Impact of the model‐building strategy on inference about nonlinear and time‐dependent covariate effects in survival analysis

Abstract

Cox's proportional hazards (PH) model assumes constant-over-time covariate effects. Furthermore, most applications assume linear effects of continuous covariates on the logarithm of the hazard. Yet, many prognostic factors have time-dependent (TD) and/or nonlinear (NL) effects, that is, violate these conventional assumptions. Detection of such complex effects could affect prognosis and clinical decisions. However, assessing the effects of each of the multiple, often correlated, covariates in flexible multivariable analyses is challenging. In simulations, we investigated the impact of the approach used to build the flexible multivariable model on inference about the TD and NL covariate effects. Results demonstrate that the conclusions regarding the statistical significance of the TD/NL effects depend heavily on the strategy used to decide which effects of the other covariates should be adjusted for. Both a failure to adjust for true TD and NL effects of relevant covariates and inclusion of spurious effects of covariates that conform to the PH and linearity assumptions increase the risk of incorrect conclusions regarding other covariates. In this context, iterative backward elimination of nonsignificant NL and TD effects from the multivariable model, which initially includes all these effects, may help discriminate between true and spurious effects. The practical importance of these issues was illustrated in an example that reassessed the predictive ability of selected biomarkers for survival in advanced non-small-cell lung cancer. In conclusion, a careful model-building strategy and flexible modeling of multivariable survival data can yield new insights about predictors' roles and improve the validity of analyses.

Flexible estimation of survival curves conditional on non‐linear and time‐dependent predictor effects

“…We rely on the backward elimination to select the NL and TD effects into the final flexible multivariable model. Whereas this may raise concerns about potential inflation of type I error and over‐fit bias , both our sensitivity analyses (Table A4) and previous simulations indicated only a negligible impact on our estimates and tests. However, future research should implement and compare alternative strategies for selection of flexible covariate effects .…”

“…NL model diverged from our model even more substantially as it (i) failed to detect a significant NL effect of delay and, thus, to select this important prognosticator into the final model and (ii) identified a ‘significant’ NL effect of the SOFA score, which was non‐significant ( p = 0.986) in our model , once adjusted for its significant TD effect (third and fifth columns of Table ). The latter finding reflects the residual confounding between NL and TD effects of the same continuous predictor . As a result, the NL‐only model fits the data much worse than our model ( p = 1.8 * 10 −7 ).…”

Validation of the alternating conditional estimation algorithm for estimation of flexible extensions of Cox's proportional hazards model with nonlinear constraints on the parameters

“…This method does not involve estimating both effects simultaneously and, thus, avoids the issue of nonidentifiability. However, both real‐life analyses and simulation studies showed that the power for testing the NL effects may be substantially reduced if the tests are based on the misspecified model that fails to account for the TD effects of the relevant continuous covariates (Abrahamowicz and MacKenzie, ; Binquet et al., ; Gagnon et al., ; Wynant and Abrahamowicz, ). Indeed, some simulation results in Section of the current manuscript indicate that the covariate effects may be biased if the selection of the NL effects is conditional on an incorrect a priori PH assumption.…”

Multiple imputation in Cox regression when there are time‐varying effects of covariates

“…See also chapter 11 in Royston and Sauerbrei . Abrahamowicz et al and Wynant and Abrahamowicz also described methods for joint estimation of time‐varying and nonlinear effects based on splines. Areas for further work include the extension of the methods proposed in this paper to a setting in which TVEs are modelled using fractional polynomials and to allow selection of functional forms for continuous variables and covariate interactions.…”

“…Several authors have proposed algorithms for model selection involving both TVEs and transformation of covariates, though all assume fully observed data. We propose an algorithm (the MI‐MTVE algorithm), which provides a model selection procedure for identifying TVEs using multiply imputed data.…”