narrower total range than those produced by the comparison of variances (0.07 to 0.23 as against -0.04 to 0.69). There is some consistency of the estimates across methods, in that Leahy’s study yields the highest estimates of GE correlation by both methods and Horn and co-workers’ study the lowest, but the other three studies show no consistent pattern. On the whole Table II yields slightly lower estimates of the GE correlation for IQ than does Table I, centering at about 0.15 rather than at about 0.25 for Table I.
It is probably inappropriate to attempt to attach precise standard errors to rough estimates such as these. Nonetheless, some indication of the likely effects of sampling fluctuation on the estimates would seem essential if one is to evaluate differences across studies or methods of estimation. In order to provide an indication of this sort, the analysis was repeated with each of the original correlations or standard deviations increased or decreased by approximately one standard error, in such a way as to yield a relatively high and a relatively low estimate of rGE.
For the estimates based on standard deviations, the large-sample standard error for the standard deviation in a normal population, S/SQRT(2n), was used. For the high estimate, this was added to SN and subtracted from SA; for the low estimate, the reverse. The estimates based on correlations used Fisher’s z transformation and its standard error, 1/SQRT(n-3). In most cases, this was added to both correlations to yield high values and subtracted from both to yield low values. In two samples, however, the use of another quantity in the range ±1S to adjust the adoptive correlation gave a more extreme estimate, and this was used instead. To simplify the calculations, the adjustment of the adopted-child correlation for reduced variability was omitted.
The high and low estimates of hse were each converted to a range of rGE values as before, although for economy of presentation only the midpoint of each such range is shown in Table III. Table III gives these high and low estimates of rGE, along with central values, the midpoints of the estimates from Tables I and II.
In interpreting the ranges given in Table III, one might note that the probability of obtaining by chance an estimate greater than the high value is approximately the square of the probability of obtaining a single statistic more than one standard error from its mean; for a normal distribution this would be 0.1597², or 0.0252. This is also the probability of obtaining a value lower than the low value. Thus the range from high value to low value might be thought of as something like a 95% confidence interval around the central value, although it is not strictly such.
What conclusions, then, might one draw from Table III?
Across studies within methods, the ranges of uncertainty generally overlap from one study to another. There is one exception – the range of estimates of rGE from S in the study by Horn et al. does not overlap with that from the Leahy study. However, one failure of overlap in 20 pairwise comparisons should probably not be taken as strong evidence of heterogeneity of values of raE across studies.
Across methods, there is a striking difference in precision. The estimates based on S have a much larger range of uncertainty than the methods based on r – the range for the former is typically three or four times the range for the latter.
Thus, the comparison of parent-child correlations rather than children’s variances would seem to be the method of choice for estimating passive GE covariance in an adoption study, unless extremely large samples are available. The latter method does, of course, have the advantage of being usable in certain conditions – e.g., if data are available only for children – when the comparison of parent-child correlations is not.
A number of simplifying assumptions were made in obtaining the GE correlation estimates in Tables I to III. For example, the effect of measurement error was neglected. How would taking it into account have affected the estimates?
Not all the studies report reliability estimates for their IQ measures, but Burks and Horn et al. do. Burks (1928) estimated the reliability of her children’s IQ measure to be 0.83 and that of her parental mental age (MA) measure to be 0.79. Horn et al. (1979, Table 5) gave estimates of 0.86 and 0.85 for fathers and mothers, and 0.78 and 0.77 for sons and daughters, for the reliability of their IQ measures in their sample. How would allowing for measurement error of the order of 20% affect the estimates reported above?
Adjusting for measurement errors, it turns out, makes quite a lot of difference in the estimate of rGE.
Briefly, the estimates of hse will be increased by a factor of the order of 1/rtt² for the estimates based on correlations and by a factor of 1/rtt for the estimates based on variances, where rtt is the reliability of the measure. For low values of hse, there will be a further increase of approximately 1/rtt in going from hse to rGE. (For details of these corrections, see the Appendix.) This means that with test reliabilities of the order of 0.8, the estimate of rGE from correlations will be nearly doubled if error of measurement is taken into account (1/0.83 = 1.95), while the estimate from variances will be increased by about half (1/0.8² = 1.56).
The greater effect of errors of measurement in lowering the estimate from correlations than in lowering the estimate from variances may partially account for the somewhat lower values of rGE obtained by the former method in Tables I through III.
On the whole, we may conclude that the comparison of nonadoptive and adoptive parent-child correlations is a reasonably effective way of estimating passive genotype-environment correlation. It appears to be a much more powerful approach than the comparison of the variances of nonadoptive and adoptive children, contrary to the view that has been expressed that the latter is the “most direct and powerful” test of the presence of GE covariance (Eaves, 1976a). (Eaves does, however, emphasize the need for large samples for such a test.)
One can, of course, use both. In situations where the data are available for the comparison of parent-child correlations, the standard deviations for the children will presumably usually also be available, and the two estimates can be combined. Or one may fit a single parameter to both kinds of data in a covariance model (e.g., Futker and DeFries, 1983). If one is combining separate estimates, the proper weights to assign the two are in proportion to the information they convey, i.e., the inverse of their variances (e.g., Mather, 1972, sect. 56). The roughly 4-to-1 ratio of standard errors implies a relative weight of 16 to 1; i.e., the addition of the information from variances will not contribute much to the final estimate and may hardly be worth the trouble.
A second conclusion from these results is that errors of measurement may be expected to have large effects on estimates of GE correlation and that substantial underestimates of rGE are likely unless unreliability of measurement is allowed for.
The approaches outlined in this paper suggest an appreciable value of passive GE correlation for IQ, perhaps in the neighborhood of 0.30 when corrected for measurement error. This implies that a fraction of the IQ variance of nearly this magnitude will be attributable to the covariation of genotypes and environments (2hse approaches s as the product h*e approaches 1/2; h*e will be only slightly less than this for a considerable range of relative sizes of h and e, unless GE covariance is quite large). Such a GE component is likely to lead to some ambiguity in heredity-environment analyses, thus its presence should not be ignored.
It remains to consider why some other, different approaches have not obtained the same results and to comment briefly on some other strategies for exploring GE correlation.
First, Jensen’s low estimate of the GE correlation from twin data on IQ seems to have been a consequence of a limited exploration of the potential parameter space (cf. Goldberger, 1976). For example, Jensen considered values of the correlation of MZ twin environments down to only 0.70, and DZ environments down to only 0.60. Had he entertained the possibility that such environmental correlations might go down to 0.60 and 0.50, respectively, he would have found values of rGE well above 0.08, ranging up to 0.25 within the limits of other parameters examined. If one considers that accidental as well as systematic environmental events may affect intellectual development, exploration of a lower range of environmental correlations would not seem unreasonable.
Cattell’s negative GE correlations for general ability are more problematic, but it appears that they may stem from a debatable feature of the MAVA equations. For the analysis of fluid and crystallized intelligence measures, Cattell used only MZ and DZ twins and ordinary siblings, plus a general population sample, i.e., the “Most Limited Design” of MAVA (Cattell, 1982, p. 103). In equations 1 and 4 of this design, for variances within and between identical twin pairs, a lower amount of GE covariance is implied for identical twins than for fraternal twins, ordinary siblings, or individuals in the general population, by a ratio of 2*SQRT(2) to 4. This has been controversial (Cattell, 1982, p. 97). Since GE covariance for MZ twins necessarily all lies between pairs, this also shortchanges the expected covariance of MZ twins if (positive) GE covariance is present. Thus, if either the observed variance or the observed covariance of MZ twins is low relative to expectations based on the other groups and the rest of the model, the assumption of positive GE correlation will improve the fit of the model. However, if the observed variance or covariance of MZ twins is high relative to expectations, the assumption of negative GE correlation will improve the fit. If the variance and covariance are roughly in line with expectations, the model should estimate a zero GE correlation, neutralizing the discrepant coefficients.
Is there any reason why an observed MZ variance or covariance might be relatively too high, thus leading Cattell’s model to estimate negative GE correlations? There are at least two plausible reasons why this might be the case for MZ covariances” a higher degree of shared environments of MZ twins and the presence of nonadditive genetric variance affecting the trait. Cattell allows in principle for the former possibility, but in the actual solution of the equations he assumes equal environmental similarity for MZ and DZ twins. His equations also assume genetic additivity. To the extent that dominance and epistasis affect the trait in question, the genetic covariance of DZ twins and ordinary siblings will be lowered relative to that of MZ twins. It has been argued that genetic dominance, at least, may be important for the trait of general intelligence (Jinks and Fulker, 1970). Thus, Cattell’s negative estimates of GE correlation might stem from this feature of his equations.
We have largely confined ouselves in this paper to a discussion of passive GE correlation. How might active and reactive GE correlation be approached? One possibility is to locate groups of children who may reasonably be assumed to have more or less opportunity to select environments, or with environments more or less responsive to their behavior, and compare their variances (Loehlin, 1965). Apparently no one has yet tried this method, which would require large samples.
Another strategy, intended primarily as a screening approach to detect active or reactive GE covariance, is to correlate the traits of the birth parents of adopted children with features of their children’s adoptive environments (Plomin et al., 1977). Assuming that one could rule out selective placement with respect to the environmental features, significant correlations of this kind would imply a link via the child’s genotype, a GE covariance of an active or reactive kind.
A third strategy is to model specific mechanisms giving rise to GE covariance. An example would be Eaves’ (1976b) treatment of sibling effects. Eaves postulates that a gene may have two effects: a direct effect on the individual and an indirect effect on his/her sibling, which may be in the same direction, called “cooperation,” or in the opposite direction, called “competition.” Eaves works out the implications of the model for twin variances and covariances: for example, when the mechanism called competition is present, the total variance of MZ twins will be less than that of DZ twins, and the covariance of DZs will be less than half that of MZs – and may in fact be negative. An approach such as Eaves’ might profitably be extended to modeling specific hypotheses about the environment as well as about the genetic aspects of genotype-environment correlation. Anyone contemplating such a step might find it helpful to read Cattell’s (1982, pp. 247-250) discussion of different ways in which genetic and environmental mechanisms might lead to correlations of genotypes and environments within and between families.
First we consider correction for measurement error in estimating hse from variances. The equation for estimating hse is
hse = 1/2*(SN² – SA²)/SN².
If error variance is the same for natural and adoptive children, the numerator will not change, and the denominator will be multiplied by rtt. Thus the error-corrected estimate of hse will be greater by a factor of 1/rtt. For example, for the Barks data, the estimate of hse will be 1/0.83 = 1.205 times greater.
The corresponding equation using correlations is
hse = (rPCN-rPCA)*rPCA.
The error correction of 1/(rtt1*rtt2)1/2 applies separately to the difference rPCN-rPCA and to the rPCA on the right, so the overall correction would be the square of this. However, the midparent reliability will be higher because it involves the mean of the two parents, rather than a single individual. An expression for the reliability of a midparent score is (rtt+m)/(1+m), where rtt is the single-parent reliability and m is the correlation between spouses. For example, for the Burks data the single-parent reliability is 0.79, the child’s rtt is 0.83, and the spouse correlation is 0.42; the correction for rPCA would be 1/[(0.79 + 0.42)/(1 + 0.42)]1/2 x 1/(0.83)1/2 = 1.19, and that for hse the square of this, or 1.414.
Finally, in both cases above, measurement error enters in the conversion of hse to rGE*rGE = hse/he, where h*e is based on h² + e², and obtained as rtt – 2hse, instead of 1 – 2hse. For example, in the Burks data h² + e² would be 0.83 – 2hse.
Applying the full set of error corrections to the Burks data, for the variance-based estimate we obtain hse = 0.00855 x 1.205 = 0.0103; h² + e² = 0.83 – 0.0206 = 0.8094; h²*e² goes from 0.0910 to 0.1637, and h*e from 0.3016 to 0.4047, yielding final estimates of rGE of 0.025 to 0.034, instead of 0.017 to 0.023, up by about 50%, although still small in absolute terms.
For the correlation-based estimate, we obtain hse = 0.0623 x 1.414 = 0.0881; h² + e² = 0.83 – 0.176 = 0.654; h²*e² goes from 0.0594 to 0.1069, and h*e from 0.2437 to 0.3270, yielding final estimates of rGE of 0.27 to 0.36, instead of 0.14 to 0.19, or nearly double.