A Note on False Positives and Power in G × E Modelling of Twin Data
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DOI: 10.1007/s10519-011-9480-3
- Cite this article as:
- van der Sluis, S., Posthuma, D. & Dolan, C.V. Behav Genet (2012) 42: 170. doi:10.1007/s10519-011-9480-3
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Abstract
The variance components models for gene–environment interaction proposed by Purcell in 2002 are widely used. In both the bivariate and the univariate parameterization of these models, the variance decomposition of trait T is a function of moderator M. We show that if M and T are correlated, and moderator M is correlated between twins as well, the univariate parameterization produces a considerable increase in false positive moderation effects. A simple extension of this univariate moderation model prevents this elevation of the false positive rate provided the covariance between M and T is itself not also subject to moderation. If the covariance between M and T varies as a function of M, then moderation effects observed in the univariate setting should be interpreted with care as these can have their origin in either moderation of the covariance between M and T or in moderation of the unique paths of T. We conclude that researchers should use the full bivariate moderation model to study the presence of moderation on the covariance between M and T. If such moderation can be ruled out, subsequent use of the extended univariate moderation model, as proposed in this paper, is recommended as this model is more powerful than the full bivariate moderation model.
Keywords
G × E interaction Moderation TwinsIntroduction
In the classical twin model, phenotypic variance is decomposed into genetic and environmental variance components, which are usually assumed to be homoskedastic, i.e., constant across relevant environmental or genetic conditions. Heteroskedasticity will arise if the genetic and/or environmental variance components vary in size as a function of a given variable, or moderator. Such a moderator can be truly environmental in nature (e.g., exposure to radiation from a nuclear plant, the level of iodine in soil or drinking water^{1}), or be a trait that itself is subject to genetic influences (e.g., eating or exercise habits, educational attainment level, personality traits). If moderators have a limited number of levels, their effects can be modelled in a multi-group design. However, a multi-group approach does not naturally account for group order, and quickly becomes impractical if the moderator is characterized by many levels (i.e., continuous in the extreme case). As few as, say, 3 or 4 levels may already require a challenging number of groups, especially if the moderator differs within twin pairs (i.e., is not ‘shared’), and the sample includes additional family members (e.g., parents, siblings, partners). In such circumstances, behavioural geneticists often turn to the moderation models proposed by Purcell (2002). The popularity of these model is evident given its frequent use in twin studies on moderation in the context of, for instance, cognitive ability (Bartels et al. 2009; Grant et al. 2010; Harden et al. 2007; Johnson et al. 2009a; Turkheimer et al. 2003; van der Sluis et al. 2008), personality (Bartels and Boomsma 2009; Brendgen et al. 2009; Distel et al. 2010; Hicks et al. 2009a, b; Johnson et al. 2009b; Tuvblad et al. 2006; Zhang et al. 2009), health (Johnson and Krueger 2005; Johnson et al. 2010; McCaffery et al. 2008, 2009), and brain morphology (Lenroot et al. 2009; Wallace et al. 2006).
It is important to realize that the bivariate moderation model considers the joint distribution of M and T, while the univariate moderation model considers moderation of the variance decomposition of T conditional on M. With M included in the means model of T, the univariate moderation model does not allow further investigation of the nature of the covariance between M and T but specifically focuses on the question whether the decomposition of the variance unique to T depends on M. Entering M in the means model of T to allow for a main effect is believed to effectively remove from the covariance model any (genetic) effects that are shared between trait and moderator (Purcell 2002, p. 563). In essence, the variance common to M and T is partialled out, and the moderator effects of M are modeled on the residual variance of T, T′, i.e., the variance of T that was not shared with M. As a result, the effects that M has on the variance decomposition of the residual T′ are believed to be independent of (i.e., not due to) any (unmodeled) (genetic) correlation between M and T (Purcell 2002, p. 563).
However, although M_{1} is indeed unrelated to the for-M_{1}-corrected residual \( {\text{T}}_{1}^{\prime } \), this residual \( {\text{T}}_{1}^{\prime } \) is not necessarily uncorrelated to the moderator M_{2} of the co-twin. In this paper, we first show that non-zero semi-partial correlations between \( {\text{T}}_{1}^{\prime } \) and M_{2} can result in a considerable increase in false positive moderation effects on variance components A and C, especially if the correlation between T and M runs fully (or predominantly) via E (rather than via A and/or C). We subsequently study whether a simple extension of the univariate moderation model prevents this increase of false positive rate. In the first part of this paper, we focus on illustrations and simulations in which the correlation between trait T and moderator M runs either exclusively via A, or via C or via E. Although these settings may be considered quite special, they conveniently simplify the explanation of the problem of non-zero semi-partial correlations in the univariate moderation model proposed by Purcell, and clarify how this model would need to be extended. In the subsequent investigation of the usefulness of this extended version of the univariate model, the simulations are extended to more realistic conditions.
Semi-partial correlations
If the relation between T and M runs exclusively via E, then in both MZ and DZ twins, the cross-trait cross-twin correlation between M_{2} (the moderator of twin 2) and T_{1} (the trait of twin 1) is 0, just as the correlation between M_{1} and T_{2} is 0, i.e., r_{m2,t1} = r_{m1,t2} = 0. If the correlation between T and M runs via C, then the correlation between M_{2} and T_{1}, and between M_{1} and T_{2}, is in both MZ and DZ twins calculated as \( \sqrt {.3*.2} = .24 \). Finally, if the correlation between T and M runs via A, then this correlation is \( \sqrt {.4*.15} = .24 \) in MZ twins and \( \sqrt {.4} *.5*\sqrt {.15} = .12 \) in DZ twins.
Correlations and semi-partial correlations between M_{2} and T_{1} if the within-twin correlation between T and M runs via A, via C, or via E
r_{m2,t1} | r_{m2(t1·m1)} | |
---|---|---|
T and M correlated via A | ||
MZ | .24 | .074 |
DZ | .12 | 0 |
T and M correlated via C | ||
MZ | .24 | .074 |
DZ | .24 | .124 |
T and M correlated via E | ||
MZ | 0 | −.173 |
DZ | 0 | −.124 |
Clearly, as a result of partialling out M_{1} from T_{1}, the semi-partial correlation between \( {\text{T}}_{1}^{\prime } \) and M_{2} is lower than the correlation between T_{1} and M_{2}. However, the semi-partial correlation between \( {\text{T}}_{1}^{\prime } \) and M_{2} is often not equal to zero: especially if the correlation between T_{1} and M_{2} was zero to begin with (i.e., if T and M are correlated via E), the semi-partial correlation between \( {\text{T}}_{1}^{\prime } \) and M_{2} is quite large and negative. Estimated across an entire study sample (while weighing for the MZ/DZ ratio), these non-zero semi-partial correlations can be quite considerable (e.g., in the case that T and M are correlated via E), and are likely to cause problems in the univariate moderation model. After all, these non-zero semi-partial correlations, whether positive or negative, will somehow need to be accommodated in the model. Considering the univariate moderation model as depicted in Fig. 1b, a non-zero semi-partial correlation between \( {\text{T}}_{1}^{\prime } \) and M_{2} is most likely to be accommodated via the effects that M has on the variance components A and C, i.e., via β_{a} and β_{c}, as these are the only links between M_{2} and \( {\text{T}}_{1}^{\prime } \), and M_{1} and \( {\text{T}}_{2}^{\prime } \), respectively. In Simulation study 1, we investigated first whether these non-zero semi-partial correlations do indeed cause problems in the univariate moderation model. We expect problems to be greatest if the semi-partial correlation deviates more from zero (i.e., in the case that T and M are correlated via E). Second, we investigated whether these problems indeed manifest themselves mostly through β_{a} and β_{c}.
Simulation study 1
To investigate the effect of partialling out M on T within each individual twin on the significance of parameters β_{a}, β_{c}, and β_{e}, we simulated data according to the models shown in Figs. 2a–c, i.e., with correlations between T and M running either exclusively via A, exclusively via C, or exclusively via E. In these simulated data, T and M were both standard normally distributed. Also, T and M were correlated, but moderation effects of M on the cross paths and on the variance components of the residual of T were absent. For each scenario, we simulated 2000 datasets each comprising Nmz = Ndz = 500 pairs. We then fitted to these datasets the standard univariate moderation model with moderator M modeled on the means (Fig. 1b), and then constrained either β_{a}, β_{c}, or β_{e} to zero to test for the significance of each parameter individually, i.e., a 1-df test. The difference between the −2 log-likelihood of the full model (the specific moderation parameter estimated freely) and the −2 log-likelihood of the restricted model (the specific moderation parameter fixed to 0), denoted as χ_{diff}^{2}, is χ^{2}-distributed. Since moderation parameters β_{a}, β_{c}, and β_{e} were zero in these simulated data, we expect the distribution of the χ_{diff}^{2} as calculated across all 2000 data sets to follow a central χ^{2}(1) distribution. Given an nominal α of .05, we expected 5% of χ_{diff}^{2} test to be significant, i.e., larger than the critical value of 3.84.
Results Simulation study 1: false positive rates under Purcell’s univariate moderation model
Drop β_{a} | Drop β_{c} | Drop β_{e} | ||
---|---|---|---|---|
Settings Simulation study 1 | ||||
r(T,M) = .24 via A only | % hits | .07 | .07 | .05 |
r(T,M) = .24 via C only | % hits | .08 | .09 | .05 |
r(T,M) = .24 via E only | % hits | .53 | .55 | .08 |
Additional simulations | ||||
r(T,M) = .24 via A, C and E in equal proportions^{a} | % hits | .04 | .05 | .07 |
r(T,M) = .62 via A, C and E in equal proportions^{b} | % hits | .05 | .05 | .06 |
r(M_{1},M_{2}) = 0^{c} | % hits | .05 | .05 | .04 |
r(M_{1},M_{2}) = 1^{d} | % hits | .03 | .03 | .05 |
Summarizing, Simulation study 1 shows that under the univariate moderation model, in which T_{1} is corrected for M_{1} only, and T_{2} is corrected for M_{2} only, the false positive rate can be (much) higher than the nominal α-level, especially if the correlation between T and M runs predominantly via E.
Solution: extension of the univariate moderation model?
In this section we aim to investigate whether we can solve the problems that can result from non-zero semi-partial correlations by extending the univariate moderation model. An obvious solution is to extend the means model such that the trait value of twin 1 is not only corrected for the moderator value of twin 1, but also for any residual association to the moderator value of the co-twin, as this would result in a residual \( T_{1}^{\prime \prime } \) (\( T_{2}^{\prime \prime } \)) that is uncorrelated to both M_{1} and M_{2}. Taking into account the way regression coefficients in a multiple regression model with two predictors are calculated, it is easy to show that the parameters in the means models should generally also differ across zygosity.
With 12 parameters (6 to describe the variance part of the model: 3 related, and 3 unrelated to the moderator; and 6 parameters to describe the means models: 3 for MZ twins, and 3 for DZ twins), this extended univariate moderation model is still more parsimonious than the bivariate moderation model (17 parameters of which 15 concern the variance decomposition). We conducted Simulation study 2 to investigate whether the false positive rate of this extended univariate moderation model is correct, and comparable to the false positive rate of the full bivariate moderation model.
Simulation study 2
To investigate whether the extended univariate moderation model results in the correct false positive rate of 5%, we re-analyzed the data-sets created under Simulation study 1 using the extended univariate moderator model. Again we tested the significance of moderator parameters β_{a}, β_{c} and β_{e}. As these parameters were simulated to be zero, we expected the χ_{diff}^{2} to be χ(1) distributed across the 2000 datasets, independent of whether T and M were correlated via A, via C, or via E. In addition, these same data sets were analyzed using the full bivariate moderation model (as depicted in Fig. 1a) in which all 17 parameters were estimated freely (Note that we did not use the full bivariate moderation model to analyze the data with r_{M1,M2} = 1 or r_{M1,M2} = 0 because in practice one would never choose a bivariate parameterization under these circumstances).
Results Simulation study 2: false positive rates under the extended univariate moderation model and the full bivariate moderation model when the covariance between T and M is not moderated
Drop β_{a} | Drop β_{c} | Drop β_{e} | ||||
---|---|---|---|---|---|---|
% hits | nsim | % hits | nsim | % hits | nsim | |
r_{M,T} .24 via A | ||||||
Ext univariate | .04 | 2000 | .04 | 2000 | .04 | 1998 |
Full bivariate | .01 | 2000 | .01 | 2000 | .04 | 2000 |
p < .001 | p < .001 | p = .88 | ||||
r_{M,T} .24 via C | ||||||
Ext univariate | .04 | 2000 | .04 | 2000 | .05 | 2000 |
Full bivariate | .01 | 1999 | .01 | 1998 | .04 | 2000 |
p < .001 | p < .001 | p = .08 | ||||
r_{M,T} .24 via E | ||||||
Ext univariate | .05 | 1998 | .04 | 1999 | .05 | 1999 |
Full bivariate | .01 | 1996 | .01 | 1996 | .05 | 2000 |
p < .001 | p < .001 | p = .65 | ||||
r_{M,T} = .24 via A, C and E in equal proportions | ||||||
Ext univariate | .04 | 2000 | .05 | 2000 | .05 | 2000 |
Full bivariate | .02 | 2000 | .01 | 2000 | .05 | 2000 |
p < .005 | p < .001 | p = .43 | ||||
r_{M,T} = .62 via A, C and E in equal proportions | ||||||
Ext univariate | .05 | 1998 | .05 | 2000 | .05 | 2000 |
Full bivariate | .02 | 2000 | .02 | 2000 | .05 | 2000 |
p < .001 | p < .001 | p = .94 | ||||
r_{M1,M2} = 0 (fully E), r_{M,T} = .24 | ||||||
Ext univariate | .05 | 1999 | .05 | 2000 | .04 | 1999 |
r_{M1,M2} = 1 (fully C), r_{M,T} = .24 | ||||||
Ext univariate | .03 | 2000 | .03 | 2000 | .05 | 1999 |
Summarizing the results of Simulation study 2, we conclude that the extension of the univariate moderation model avoids the inflated false positive scores that were observed for the standard univariate moderation model, while the full bivariate moderation model actually proved too conservative. However, Simulation studies 1 and 2 concerned scenarios in which the covariance between M and T was itself not subject to moderation, i.e., β_{ac}, β_{cc}, and β_{ec} on the cross paths between M and T in Fig. 1a were fixed to 0. That is, the covariance between M and T did not dependent on the level of M. In practice, however, it is possible that the covariance between M and T fluctuates as a function of M. Simulation study 3 was conducted to investigate the false positive rate of the extended univariate and the full bivariate moderation models in the context of data in which the covariance between M and T is moderated. These simulations are of specific interest since moderator-dependent variation in the strength of the covariance between M and T is not well accommodated by the estimated regression parameters β_{0,MZ}, β_{1,MZ}, β_{2,MZ}, β_{0,DZ}, β_{1,DZ}, and β_{2,DZ} in Eqs. 8 and 9, and problems are therefore to be expected for the extended univariate moderation model.
Simulation study 3
We again simulated data for standard normally distributed moderator M and trait T in 500 MZ and 500 DZ twin pairs. Suppose again that A, C, and E account for 40%, 30%, and 30%, respectively, of the variance in M. The parts of the cross paths between M and T that do not depend on M (a_{c}, c_{c}, and e_{c} in Fig. 1a) are all set to .05, and A, C, and E unique to T (a_{u}, c_{u}, and e_{u} in Fig. 1a) are set to .35, .25 and .25, respectively. That is, if moderation is fully absent, the correlation between M and T equals .39, while genetic and (common) environmental effects explain 40%, 30% and 30% of the variance in T, respectively. We now introduce moderation on the cross paths by setting either β_{ac}, β_{cc}, or β_{ec} to .10. Moderation on the unique parts of T is, however, absent (i.e., β_{a}, β_{c}, and β_{e} in Fig. 1a are set to 0). For each of these settings we simulated 2000 data sets. Note that we deliberately choose the moderation parameters on the cross paths to be quite substantial: if the false positive rate of the extended univariate model is affected by moderation of the cross paths, then we are sure to pick it up. If the false positive rate of the extended univariate model is not affected by moderation of the cross paths, then the size of this moderation should not matter.
We then fitted to these datasets a) the full bivariate moderation model including all 17 parameters, and b) the extended univariate moderation model in which both moderators M_{1} and M_{2} are modeled on the means with means parameters differing across zygosity (Eqs. 8 and 9). Within these models we constrained either β_{a}, β_{c}, or β_{e} to zero to test for the significance of each parameter individually, i.e., a 1-df test. Since moderation parameters β_{a}, β_{c}, and β_{e} were simulated as 0 in these data, we expect the distribution of the χ_{diff}^{2} as calculated across all 2000 data sets to follow a central χ^{2}(1) distribution. Given an nominal α of .05, we expected 5% of χ_{diff}^{2} test to be significant, i.e., larger than the critical value of 3.84.
Results Simulation study 3: false positive rates under the extended univariate moderation model and the full bivariate moderation model when the covariance between T and M is moderated
Drop β_{a} | Drop β_{c} | Drop β_{e} | ||||
---|---|---|---|---|---|---|
% hits | nsim | % hits | nsim | % hits | nsim | |
Baseline: β_{ac} = β_{cc} = β_{ec} = 0 | ||||||
Ext univariate | .05 | 1999 | .05 | 2000 | .05 | 1999 |
Full bivariate | .02 | 2000 | .01 | 2000 | .05 | 2000 |
p < .001 | p < .001 | p = .42 | ||||
β_{ac} = .10; β_{cc} = β_{ec} = 0 | ||||||
Ext univariate | .18 | 1998 | .18 | 1999 | .08 | 1996 |
Full bivariate | .03 | 2000 | .02 | 2000 | .06 | 2000 |
p < .001 | p < .001 | p < .01 | ||||
β_{cc} = .10; β_{ac} = β_{ec} = 0 | ||||||
Ext univariate | .13 | 2000 | .13 | 1999 | .07 | 1999 |
Full bivariate | .02 | 2000 | .02 | 2000 | .05 | 2000 |
p < .001 | p < .001 | p < .01 | ||||
β_{ec} = .10; β_{ac} = β_{cc} = 0 | ||||||
Ext univariate | .23 | 2000 | .23 | 1998 | .08 | 1997 |
Full bivariate | .02 | 2000 | .01 | 2000 | .04 | 2000 |
p < .001 | p < .001 | p < .001 |
In summary, the results of Simulation study 3 show that the extended univariate moderation model can be used as a moderation detection method, but is not very suited to establish the exact location of the moderation as it cannot distinguish moderation on cross paths from moderation on the unique paths of T.
False negatives
We have shown that the false positive rate (i.e., type I error rate) is correct under the extended univariate moderation model, but only if the covariance between M and T is free of moderation by M. We now address the false negative rate, i.e., the type II error, of the extended univariate moderation model compared to the full bivariate model. In a fourth and fifth simulation study, we investigate whether the false negative rate of the extended univariate moderation model is comparable to the false negative rate of the bivariate moderation model when the covariance between M and T is not subject to moderation (Simulation study 4) or when this covariance is subject to moderation as well (Simulation study 5).
Simulation study 4
Results Simulation study 4: false negative rates under the extended univariate moderation model and the full bivariate moderation model when the covariance between T and M is not moderated
Drop β_{a} | Drop β_{c} | Drop β_{e} | ||||
---|---|---|---|---|---|---|
% hits | nsim | % hits | nsim | % hits | nsim | |
T and M correlated via A | ||||||
Ext univariate | .20 | 1999 | .31 | 1998 | .48 | 2000 |
Full bivariate | .07 | 1998 | .21 | 2000 | .50 | 2000 |
p < .001 | p < .001 | p = .47 | ||||
T and M correlated via C | ||||||
Ext univariate | .27 | 1999 | .16 | 1999 | .47 | 2000 |
Full bivariate | .25 | 2000 | .03 | 1998 | .48 | 2000 |
p = .18 | p < .001 | p = .95 | ||||
T and M correlated via E | ||||||
Ext univariate | .48 | 1999 | .47 | 1999 | .91 | 1997 |
Full bivariate | .25 | 1998 | .28 | 1989 | .91 | 2000 |
p < .001 | p < .001 | p = .83 |
For each of these 9 settings (r_{t,m} runs via A, C or E, and moderation is present on either A, C, or E) we simulated 2000 datasets and analyzed these using either the full bivariate moderation model (estimating moderation parameters on the cross paths as well as on the paths unique to T) or the extended univariate moderation model (estimating moderation on the variance components of the residual T″). We then tested whether the moderation parameter of interest (either β_{a}, or β_{c}, or β_{e}) was significant given α = .05.
The results of these simulations are presented in Table 5. In 5 out of 9 scenarios, the power of the extended univariate moderation model was significantly higher than the power of the full bivariate moderation model. Note that we can indeed speak of higher power because we know from the results of Simulation study 2 that the false positive rate of neither models is inflated. The lower power of the full bivariate moderation model is probably due to the variance being decomposed into as many as 15 parameters, compared to the 6 of the extended univariate moderation model: misfit resulting from fixing one of the moderation parameters to zero can more easily be absorbed by the remaining 14 parameters.
Simulation study 5
Results Simulation study 5: false negative rates under the extended univariate moderation model and the full bivariate moderation model when the covariance between T and M is moderated
Drop β_{a} | Drop β_{c} | Drop β_{e} | ||||
---|---|---|---|---|---|---|
% hits | nsim | % hits | nsim | % hits | nsim | |
Baseline: β_{ac} = β_{cc} = β_{ec} = 0 | ||||||
Ext univariate | .29 | 2000 | .34 | 1996 | .56 | 1999 |
Full bivariate | .25 | 2000 | .26 | 1999 | .57 | 2000 |
p < .01 | p < .001 | p = .72 | ||||
β_{ac} = .10; β_{cc} = β_{ec} = 0 | ||||||
Ext univariate | .35 | 1989 | .17 | 1997 | .07 | 1998 |
Full bivariate | .23 | 2000 | .03 | 2000 | .05 | 2000 |
p < .01 | p < .001 | p = .01 | ||||
β_{cc} = .10; β_{ac} = β_{ec} = 0 | ||||||
Ext univariate | .14 | 1996 | .45 | 1995 | .07 | 1998 |
Full bivariate | .04 | 2000 | .25 | 2000 | .05 | 2000 |
p < .01 | p < .001 | p = .03 | ||||
β_{ec} = .10; β_{ac} = β_{cc} = 0 | ||||||
Ext univariate | .22 | 1996 | .22 | 1998 | .50 | 1995 |
Full bivariate | .02 | 2000 | .01 | 2000 | .57 | 2000 |
p < .01 | p < .001 | p < .001 |
The results of these simulations are presented in Table 6. In all scenarios, the extended univariate moderation model picks up moderation more often than the full bivariate moderation model. However, these results can, except for the Baseline model, not be interpreted as the extended univariate moderation model having more power than the full bivariate moderation model. Given the results of Simulation study 3, which showed that the extended moderation model picks up the moderation on the cross paths as if it is moderation on the unique paths, we conclude that the power of the extended univariate moderation model is too high, or at least that the location of the moderation that is detected, is uncertain. That is, β_{a}, β_{c}, and β_{e} are biased in the extended univariate moderation model because the moderation on the cross paths (β_{ac}, β_{cc}, and β_{ec}) is not adequately accommodated by the regression coefficients in the means part of the model.
Discussion
In this paper, we showed that the univariate moderation model proposed by Purcell (2002) produces (highly) inflated false positive rates if the moderator M is correlated between twins, and M and T are correlated as well. We investigated an extension of this model as a solution to this problem, and conclude that the extended univariate moderation model works well, but only if moderation on the covariance between M and T is absent. Moderation of the covariance between M and T is, however, not accommodated adequately in the extended univariate moderation model, and as a result, moderation of the covariance is picked up as moderation on the variance components unique to T. In the absence of moderation of the covariance between M and T, the extended univariate moderation model is actually more powerful than the full bivariate moderation model, but in the presence of moderation of the covariance between M and T, the extended univariate moderation model detects moderation of the variance components unique to T, as such misspecifying the actual location of the moderation.
Fortunately, most published papers in which the univariate moderation model was used concern moderation effects of family-level moderators such as SES, parental educational attainment level, or the age of the twins, i.e., variables that are by definition equal in both twins. As we have shown, non-zero semi-partial correlations are not a problem in that case and the false positive rate is rather too low than too high (i.e., the model is slightly conservative). In a few published papers, however, moderators were studied that did show variation between twins (e.g., McCaffery et al. 2008, 2009; Timberlake et al. 2006: moderators under study were educational attainment level of the twins, exercise level of the twins, and the twins’ self-reported religiosity, respectively). Whether the moderation effects reported in these papers are genuine or spurious (i.e., the result of non-zero semi-partial cross-trait cross-twin correlations) depends, as we have shown, on the nature of the correlation between T and M, on the nature of the correlation between M_{1} and M_{2}, and on the absence or presence of moderation of the covariance between M and T. Re-analysis of these data using the full bivariate moderation model, or the extended univariate moderation model if the presence of moderation of the covariance has been excluded, is advised. Overall, we conclude that researchers should use the full bivariate moderation model to study the presence of moderation on the covariance between M and T. If such moderation can be ruled out, subsequent use of the extended univariate moderation model is recommended as this model is more powerful than the full bivariate moderation model.
Note that such environmental factors could indeed be purely environmental, but could also in part be subject to genetic influences. For example, the chance of exposure to radiation may depend on one’s occupation or residential area, and such social-economic factors may again be under genetic influence.
To ease presentation, we limit ourselves to the linear moderation model. We note, however, that non-linear effects of the moderator on variance components A, C and E are discussed by Purcell (2002).
Discretizing either variable to render T and M comparable in scale entails a loss of information and is therefore undesirable.
Note the difference between a partial correlation and a semi-partial correlation. The partial correlation between X and Y given Z, is the correlation between the residual X′ and the residual Y′, where Z is partialled out in both variables. The semi-partial correlation is the correlation between the residual X′ and the uncorrected variable Y, i.e., Z is partialled out only in X but not in Y.
Acknowledgement
Sophie van der Sluis (VENI-451-08-025 and VIDI-016-065-318) and Danielle Posthuma (VIDI-016-065-318) are financially supported by the Netherlands Scientific Organization (Nederlandse Organisatie voor Wetenschappelijk Onderzoek, gebied Maatschappij-en Gedragswetenschappen: NWO/MaGW). Simulations were carried out on the Genetic Cluster Computer which is financially supported by an NWO Medium Investment grant (480-05-003), by the VU University, Amsterdam, The Netherlands, and by the Dutch Brain Foundation.
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