# Can GE-Covariance Originating in Phenotype to Environment Transmission Account for the Flynn Effect?

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## Abstract

**:**

## 1. Introduction

_{t}) is predictive of environmental (relevant to the phenotype) influences at time t + 1, E

_{t+1}. The direct regression of E

_{t+1}on Ph

_{t}is compatible with active and passive GE-cov. Ph->E transmission gives rise to the correlation of genetic influences at time t with the environmental influences at time t + 1, through the mediation of the phenotype at time t. Dickens and Flynn considered the influence of GE-cov in four different effects: Matching, averaging, and multiplication either within the individual, or within the social context [1]. The former three pertain to the individual environment, whereas the latter pertains to the social environment. The longitudinal twin model presented below includes the effects of matching, averaging and multiplication, but does not include the social multiplier. However the proposed model does have the virtue of being operational, i.e., it can be applied to longitudinal twin data. Fitting the model provides empirical estimates of GE-cov representative of Dutch children born between 1985 and 1996, and so shines some light on the plausibility of the essential role of GE-cov within the Dickens and Flynn model. Furthermore, using the results obtained, we can assess the degree to which hypothetical changes in environmental (latent) means and Ph->E transmission parameters account for the Flynn effect. Specifically, we determine the degree to which increases in the latent environmental means in combination with changes in Ph->E transmission parameters give rise to changes in phenotypic means. The changes in environmental means are supposed to be due to environmental improvements relevant to the development of cognitive abilities (these may in principle include social multiplier effects).

## 2. Longitudinal Genetic Modeling

#### 2.1. The Standard ACE Simplex Model Applied to Twin Data

_{it}denote the FSIQ or perfIQ test score of twin member i (i = 1, 2) at time t (t = 1, ..., T; indexed by age, see below), where the phenotype y

_{it}is decomposed into a part, y*

_{it}, that is subject to the longitudinal model, and the time specific part, z

_{it}, which may include zero mean occasion specific influences

_{it}= β

_{0t}+ y*

_{it}+ z

_{it},

_{0t}is the phenotypic mean at time t. The score z

_{it}is a zero mean time specific residual, which is subject to its own decomposition:

_{it}= a

_{it}+ c

_{it}+ e

_{it},

_{it}, c

_{it}and e

_{it}are time specific effect, i.e., uncorrelated over time. The term e

_{it}includes measurement error. The score y*

_{it}is decomposed into additive genetic (A), shared environmental (C), and unshared environmental (E) scores

_{it}=A

_{it}+ C

_{it}+ E

_{it}

_{it}, C

_{it}, and E

_{it}are the zero mean additive genetic, shared environmental and unshared environmental scores at time t, respectively. Each latent source of individual differences is subject to the following longitudinal model (see Figure 1):

_{it}= β

_{At,t-1}A

_{it-1}+ ζ

_{Ait}

_{it}= β

_{Ct,t-1}C

_{it-1}+ ζ

_{Cit}

_{it}= β

_{Et,t-1}E

_{it-1}+ ζ

_{Eit}

**Figure 1.**The standard additive genetic, common (shared) and unshared environment (ACE) simplex model depicted for both twins, where y*’s represent the series of observed phenotypes, A’s represent the underlying additive genetic series, C’s represent the underlying shared environment series, and E’s represent the underlying unique environment series. Within each series, ζ represents innovation and β represents stability. Conform biometric theory, the covariance between the A’s at each time point is set equal to the var(A

_{t}) (for MZ) or 0.5 var(A

_{t}) (for DZ twins). For clarity reasons, time specific effects (i.e., a, c, and e) and phenotypic means are omitted. Note that the scaling and parameter labels are used throughout the series.

_{i1}= ζ

_{Ai1}, C

_{i1}= ζ

_{Ci1}, and E

_{i1}= ζ

_{Ei1}. Note that the latent variables A, C, and E at time t are regressed on the variables at time t-1. The variables ζ

_{Ait}, ζ

_{Cit}, and ζ

_{Eit}are uncorrelated with each other, but correlated over time (Equations (4)–(6)). As such, this model provides a decomposition into uncorrelated genetic and environmental components of phenotypic variance at each time point (Equation (2)), and of phenotypic stability (covariance) between time points (Equations (3)–(6)). The phenotypic stability is a function of the genotypic and environmental stabilities, which are quantified as explained variances of the latent regressions (Equations (4)–(6)). These depend both on the magnitude of the regression coefficients (β

_{At,t-1}, β

_{Ct,t-1}, β

_{Et,t-1}) and the (innovation) residual variances (var(ζ

_{Ait)}, var(ζ

_{Cit}), var(ζ

_{Eit})).

_{A1t}ζ

_{A2t}) = cor(a

_{1t}a

_{2t}) = 0.5 in DZ twins, and cor (ζ

_{A1t}ζ

_{A2t}) = cor(a

_{1t}a

_{2t}) = 1 in MZ twins, and by constraining cor(ζ

_{C1t}ζ

_{C2t}) = cor(c

_{1t}c

_{2t}) = 1 (note that cor(ζ

_{E1t}ζ

_{E2t}) = cor(e

_{1t}e

_{2t}) = 0, by definition of these environmental influences being unshared). Note that at t = 1 and t = T, the time specific variances are not identified [16]. These variances are identified by imposing the over-identifying constraint that the variance components are equal over time (i.e., var(a

_{t}) = var(a), var(c

_{t}) = var(c), var(e

_{t}) = var(e), t = 1...4).

_{Ait}) (t = 2, ..., T)) tends to zero in the presence of non-zero autoregression, this part of the model tends to a common factor model. We presented the ACE model here, as it counts as a standard model for repeated measures in GCSM of twin data and can therefore serve as a reference model. In the next section, we present our alternative AE* model, which includes Ph->E transmission.

#### 2.2. The AE* Simplex Model with Ph->E Transmission Applied to Twin Data

_{it}). We introduce the environmental variables E*

_{it}, where we add an asterisk to emphasize that E*

_{it}differs from the variable E

_{it}in the standard model. Within the alternative model, E*

_{it}represents the total environmental influence, which is partially due to shared environmental factors (hence, the correlation between ζ

_{E*1t}and ζ

_{E*2t}), and partially due to unique environmental variance in ζ

_{E*it}. The ACE model collapses into the AE* model by imposing the constraint: β

_{Ct,t-1}= β

_{Et,t-1 }(i.e., the values of the autoregression series for the shared and unshared environmental series are set to be equal). As we explain below, we adopt this alternative treatment of environmental effects (E vs. E*) for reasons of (empirical) identification.

_{1t}and E*

_{2t}on the twins’ phenotypes at the preceding time y

_{1t-1}and y

_{2t-1}(the relevant parameters are denoted α

_{t}and γ

_{t}). As above, the phenotype at each time point is related to the intercept β

_{0t}(the phenotypic mean at time t) the time specific residual (z

_{it}), and the zero mean additive genetic (A

_{it}) and environmental variables (E*

_{it}):

_{it}= β

_{0t}+ y

_{it}* + z

_{it},

_{it}* = A

_{it}+ E*

_{it},

_{it}= β

_{At,t-1}A

_{it-1}+ ζ

_{Ait}

_{1t}= β

_{Et,t-1}E*

_{1t-1}+ α

_{t}y*

_{1t-1}+ γ

_{t}y*

_{2t-1}+ ζ

_{E*1t}

_{2t}= β

_{Et,t-1}E*

_{2t-1}+ γ

_{t}y*

_{1t-1}+ α

_{t}y*

_{2t-1}+ ζ

_{E*2t},

_{t}(transmission within a twin member, e.g., α

_{t}y*

_{1t-1}) and γ

_{t}(transmission across twin members, e.g., γ

_{t}y*

_{2t-1}). Again at t = 1, we set A

_{i1}= ζ

_{Ai1}and E*

_{i1}= ζ

_{E*i1}. By including the α

_{t}parameter, we recognize that the phenotype of each twin member may contribute directly to his or her own environment (i.e., niche picking, [4,5]). By including the γ

_{t}parameter, we recognize that the phenotype of one twin member may also contribute to the environment of the other twin member (i.e., sibling effects) [14,17,18]. Both niche picking and sibling effects are concepts which match the phenotype to the environment, which plays an important role in D&F model [1]. By substituting y*

_{it}in Equations (10a) and (11a) for y*

_{it}given in Equations (8), we have:

_{1t}= β

_{Et,t-1}E*

_{1t-1}+ α

_{t}(A

_{1 t-1}+ E*

_{1 t-1})+ γ

_{t}(A

_{2 t-1}+ E*

_{2 t-1})+ ζ

_{E*1t}

_{2t}= β

_{Et,t-1}E*

_{2t-1}+ γ

_{t}(A

_{1 t-1}+ E*

_{1 t-1})+ α

_{t}(A

_{2 t-1}+ E*

_{2 t-1})+ ζ

_{E*2t}

_{t}(A

_{i t-1}) and γ

_{t}(A

_{i t-1}). Additionally, when writing out the equation for consecutive time points, it will become evident that due to the interplay between the phenotype and environment, a higher IQ leads to a better environment, which, in turn, leads to a higher IQ. This interplay thus functions as the multiplier.

_{t}and γ

_{t}) is extremely poor in the presence of an independent C simplex (i.e., Equation (5)). This is because Ph->E transmission itself gives rise to correlated environmental variables (see below; Table 4 and Table 5). Within the AE* model, the resolution of the Ph->E transmission parameters improves considerably. In presence of Ph->E transmission, the correlations between ζ

_{E*1t}and ζ

_{E*2t}are due to shared environmental influences in addition to the correlated environment originating in the Ph->E transmission [19]. Dolan et al. established that, given four time points, the parameters α

_{t}and γ

_{t}are identified subject to equality constraints (e.g., α

_{2}= α

_{3}and γ

_{2}= γ

_{3}) [20]. Here, we impose the over-identifying constraint α

_{t}= α

_{t-1}and γ

_{t}= γ

_{t-1}, t = 2, 3, 4, and therefore drop the time subscript (i.e., the parameters are denoted α and γ). As above, remaining identification issues are resolved by cor(ζ

_{A1t}ζ

_{A2t}) = cor(a

_{1t}a

_{2t}) = 0.5 in DZ twins, cor(ζ

_{A1t}ζ

_{A2t}) = cor(a

_{1t}a

_{2t}) = 1 in MZ twins, and cor(c

_{1t}c

_{2t}) = 1. As depicted in Figure 2 and discussed above, at t = 1, we assume that GE-cov is zero. This is a strong assumption, to which we return below.

**Figure 2.**The AE* simplex model with Ph->E transmission, where parameters α can be interpreted as within twin Ph->E transmission and γ as between twin Ph->E transmission, and where y’s are the series of observed variables, A’s are the underlying additive genetic series, E’s are the underlying unique environment series and * denotes the time point specific environmental covariance (indicative of C). Conform biometric theory, the covariance between the A’s at each time point is set equal to the var(A

_{t}) (for MZ) or 0.5 var(A

_{t}) (for DZ twins). For clarity reasons, time specific effects and phenotypic means are omitted. Note that the scaling and parameter labels are used throughout the series.

## 3. Method

^{2}goodness of fit index, incremental fit index TLI (>0.95 acceptable, >0.97 good), the BIC and the sample size adjusted Bayesian Information criterion (adjBIC; the smaller the better), and the RMSEA, with 95% confidence intervals (<0.08 acceptable; <0.05 good). The criteria given in parentheses are based on Schermelleh-Engel, Moosbrugger, and Müller [31]. The FIML estimates of the means, standard deviations, and correlation matrices are given in the appendix. We tested for sex differences in both FSIQ and perfIQ. At none of the time points, sex differences were found to be significant (given α = 0.05/4). As such, we excluded sex from the analyses.

## 4. Results

#### 4.1. Standard Model: ACE Simplex

_{t}) = var(a), var(c

_{t}) = var(c), var(e

_{t}) = var(e), t=1,...,4). For FSIQ, the standard simplex model fitted the data well (χ

^{2}(60) = 76.8; TLI = 0.99; adjBIC = 21,470; BIC = 21,559; RMSEA = 0.032 (CI95: 0.0–0.051)). The time-specific genetic variance (denoted var(a) above) (1 parameter) and the variances var(ζ

_{A3}) and var(ζ

_{A4}) (two parameters) were estimated at zero. They were therefore fixed to zero, and the model refitted: χ

^{2}(63) = 76.8; TLI = 0.99; adjBIC = 21,460; BIC = 21,540; RMSEA = 0.028 (CI95: 0.0–0.048). The fact that the χ

^{2}is unchanged is consistent with the fact that the fixed variances were estimated as zero. Although some parameters were not significant (at α = 0.01, say) in this model, we made no effort to further prune the model. In the analysis of perfIQ, we obtained similar results. The standard simplex model fitted the data well (χ

^{2}(60) = 56.9; TLI > 0.99; adjBIC = 18,496; BIC = 18,585; RMSEA = 0.0 (CI95: 0.0–0.035)). The time-specific genetic variance was zero (one parameter), as were var(ζ

_{A2}), var(ζ

_{A3}) (two parameters), and var(ζ

_{C2}), var(ζ

_{C3}), var(ζ

_{C4}), and var(ζ

_{E3}) (four parameters). We refitted the model with these parameters fixed to zero, and obtained χ

^{2}(67) = 56.9, TLI>0.99, adjBIC = 18,475, BIC = 18,541, and RMSEA = 0.0 (CI95: 0.0–0.025).

**Table 1.**Standard ACE simplex model results of full scale IQ (FISQ) and performance IQ (perfIQ): Correlations among A, E, and C, standard deviations, variances and standardized variance. Note in this model A, C, and E are uncorrelated.

FSIQ | perfIQ | |||||||
---|---|---|---|---|---|---|---|---|

Age in years: | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 |

A | A | |||||||

Correlations | 1.000 | 1.000 | ||||||

0.801 | 1.000 | 1.000 | 1.000 | |||||

0.801 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |||

1.000 | 1.000 | 1.000 | 0.959 | 0.959 | 0.959 | 1.000 | ||

Standard deviation | 7.69 | 10.21 | 11.84 | 10.52 | 7.58 | 11.67 | 12.38 | 12.37 |

Variance | 59.1 | 104.3 | 140.1 | 110.8 | 57.4 | 136.2 | 153.3 | 152.9 |

Standardized variance | 0.27 | 0.48 | 0.60 | 0.54 | 0.29 | 0.50 | 0.60 | 0.53 |

E | E | |||||||

Correlations | 1.000 | 1.000 | ||||||

0.112 | 1.000 | 0.077 | 1.000 | |||||

0.056 | 0.212 | 1.000 | 0.048 | 0.205 | 1.000 | |||

0.115 | 0.119 | 1.000 | 0.002 | 0.008 | 0.005 | 1.000 | ||

Standard deviation | 7.02 | 7.93 | 6.80 | 6.78 | 8.35 | 9.41 | 8.28 | 9.18 |

Variance | 49.3 | 62.9 | 46.2 | 46.0 | 69.6 | 88.5 | 68.5 | 84.2 |

Standardized variance | 0.23 | 0.29 | 0.20 | 0.22 | 0.35 | 0.32 | 0.27 | 0.29 |

C | C | |||||||

Correlations | 1.000 | 1.000 | ||||||

0.756 | 1.000 | 0.760 | 1.000 | |||||

0.457 | 0.478 | 1.000 | 0.697 | 0.658 | 1.000 | |||

0.418 | 0.675 | 1.000 | 0.768 | 0.724 | 0.665 | 1.000 | ||

Standard deviation | 10.39 | 7.08 | 6.78 | 7.05 | 8.39 | 6.96 | 5.88 | 7.16 |

Variance | 107.8 | 50.1 | 46.0 | 49.8 | 70.5 | 48.5 | 34.6 | 51.3 |

Standardized variance | 0.50 | 0.23 | 0.20 | 0.24 | 0.36 | 0.18 | 0.14 | 0.18 |

#### 4.2. Alternative Model: AE* Simplex with Ph->E Transmission

^{2}(61) = 73.3; TLI>0.99; adjBIC = 21,463; BIC = 21,549; RMSEA = 0.027 (CI95: 0.0–0.047)). However, in this model, some variances were estimated at zero. We reduced the model in the following way. First, as in the standard model, the variance var(a), and the variances var(ζ

_{A3}) and var(ζ

_{A4}) were estimated at zero, so we fixed these to zero, (three parameters). In addition, the unshared environmental auto-regression coefficients (β

_{Et,t-1}) were near (one time) zero or (two times) negative (i.e., hard to interpret), and none were significant, so we fixed these to zero (three parameters). Finally, the correlations between ζ

_{E*1t}and ζ

_{E*2t}at t = 2, 3, 4 were not significant, so we fixed these to zero (three parameters). The goodness of fit indices of the resulting model are χ

^{2}(70) = 74.9, LI>0.99, adjBIC = 21,463, BIC = 21,494, and RMSEA = 0.016 (CI95: 0.0–0.039). The small increase in χ

^{2}(9) =1.6) underlines the non-significance of the dropped parameters. The genetic autoregressive coefficients equaled 0.761 (s.e. 0.276), 1.062 (s.e. 0.133), and 0.772 (s.e. 0.124). We constrained these to be equal merely to facilitate subsequent calculations (see below). This resulted in χ

^{2}(72) = 77.379, TLI>0.99, adjBIC = 21,432, BIC = 21,483, and RMSEA = 0.016 (CI95: 0–0.039), and a test for the equality constraints of χ

^{2}(2) = 2.48 (p = 0.29), which suggests that the equality constraint is tenable. Table 2 contains the parameter estimates of this model, Table 3 contains the standardized variance components, and Table 4 contains the covariance matrices of A and E*, based on the estimates in Table 2. Figure 3 shows the model as fitted. This final model gives rise to an AE* correlation equaling 0 (t = 1), 0.32 (t = 2), 0.56 (t = 3) and 0.66 (t = 4). As formulated, at time point 1, we assume GE-cov = 0. However, within the final model, the within twin member correlation between A

_{i1}and E*

_{i1}is identified. To evaluate whether our initial assumption is correct, we refitted the model to estimate cor(A

_{1},E*

_{1}). Based on the Wald test, this parameter showed not to be significant (cor(A

_{1},E*

_{1}) = −0.018, s.e. 0.032).

**Table 2.**FIML parameters and standard errors (s.e.) of the AE* model with Ph->E transmission as fitted to FSIQ and perfIQ. 0* indicates that the parameter value was fixed to zero.

Parameter | FSIQ ML Estimate (s.e.) | perfIQ ML Estimate (s.e.) |
---|---|---|

α_{1} = α_{2} = α_{3} | 0.385 (0.13) | 0.482 (0.147) |

γ_{1} = γ_{2} = γ_{3} | 0.070 (0.03) | 0.037 (0.028) |

var(ζ_{E*1})^{0.5} = var(E*_{1})^{0.5} | 10.18 (0.74) | 5.68 (1.301) |

var(ζ_{E*2})^{0.5} | 5.644 (1.39) | 5.46 (1.010) |

var(ζ_{E*3})^{0.5} | 4.197 (2.00) | 0* |

var(ζ_{E*4})^{0.5} | 3.857 (2.00) | 4.819 (0.974) |

cor(E*_{11} E*_{21}) | 0.804 (0.12) | 0.765 (0.249) |

var(ζ_{A1})^{0.5} = var(A_{1})^{0.5} | 7.964 (0.80) | 9.041 (0.813) |

var(ζ_{A2})^{0.5} | 4.185 (1.10) | 0* |

var(ζ_{A3})^{0.5} | 0* | 0* |

var(ζ_{A4})^{0.5} | 0* | 4.482 (1.471) |

β_{At,t-1 (t = 2,3,4)} | 0.895 (0.06) | 0.877 (0.095) |

var(e)^{0.5} | 5.350 (1.32) | 7.763 (0.340) |

var(c)^{0.5} | 4.277 (0.35) | 3.620 (0.620) |

**Table 3.**Decomposition of phenotypic variance into A, E*, AE*-cov and C components. The standardized components are given in parentheses. Note that here var(E*) includes both var(ζ

_{E}) and var(e) and C is limited to time specific effects (var(c)). Additive genetic time specific effects are absent.

Age | var(FSIQ) | var(A) | var(E*) | var(C) (Time Specific) | 2cov(AE*) | 2cov(AE*) + var(A) |
---|---|---|---|---|---|---|

5.5 years | 214.1 | 63.43 (0.30) | 132.40 (0.62) | 18.29 (0.09) | 0 (0) | 63.43 (0.30) |

6.8 years | 226.5 | 68.3 (0.30) | 92.29 (0.41) | 18.29 (0.08) | 47.6 (0.21) | 115.9 (0.51) |

7.9 years | 221.3 | 54.7 (0.25) | 78.63 (0.36) | 18.29 (0.08) | 69.6 (0.31) | 124.3 (0.56) |

12.2 years | 205.5 | 43.8 (0.21) | 75.39 (0.37) | 18.29 (0.09) | 67.8 (0.33) | 111.6 (0.54) |

Age | var(perfIQ) | var(A) | var(E*) | var(C) (Time Specific) | 2cov(AE*) | 2cov(AE*) + var(A) |

5.5 years | 187.4 | 81.7 (0.44) | 92.6 (0.49) | 13.1 (0.07) | 0 (0) | 81.7 (0.44) |

6.8 years | 266.8 | 62.8 (0.24) | 119.1 (0.45) | 13.1 (0.05) | 71.7 (0.27) | 134.5 (0.51) |

7.9 years | 256.9 | 48.3 (0.19) | 108.7 (0.42) | 13.1 (0.05) | 86.8 (0.34) | 135.1 (0.53) |

12.2 years | 281.1 | 57.3 (0.20) | 130.0 (0.46) | 13.1 (0.05) | 80.7 (0.29) | 138.0 (0.49) |

_{A2}), var(ζ

_{A3}) (two parameters), and the correlations between ζ

_{E*1t}and ζ

_{E*2t}at t = 2,3,4 fixed to zero. The model fitted well (χ

^{2}(67) = 59.7; TLI > 0.99; adjBIC = 18,477; BIC = 18,544;RMSEA = 0.0 (CI95: 0.0–0.029)). However, as in FSIQ, the unshared environmental auto-regression coefficients (β

_{Et,t-1}) were not significant, and var(ζ

_{E3}) was zero. We dropped these parameters to arrive at χ

^{2}(71) = 60.43, TLI > 0.99, adjBIC = 18,466, BIC = 18,520, and RMSEA = 0.0 (CI95: 0.0–0.024). The genetic autoregressive coefficients equaled 0.791 (s.e. 0.189), 0.787 (s.e. 0.194), and 0.977 (s.e. 0.178). As above, we constrained these to be equal to facilitate subsequent calculations (see below). This resulted in χ

^{2}(73) = 61.1, TLI>0.99, adjBIC = 18,460, BIC = 18,508, and RMSEA = 0.0 (CI95: 0–0.022), and a test for the equality constraints of χ

^{2}(2) = 0.75 (p = 0.69). The results are given in Table 2, Table 3, and Table 5. Figure 4 depicts the model as fitted. This final model gives rise to an AE* correlation equaling 0 (t = 1), 0.41 (t = 2), 0.60 (t = 3) and 0.47 (t = 4). We estimated cor(A

_{1}, E*

_{1}), and found this correlation to be zero (cor(A

_{1}, E*

_{1}) = −0.002, s.e. 0.078).

**Figure 3.**The AE* simplex model with Ph->E transmission as fitted to FSIQ. Time specific effects are not shown. Note that in FSIQ, time specific effects were found to be environmental (time specific genetic variance was zero).

**Figure 4.**The AE* simplex model with Ph->E transmission as fitted to perfIQ. Time specific effects are not shown. Note that in perfIQ, time specific effects were found to be environmental (time specific genetic variance was zero).

**Table 4.**MZ covariance matrix of A and E* in the AE* model of FSIQ with Ph->E transmission. These include the AE* covariance and AE* correlation within twins (bold) and between twins (bold italics) and the covariance and correlations among the twin environments (italics). Age is given in years.

Correlation | Covariance | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A Twin 1 | E* Twin 1 | A Twin 2 | E* Twin 2 | |||||||||||||

Age in years | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 |

A twin 1 | 1 | 31.71 | ||||||||||||||

0.86 | 1 | 28.38 | 34.16 | |||||||||||||

0.86 | 1.00 | 1 | 25.40 | 30.57 | 27.36 | |||||||||||

0.86 | 1.00 | 1.00 | 1 | 22.74 | 27.36 | 24.49 | 21.92 | |||||||||

E twin 1 | 0.00 | 0.00 | 0.00 | 0.00 | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 83.44 | ||||||

0.35 | 0.30 | 0.30 | 0.30 | 0.41 | 1 | 16.65 | 14.90 | 13.34 | 11.94 | 39.39 | 26.64 | |||||

0.50 | 0.53 | 0.53 | 0.53 | 0.20 | 0.43 | 1 | 23.18 | 25.34 | 22.68 | 20.30 | 18.37 | 22.12 | 23.56 | |||

0.53 | 0.59 | 0.59 | 0.59 | 0.09 | 0.30 | 0.47 | 1 | 24.73 | 28.53 | 25.53 | 22.85 | 8.50 | 17.71 | 23.74 | 24.14 | |

A twin 2 | 0.50 | 63.43 | ||||||||||||||

0.43 | 0.50 | 56.77 | 68.32 | |||||||||||||

0.43 | 0.50 | 0.50 | 50.81 | 61.15 | 54.73 | |||||||||||

0.43 | 0.50 | 0.50 | 0.50 | 45.47 | 54.73 | 48.98 | 43.84 | |||||||||

E twin 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 132.40 | ||||||

0.22 | 0.19 | 0.19 | 0.19 | 0.36 | 0.29 | 26.64 | 23.84 | 21.34 | 19.10 | 45.79 | 92.29 | |||||

0.33 | 0.35 | 0.35 | 0.35 | 0.18 | 0.26 | 0.30 | 35.26 | 38.92 | 34.83 | 31.17 | 20.39 | 36.60 | 78.63 | |||

0.36 | 0.40 | 0.40 | 0.40 | 0.09 | 0.21 | 0.31 | 0.32 | 36.54 | 42.44 | 37.98 | 33.99 | 9.14 | 24.79 | 35.90 | 75.39 |

**Table 5.**DZ covariance matrix of A and E* in the AE* model of perfIQ with Ph->E transmission. These include the AE* covariance and AE* correlation within twins (bold italics) and between twins (bold italics) and the covariance and correlations among the twin environments (italics), which mimics the presence of shared variance. Age is given in years.

Correlation | Covariance | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A Twin 1 | E* Twin 1 | A Twin 2 | E* Twin 2 | |||||||||||||

Age in years | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 |

A twin 1 | 1.00 | 40.87 | ||||||||||||||

1.00 | 1.00 | 35.84 | 31.43 | |||||||||||||

1.00 | 1.00 | 1.00 | 31.43 | 27.57 | 24.18 | |||||||||||

0.81 | 0.81 | 0.81 | 1.00 | 27.57 | 24.18 | 21.20 | 28.64 | |||||||||

E twin 1 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 24.74 | ||||||

0.41 | 0.41 | 0.41 | 0.33 | 0.16 | 1.00 | 22.72 | 19.93 | 17.48 | 15.33 | 13.12 | 19.40 | |||||

0.60 | 0.60 | 0.60 | 0.48 | 0.08 | 0.41 | 1.00 | 32.40 | 28.41 | 24.92 | 21.85 | 6.94 | 22.46 | 28.10 | |||

0.58 | 0.58 | 0.58 | 0.47 | 0.04 | 0.32 | 0.39 | 1.00 | 35.18 | 30.85 | 27.06 | 23.73 | 3.66 | 22.16 | 28.95 | 30.41 | |

A twin 2 | 0.50 | 81.74 | ||||||||||||||

0.50 | 0.50 | 71.69 | 62.87 | |||||||||||||

0.50 | 0.50 | 0.50 | 62.87 | 55.14 | 48.35 | |||||||||||

0.40 | 0.40 | 0.40 | 0.50 | 55.14 | 48.35 | 42.41 | 57.28 | |||||||||

E twin 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.27 | 0.00 | 0.00 | 0.00 | 0.00 | 92.61 | ||||||

0.23 | 0.23 | 0.23 | 0.19 | 0.12 | 0.16 | 40.91 | 35.88 | 31.47 | 27.60 | 16.50 | 119.1 | |||||

0.34 | 0.34 | 0.34 | 0.28 | 0.07 | 0.20 | 0.26 | 56.44 | 49.50 | 43.41 | 38.07 | 8.44 | 47.12 | 108.7 | |||

0.34 | 0.34 | 0.34 | 0.27 | 0.03 | 0.18 | 0.24 | 0.23 | 59.87 | 52.50 | 46.05 | 40.38 | 4.32 | 39.36 | 46.24 | 130.0 |

## 4. Can Ph->E Transmission and Changes in Environmental Means Account for the Flynn Effect?

_{E*it}) is assumed to be zero. To incorporate the influence of environmental means in the extended AE* model with Ph->E transmission, we include structured environmental means [38]. We can do this by assigning to the means of the environmental variables E*

_{1}and ζ

_{E*t}non-zero values. We retain zero mean genetic variables, but will return to this assumption in the discussion. Given y

_{it}* = A

_{it}+ E*

_{it}, the mean of y

_{t}* is m(y

_{t}*) = m(E*

_{t}). The means are modeled in the MZs and DZs as follows:

_{1t}) = β

_{Et,t-1}m(E*

_{1t-1}) + α

_{t}m(y*

_{1t-1}) + γ

_{t}m(y*

_{2t-1}) + m(ζ

_{E*1t})

_{2t}) = β

_{Et,t-1}m(E*

_{2t-1}) + γ

_{t}m(y*

_{1t-1}) + α

_{t}m(y*

_{2t-1}) + m(ζ

_{E*2t}),

_{it}) = m(E*

_{i1}) = m(ζ

_{E*i1})). However, due to the impact of IQ on the environment, as defined by via the terms α

_{t}m(y*

_{1t-1}) and γ

_{t}m(y*

_{1t-1}), the impact of IQ on the environment (Ph->E transmission) will enhance the environmental effect, via multiplication of the environmental mean. Using Equations (12) and (13), we can evaluate how a shift in environmental means leads to a shift in phenotypic means.

_{Et,t-1}= 0. Therefore, within the following simulation, the environmental means are defined to be

^{*}

_{1t}) = α

_{t}m(y

^{*}

_{1t-1}) + γ

_{t}m(y

^{*}

_{2t-1}) + m(ζ

_{E*1t})

^{*}

_{2t}) = γ

_{t}m(y

^{*}

_{1t-1}) + α

_{t}m(y

^{*}

_{2t-1}) + m(ζ

_{E*2t}).

_{E*1t}) = m(ζ

_{E*2t}) (t = 1…4), and m(ζ

_{E*it}) = m(ζ

_{E*it+1}) (t = 1...3).

_{αγ}= 1/4, w

_{αγ}= 2/4, w

_{αγ}= 3/4, or w

_{αγ}= 4/4 of their 1990s values, where we take the 1990s value to be those observed in the present analysis of FSIQ (α = 0.385 and γ = 0.07) and perfIQ (α = 0.482 and γ = 0.037). We chose values of the mean m(E*

_{1}) equal to 1/4, 2/4, or 3/4 of the average phenotypic standard deviation as observed in the analysis of FSIQ (14.7) and perfIQ (15.7), and choose the means m(ζ

_{E*t}) such that the phenotypic mean at age 12.2 years equals to the chosen m(E*

_{1}), given w

_{αγ}= 1/4. The values chosen serve only to obtain an indication of the effect on the phenotypic mean given a relatively large variation in the environmental means and w

_{αβ}.

_{αγ}is immaterial). We then set the mean of m(E*

_{1}) to equal 14.7/4 = 3.675, one fourth of the observed phenotypic standard deviation, and choose the mean m(ζ

_{E*t}) to equal 3.26, so that the phenotypic mean at t = 4 (age 12) equals 14.7/4. So, given α = 0.386/4 and β = 0.07/4, changing the environmental means from m(E*

_{1}) = 0 and m(ζ

_{E*t}) = 0 to m(E

^{*}

_{1}) = 14.7/4 and m(ζ

_{E*t}) = 3.675, results in an increase in phenotypic mean from 0 to 3.67 (1/4 of 14.7 std) at age 12.2 y. We then ask: Given the values of m(E*

_{1}) = 3.675 and m(ζ

_{E*t}) = 3.675, what is the effect of increasing the parameters α and γ (increasing w

_{αγ}from ¼ to 1). The answer to this question is contained in the subsequent rows, where we see that the phenotypic mean increases from 0.25 (as chosen) to 0.39 in standard deviation units (std = 14.7). Note that this increase is due to changes in Ph->E transmission only. So, the combination of increasing the environmental means (from zero to 3.67 and 3.26), and increasing the transmission parameters from 0.096 and 0.017 (w

_{αγ}= 1/4) to the observed values of 0.385 and 0.070 (w

_{αγ}= 1) increases the phenotypic means by 0.39 standard deviation units. The results in the rest of the table pertain to the greater increases in environmental means (from 1/4 to 2/4 and 3/4). For instance, the last four rows pertaining to the largest changes in environmental mean indicate that the increase in environmental means with increasing Ph->E transmission parameters results in a maximal increase in the phenotypic means of 1.18 standard deviation units. This demonstrates that changes in the environmental means, reflecting improvements in the environment relevant to FSIQ are boosted by a factor of about 1.57 by the increase in the Ph->E transmission parameters (w

_{αγ}= 1/4 to 1). Table 7 contains similar results pertaining to perfIQ, where the effects are given in average standard deviation units of perfIQ (std = 15.7). Here, the increase in w

_{αγ}from ¼–1 boosts the environmental mean effect by a factor 1.7. These results demonstrate the possible role of Ph->E transmission (GE-cov) in the Flynn effect.

**Table 6.**Raw phenotypic mean changes (std units) of FSIQ as a function of weight given to the Ph->E transmission parameters (w

_{αγ}= 0.25, 0.50, 0.75) and different environmental means. Environmental means at t = 1 is defined as a function of standard deviation of FSIQ (sd

_{FSIQ}= 14.7, Small = sd

_{FSIQ}* 0.25, Medium = sd

_{FSIQ}* 0.50 and Large = sd

_{FSIQ}* 0.75). Environmental means at t + 1 are calculated according to Equations (14) and (15).

Weight Ph- > E Parameters | Increase Environment | ||
---|---|---|---|

Small | Medium | Large | |

0.25 | 3.68 (0.25) | 7.36 (0.50) | 11.04 (0.75) |

0.50 | 4.22 (0.29) | 8.43 (0.57) | 12.65 (0.86) |

0.75 | 4.90 (0.33) | 9.80 (0.67) | 14.70 (1.00) |

1 | 5.77 (0.39) | 11.53 (0.78) | 17.30 (1.18) |

**Table 7.**Raw phenotypic mean changes (std units) as a function of weight given to the Ph->E transmission parameters (w

_{αγ}= 0.25, 0.50, 0.75) and different environmental means. Environmental means at t = 1 is defined as a function of standard deviation of perf IQ (sd

_{perfIQ}= 15.71, Small = sd

_{perfIQ}* 0.25, Medium = sd

_{perfIQ}* 0.50 and Large = sd

_{perfIQ}* 0.75). Environmental means at t + 1 are calculated according to Equations (14) and (15).

Weight Ph->E Parameters | Increase Environment | ||
---|---|---|---|

Small | Medium | Large | |

0.25 | 3.93 (0.25) | 7.85 (0.50) | 11.78 (0.75) |

0.50 | 4.60 (0.29) | 9.21 (0.59) | 13.81 (0.88) |

0.75 | 5.49 (0.35) | 11.00 (0.70) | 16.49 (1.05) |

1 | 6.66 (0.42) | 13.32 (0.85) | 19.98 (1.27) |

## 5. Discussion

_{αγ}= 1/4 to w

_{αγ}= 1). We are satisfied, based on our results, that GE-cov can contribute to the Flynn effect, but we cannot be sure about the ultimate effect size (our numerical values are arbitrary). In our numerical exploration, we have assumed that measurement invariance (MI) holds with respect to cohort. However, Wicherts et al. demonstrated that common IQ tests do not display measurement invariance [43]. This poses a psychometric problem, if one wishes to interpret cohort or generation related mean differences in terms of latent variables. However, the lack of MI has no bearing on the possible role of GE-cov in the Flynn effect: Increases in latent environmental (and genetic means) in combination with changes in Ph->E transmission parameters may contribute to the Flynn effect, even if the lack of MI complicates the interpretation of increases, over time, in phenotypic means.

## Acknowledgments

## Author Contributions

## Appendix I. FIML Estimates of Correlations, Standard Deviations, Means

FSIQ MZ | ||||||||
---|---|---|---|---|---|---|---|---|

Twin 1 | Twin 2 | |||||||

Age in years | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 |

Correlations | 1 | |||||||

0.661 | 1 | |||||||

0.486 | 0.725 | 1 | ||||||

0.419 | 0.603 | 0.781 | 1 | |||||

0.770 | 0.669 | 0.584 | 0.552 | 1 | ||||

0.450 | 0.674 | 0.752 | 0.736 | 0.521 | 1 | |||

0.453 | 0.641 | 0.840 | 0.753 | 0.534 | 0.797 | 1 | ||

0.453 | 0.577 | 0.759 | 0.802 | 0.612 | 0.665 | 0.760 | 1 | |

Standard deviations | 14.13 | 15.03 | 17.06 | 15.35 | 14.56 | 15.59 | 17.37 | 15.05 |

Means | 109.76 | 104.60 | 104.06 | 98.24 | 108.92 | 101.63 | 104.68 | 99.57 |

FSIQ DZ | ||||||||

Correlations | 1 | |||||||

0.603 | 1 | |||||||

0.477 | 0.662 | 1 | ||||||

0.471 | 0.673 | 0.737 | 1 | |||||

0.641 | 0.299 | 0.284 | 0.259 | 1 | ||||

0.421 | 0.482 | 0.366 | 0.333 | 0.505 | 1 | |||

0.225 | 0.322 | 0.481 | 0.357 | 0.383 | 0.623 | 1 | ||

0.246 | 0.396 | 0.470 | 0.500 | 0.345 | 0.626 | 0.706 | 1 | |

Standard deviations | 14.37 | 14.66 | 13.92 | 14.03 | 15.54 | 14.27 | 14.34 | 13.69 |

Means | 108.69 | 104.12 | 105.56 | 100.21 | 108.65 | 104.54 | 105.89 | 99.45 |

perfIQ MZ | ||||||||
---|---|---|---|---|---|---|---|---|

Twin 1 | Twin 2 | |||||||

Age in years | 5.5 | 6.8 | 9.7 | 12.2 | 5.5 | 6.8 | 9.7 | 12.2 |

Correlations | 1 | |||||||

0.723 | 1 | |||||||

0.667 | 0.726 | 1 | ||||||

0.674 | 0.635 | 0.696 | 1 | |||||

0.697 | 0.643 | 0.625 | 0.603 | 1 | ||||

0.628 | 0.722 | 0.733 | 0.654 | 0.655 | 1 | |||

0.611 | 0.653 | 0.754 | 0.670 | 0.573 | 0.762 | 1 | ||

0.722 | 0.649 | 0.709 | 0.741 | 0.640 | 0.690 | 0.669 | 1 | |

Standard deviations | 14.35 | 16.92 | 17.83 | 16.72 | 15.68 | 18.34 | 16.51 | 18.65 |

Means | 99.52 | 102.72 | 103.60 | 98.52 | 97.88 | 102.21 | 104.74 | 99.11 |

perfIQ DZ | ||||||||

Correlations | 1 | |||||||

0.573 | 1 | |||||||

0.616 | 0.661 | 1 | ||||||

0.571 | 0.685 | 0.618 | 1 | |||||

0.491 | 0.261 | 0.239 | 0.344 | 1 | ||||

0.435 | 0.386 | 0.374 | 0.320 | 0.492 | 1 | |||

0.416 | 0.305 | 0.430 | 0.334 | 0.517 | 0.708 | 1 | ||

0.353 | 0.376 | 0.325 | 0.396 | 0.439 | 0.535 | 0.601 | 1 | |

Standard deviations | 13.40 | 15.97 | 14.66 | 16.80 | 13.55 | 15.92 | 15.86 | 16.07 |

Means | 99.36 | 104.88 | 105.26 | 101.03 | 99.45 | 104.89 | 105.10 | 100.58 |

## Conflicts of Interest

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De Kort, J.M.; Dolan, C.V.; Kan, K.-J.; Van Beijsterveldt, C.E.M.; Bartels, M.; Boomsma, D.I.
Can GE-Covariance Originating in Phenotype to Environment Transmission Account for the Flynn Effect? *J. Intell.* **2014**, *2*, 82-105.
https://doi.org/10.3390/jintelligence2030082

**AMA Style**

De Kort JM, Dolan CV, Kan K-J, Van Beijsterveldt CEM, Bartels M, Boomsma DI.
Can GE-Covariance Originating in Phenotype to Environment Transmission Account for the Flynn Effect? *Journal of Intelligence*. 2014; 2(3):82-105.
https://doi.org/10.3390/jintelligence2030082

**Chicago/Turabian Style**

De Kort, Janneke M., Conor V. Dolan, Kees-Jan Kan, Catharina E. M. Van Beijsterveldt, Meike Bartels, and Dorret I. Boomsma.
2014. "Can GE-Covariance Originating in Phenotype to Environment Transmission Account for the Flynn Effect?" *Journal of Intelligence* 2, no. 3: 82-105.
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