On the Use of Cause-Specific Failure and Conditional Failure Probabilities: Examples from Clinical Oncology Data

Abstract

Nonparametric maximum likelihood estimation of the probability of failing from a particular cause by time t in the presence of other acting causes (i.e., the cause-specific failure probability) is discussed. A commonly used incorrect approach is to take 1 minus the Kaplan-Meier (KM) estimator (1 – KM), whereby patients who fail of extraneous causes are treated as censored observations. Examples showing the extent of bias in using the 1-KM approach are presented using clinical oncology data. This bias can be quite large if the data are uncensored or if a large percentage of patients fail from extraneous causes prior to the occurrence of failures from the cause of interest. Each cause-specific failure probability is mathematically defined as a function of all of the cause-specific hazards. Therefore, nonparametric estimates of the cause-specific failure probabilities may not be able to identify categorized covariate effects on the cause-specific hazards. These effects would be correctly identified by cause-specific cumulative hazard or KM plots in which the extraneous causes of failure are treated as censored observations. Examples are provided. Finally, nonparametric graphical representation of the two distinct cause-specific failure components of the mixture model (i.e., the probability of ever failing from a particular cause and the time-to-failure distribution given that a patient will fail of that cause) are presented. The difficulty in extrapolating the nonparametric estimates beyond the range of observed failure times is highlighted. In addition, the mathematical relationship of a single covariate z and the two cause-specific failure components is shown for the case where z acts multiplicatively on the cause-specific hazards. Examples are considered using the important prognostic factors in adult soft tissue sarcoma (STS) of the extremity and adult acute lymphoblastic leukemia (ALL). In each of the mathematical and actual examples considered, the characteristic that was associated with a higher hazard rate of failure from the disease was also associated with a higher probability of ever failing from the disease as well as a shorter time-to-failure distribution given that failure due to the disease will occur. Thus it may be quite common in chronic disease situations to find prognosticators that are associated with both cause-specific failure components.

Key Words:

• Cause-specific failure probability
• Cause-specific hazard
• Conditional failure probability
• Cumulative hazard
• Mixture model
• Nonparametric estimates