Introduction 2005 2002 2001 2001 2000 2005 2004 2001 1998 2004 1989 2001 1996 1989 2007 2002 2003 2002 2007 The aim of this twin study was to model birth weight across gestational ages and to quantify the genetic and environmental components in order to explain the effects of covariates on the variance of birth weight. We hypothesize that heritability changes during gestation and that covariates mainly explain the common environmental and residual factors of the variance. By entering these covariates in the means model and in the covariance matrix, heritability will increase and, as a consequence, the power to find genes by linkage and association studies will increase, although the effects of each covariate on the genetic and environmental component may be different. Materials and methods Subjects 1998 2006 2007 2005 1990 1986 2001 2007 Statistical analysis 2 2 Means model \documentclass[12pt]{minimal}
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\begin{document}$$ \mu = \beta _{1} /1 + {\text{e}}^{{\beta _{2} - \beta _{3} {\text{Time}}}}$$\end{document} 1 3 2000 \documentclass[12pt]{minimal}
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\begin{document}$$\mu=\beta _{1}/1 +\hbox{e}^{\beta_{2}-\beta_{3}\rm{Time}+\beta_{3}\beta_{4}(\rm{Time} > \beta_{5})(\rm{Time}-\beta_5)}$$\end{document} β 4 β 5 2007 \documentclass[12pt]{minimal}
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\begin{document}$$ \mu = \beta _{1} /1 + {\text{e}}^{{ \beta _{2} - \beta _{3} {\text{Time}} + \beta _{3}\beta _{4} ({\text{Time}} > \beta _{5} )({\text{Time}} - \beta_{5} ) - {\varvec{\beta}} {\bf X} }} $$\end{document}  X β \documentclass[12pt]{minimal}
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\begin{document}$$ \mu = \beta _{1} /1 + {\text{e}}^{(\beta_{2}-\beta _{3})\text{Time}+\beta_{3}\beta_{4}({\text{Time}} > \beta _{5})({\text{Time}}-\beta _{5})+\beta_{3}\beta_{6}({\text{Time}} > \beta_{7})({\text{Time}}-\beta_{7})\bf{Z}- {\varvec{\beta}} \bf{X} }$$\end{document}  Z 6 7  Z 2007 Covariance matrix 2002 \documentclass[12pt]{minimal}
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\begin{document}$$ \Upsigma ^{2}_{{{\text{MZ}}}} = {\left( {\begin{array}{*{20}c} {{A + C + E}} & {{A + C}} \\ {{A + C}} & {{A + C + E}} \\ \end{array} } \right)} $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ \Upsigma ^{2}_{{{\text{DZ}}}} = {\left( {\begin{array}{*{20}c} {{A + C + E}} & {{A/2 + C}} \\ {{A/2 + C}} & {{A + C + E}} \\ \end{array} } \right)} $$\end{document} 2007 α1 α2 δ1 δ2 E 1 2 E 1 2 3 3 2002 For example, let us assume a model in which genetic, common and unique environmental components are defined as  7.403 1 6.688 1 E \documentclass[12pt]{minimal}
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\begin{document}$$ \frac{{{\text{e}}^{{7.043}} \times 25}} {{{\text{e}}^{{7.043}} \times 25 + {\text{e}}^{{6.688}} \times 25 + {\text{e}}^{{5.4054 + 0.1435 \times 25 - 0.163 \times 1}} }} = 0.515, $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ \frac{{{\text{e}}^{{7.043}} \times 40}} {{{\text{e}}^{{7.043}} \times 40 + {\text{e}}^{{6.688}} \times 40 + {\text{e}}^{{5.4054 + 0.1435 \times 40 - 0.163 \times 1}} }} = 0.335 $$\end{document} Model selection 1973 1998 Effect of each separate covariate on the genetic and environmental component To explain the effect of each covariate on the (co-)variance of birth weight at different gestational ages, we removed a significant covariate completely from the full model (out of the mean regression and out of the covariance matrix) and compared the (co-)variances, so that the influence of that specific covariate became apparent. 1996 2001 Results Heritability decreases during gestation 1 Table 1 Model fit statistics of ACE and ADE models for the unadjusted model with only gestational age in the mean regression, the adjusted model with covariates in the mean regression, and the full model with covariates in the mean regression and in the covariance matrix. The best fitting model (lowest AIC) is presented in bold  Unadjusted model  Adjusted model −Log likelihood df AIC −Log likelihood Df AIC ACE 61696.84 8457 61703.84 61350.49 8441 61373.49 ADE 61727.45 8457 61734.45 61380.6 8441 61403.6 CE 61711.88 8458 61717.88 61364.28 8442 61386.28 AE 61727.45 8458 61733.45 61380.58 8442 61402.58 E 62189.72 8459 62194.72 61826.93 8443 61847.93 ACE; A and C allowed to change over time 61688.44 8457 61695.44 61343.61 8441 61366.61     Full model   ACE; A and C allowed to change over time and covariates in covariance matrix    61297.26 8437 61324.26 A: Additive genetic effects, D: Dominance genetic effects, C: Effects of common environment, E: Effects of unique environment 1 1 1 2 Fig. 1 Genetic (A), common environmental (C) and unique environmental (E) variance and standardized variance of birth weight of the unadjusted (unadj) model and the adjusted model (adj). E-variance was allowed to change over time, but the A- and C-variance not  Table 2 Regression coefficients of the covariance matrix of the unadjusted model, the adjusted model, and the full model  A  C  Log(E)  Log(estimate) SE Log(estimate) SE estimate SE b Intercept 10.3700  0.1735 10.5800  0.1154 6.0055 0.3462 Gestational age –  –  0.1405 0.0094 a Intercept 0.0000 – 0.0000 – 6.3061 0.3468 Gestational age 6.8930 0.1604 6.942 0.1250 0.1317 0.0094 b Intercept 10.2800 0.1857 10.4800 0.1222 6.1436 0.3495 Gestational age –  –  0.1347 0.0094 a Intercept 0.0000 – 0.0000 – 6.4552 0.3489 Gestational age 6.7830 0.1737 6.8600 0.1297 0.1256 0.0094 Full model Intercept 0.0000 – 0.0000 – 5.4054 0.3963 Gestational age 7.0430 0.1421 6.6880 0.1571 0.1435 0.0103 Second born     0.4324 0.0642 Neonatal death     0.7240 0.1981 Multiparity     −0.1630 0.0491 One placental mass     0.2441 0.0636 Unadjusted:model with only gestational age in the mean regression Adjusted:model with covariates in the mean regression Full model:model with covariates in the mean regression and in the covariance matrix a b \documentclass[12pt]{minimal}
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\begin{document}$$ A = {\text{e}}^{{\alpha _{1} }} + {\text{e}}^{{\alpha _{2} }} {\text{Time}}, $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ C = {\text{e}}^{{\delta _{1} }} + {\text{e}}^{{\delta _{2} }} {\text{Time}}, $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ {\text{ln}}(E) = \Upsigma _{1} + \Upsigma _{2} {\text{Time}} + {\varvec{\beta}} \bf{X} , $$\end{document}  X β Effect of covariates in the covariance matrix on heritability 1 3 3 2 2 3 2 2 Table 3 Regression coefficients of the full model with covariates in the mean regression and in the covariance matrix  Estimate SE Means model Coefficients of the logistic growth curve b1 = saturation level at which growth stops 3688.0980 84.9100 b2 = growth rate 5.8061 0.1762 b3 = location parameter −0.1842 0.0068 b4 = decrease in weight after inflection point 0.9563 0.3017 a 37.5934 0.3216 1 b 0.1054 0.0021 1 a 30.7221 0.8399 2 b 0.5733 0.1846 2 a 29.0000 1.098 3 b 1.1438 0.5059 3 a 37.7869 0.4227 Covariates Multiparity 0.1165 0.0148 Male 0.1405 0.0118 Birth year (form 0 [1964] to 38 [2002]) 0.0018 0.0007 One placental mass −0.0525 0.0127 Peripheral umbilical cord insertion −0.0593 0.0227 Maternal age 0.0046 0.0014 Interaction terms a −0.0027 0.0009 a  0.0317 0.0131 a b \documentclass[12pt]{minimal}
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\begin{document}$$ \mu = \beta _{1} /1+ {\text{e}}^{{\beta _{2} - \beta _{3} {\text{Time}}+ \beta _{3} \beta _{4} ({\text{Time}} > \beta _{5} )({\text{Time}}- \beta _{5} )+ \beta _{3} \beta _{6} ({\text{Time}} > \beta _{7} )({\text{Time}}- \beta _{7} ) \bf{Z} - {\varvec{\beta}} \bf{X} }} $$\end{document} Z X β Fig. 2 T1  2 3 4 3 4 3 4 Fig. 3 Genetic (A), environmental (C) and residual (E) variance of birth weight of the adjusted (Adj) model and the full model (Full). Legend: Lowest: E-variance for a first-born twin of a multipara with two separate placentas who is staying alive. Primipara: E-variance for a first-born twin of a primipara with two separate placentas who is staying alive. One placenta: E-variance for a first-born twin of a multipara with one placental mass who is staying alive. Second born: E-variance for a second born twin of a multipara with two separate placentas who is staying alive. Neonatal death (ND): E-variance a first-born twin of a multipara with two separate placentas who died neonatally. Highest: E-variance for a second born twin of a primipara with one placental mass who died neonatally  Fig. 4 3  Effect of each separate covariate on the genetic and environmental component Comparing the variances of the full model with and without a specific covariate showed that maternal factors (age and parity) had minor influence on the genetic variance, but no influence on heritability. Removing sex, birth order and neonatal survival gave an increase in genetic variance. However, only for sex, heritability increased if removed from the model. Removing the placental factors from the full means model resulted in a decrease of the genetic variance and a decrease of heritability. Insertion of the umbilical cord (unique for each twin) explained mainly the unique environmental variance, whereas number of placentas also explained common environment. Chorionicity or other placental factors 5 5 Fig. 5 Effect of either placental factors (site of insertion of umbilical cord and number of placentas), or chorionicity on the variance and heritability of birth weight The curves represent the differences between the full model and the model without the specific covariate. The presented curves are the curves for the first born twin of a primipara, who is staying alive  Discussion The aim of this study was to model birth weight at different gestational ages and quantify the genetic and environmental components. To our knowledge we are the first to show that heritability of birth weight seems to decrease during gestation, in our case from 38% at 25 weeks to 15% at 42 weeks. Entering the covariates of the mean regression as moderators in the covariance matrix, explained more of the common environmental variance than of the genetic variance. However, the unique environmental variance depended upon the representative state of the covariates and was lowest for first-born twins of multipara with two separate placentas, who stayed alive neonatally. Therefore these twins have the highest heritability: from 52% at 25 weeks to 30% at 42 weeks. 2001 2005 1993 2006 2006 2005 1996 2007 2001 1996 1996 2006 2007 1999 2004 2005 α β n 2006 2006 2002 2006 ,  2002 2006 GE 2002 2002 2006 Covariates also contain useful information in order to identify genes, which may be associated with the trait. In this study, only sex contained latent genetic information. For all other covariates used in this study, either heritability did not change, or environmental factors were explained with heritability depending on the representative state of the covariate. Since sex explained part of heritability, entering sex in the means model will mask genes that act differently according to sex and it is better to have these genes in the model instead of sex. 1989 2007 2006 2000 1984 1983 2005 1995 1996 2001 2004 2004 In conclusion, modeling genetic and environmental factors gives a better insight in factors influencing growth during gestation, and therefore birth weight. Heritability of birth weight changed during gestation with a decrease from 25 to 42 weeks with highest heritability for twins with the least environmental restriction (first-born twins of multipara with two separate placentas who stayed alive neonatally). For this complex trait, modeling the covariance matrix with covariates, mainly environmental factors were explained, resulting in an increase of the heritability and subsequently the chance of finding genes by linkage and association studies.