Spline Estimation For Panel Count Data

We proposed a panel count data model with both time-varying and time-invariant coefficients.We proposed four regression spline estimators,to compare the pseudo-likelihood with the full-likelihood,and to compare estimating the non-negative function itself directly using spline with estimating the log-form of the non-negative function using spline.We newly proposed the penalized spline estimation based on likelihood method and compared the penalized spline estimation with the best regression spline estimation method among the four methods.We developed a cross-validated log-likelihood(CVL)score to select the smoothing parameters for penalized spline estimation and deduced an approximation to the score which was easy to compute.We can select more than one smoothing parameters and the number of the knots simultaneously using the CVL score.We compared four algorithms.The first one was called Projected Newton-Raphson algorithm,and we verified its global convergence,but the other three algorithms were not global convergent.By extensive simulations,we found that the differences between estimating the non-negative function itself and its log-form using splines depended on the differences between the complexity of their likelihood functions;the spline estimators based on the fulllikelihood method often had smaller bias and mean square errors than those based on pseudo-likelihood method but with longer computing time;we recommend estimating the log-form of the non-negative function using spline based on pseudo-likelihood when considering both accuracy and computation time;the penalized spline estimators had about 25% smaller mean square errors than regression spline estimators and with smaller bias sometimes;the estimated variances of the penalized spline estimators using bootstrap method were close to the Monte Carlo variances;the penalized spline estimation method was robust against model misspecification;finally,we found the time-varying effect of IL5 on the risk of wheezing after applying the penalized spline estimation method to the childhood wheezing data.The main innovation points were listed as follows: we firstly studied the spline estimators based on the full-likelihood under panel count data model with time-varying coefficients,we firstly proposed to estimate the non-negative function itself(rather than its log-form)using spline directly,and we newly proposed the penalized spline estimation based on likelihood method and showed that it performed better than regression spline estimation.