Sampling adjusted analysis of dynamic additive regression models for longitudinal data (Q2739867)

Data of repeated measurements \((T_i^k, Z_i(T_i^k), X_i(t))\) are considered, where \(T_i^k\) is the time-point of the \(k\)-th measurement of the \(j\)-th object, \(Z_i(t)\) is the response variable, and \(X_i(t)\) is a time-dependent \(p\)-dimensional covariate. The data are modeled by NEWLINE\[NEWLINEZ_i(T_i^k)=\beta^T(T_i^k)X_{\beta i}(T_i^k)+\gamma^T X_{\gamma i}(T_i^k)+\varepsilon_i^k,NEWLINE\]NEWLINE with \(E(\varepsilon_i^k |X_i(T_i^k))=0\), \({\text Var}(\varepsilon_i^k |X_i(T_i^k))=\sigma^2(T_i^k)\), where \(\beta(t)\) and \(\gamma\) are unknown regression coefficients, and \(X_{\alpha i}, X_{\beta i}, X_{\gamma i}\) are some parts of \(X_i\). The measurement times \(T_i^k\) are modelled using a counting process with intensity \(\lambda_i(t)=Y_i(t)\alpha^T(t)X_{\alpha i}(t)\), where \(\alpha(t)\) is an unknown deterministic function and \(Y_i(t)\) is a predictable at risk indicator. Using a martingale decomposition of the observations considered as a marked point process, the authors derive estimators for \(\alpha(t)\), \(\beta(t)\), \(\gamma\) and \(\sigma^2(t)\). These estimators are applied to measurements of height of patients with cystic fibrosis and measurements of the prothrombin index for liver cirrhosis patients. Results of simulation studies are also presented.