Variable selection and model choice in structured survival models

Abstract

We aim at modeling the survival time of intensive care patients suffering from severe sepsis. The nature of the problem requires a flexible model that allows to extend the classical Cox-model via the inclusion of time-varying and nonparametric effects. These structured survival models are very flexible but additional difficulties arise when model choice and variable selection are desired. In particular, it has to be decided which covariates should be assigned time-varying effects or whether linear modeling is sufficient for a given covariate. Component-wise boosting provides a means of likelihood-based model fitting that enables simultaneous variable selection and model choice. We introduce a component-wise, likelihood-based boosting algorithm for survival data that permits the inclusion of both parametric and nonparametric time-varying effects as well as nonparametric effects of continuous covariates utilizing penalized splines as the main modeling technique. An empirical evaluation of the methodology precedes the model building for the severe sepsis data. A software implementation is available to the interested reader.

Appendices

Appendix A: Detailed simulation results

In this section, more details on the simulation results are given. To explore the accuracy of model choice and thus of variable selection we examine the relative frequency of selected base-learners. This means we count the models (simulation replicates) where the base-learner was included and ignore how often and in which boosting iteration(s) the base-learner was selected. Furthermore, the magnitude of the estimated effects is neglected.

Table 1 (left) shows the selection frequencies of the base-learners in the first simulation scheme. Only the variables $x_1$ to $x_6$ have an effect on the hazard rate and thus, on the survival time. These effective covariates are presented in the upper half of the table. We see that almost all effective base-learners have a selection frequency close to one or exactly one except for the linear base-learners $\mathtt bols( x_2\mathtt )$ and bols( $x_6$ ). If we look at the true influence of $x_2$ it shows that this is a good result, as we have a quadratic influence of this covariate and hence no linear effect is required. The low selection frequency of $\mathtt bols( x_6\mathtt )$ can be attributed to the size of the effect of $x_6$, which is very small. It is $20$ times smaller than the effect of the other categorical covariate $x_5$. Hence, the low selection frequency seems very plausible. For $x_4$ (which has in reality a linear effect) the algorithm selected in $26~\%$ of the replicates a flexible deviation from linearity. Thus, in some models the (wrong) impression of an underlying nonlinear effect of $x_4$ is given. However, compared to the selection frequencies of the other effects, this is only of minor importance. In addition, at further inspection we see that the departures from linearity are only very small.

For non-effective covariates we expect the selection frequency of a base-learner to be close to zero or at least substantially smaller than for effective covariates. Looking at the lower part of Table 1 (left) this can be confirmed.

Note that in our simulation the number of base-learners is not equal to the number of variables. A variable is selected if any of the base-learners of this variable is selected. Using this definition, we see that on average we selected 9.8 variables with 5.4 effective variables and 4.4 non-effective variables. Compared to a scheme where we assign only one base-learner for each variable (e.g., a flexible base-learner per covariate with 4 degrees of freedom) we realize that the model choice scheme tends to select more variables and to select more non-effective variables. Perhaps, this is due to an increased number of possible base-learners per covariate. This argument is backed by the finding that we selected on average 12.8 out of 25 base-learners. We selected (on average) about 5.3 non-effective base-learners, which corresponded on average to 4.4 non-effective variables. Thus, almost every non-effective base-learner is based on another variable.

See Figs. 4, 5, 6.

Fig. 4

Simulation scheme 1—remaining estimates of effects of influential covariates from 20 models (gray lines) and real effects (dashed lines). Effect estimates and real effects are centered

Fig. 5

Simulation scheme 1—remaining estimates of covariate effects for non-effective covariates from 20 models (gray lines) and real effects (dashed lines). Effect estimates and real effects are centered

Fig. 6

Simulation scheme 2—estimation of covariate effects from 20 models (gray lines) and real effects (dashed lines). Effect estimates and real effects are centered

Appendix B: Model for surgical patients: further tables and graphics

See Table 2 and Fig. 7.

 

Fig. 7

Remaining estimated effects for the complete Großhadern data set together with pointwise 80 % variability intervals based on sub-samples. The upper three rows represent time-varying effects (for binary covariates except creatinine concentration). The fourth row shows the time-constant effects for continuous covariates and the last row the same for binary covariates