Dynamic prediction of competing risk events using landmark sub-distribution hazard model with multiple longitudinal biomarkers

Abstract

The cause-specific cumulative incidence function quantifies the subject-specific disease risk with competing risk outcome. With longitudinally collected biomarker data, it is of interest to dynamically update the predicted cumulative incidence function by incorporating the most recent biomarker as well as the cumulating longitudinal history. Motivated by a longitudinal cohort study of chronic kidney disease, we propose a framework for dynamic prediction of end stage renal disease using multivariate longitudinal biomarkers, accounting for the competing risk of death. The proposed framework extends the local estimation-based landmark survival modeling to competing risks data, and implies that a distinct sub-distribution hazard regression model is defined at each biomarker measurement time. The model parameters, prediction horizon, longitudinal history and at-risk population are allowed to vary over the landmark time. When the measurement times of biomarkers are irregularly spaced, the predictor variable may not be observed at the time of prediction. Local polynomial is used to estimate the model parameters without explicitly imputing the predictor or modeling its longitudinal trajectory. The proposed model leads to simple interpretation of the regression coefficients and closed-form calculation of the predicted cumulative incidence function. The estimation and prediction can be implemented through standard statistical software with tractable computation. We conducted simulations to evaluate the performance of the estimation procedure and predictive accuracy. The methodology is illustrated with data from the African American Study of Kidney Disease and Hypertension.

Keywords: Competing risks; Fine-Gray model; dynamic prediction; landmark analysis; longitudinal biomarkers; prediction model.

Figure 1.

Estimated surface of the cumulative incidence function over the landmark time and prediction horizon. This shows an examplary population with age = 55, $\text{eGFR}=45ml/min/1.73m^2$, eGFR. $\text{slope}=0$, $\text{UP}/\text{Cr}=0.3g/g$, $\text{albumin}=4g/\mathit{\text{dL}}$, and hospitalization within the past year.

Figure 2.

Individual risk predictions for three selected subjects: subject 1 was censored (dotted vertical green line), subject 2 had ESRD (dotted vertical red line) and subject 3 died (dotted vertical black line). Three biomarkers are plotted over time: “G” is eGFR $\left(\mathit{\text{ml}}/\mathit{\text{min}}/1.73m^2\right)$, “R” is log-urine protein-to-creatinine ratio $(g/g)$, and “A” is albumin $(g/dL)$. The connected red dots are predicted probabilities of ESRD within a horizon of ${\tau }_1=3$ years. The gray vertical bars represent episodes of hospitalization, with the two vertical borders being admission and discharge dates. The connected black dots are the predicted probability of death within ${\tau }_1=3$ years. The y-axis to the left is the scale of eGFR, and the y-axis to the right is the scale of predicted probabilities (0 to 1). The other two biomarkers, log-UP/CR and albumin, are re-scaled to be displayed in the same plot with eGFR but their respective scales are not shown. The dynamic predicted probabilities of ESRD are calculated using the dynamic SDH model with four predictors: eGFR, eGFR slope in the past three years, log-UP/CR and phosphorus. The dynamic predicted probabilities of death are calculated using the dynamic SDH model with four predictors: current age, serum albumin, any hospitalization within the past year, and log urine potassium.

Figure 3.

Individual dynamic predicted CIF for the three selected subjects in Figure 2. Each row in the panel represents one subject, the three columns are the predictions made at landmark years $s=\mathrm{3,5},7$. The prediction is made at the vertical blue dashed lines. The predicted CIFs up to ${\tau }_1=3$ years are plotted for the event of ESRD (red curve) and death (black curve). Symbols in the figure are similar to Figure 2.