vignettes/hglmbc2_Documentation.Rmd
hglmbc2_Documentation.RmdThe hglmbc2 package vignette illustrates a series of applicaitons that may be interest in Small Area Estimation (SAE) based on the calibrated \(h-\)likelihood approach. In SAE a random effect or the laetent variable \(u_i\) described the between area estimation. The proposed calibrated hierarchical \(h-\)likelihood in SAE obtain SAE through hierarchical estimation of fixed effects and random effects with bias correction using Regression Calibration Methodd (RCM). This apparoach allows the conditional distribution of y given random effect\((u)\) from any exponential family distributions (GLM family) and the random effect \(u \sim N(0,\sigma^2)\) which is called the calibrated \(h-\) likelihood with bias correction. This hglmbc2 is mainly built to fit exponential family distributions (GLM family) (2005) via \(h-\)likelihood with \(Bias-Correction\) in SAE. This package also can be applied to Hierarchical Generalized Linear Models (HGLM) where \(y|u\) from the exponential family distribution and \(u\) may also be from the exponential family distribution, without bias correction approach (Lee and Nelder 1996; Lee, Nelder, and Pawitan 2006). The CPH with bias correction approach is described in detail in the PhD dissetation report, (See the dissertation). The \(h-\)likelihood is the sum of conditional log likelihood of \(y|u\) and the log likelihood of \(u\),
\[ h = \sum_{i=1}^m\sum_{j=1}^{n_i} \ell_{y_{ij}|u_i} + \sum_{i=1}^m \ell_{u_i}, \]
\[ h = \sum_{i=1}^m\sum_{j=1}^{n_i} (y_{ij}(x_{ij}^T\beta +u_i) - log(1+exp(x_{ij}^T\beta +u_i))) - \sum_{i=1}^m \frac{m}{2}log(\sigma^2) -\frac{1}{2\sigma^2}\sum_{i=1}^mu_i^2-\frac{m}{2}log(2\pi) \]
The estimated model parameters through the proposed method are called the maximum hierarchical likelihood estimators (MHLEs). Let \(\tau=(\beta,u)\) be the unknown parameters. Given the variance parameter \(\theta=(\theta_1,…,\theta_q )^T=\sigma^2\), MHLE of \(\hat \tau=(\hat \beta, \hat u)\) are obtained by solving the score function \(\partial h/\partial \tau = 0\). When the solution does not have a closed form, the MHLE for \((\beta,u)\) are obtained using Newton-Raphson approximation through an iterative procedure using equation. Unlike in the standard linear mixed model, the model (1) does not have a closed-form for the joint log likelihood, hence it is very challenging to estimate BLUP or EBLUP of model parameters. In such situations, \(h-\)likelihood plays a major role in simplifying the parameter estimation procedure. Mostly, the MHLEs are often obtained via numerical approximation methods due to intractable integrals in the joint log-likelihood function.
The score function \(\mathcal{S}\) and the hessian matrix J of the Newton Raphson iterative algorithm shown below,
\[ \pmb{\mathcal{S}}= \left(\begin{array}{cc} \frac{\partial h}{\partial \beta_r}\\ \frac{\partial h}{\partial u_i} \end{array}\right) , \pmb{\mathcal{J}}= \left(\begin{array}{cc} -\frac{\partial^2 h}{\partial \beta_r \partial \beta_s} & - \frac{\partial^2 h}{\partial \beta_r \partial u_i}\\ -\frac{\partial^2 h}{\partial u_i \partial \beta_s} & - \frac{\partial^2 h}{\partial u_i \partial u_l} \end{array}\right) \]
The estimated \(\pmb{\mathcal{J}}\) is the asymptotic variance-covariance matrix of \(\pmb{\hat \beta}\) and \(\pmb{\hat u}\). The variance-covariance matrix for variance component \(\pmb{\hat \theta}\) will be estimated by MHLE of \(\pmb{\theta}\) using \(h_A\) through an iterative procedure.
we consider bias correction of random effects within the parameter estimation process to mitigate the biasness that could occur due to the use of current estimates of random effects \((\pmb{\hat u}^{(t)})\) to obtain estimations for variance parameter \(\theta(\sigma^2)\). The MHLE of \((\pmb{\hat u})\) will lead to bias and inconsistent estimations of regression coefficients and parameters in the random effects, such as variance parameters. We apply the Regression Calibration Method (RCM) which corrects the biasness in the estimators introducing a correction factor to adjust the current estimators based on the variance of the previous and current estimators of \(\pmb{u}\). Under RCM, the current estimate of \(\pmb{u}, \hat{\pmb{u}}\) is replaced by the expectation of conditional expectation of \(\pmb{u}|\hat{\pmb{u}}\). Given \(\pmb{u} \sim N(0,\sigma^2)\),
\[ E[\pmb{u}|\hat{\pmb{u}}] = \pmb{\zeta}\hat{\pmb{u}}=\frac{\sigma^2}{\sigma^2+\pmb{\gamma}^2} \hat{\pmb{u}}, \]
\[ E[exp(\pmb{u})|\hat{\pmb{u}}] = exp \biggl( \pmb{\zeta}\hat{\pmb{u}}+\frac{\sigma^2(1-\pmb{\zeta})}{2} \bigg), \]
where \(\pmb{\zeta}=\sigma^2/(\sigma^2+\pmb{\gamma}^2)\) is the correction factor, and \(\pmb{\gamma}=diag(\Sigma_{\hat{\pmb{u}}}),\Sigma_{\hat{\pmb{u}}}=(\pmb{\mathcal{J}}^{-1})_{22}\) is a vector of \(m\) elements obatined from the variance-covariance matrix of \(\hat{\pmb{u}}\). The dispersion parameters are obained after correcting to the current \(\hat{\pmb{u}}\). The dispersion parameters will be estimated iteratively based on the adjusted \(h-\)likelihood function.
Lee, Youngjo, and John A Nelder. 1996. “Hierarchical Generalized Linear Models.” Journal of the Royal Statistical Society: Series B (Methodological) 58 (4): 619–56.
Lee, Youngjo, John A Nelder, and Yudi Pawitan. 2006. Generalized Linear Models with Random Effects: Unified Analysis via H-Likelihood. Chapman; Hall/CRC.
McCulloch, Charles E, and John M Neuhaus. 2005. “Generalized Linear Mixed Models.” Encyclopedia of Biostatistics 4.
University of Nebraska Medical Center, niro.uno@gmail.com, https://niroshar.github.io/My-Profile/↩