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J Thorac Cardiovasc Surg 2001;122:216-219
© 2001 The American Association for Thoracic Surgery
Statistics for the Rest of Us (STATS) |
From the Providence Health System, Portland, Ore.
Presented at The American Association for Thoracic Surgery Symposium on Adult Cardiac Surgery, May 6, 2001, San Diego, Calif.
Received for publication April 30, 2001. Accepted for publication May 21, 2001. Address for reprints: Gary Grunkemeier, PhD, 9155 SW Barnes, #33, Portland, OR 97225 (E-mail: ggrunkemeier{at}providence.org).
Abstract
Objective: To describe the cumulative incidence (actual) method of analysis and to contrast it with the Kaplan-Meier method.
Method: We use data on porcine valve replacement to illustrate these two statistical techniques.
Results: The "actual" analysis estimates the percentage of events expected to occur. The percentage given by the Kaplan-Meier method is much larger.
Conclusion: Actual (cumulative incidence) analysis is preferred for estimating the probability of occurrence of a nonfatal time-related event.
See related editorial on page 207.
It is not usually appreciated that the Kaplan-Meier (KM; "actuarial") estimate of event-free percentages, when used for nonfatal time-related events, gives overly pessimistic predictions. Another method, called cumulative incidence analysis, provides the "actual" probabilities of event occurrences.
Survival Curves
Follow-up studies involve the collection of information regarding time-related events, including death and nonfatal events. Instead of the usual summary statistics (eg, mean, standard deviation), a complete description of such time-to-event variables is usually of interest. The proportion of event times that are greater than a given time are plotted as a function of time to produce a survival (or event-free) curve. The result is a curve starting at 100% at time zero and decreasing steadily to 0% at the longest survival time. For a completed series, when all death times are known, the (empirical) survival curve can be readily calculated as the complement of the cumulative sum of deaths at each point in time.
However, in ongoing studies, the investigator cannot know the complete lifetimes of patients who are still alive at the time of the study, and survival curves must be estimated before all of the patients have died. KM analysis solves this problem by assigning to each person who is still alive (called a "censored" observation) a probability of death at each future time equal to the fraction of patients who have already died at that time. This assumes that the distribution of the times to death for currently alive patients will follow the pattern of those who have already died. Censored patients are added back into the calculation as future deaths distributed over future time.
Nonfatal Events
The KM method is also used for events other than death, for example, structural valve deterioration (SVD) or thromboembolism (TE). When applied to an event other than death, this method includes as "censored" any patients who have not yet experienced that event, including those who have died and will never have the event. The resulting event-free estimate attempts to answer the hypothetical question, "What is the risk of the event if no patient ever died?" An alternative method, recently called "actual" analysis in the cardiac literature, modifies this estimate to exclude future events attributed to already deceased patients and answers the more direct question, "What is the risk of the event?"
Example
To illustrate these concepts, we used data from a previous investigation of porcine valve SVD, with 4895 operative survivors of isolated valve replacement and 29,610 valve-years of follow-up.
1 The Gompertz distribution was used to model survival and the Weibull distribution to model SVD. This series contained patients from the beginning of porcine valve usage, using first-generation porcine valves and a mean age of only 60 years. Current results would be expected to be improved, but these suffice for our expository purposes.
Survival (Two-State Model)
A useful technique for describing time-related processes, in this and more complex situations, is the multistate model. For survival analysis, there are 2 states, alive and dead (Figure 1). All patients begin in the alive state and make a transition from alive to dead at varying times according to some statistical distribution. The transition is inevitable and final: death is called an absorbing state because once a patient reaches that state his process is over. At any point in time, the sum of alive plus dead is 100% and, at the end of the process, 100% are dead. The survival curve is the plot of the percentage alive over time that corresponds to this process.
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describes the result for the average (60-year-old) patient in this series; for older patients the difference between KM and actual estimates is even greater.
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We recently provided an empirical validation of the cumulative incidence method using a series of Starr-Edwards valves implanted from 1965 through 1977 and prospectively followed up through 1998, using TE as the nonfatal event. 2 This almost complete series provided an opportunity to compare the KM and actual estimates with the percentage of patients who really had a TE.
For the aortic series, the 30-year KM estimate of TE was 63% and the actual estimate was 34%, with only 33% of the patients having had a TE to date. For the mitral series, the 30-year actuarial and actual TE estimates were 58% and 42%, respectively, and 41% had had a TE. Thus, in both positions, the actual estimates were much lower than the KM estimates and only slightly higher than the percentage of patients who had had a TE to date. Only a few additional TEs would be expected to occur before 30 years, because only a few patients remained at risk.
Conclusion
For event-free analysis, the cumulative incidence (actual) method, unlike the KM method, assumes that only living patients continue to be at risk for a future event and thus estimates the events actually sustained. The result is that the actual event percentages are smaller than the KM estimates, and more so in older patients.
The actual method provides a valid probability of failure and is essential for individual patient counseling and population management. For example, for management of patients with failure-prone implanted cardiac devices, it is the actual estimate of failure that must be weighed against the risk of prophylactic explantation.
Appendix: A Note on Terminology
Actuarial analysis originally referred to the life-table method, where events are grouped into intervals. 3 Since the availability of computers, the KM method 4 is used almost universally. It is called the product-limit method because it can be considered a life-table method in which the intervals are so small that they contain only one event time. In this sense, it could be considered part of the actuarial family. The essence of the 2 methods is the same—incorporating partially completed lifetimes by the technique of censoring. In the case of survival, the censoring refers to patients still alive. In the case of nonfatal events, censoring is done for patients still free of the event, including deaths.
Actual analysis is a relatively new designation. Many terms have been used in the statistical literature. Cumulative incidence is generally preferred, 5,6 but several other adjectives have been used as well, including crude, 7 unadjusted, 8 absolute, 9 influenced, 10 and observable. 11 Recently the term actual was used to refer to the cumulative incidence method, 12 and this term seems to be accepted in the cardiac literature. 13-18
Acknowledgments
We appreciate the interest and encouragement of Dr D. Craig Miller in advocating the use of proper statistical methods and in his continued endorsement of the value of "actual" analysis. 1,15,19
References
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