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Competing risks analysis for discrete time‐to‐event data

Advanced Review

Moritz Berger, Matthias Schmid

Published Online: Sep 08 2020

DOI: 10.1002/wics.1529

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Abstract This article presents an overview of statistical methods for the analysis of discrete failure times with competing events. We describe the most commonly used modeling approaches for this type of data, including discrete versions of the cause‐specific hazards model and the subdistribution hazard model. In addition to discussing the characteristics of these methods, we present approaches to nonparametric estimation and model validation. Our literature review suggests that discrete competing‐risks analysis has gained substantial interest in the research community and is used regularly in econometrics, biostatistics, and educational research. This article is categorized under: Statistical Models > Survival Models Statistical Models > Semiparametric Models Statistical Models > Generalized Linear Models

Analysis of the CRASH‐2 data. The left panel presents the estimated CIF for death due to bleeding of a randomly selected female patient aged 41 years (Itime = 0.5, Itype = bp, Spb = 120, Hr = 70, Rr = 15, CCrt = 3, Gcs = 15), as obtained from the discrete cause‐specific hazards model in Table 2. The right panel presents the corresponding CIF obtained from the discrete subdistribution hazard model in Table 3. Gray lines refer to the true sample data (Sbp = 120) whereas the black lines were obtained by switching the Sbp value from 120 to 80 mmHg. The estimates presented in the left panel were computed with the help of Formula 1, that is, by using all cause‐specific hazard estimates shown in Table 2. The estimates presented in the right panel were computed with the help of the formula , see also p. 8. Hence the calculations for the left panel involved three times as many coefficients as the calculations for the right panel, which were based on the values in Table 3

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Analysis of the CRASH‐2 data. The figure depicts a calibration plot for the predicted cause‐specific hazard of the target event “death due to bleeding” (log scale). Predicted cause‐specific hazard values were obtained by applying the model in Table 2 to the 6,346 patients that were not used for model fitting (one‐third of the analysis data set). The plot is based on G = 10 subsets of the patients. A constant value of 0.0001 was added to the predicted and observed cause‐specific hazard values before application of the log transformation. The p‐value of the likelihood ratio test (joint null hypothesis “H₀ : a_j = 0, b_jj = 1 ∀ j”, Heyard et al., 2020) was 0.247, indicating no significant deviations from a well‐calibrated model

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Analysis of the CRASH‐2 data. The figure visualizes the results of the CART‐based procedure by Berger et al. (2019) when applied to two‐thirds of the analysis data. Hellinger's distance was used for node splitting. Panel (a) depicts the tree structure that resulted from pruning with the BIC. Part of the left branch of the tree (corresponding to time points t ≤ 3) is illustrated in Panel (b). Estimates of the cause‐specific hazards are presented in each of the terminal nodes (1 = death due to bleeding, 2 = death due to head injury, 3 = death due to other causes). The tree‐based CIF estimate (referring to death due to bleeding) for the example patient from Figure 1 is visualized in Panel (c)

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