Bayesian Joint Semiparametric Mean–Covariance Modeling for Longitudinal Data

Abstract

Joint parsimonious modeling the mean and covariance is important for analyzing longitudinal data, because it accounts for the efficiency of parameter estimation and easy interpretation of variability. The main potential risk is that it may lead to inefficient or biased estimators of parameters while misspecification occurs. A good alternative is the semiparametric model. In this paper, a Bayesian approach is proposed for modeling the mean and covariance simultaneously by using semiparametric models and the modified Cholesky decomposition. We use a generalized prior to avoid the knots selection while using B-spline to approximate the nonlinear part and propose a Markov Chain Monte Carlo scheme based on Metropolis–Hastings algorithm for computations. Simulation studies and real data analysis show that the proposed approach yields highly efficient estimators for the parameters and nonparametric parts in the mean, meanwhile providing parsimonious estimation for the covariance structure.

Appendix

Here we provide a detailed derivation of the posterior full conditional distributions that were used for the developments in Sect. 3. For (3.1), it is easy to know that

$$\begin{aligned}&\pi (\beta |Y,X,f_0,\Sigma )\propto f(Y|X,f_0,\beta ,\Sigma )\pi (\beta )\\&\quad \propto \mathrm{exp}\left\{ -\frac{1}{2}\sum _{i=1}^m (Y_i-X_i\beta -f_0(t_i))'\Sigma _i^{-1}(Y_i-X_i\beta -f_0(t_i))\right. \\&\left. \qquad -\frac{1}{2}(\beta -\beta _0)'\Delta _0^{-1}(\beta -\beta _0)\right\} \\&\quad = N_p\left( \Delta _1^{-1}\left( \Delta _0^{-1}\beta _0+\sum _{i=1}^mX'_i\Sigma _i^{-1}(Y_i-f_0(t_i))\right) ,\Delta _1^{-1}\right) , \end{aligned}$$

where $\Delta _1=\Delta _0^{-1}+\sum _{i=1}^mX'_i\Sigma _i^{-1}X_i$.

Noting that $f_0=B_0\psi _0$, thus for (3.2) we have

$$\begin{aligned}&\pi (\psi _0|\beta ,\gamma ,\lambda ,f_1)\propto f(Y|X,f_0,\beta ,\Sigma )p(\psi _0|\tau _0)\\&\quad \propto \mathrm{exp}\left\{ -\frac{1}{2}\sum _{i=1}^m (Y_i-X_i\beta -f_0(t_i))'\Sigma _i^{-1}(Y_i-X_i\beta -f_0(t_i))-\frac{1}{2\tau _0^2}\psi _0'\Omega ^{-1}\psi _0\right\} \\&\quad = N_L\left( P_0^{-1}\sum _{i=1}^mB_{0i}'\Sigma _i^{-1}(Y_i-X_i\beta ),P_0^{-1}\right) , \end{aligned}$$

where $P_0=\sum _{i=1}^mB_{0i}'\Sigma _i^{-1}B_{0i}+\Omega /\tau _0^2$.

By (3.3), $\epsilon _i\sim N(0,D_i)$ and (2.8), we have the posterior full conditional distribution of $\gamma$ as follows

$$\begin{aligned}&\pi (\gamma |Y,X,\beta ,\lambda , f_0, f_1)\propto \prod _{i=1}^m f(Y_i-\mu _i|X_i,\beta ,\lambda ,f_0,f_1)\pi (\gamma ) \\&\quad \propto \prod _{i=1}^m \prod _{j=1}^{n_i}f(Y_{ij}-\mu _{ij}|X_i,\beta ,\lambda ,f_0,f_1, Y_{i1}-\mu _{i1},\ldots ,Y_{i(j-1)}-\mu _{i(j-1)})\pi (\gamma ) \\&\quad \propto \mathrm{exp}\left\{ -\frac{1}{2}\sum _{i=1}^m (Y_i-\mu _i-W_i\gamma )'D_i^{-1}(Y_i-\mu _i-W_i\gamma )\right. \\&\left. \qquad -\frac{1}{2}(\gamma -\gamma _0)'\Gamma _0^{-1}(\gamma -\gamma _0)\right\} \\&\quad = N_q\left( \Gamma _1^{-1}\left( \Gamma _0^{-1}\gamma _0+\sum _{i=1}^mW'_iD_i^{-1}(Y_i-\mu _i)\right) ,\Gamma _1^{-1}\right) , \end{aligned}$$

where $\Gamma _1=\Gamma _0^{-1}+\sum _{i=1}^mW'_iD_i^{-1}W_i$.

For (3.5) and (3.6), by (3.3), $\epsilon _i\sim N(0,D_i)$ and priors (2.8) it directly obtains that

$$\begin{aligned}&\pi (\lambda |\beta ,\gamma ,f_0,f_1)\propto f(Y-\mu | \mu ,f_0,f_1)\pi (\lambda )\\&\quad \propto \prod _{i=1}^m|D_i|^{-1/2}\mathrm{exp}\left\{ -\frac{1}{2}\sum _{i=1}^m(y_i-\mu _i-W_i\gamma )'D_i^{-1}(y_i-\mu _i-W_i\gamma )\right. \\&\left. \qquad -\frac{1}{2}(\lambda -\lambda _0)'\Lambda _0^{-1}(\lambda -\lambda _0)\right\} \\&\pi (\psi _1|\beta ,\gamma ,\lambda ,f_0)\propto f(Y-\mu |\mu ,f_0,f_1)p(\psi _1|\tau _1)\\&\quad \propto \prod _{i=1}^m|D_i|^{-1/2}\mathrm{exp}\left\{ -\frac{1}{2}\sum _{i=1}^m(y_i-\mu _i-W_i\gamma )'D_i^{-1}(y_i-\mu _i-W_i\gamma )\right. \\&\left. \qquad -\frac{1}{2\tau _1^2}\psi _1'\Omega \psi _1\right\} , \end{aligned}$$

respectively.