Predictive accuracy of novel risk factors and markers: A simulation study of the sensitivity of different performance measures for the Cox proportional hazards regression model

Abstract

Predicting outcomes that occur over time is important in clinical, population health, and health services research. We compared changes in different measures of performance when a novel risk factor or marker was added to an existing Cox proportional hazards regression model. We performed Monte Carlo simulations for common measures of performance: concordance indices ( c, including various extensions to survival outcomes), Royston's D index, R²-type measures, and Chambless' adaptation of the integrated discrimination improvement to survival outcomes. We found that the increase in performance due to the inclusion of a risk factor tended to decrease as the performance of the reference model increased. Moreover, the increase in performance increased as the hazard ratio or the prevalence of a binary risk factor increased. Finally, for the concordance indices and R²-type measures, the absolute increase in predictive accuracy due to the inclusion of a risk factor was greater when the observed event rate was higher (low censoring). Amongst the different concordance indices, Chambless and Diao's c-statistic exhibited the greatest increase in predictive accuracy when a novel risk factor was added to an existing model. Amongst the different R²-type measures, O'Quigley et al.'s modification of Nagelkerke's R² index and Kent and O'Quigley's [Formula: see text] displayed the greatest sensitivity to the addition of a novel risk factor or marker. These methods were then applied to a cohort of 8635 patients hospitalized with heart failure to examine the added benefit of a point-based scoring system for predicting mortality after initial adjustment with patient age alone.

Keywords: Cox proportional hazards model; Monte Carlo simulations; Survival analysis; discrimination; model performance; predictive accuracy; predictive models; risk factors.

Figure 1.

Relationship between change in c-statistic and c-statistic of univariate model (low event rate – uncorrelated binary).

Figure 2.

Relationship between change in D index and D index of univariate model (low event rate – uncorrelated binary).

Figure 3.

Relationship between change in R2 and R2 of univariate model (low event rate – uncorrelated binary).

Figure 4.

Relationship between change in c-statistic and c-statistic of univariate model (high event rate – uncorrelated binary).

Figure 5.

Relationship between change in D index and D index of univariate model (high event rate – uncorrelated binary).

Figure 6.

Relationship between change in R2 and R2 of univariate model (high event rate – uncorrelated binary).

Figure 7.

Relationship between change in model accuracy and model accuracy of univariate model (low event rate – uncorrelated continuous).

Figure 8.

Relationship between change in model accuracy and model accuracy of univariate model (high event rate – uncorrelated continuous).