2 Main Methodology
Denote the response vector for the \(i\)th subject by \(\boldsymbol{y}_i=(y_{i1},\cdots,y_{it},\cdots,y_{in_i})'\) where \(y_{it}\) is a response at time period \(t\) (\(i=1,\cdots, N\); \(t=1,\cdots,n_i\)). Note that the model and associated methodology can be applicable to the unequally spaced times and the distinct number of observations from subject to subject.We assume that the responses on different subjects are independent. Also, we assume that \(y_{it}\)’s are conditionally independent given a random vector \(b_{i}\), and that \(y_{it}\)’s. For categorical responses, we assume that \(y_{it}\) has an exponential family distribution so that generalized linear models (GLMs) can be specified by \[\begin{eqnarray} &&g\left\{E(y_{it})\right\}=x_{it}^T\beta+z_{it}^Tb_{i}, \tag{2.1}\\ &&b_i=(b_{i1},\ldots,b_{iq})^T\stackrel{\mbox{indep.}}{\sim} N(0,v_i^{-1}\Sigma), \notag\\ &&v_i \stackrel{\mbox{indep.}}{\sim} \Gamma\left(\nu/2,\nu/2\right), \notag \end{eqnarray}\] where \(\beta\) is a \(p\times 1\) unknown mean parameter vectors, \(x_{it}\) is a \(p\times 1\) corresponding vector of covariates, \(z_{it}\) is a \(q\times 1\) vector, \(0\) is a \(n_i\times 1\) zero vector, \(\Sigma\) is a \(q\times q\) covariance matrix reflecting the subject variations, and \(\Gamma(a,b)\) denotes the gamma distribution with shape parameter \(a\) and scale parameter \(b\). In this paper, we consider the normal and binary responses and the corresponding links are identity and probit, respectively.
To employ Markov Chain Monte Carlo algorithm techniques for Bayesian estimates and reduce the computational complexity, we introduce a latent variable latent variable \(y_{it}^*\) to associate with the binary or ordinal outcome \(y_{it}\) as follows, respectively.
Binary outcome: The unobservable latent variable \(y_{it}^*\) and the observed binary outcome \(y_{it}\) are connected by: \[ y_{it}=\mathbf{I}_{(y_{it}^*>0)}, \quad t = 1, \ldots, n_i, \] where \(\mathbf{I}_{A}\) is the indicator of event \(A\). Note that \(y_{it}\) is 1 or 0 according to the sign of \(y_{it}^*\).
Ordinal outcome: The atent variable \(y_{it}^*\) is associated with each ordinal response \(y_{it}\). Hence, the probaility of \(y_{it}=k\) is modeled through the probability of \(y_{it}^*\) falling into the interval of \((\alpha_{k-1},\alpha_k]\), that is, given the random effect \(b_i\), \[\begin{eqnarray}\label{model-3} y_{it}=k \mbox{ if } \alpha_{k-1} < y_{it}^* \leq \alpha_k \mbox{ for }k=1,\ldots, K, \end{eqnarray}\] where \(-\infty=\alpha_0<\alpha_1<\cdots<\alpha_K=\infty\). As consequence, we have the following result: \[\begin{eqnarray*} p(y_{it}=k | b_i)=p(\alpha_{k-1}< y_{it}^* \leq \alpha_{k} | b_i), \end{eqnarray*}\] for \(k=1,\ldots,K\).
We assume that the latent variable is associated with explanatory variable \(x_{it}\) and random effect \(z_{it}\) with two different approaches to explaining the correlation of the repeated measures within a subject in next two sections.
2.1 Modified Cholesky Decomposition with Hypersphere Decomposition
We assume \[ y_{it}^*=x_{it}^T\beta+z_{it}^Tb_i+\epsilon_{it}, \] where \(\epsilon_{it}\)’s are prediction error and are assumed as \[ \boldsymbol{\epsilon}_i=(\epsilon_{i1},\ldots,\epsilon_{in_i})^T \stackrel{indep.}{\sim} N(0,R_i) \] with a correlation matrix \(R_i\). Then the model (2.1) is equivalent to, for \(i=1, \ldots, N\) and \(t=1, \ldots, n_i\), \[\begin{equation} \begin{aligned} y_{it} &= \begin{cases} 1, & y_{it}^*>0; \\ 0 & \mbox{otherwhise}. \end{cases} \end{aligned}\tag{2.2} \end{equation}\] Let \(\boldsymbol{y}_i^* = (y_{i1}, \ldots, y_{in_i})'\) and rewrite (2.2) in matrix form as \[\begin{eqnarray*} \boldsymbol{y}_i^*=X_i\beta+Z_i b_i +\boldsymbol{\epsilon}_i, \end{eqnarray*}\] where \(X_i\) and \(Z_i\) are \(n_i\times p\) and \(n_i\times q\) matrices and defined as follows, respectively, \[\begin{eqnarray*} X_i=\left( \begin{array}{c} x_{i1}^T \\ \vdots \\ x_{in_i}^T \\ \end{array} \right), Z_i=\left( \begin{array}{c} z_{i1}^T \\ \vdots \\ z_{in_i}^T \\ \end{array} \right) . \end{eqnarray*}\]
On account of identifiability, \(R_i\) is restricted as a correlation matrix. In addition to the diagonal elements equal to 1 and off-diagonal elements between -1 and 1, \(R_i\) is required to be a positive definite matrix. Moreover, the number of parameters to be estimated increases quadratically with the dimension of the matrix. In order to model \(R_{i}\) being positive definite, while alleviating the computational expensive, we propose a modeling of the correlation matrix using the hypersphere decomposition (HD) approach (Zhang, Leng, and Tang 2015). The correlation matrix \(R_i\) is reparameterized via hyperspherical coordinates (Zhang, Leng, and Tang 2015) by the following decomposition: \[\begin{eqnarray*} R_i=F_iF_i^T, \end{eqnarray*}\] where \(F_i\) is a lower triangular matrix with the \((t, j)\)th element \(f_{itj}\) given by \[\begin{eqnarray*} f_{itj}=\left\{ \begin{array}{ll} 1, & \hbox{for $t=j=1$;}\\ \cos(\omega_{itj}), & \hbox{for $j=1$, $t=2,\cdots,n_i$;} \\ \cos(\omega_{itj})\prod_{r=1}^{j-1}\sin(\omega_{itr}), & \hbox{for $2\leq j<t\leq n_i$;} \\ \prod_{r=1}^{j-1}\sin(\omega_{itr}), & \hbox{for $t=j;~ j=2,\cdots,n_i$.} \end{array} \right. \end{eqnarray*}\] Here \(\omega_{itj}\)’s \((\in (0,\pi))\) are angle parameters for trigonometric functions, and the angle parameters are referred to hypersphere (HS) parameters.
As in Zhang, Leng, and Tang (2015), we consider the modeling of the angles \(\omega_{itj}\)’s instead of the direct modeling of the correlation matrix, and the modeling can be directly interpreted for the correlation (Zhang, Leng, and Tang 2015). In order to obtain the unconstrained estimation of \(\omega_{itj}\) and to reduce the number of parameters for facilitating the computation, we model the correlation structures of the responses in terms of the generalized linear models which are given by: \[\begin{eqnarray} &&\log\left(\frac{\omega_{itj}}{\pi-\omega_{itj}}\right)=u_{itj}^T\delta,\tag{2.3} \end{eqnarray}\] where \(\delta\) is \(a \times 1\) vector of unknown parameter vector to model the HS parameters. Importantly, the proposed method reduces the model complexity and obtain fast-to-execute models without loss of accuracy. In addition, note that the design vector \(u_{itj}\) in (2.3) is used to model the HS parameters as functions of subject-specific covariates (Zhang, Leng, and Tang 2015; Pan and Mackenzie 2015). As a result, the design vector is specified in a manners similar to those used in heteroscedastic regression models. For example, time lag, \(|t - j|\), in the design vector \(u_{itj}\) specifies higher lag models. We will introduce the priors of parameters in the model in Section 3.
2.2 Generalized Autoregressive and Moving-Averaging Model
In order to give the complete specification of the joint distribution, the latent random vectors \(\boldsymbol{y}_{i}^*=(y_{i1}^*,\ldots,y_{in_i}^*)^T\) are jointly normally distributed given by: \[\begin{equation} \begin{aligned} y_{i1}^*&= x_{i1}^T\beta + \epsilon_{i1}, \\ y_{it}^*&=x_{it}^T\beta+z_{it}^Tb_i+\sum_{j=1}^{u-1}\phi_{ij}(y_{i,t-j}^* - x_{i,t-j}^T\beta)+ \sum_{s=1}^{v-1} \psi_{i,t-s}\epsilon_{i,t-s}+\epsilon_{it}, t=1, \ldots, n_i, \end{aligned}\tag{2.4} \end{equation}\] where \(\phi_{ij}\)’s are generalized autoregressive parameters (GARPs) and \(\psi_{is}\)’s are generalized moving-average parameters (GMAPs). In addition, \(\epsilon_{it}\)’s are prediction error and are assumed as \[ \boldsymbol{\epsilon}_i=(\epsilon_{i1},\ldots,\epsilon_{in_i})^T \stackrel{indep.}{\sim} N(0,I_i), \] where \(I_i\) is an \(n_i\times n_i\) identity matrix. We can rewrite (2.4) in matrix form as \[\begin{equation*} \Phi_i (\boldsymbol{y}_i^*-X_i\beta) = Z_i b_i + \Psi_i \boldsymbol{\epsilon}_i, \end{equation*}\] where \(X_i\), \(n_i\times p\), \(Z_i\), \(n_i\times q\), \(\Phi_i\), \(n_i\times n_i\), \(\Psi_i\), \(n_i\times n_i\), are matrices and defined as follows, respectively, \[\begin{eqnarray*} X_i=\left( \begin{array}{c} x_{i1}^T \\ \vdots \\ x_{in_i}^T \\ \end{array} \right),\quad Z_i=\left( \begin{array}{c} z_{i1}^T \\ \vdots \\ z_{in_i}^T \\ \end{array} \right) \end{eqnarray*}\] \[\begin{eqnarray*} \Phi_i =\left( \begin{array}{cccccc} 1 & 0 & 0 & \ldots & 0&0\\ -\phi_{i1} & 1 & 0 & \ldots &0&0\\ -\phi_{i2} & -\phi_{i1} & 1 & \ldots & 0&0 \\ \vdots & \vdots & \vdots & \ddots & \vdots &\vdots \\ 0& \ldots & -\phi_{i,u-2} & \ldots & 1 &0 \\ 0 & \ldots &-\phi_{i,u-1} & \ldots & -\phi_{i1} & 1 \end{array} \right), \quad \Psi_i =\left( \begin{array}{cccccc} 1 & 0 & 0 & \ldots & 0&0\\ \psi_{i1} & 1 & 0 & \ldots &0&0\\ \psi_{i2} & \psi_{i1} & 1 & \ldots & 0&0 \\ \vdots & \vdots & \vdots & \ddots & \vdots &\vdots \\ 0& \ldots & \psi_{i,v-2} & \ldots & 1 &0 \\ 0 & \ldots &\psi_{i,v-1} & \ldots & \psi_{i1} & 1 \end{array} \right) \end{eqnarray*}\] Note that \(\Phi_i\) and \(\Psi_i\) uniquely exist and are respectively called the generalized autoregressive parameter matrix (GARPM) and moving-average parameter matrix (GMAPM).
The density of the latent variable \(\boldsymbol{y}^*\) conditional on the random effect \(b=(b_1, \ldots, b_q)\) is given by \[ p(\boldsymbol{y}^*|\boldsymbol{b}, \theta) = \prod_{i=1}^N\prod_{t=1}^{n_i} f(y^*_{it}; \mu_{it}, I_i), \] where \(\theta = (\beta, \nu, \Sigma, \phi, \psi)\) denote the collection of model parameters, \(\mu_{it} = x_{it}^T\beta+z_{it}^Tb_i\) and \(f(\cdot)\) is the multivariate normal density function.