Survival analysis of patients with heart failure: implications of time-varying regression effects in modeling mortality

Abstract

Background: Several models have been designed to predict survival of patients with heart failure. These, while available and widely used for both stratifying and deciding upon different treatment options on the individual level, have several limitations. Specifically, some clinical variables that may influence prognosis may have an influence that change over time. Statistical models that include such characteristic may help in evaluating prognosis. The aim of the present study was to analyze and quantify the impact of modeling heart failure survival allowing for covariates with time-varying effects known to be independent predictors of overall mortality in this clinical setting.

Methodology: Survival data from an inception cohort of five hundred patients diagnosed with heart failure functional class III and IV between 2002 and 2004 and followed-up to 2006 were analyzed by using the proportional hazards Cox model and variations of the Cox's model and also of the Aalen's additive model.

Principal findings: One-hundred and eighty eight (188) patients died during follow-up. For patients under study, age, serum sodium, hemoglobin, serum creatinine, and left ventricular ejection fraction were significantly associated with mortality. Evidence of time-varying effect was suggested for the last three. Both high hemoglobin and high LV ejection fraction were associated with a reduced risk of dying with a stronger initial effect. High creatinine, associated with an increased risk of dying, also presented an initial stronger effect. The impact of age and sodium were constant over time.

Conclusions: The current study points to the importance of evaluating covariates with time-varying effects in heart failure models. The analysis performed suggests that variations of Cox and Aalen models constitute a valuable tool for identifying these variables. The implementation of covariates with time-varying effects into heart failure prognostication models may reduce bias and increase the specificity of such models.

Figure 1. Graphical checks of the proportional hazards assumption.

A. Scaled Schoenfeld residuals against time plotted for each covariate in the proportional hazards Cox model. B. Observed test process plotted along with 50 processes simulated for each covariate in the proportional hazards Cox model under the hypothesis of time-invariant effects.

Figure 2. Cumulative coefficients obtained from the extended Cox’s model.

Estimates from 1 to 750 days are for the covariates considered in the model as having time-varying effects. Curves along with the estimates are 95% confidence limits.

Figure 3. Cumulative coefficients obtained from the additive hazards model.

Estimates from 1 to 750 days are for the covariates considered in the model as having time-varying effects. Curves along with the estimates are 95% confidence limits.

Figure 4. Graphical checks of the overall fit of the Cox model.

A. Survival probabilities obtained from the Cox-Snell residuals by considering the unit exponential distribution and the Kaplan-Meier estimator. B. Survival curves obtained from the Cox-Snell residuals by considering the Kaplan-Meier estimator and the unit exponential distribution.

Figure 5. Cumulative martingale residuals plotted for each covariate.

A. Cumulative residuals from the Cox́s model with time-varying effects. B. Cumulative residuals from the additive hazards model with time-varying effects.

Figure 6. Non-parametric and model-based survival curves for two scenarios.

A. Curves predicted for patients with mean values for all covariates (mean values in Table 1). B. Curves predicted for hypothetical patients aged 32 yrs old, serum sodium  = 137, Hb  = 13.7, creatinine  = 1.0 and LV ejection fraction  = 0.37. A subset of 39 patients who provided mean values equal to those considered in this scenario was used to estimate the Kaplan-Meier curve.