Genetic Mixed Linear Models for Twin Survival Data

Abstract

Twin studies are useful for assessing the relative importance of genetic or heritable component from the environmental component. In this paper we develop a methodology to study the heritability of age-at-onset or lifespan traits, with application to analysis of twin survival data. Due to limited period of observation, the data can be left truncated and right censored (LTRC). Under the LTRC setting we propose a genetic mixed linear model, which allows general fixed predictors and random components to capture genetic and environmental effects. Inferences are based upon the hierarchical-likelihood (h-likelihood), which provides a statistically efficient and unified framework for various mixed-effect models. We also propose a simple and fast computation method for dealing with large data sets. The method is illustrated by the survival data from the Swedish Twin Registry. Finally, a simulation study is carried out to evaluate its performance.

Appendices

Appendix A

Derivation of model (2)

From v _ij  = g _ij  + c _i0 for j = 1, 2, the model (1) can be expressed as a simple matrix form:

$$ \hbox{log}\,T_{i}=X_{i} \beta + Z_i v_i + \epsilon _{i}, $$

(A.1)

where T _i  = (T _i1, T _i2)^T, X _i  = (x _i1, x _i2)^T is the 2 × p model matrix of β, Z _i is the model matrix of v _i , ε _i  = (ε _i1, ε _i2)^T ∼ N(0,σ ²_ε I ₂), and I ₂ is the 2 × 2 identity matrix. For the MZ _i , Z _i  = (1,1)^T and v _i (= v _i1 = v _i2) ∼ N(0,σ ² _v ), but for the DZ _i , Z _i  = I ₂ and v _i  = (v _i1, v _i2)^T ∼ N(0,σ ² _v Σ _i ) with a compound symmetric structure

$$ \Sigma_i=\left( \begin{array}{cc} {1} & {\rho} \\ & \\ {\rho} & {1} \end{array} \right). $$

Here

$$ \sigma_v^2=\sigma_g^2+\sigma_c^2 $$

(A.2)

and

$$ \rho=\hbox{corr}(v_{i1}, v_{i2})=\frac{0.5\sigma_g^2+\sigma_c^2}{\sigma_g^2+\sigma_c^2}, $$

(A.3)

where ρ ∈ [0.5, 1.0]. The use of ρ leads to useful results. From (A.3) we see that σ ² _g is much larger than σ ² _c (i.e., σ ² _g ≫ σ ² _c ) if ρ goes to 0.5, but σ ² _g ≪ σ ² _c if ρ goes to 1.0. In particular, the model (A.1) reduces to model (1) without random-environment effects c _ij if ρ = 0.5 (i.e., σ ² _c  = 0), while it becomes model (1) without random-genetic effects g _ij if ρ = 1.0 (i.e., σ ² _g  = 0).

Following Lee and Nelder (2001a), the random effects v _i for DZ _i are assumed to have the form L _i (ρ)u _i , where u _i  ∼ N(0,σ ² _v I ₂). For the DZ _i , using the Cholesky decomposition we have a lower triangular matrix L _i such that Σ _i  = L _i L ^T _i . Here, we choose

$$ L_i(\rho)=\left( \begin{array}{cc} {1} & {0} \\ {\rho} & \sqrt{1-\rho^2} \end{array} \right), $$

and so the random effects v _i  = L _i u _i  ∼ N(0,σ ² _v L _i L ^T _i ).

Thus, model (A.1) can be written as

$$ \hbox{log}\,T_{i}=X_{i} \beta + Z_i^{\ast} u_i + \epsilon _{i}, $$

(A.4)

where u _i  ∼ N(0,σ ² _v I _k ), and Z ^* _i  = (1,1)^T and I _k  = 1 for the MZ _i , and Z ^* _i  = L _i (ρ) and I _k  = I ₂ for the DZ _i . Note that from (A.2) and (A.3) we obtain σ ² _g and σ ² _c as follows:

$$ \sigma_g^2=\sigma_v^2 - \sigma_c^2\quad\hbox{and}\quad \sigma_c^2=2(\rho-0.5)\sigma_v^2. $$

(A.5)

Then the jth element of model (A.4) becomes the model (2).

Appendix B

Proofs of score equations of (7) and (8) and the computation of variance of ${\widehat \beta}$

Let μ be the n × 1 vector with ijth element μ _ij ,

$$ \mu=X \beta + Z^{\ast} u, $$

where X = (X ^T₁ , …, X ^T _q )^T is the n × p model matrix for the p × 1 fixed effects β and Z ^* = blockdiag(Z ^*₁ , …, Z ^* _q ) is the n × q ^* block diagonal matrix for the q ^* × 1 random effects u = (u ₁, …, u _q )^T. Here, q ^* = q ₁ + 2q ₂, q ₁ is the number of MZ twin pairs and q ₂ is that of DZ twin pairs. Note that q = q ₁ + q ₂. Let y ^* = (y ^{* T}₁ …, y ^{* T} _q )^T be the n × 1 vector with the ith vector y ^* _i  = (y ^* _i1 , y ^* _i2 )^T. Assume that ρ is known. Given θ = (σ ²_ε ,σ ² _v )^T and y ^*, from (5) and (6) the score equations for the MHLEs of τ = (β^T, u ^T)^T become Henderson’s (1975) mixed-model equations with pseudo-response variables y ^*:

$$ \left( \begin{array}{cc} {X^{T}X} & {X^{T}Z^{\ast}} \\ {Z^{\ast T}X} & {Z^{\ast T}Z^{\ast}+\Lambda } \end{array} \right) \left( \begin{array}{c} {\widehat{\beta }} \\ {\widehat{u}} \end{array} \right) =\left( \begin{array}{c} X^{T}y^{\ast } \\ Z^{\ast T}y^{\ast } \end{array} \right), $$

(B.1)

where ${\Lambda=\lambda I_{q^{\ast}}}$, λ = σ ²_ε /σ ² _v and ${I_{q^{\ast}}}$ is the q ^* × q ^* identity matrix. Equation (B.1) can be expressed as the two equations:

$$ (X^T X) \widehat{\beta}+(X^T Z^{\ast}) \widehat{u}=X^T y^{\ast}, $$

(B.2)

$$ (Z^{\ast T} X) \widehat{\beta}+(Z^{\ast T} Z^{\ast} +\lambda I_{q^{\ast}}) \widehat{u}=Z^{\ast T} y^{\ast}. $$

(B.3)

Substituting X = (X ^T₁ , …, X ^T _q )^T, Z ^* = blockdiag(Z ^*₁ , …, Z ^* _q ) and y ^* = (y ^{* T}₁ , …, y ^{* T} _q )^T into (B.2) and (B.3) reduces them to Eqs. (7) and (8).

The asymptotic covariance matrix for ${\widehat{\tau }-\tau }$ is given by D ⁻¹ with

$$ D(h, \tau)=- \frac{\partial ^{2}h}{\partial \tau ^{2}}=\frac{1}{\sigma _{\epsilon }^{2}}H, $$

(B.4)

where

$$ H={\left( \begin{array}{cc} {X^{T}WX} & {X^{T}WZ^{\ast}} \\ {Z^{\ast T}WX} & {Z^{\ast T}WZ^{\ast}+\Lambda} \end{array} \right) }. $$

Here, W = diag(w _ij ) is the n × n diagonal matrix with the ijth element w _ij  = δ _ij  + (1 − δ _ij )ξ(m _ij )  − ξ(m ^* _ij ) and ξ(x) = V(x){V(x) − x}. So, the upper left-hand corner of D ⁻¹ in (B.4) gives the variance matrix of ${\widehat{\beta }}$, which is also easily computed for the large samples as follows. Let H ¹¹ be the upper left-hand corner of H ⁻¹ in (B.4). Then we have that

$$ \hbox{var}(\widehat{\beta})=\sigma_\epsilon^2 {H}^{11}, $$

(B.5)

where

$$ \begin{array}{lll} {H}^{11} &= & \left\{ (X^T W X)-(X^T W Z^{\ast})(Z^{\ast T} W Z^{\ast} + \lambda I_{q^{\ast}})^{-1} (Z^{\ast T} W X) \right\}^{-1} \\ & = & \left\{ \sum\limits_i X_i^T W_i X_i-\sum\limits_i (X_i^T W_i Z_i^{\ast})(Z_i^{\ast T} W_i Z_i^{\ast} + \lambda I_{k})^{-1} (Z_i^{\ast T} W_i X_i) \right\}^{-1}. \end{array} $$

Here W _i is the ith component matrix of W.