Inference for mutually exclusive competing events through a mixture of generalized gamma distributions

Abstract

Background: Time-to-event data with 2 or more types of endpoints are found in many epidemiologic settings. Instead of treating the times for one of the endpoints as censored observations for the other, we present an alternative approach where we treat competing events as distinct outcomes in a mixture. Our objective was to determine if and how the mixture was modified in response to an intervention.

Methods: We used a mixture of generalized gamma distributions to concatenate the overall frequency and distribution of the times of 2 competing events commonly observed in critical care trials, namely (1) unassisted breathing followed by discharge home alive and (2) in-hospital death. We applied our proposed methods to data from 2 randomized clinical trials of critically ill patients.

Results: Mechanical ventilation with lower tidal volumes modified the mixture (P = 0.103) when compared with traditional tidal volumes by lowering the overall frequency of death (P = 0.005), rather than through affecting either the distributions of times to unassisted breathing (P = 0.477) or times to death (P = 0.718). Likewise, use of a conservative versus a liberal fluid management modified the mixture (P < 0.001) by achieving earlier times to unassisted breathing (P < 0.001) and not through affecting the overall frequency of death (P = 0.202) or the distribution of times to death (P = 0.693).

Conclusions: A mixture approach to competing risks provides a means to determine the overall effect of an intervention and insights into how this intervention modifies the components of the mixture.

Figure 1

Parametric and non-parametric estimates of the cumulative percentage of ventilated patients in the tidal volume trial who either achieved unassisted breathing or died at various days after randomization, stratified by study group. Parametric estimates are depicted with smooth lines and non-parametric estimates with steps.

Figure 2

Estimated values for the overall probability of unassisted breathing, the conditional times to unassisted breathing, and the conditional times to death in the tidal volume trial, stratified by study group. Top panel: Percentages above and below the filled circle correspond to the overall probabilities of death (1 – π̂) and of unassisted breathing (π̂), respectively. The whisker-and-brackets correspond to the 95% CIs. Center panel: Median and interquartile range of the conditional times to unassisted breathing in a diamond-and-whiskers plot. The diamond corresponds to the median, and the whiskers correspond to the interquartile range. The amount of filling inside the diamond is proportional to the overall probability of unassisted breathing and the percentage in the center corresponds to the overall probability of unassisted breathing. Bottom panel: Similar to the center panel but for the times to death.

Figure 3

Ratio of cumulative incidences of unassisted breathing and death of the 6 ml/kg to the 12 ml/kg study groups in the NIH Acute Respiratory Distress Syndrome tidal volume trial for the first 28 days after randomization. The continuous, blue line depicts the ratio of cumulative incidences of unassisted breathing of the 6 ml/kg to 12 ml/kg study group. The dashed, blue lines correspond to a 95% bootstrap CI. A ratio of cumulative incidences of unassisted breathing of the 6 ml/kg to the 12 ml/kg study groups above 1 favors the 6 ml/kg strategy and below 1 favors the 12 ml/kg strategy. The blue circle corresponds to the RR of unassisted breathing of the 6 ml/kg to the 12 ml/kg strategy, and vertical blue line corresponds to its 95% bootstrap CI. The continuous, red line depicts the ratio of cumulative incidences of death of the 6 ml/kg to 12 ml/kg study group. The dashed, red lines correspond to a 95% bootstrap CI. A ratio of cumulative incidences of death of the 6 ml/kg to the 12 ml/kg study groups below 1 favors the 6 ml/kg strategy and above 1 favors the 12 ml/kg strategy. The red circle corresponds to the RR of death of the 6 ml/kg to the 12 ml/kg study groups, and the vertical red line corresponds to its 95% bootstrap CI.

Figure 4

Parametric and non-parametric estimates of the cumulative percentage of ventilated patients in the fluid management trial who either achieved unassisted breathing or died at various days after randomization, stratified by study group. Parametric estimates are represented with lines and non-parametric estimates are represented with steps.

Figure 5

Estimated values for the overall probability of unassisted breathing (and its complement, the overall probability of death), the conditional times to unassisted breathing, and the conditional times to death in the fluid management trial, stratified by study group. Top panel: Percentages above and below the filled circle correspond to the overall probabilities of death (1 – π̂) and of unassisted breathing (π̂), respectively. The whisker-and-brackets correspond to the 95% CIs. Center panel: Median and interquartile range of the conditional times to unassisted breathing in a diamond-and-whiskers plot. The diamond corresponds to the median, and the whiskers correspond to the interquartile range. The amount of filling inside the diamond is proportional to the overall probability of unassisted breathing and the percentage in the center corresponds to the overall probability of unassisted breathing. Bottom panel: Similar to the center panel but for the times to death.

Figure 6

Ratio of cumulative incidences of unassisted breathing of the conservative to liberal strategy of fluid management in the NIH Acute Respiratory Distress Syndrome fluid management trial for the first 28 days after randomization. The continuous, blue line depicts the ratio of cumulative incidences of unassisted breathing of the conservative to liberal strategy. The dashed, blue lines correspond to a 95% bootstrap CI. A ratio of cumulative incidences of unassisted breathing of the conservative to liberal strategy above 1 favors the conservative strategy and below 1 favors the liberal strategy. The horizontal, continuous, dark grey line corresponds to the RR of unassisted breathing of the conservative to the liberal strategy, and the light grey shaded area corresponds to its 95% bootstrap CI. The continuous, red line depicts the ratio of cumulative incidences of death of the conservative to liberal strategy. The dashed, red lines correspond to a 95% bootstrap CI. A ratio of cumulative incidences of death of the conservative to liberal strategy below 1 favors the conservative strategy and above 1 favors the liberal strategy. The red circle corresponds to the RR of death of the conservative to liberal strategy, and the vertical red line corresponds to its 95% bootstrap CI.