A New Look at Genetic and Environmental Architecture on Lipids Using Non-Normal Structural Equation Modeling in Male Twins: The NHLBI Twin Study

Abstract

This study examined genetic and environmental influences on the lipid concentrations of 1028 male twins using the novel univariate non-normal structural equation modeling (nnSEM) ADCE and ACE models. In the best fitting nnSEM ADCE model that was also better than the nnSEM ACE model, additive genetic factors (A) explained 4%, dominant genetic factors (D) explained 17%, and common (C) and unique (E) environmental factors explained 47% and 33% of the total variance of high-density lipoprotein cholesterol (HDL-C). The percentage of variation explained for other lipids was 0% (A), 30% (D), 34% (C) and 37% (E) for low-density lipoprotein cholesterol (LDL-C); 30, 0, 31 and 39% for total cholesterol; and 0, 31, 12 and 57% for triglycerides. It was concluded that additive and dominant genetic factors simultaneously affected HDL-C concentrations but not other lipids. Common and unique environmental factors influenced concentrations of all lipids.

Appendices

Appendix A

In the classical twin study using the SEM, only (first- and) second-order moments, namely covariance matrix of both MZ pairs and DZ pairs, are used. However, in the ADCE model using the nnSEM, in addition to (first- and) second-order moments, third-order moments are used. When there are two phenotypes p1 and p2 (for twin 1 and twin 2, respectively), sample third-order moments are as follows:

$$\begin{aligned} & {{S}_{p_{1}^{3}}}=\frac{1}{n}{{\sum\limits_{i=1}^{n}{\left( \text{pi1}-\overline{\text{p1}} \right)}}^{3}} \\ & {{S}_{p_{1}^{2}p2}}=\frac{1}{n}{{\sum\limits_{i=1}^{n}{\left( \text{pi1}-\overline{\text{p1}} \right)}}^{2}}\left( \text{pi2}-\overline{\text{p2}} \right) \\ & {{S}_{p1p_{2}^{2}}}\frac{1}{n}\sum\limits_{i=1}^{n}{\left( \text{pi1}-\overline{\text{p1}} \right)}{{\left( \text{pi2}-\overline{\text{p2}} \right)}^{2}} \\ & {{S}_{p_{2}^{3}}}=\frac{1}{n}{{\sum\limits_{i=1}^{n}{\left( \text{pi2}-\overline{\text{p2}} \right)}}^{3}} \\ \end{aligned}$$

where i indicates an observation (in this case, an observation pair). These sample statistics become information in the estimation of a, c, d, and e effects.

When three latent variables C, D, and E are assumed to follow non-normal distributions, the expected third-order moments for the MZ twins are as follows:

$$\begin{aligned} & E\left[ p_{1}^{3}(mz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}}+{{{e}}^{3}}{{\sigma }_{{{{E}}^{3}}}} \\ & E\left[{p}_{1}^{2}{p}2(mz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}} \\ & E\left[ {p}_{1}{p}_{2}^{2}(mz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}} \\ & E\left[{p}_{2}^{3}(mz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}}+{{{e}}^{3}}{{\sigma }_{{{{E}}^{3}}}} \\ \end{aligned}$$

In addition, the expected third-order moments for the DZ twins are as follows:

$$\begin{aligned} & E\left[ p_{1}^{3}(dz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}}+{{{e}}^{3}}{{\sigma }_{{{{E}}^{3}}}} \\ & E\left[{p}_{1}^{2}{p}_{2}(dz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D1}}^{2}{{{D2}}}}}} \\ & E\left[ {p}_{1}{p}_{2}^{2}(dz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D1}}{D2}^{2}}}} \\ & E\left[{p}_{2}^{3}(dz) \right]={{{c}}^{3}}{{\sigma }_{{{{C}}^{3}}}}+{{{d}}^{3}}{{\sigma }_{{{{D}}^{3}}}}+{{{e}}^{3}}{{\sigma }_{{{{E}}^{3}}}} \\ \end{aligned}$$

Note that A is assumed to follow a normal distribution when many loci affect the phenotype of interest. Here, ${{\sigma }_{{{\text{c}}^{3}}}}\text{, }{{\sigma }_{{{\text{d}}^{3}}}}$ and ${{\sigma }_{{{\text{E}}^{3}}}}$ are the skewness of C, D, and E, respectively. (In this case, the variances of these independent factors are fixed to 1 s. Therefore, these parameters are the skewness of the factors.) ${{\sigma }_{D{{1}^{2}}D{{2}^{2}}}}$ expresses the expected value of the second power of D of twin 1 times D of twin 2, and ${{\sigma }_{D1D{{2}^{2}}}}$ expresses the expected value of D of twin 1 times the second power of D of twin 2, respectively. ${{\sigma }_{D{{1}^{2}}D2}}$ and ${{\sigma }_{D1D{{2}^{2}}}}$ are assumed to be equal, because the order of twins is arbitrary. Therefore, ADCE model using the nnSEM estimates A, C, D, and E influences by minimizing the discrepancy function, between the sample statistics (second- and third-order moments) and expected values (second- and third-order moments) using asymptotically distribution-free (ADF) method. For more details please see Ozaki et al. (2011) (Ozaki et al. 2011).

Appendix B

By dropping some parameters in the ADCE model using the nnSEM, nested reduced ACE, CE, and E models using 2nd and 3rd order moments can be identified. For example, four nnSEM ACE models can be identified by dropping D parameter in the ADCE model and by assuming: (1) C and E are non-normally distributed and different in the skewness, (2) C and E are non-normally distributed but have the same skewness, (3) only C is non-normally distributed, and (4) only E is non-normally distributed. In all of the four models A is normal. In this way, we can identify four ACE models, four CE models (1) C and E are non-normally distributed and different in the skewness, (2) C and E are non-normally distributed and have the same skewness, (3) only C is non-normally distributed, (4) only E is non-normally distributed, and one E model (E is non-normally distributed).

Because all of the ten models use the same sample statistics and are analyzed using the same estimation method, we can compare among the nnSEM ADCE and its nested reduced models using some fit indices.