Individual Differences in Puberty Onset in Girls: Bayesian Estimation of Heritabilities and Genetic Correlations

 

We report heritabilities for individual differences in female pubertal development at the age of 12. Tanner data on breast and pubic hair development in girls and data on menarche were obtained from a total of 184 pairs of monozygotic and dizygotic twins. Genetic correlations were estimated to determine to what extent the same genes are involved in different aspects of physical development in puberty. A Bayesian estimation approach was taken, using Markov-chain Monte Carlo simulation to estimate model parameters. All three phenotypes were to a significant extent heritable and showed high genetic correlations, suggesting that a common set of genes is involved in the timing of puberty in general. However, gonadarche (menarche and breast development) and adrenarche (pubic hair) are affected by different environmental factors, which does not support the three phenotypes to be regarded as indicators of a unitary physiological factor.

APPENDIX

Estimating Heritabilities and Genetic and Environmental Correlations using MCMC

The phenotypes were measured using ordinal scales. In order to estimate heritabilities and genetic and environmental correlations, we assumed three underlying latent normally distributed variables using the so-called threshold model (Crittenden, 1961; Falconer, 1965; Lynch and Walsh, 1998). Latent traits were decomposed into additive genetic (A), shared environmental (C) and nonshared environmental (F) parts. In order to implement the model in WinBUGS, it is most efficient to parametrise the model in such a way that the nonshared environmental vector F is further decomposed into vectors U and E, where U incorporates nonshared environmental correlations between the traits and E is a vector with uncorrelated coordinates:

$$ \left(\begin{array}{c}{X_{11}} \\ {X_{12}} \\ {X_{21}} \\ {X_{22}} \\ {X_{31}} \\ {X_{32}}\end{array}\right)=\left(\begin{array}{c}{A_{11}} \\ {A_{12}} \\ {A_{21}} \\ {A_{22}} \\ {A_{31}} \\ {A_{32}}\end{array}\right)+\left(\begin{array}{c} {C_{1}} \\ {C_{1}} \\ {C_{2}} \\ {C_{2}} \\ {C_{3}} \\ {C_{3}} \end{array}\right)+\left(\begin{array}{c}{U_{11}} \\ {U_{12}} \\ {U_{21}} \\ {U_{22}} \\ {U_{31}} \\ {U_{32}} \end{array}\right)+\left(\begin{array}{c}{E_{11}} \\ {E_{12}} \\ {E_{21}} \\ {E_{22}} \\ {E_{31}} \\ {E_{32}}\end{array}\right).$$

Here, the first index i denotes the phenotype and the second j the individual in a twin pair. More precisely, we modelled the nonshared environmental vector as F=U + E with U and E independent and marginally distributed as follows:

$$ \eqalign{ {\left(\begin{array}{c}{U_{11}}\\ {U_{12}}\\ {U_{21}}\\ {U_{22}}\\ {U_{31}}\\ {U_{32}} \end{array}\right)}&\sim N_{6}{\left({{\left(\begin{array}{c}{0} \\ {0} \\ {0} \\ {0} \\ {0} \\ {0} \end{array}\right)},{\left(\begin{array}{cccccc} {\chi^{2}}&{0}&{\chi\rho_{7}\delta}&{0}&{{\chi \rho_{8}\theta}}&{0}\\ {0}&{\chi^{2}}&{0}&{\chi\rho_{7}\delta}&{0}&{\chi\rho_{8}\theta} \\ {\chi\rho_{7}\delta}&{0}&{\delta^{2}}&{0}&{\delta\rho_{9}\theta}&{0} \\ {0}&{\chi\rho_{7}\delta}&{0}&{\delta^{2}}&{0}&{\delta\rho_{9}\theta} \\{\chi\rho_{8}\theta}&{0}&{\delta\rho_{9}\theta}&{0}&{\theta^{2}}&{0} \\ {0}&{\chi\rho_{8}\theta}&{0}&{\delta\rho_{9}\theta}&{0}&{\theta^{2}} \end{array}\right)}}\right)},\hbox{and}\cr {\left(\begin{array}{c}{E_{11}}\\ {E_{12}}\\ {E_{21}}\\ {E_{22}}\\ {E_{31}}\\ {E_{31}}\end{array}\right)}&\sim N_{6}{\left({{\left(\begin{array}{c}{0} \\ {0} \\ {0} \\ {0} \\ {0} \\ {0}\end{array}\right)},{\left(\begin{array}{cccccc}{\kappa^{2}}&{0}&{0}&{0}&{0}&{0} \\ {0}&{\kappa^{2}}&{0}&{0}&{0}&{0} \\ {0}&{0}&{\gamma^{2}}&{0}&{0}&{0} \\ {0}&{0}&{0}&{\gamma^{2}}&{0}&{0} \\ {0}&{0}&{0}&{0}&{\tau^{2}}&{0} \\ {0}&{0}&{0}&{0}&{0}&{\tau^{2}}\end{array}\right)}}\right)}.} $$

The parameters in this specification are not individually identifiable from the data, but can be used to specify a prior distribution on the model used. The Bayesian approach yields a posterior distribution for all parameters, but we only report the part of the posterior distribution that concerns identifiable parameters, as the remaining part of the posterior is a result of prior specification only, without intervention of the data. For instance, the unshared environmental coefficient of correlation between the latent variables for the first and second phenotypes is given by

$$ \rho={{\chi\rho_{7}\delta}\over{{\sqrt{\chi^{2}+\kappa^{2}}} {\sqrt{\delta^{2}+\gamma^{2}}}}}. $$

Thus, although the parameters χ,ρ₇,δ,γ, and κ are individually not identifiable given the data, a function of them (ρ) is. Using the MCMC method it is straightforward to calculate the posterior distribution for this function from the sampled values of the full parameter set.

The vectors for the additive genetic effects (A) and shared environmental effects (C) were modelled in the usual way:

$$ \left(\begin{array}{c}A_{11}\\ A_{12}\\ A_{21}\\ A_{22}\\ A_{31}\\ A_{32}\end{array}\right)\sim N_{6}\left(\left(\begin{array}{c}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right),\left(\begin{array}{cccccc} \nu^{2}&\nu^{2}&\nu\rho_{1}\alpha&\nu\rho_{1}\alpha&\nu\rho_{2}\pi&\nu\rho_{2}\pi \\ \nu^{2}&\nu^{2}&\nu\rho_{1}\alpha&\nu\rho_{1}\alpha&\nu\rho_{2}\pi&\nu\rho_{2}\pi\\ \nu\rho_{1}\alpha&\nu\rho_{1}\alpha&\alpha^{2}&\alpha^{2}&\alpha\rho_{3}\pi&\alpha\rho_{3}\pi\\ \nu\rho_{1}\alpha&\nu\rho_{1}\alpha&\alpha^{2}&\alpha^{2}&\alpha\rho_{3}\pi&\alpha\rho_{3}\pi\\ \nu\rho_{2}\pi&\nu\rho_{2}\pi&\alpha\rho_{3}\pi&\alpha\rho_{3}\pi&\pi^{2}&\pi^{2} \\ \nu\rho_{2}\pi&\nu\rho_{2}\pi&\alpha\rho_{3}\pi&\alpha\rho_{3}\pi&\pi^{2}&\pi^{2} \end{array}\right)\right) $$

for MZ twin pairs and

$$ \left(\begin{array}{c} A_{11}\\ A_{12}\\ A_{21}\\ A_{22}\\ A_{31}\\ A_{32}\end{array}\right)\sim N_{6}\left(\left(\begin{array}{c}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{cccccc} \nu^{2}& 0.5\nu^{2}&\nu\rho_{1}\alpha&0.5\nu\rho_{1}\alpha&\nu\rho_{2}\pi& 0.5\nu\rho_{2}\pi \\ 0.5\nu^{2}&\nu^{2}&0.5\nu\rho_{1}\alpha&\nu\rho_{1}\alpha&0.5\nu\rho_{2}\pi&\nu\rho_{2}\pi \\ \nu\rho_{1}\alpha&0.5\nu\rho_{1}\alpha&\alpha^{2}& 0.5\alpha^{2}&\alpha\rho_{3}\pi&0.5\alpha\rho_{3}\pi \\ 0.5\nu\rho_{1}\alpha&\nu\rho_{1}\alpha&0.5\alpha^{2}&\alpha^{2}& 0.5\alpha\rho_{3}\pi&\alpha\rho_{3}\pi \\ \nu\rho_{2}\pi& 0.5\nu\rho_{2}\pi&\alpha\rho_{3}\pi& 0.5\alpha\rho_{3}\pi&\pi^{2}&0.5\pi^{2} \\ 0.5\nu\rho_{2}\pi&\nu\rho_{2}\pi& 0.5\alpha\rho_{3}\pi&\alpha\rho_{3}\pi& 0.5\pi^{2}&\pi^{2} \end{array}\right) \right) $$

for DZ twin pairs, and

$$ \left(\begin{array}{c} C_{1}\\ C_{1}\\ C_{2}\\ C_{2}\\ C_{3}\\ C_{3}\end{array}\right)\sim N_{6}\left(\left(\begin{array}{c}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right), \left(\begin{array}{cccccc}\eta^{2}&\eta^{2}&\eta\rho_{4}\beta&\eta\rho_{4}\beta&\eta\rho_{5}\psi&\eta\rho_{5}\psi \\ \eta^{2}&\eta^{2}&\eta\rho_{4}\beta&\eta\rho_{4}\beta&\eta\rho_{5}\psi&\eta\rho_{5}\psi \\ \eta\rho_{4}\beta&\eta\rho_{4}\beta&\beta^{2}&\beta^{2}&\beta\rho_{6}\pi&\beta\rho_{6}\psi \\ \eta\rho_{4}\beta&\eta\rho_{4}\beta&\beta^{2}&\beta^{2}&\beta\rho_{6}\pi&\beta\rho_{6}\psi \\ \eta\rho_{5}\psi&\eta\rho_{5}\psi&\beta\rho_{6}\psi&\beta\rho_{6}\psi&\psi^{2}&\psi^{2} \\ \eta\rho_{5}\psi&\nu\rho_{5}\psi&\beta\rho_{6}\psi&\beta\rho_{6}\psi&\psi^{2}&\psi^{2} \end{array}\right)\right), $$

Here, inference is more straightforward. For instance, ρ₁ stands for the correlation between the additive genetic effects on the first and second latent trait and ρ₄ stands for the correlation between the shared environmental effects on the first and second latent trait.

Implementation of the model in WinBUGS requires the specification of prior distributions for the parameters as well as the conditional distribution of the observed data given the model parameters. Given the genetic and non-genetic random effects A _iik C _ik and U _ijk for both individual twins from a twin pair k, the probability of a particular phenotypic state in a twin is not dependent on the respective phenotype in her co-twin nor on the other phenotypes. The conditional probability of a response in the first category for a phenotype is then

$$ \eqalign{ &P(Y_{ijk}=1|A_{11k},A_{12k},A_{21k},A_{22k},A_{31k},A_{32k},C_{1k},C_{2k},C_{3k}, U_{11k},U_{12k},U_{21k},U_{22k},U_{31k},U_{32k}) \cr &=P(X_{ijk}\le t_{i1}|A_{11k},\ldots,U_{32k}) \cr &=P(A_{ijk}+C_{ik}+U_{ijk}+E_{ijk}\le t_{i1}|A_{11k},\ldots,U_{32k}) \cr &=P(E_{ijk}\le t_{i1}-A_{ijk}-C_{ik}-U_{ijk}|A_{11k},\ldots,U_{32k})\cr &=\Phi{\left({{t_{i1}-A_{ijk}-C_{ik}-U_{ijk}}\over{\sqrt{\kappa^{2}}}}\right)},} $$

with Y _ijk denoting the the i-th phenotype in the j-th twin from the k-th twin pair, Φ(.) denoting the cumulative standard normal distribution function and t _i1 denoting the first threshold for the i-th phenotype. For the conditional probability of observing a response in the second category we have

$$ \eqalign{ &P(Y_{ijk}=2|A_{11k},A_{12k},A_{21k},A_{22k},A_{31k}, A_{32k},C_{1k},C_{2k},C_{3k},U_{11k},U_{12k},U_{21k},U_{22k},U_{31k},U_{32k})\cr &=P(t_{i1}<X_{ijk}\le t_{i2}|A_{11k},\ldots,U_{32k})\cr &=P(t_{i1}<A_{ijk}+C_{ik}+U_{ijk}+E_{ijk}\le t_{i2}|A_{11k},\ldots,U_{32k})\cr &=P(t_{i1}-A_{ijk}-C_{ik}-U_{ijk}<E_{ijk}\le t_{i2}-A_{ijk}-C_{ik}-U_{ijk}|A_{11k},\ldots,U_{32k})\cr &=\Phi{\left({{t_{i2}-A_{ijk}-C_{ik}- U_{ijk}}\over{\sqrt{\kappa^{2}}}}\right)}-\Phi {\left({{t_{i1}-A_{ijk}-C_{ik}-U_{ijk}} \over{\sqrt{\kappa^{2}}}}\right)}.} $$

All other conditional probabilities can be written out analogously. These conditional probabilities can be used in a Gibbs sampling algorithm as implemented in WinBUGS in order to estimate the thresholds and other parameters of interest. Its implementation in WinBUGS consists of a rather large script. Contact SvdB or AS for a copy.