Survival Data Analysis with Time-Dependent Covariates Using Generalized Additive Models

Abstract

We discuss a flexible method for modeling survival data using penalized smoothing splines when the values of covariates change for the duration of the study. The Cox proportional hazards model has been widely used for the analysis of treatment and prognostic effects with censored survival data. However, a number of theoretical problems with respect to the baseline survival function remain unsolved. We use the generalized additive models (GAMs) with B splines to estimate the survival function and select the optimum smoothing parameters based on a variant multifold cross-validation (CV) method. The methods are compared with the generalized cross-validation (GCV) method using data from a long-term study of patients with primary biliary cirrhosis (PBC).

Analysis of Heart Transplant Survival Data Using Generalized Additive Models

“…By extending the PLM for the grouped data based on partial likelihood as introduced by Cox [16] and Efron [17], a PLM can be proposed for ungrouped data [13, 14] having time-dependent covariates for the discrete hazard rate h l 〈 d 〉 of patient no. d at the time interval l : $\begin{array}{c}\text{PLM:}\hspace{0.17em}\ln \left(\frac{h_l^{\langle d\rangle }}{\mathrm{1}-h_l^{\langle d\rangle }}\right)={\beta }_{\mathrm{0}}+{\beta }_{\mathrm{1}}{x}_{l\mathrm{1}}^{\langle d\rangle }\\ +{\beta }_{\mathrm{2}}{x}_{l\mathrm{2}}^{\langle d\rangle }+{\beta }_{\mathrm{3}}{x}_{l\mathrm{3}}^{\langle d\rangle }\\ +\cdots +{\beta }_I{x}_{lI}^{\langle d\rangle }.\end{array}$ In recent years, a variety of powerful techniques have been developed for exploring the functional form of effects.…”

“…To avoid overfitting, such models are estimated by penalized maximum likelihood $\begin{array}{c}\ln L\left(\mathit{\beta }\right)=\sum _{d=\mathrm{normal1}}^n\{\sum _{l=\mathrm{normal1}}^{l_d-\mathrm{normal1}}\ln \left(\mathrm{1}-h_l^{\langle d\rangle }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\delta }_{l_d}^{\langle d\rangle }\ln h_{l_d}^{\langle d\rangle }+\left(\mathrm{1}-{\delta }_{l_d}^{\langle d\rangle }\right)\ln \left(\mathrm{1}-h_{l_d}^{\langle d\rangle }\right)\}\\ +\frac{\mathrm{1}}{\mathrm{2}}\sum _{i=\mathrm{normal1}}^I{\lambda }_{i}false\int {\left\{s_i^{\mathrm{\prime \prime }}\left(t\right)\right\}}^{\mathrm{2}}dt,\end{array}$ where λ i are smoothing parameters that control the trade-off between the fit and the smoothness. The functions s i ( x ) in (9) are represented by the B -spline basis functions b i ( x ); see, for details, Tsujitani et al [14]. …”

“…Applying this kind of data structure to the CV algorithm, we obtain insights into how the partition of data should be carried out. A natural extension of the v -fold CV algorithm by setting v = n is to allow the deletion of the patient with several observations; see, for details, Tsujitani et al [14]. …”

“…It is fairly straightforward to extend the model to survivor data with time-dependent covariates. Furthermore, by regarding a GAM as an extension of a partial logistic model (PLM), the unknown parameters can be estimated by maximizing the partial log-likelihood [13, 14]. …”

Prediction of Skin Lesions in Patients Undergoing Allogeneic Hematopoietic Stem Cell Transplantation Using Generalized Additive Models

“…11 We recently extended the theoretical background of GAMs. 12,13 In this study, we aimed to use GAMs to establish a model predictive of the occurrence of SLs in patients undergoing allogeneic SCT, and to identify the significant risk factors associated with SL development.…”