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J Thorac Cardiovasc Surg 2008;135:457-459
© 2008 The American Association for Thoracic Surgery
Letter to the Editor |
Lake Forest, CA 92630
To the Editor:
The recent editorial by Bodnar and Blackstone1
suggests that the so-called "actual" analysis has been misused at times and makes suggestions for limiting its future use. I generally agree with the editorial.
However, the method is mathematically valid and does have some important uses. I fear that in overreacting to some misuses we run the risk of throwing out the baby with the bath water. In this letter, I discuss some of the mathematical background and some applications in which use of competing risks analysis is critical for proper understanding of a clinical situation.
I agree with the editorial that the term "actual" is potentially misleading and that other terminology should be used. The general area of analysis is often referred to as "competing risks" analysis, and the term "cumulative incidence" seems to be widely accepted in this area. I agree with the suggestion that the term "cumulative incidence" be used.
One point that must be made is that competing risks analysis rests on a completely sound mathematical footing, and cumulative incidence is a precisely defined mathematical concept. The general setting is that there are two (or more) competing risks. Each risk will have its own probability distribution, and the concept of the first event to be observed is precisely defined. Theoretical treatments are given in Kalbfleisch and Prentice2 and Andersen and associates3; formula 4.4.19 of the latter reference includes a derivation of the standard error. A rather more readable treatment, which clearly illustrates the difference between Kaplan–Meier (actuarial) analysis and competing risks analysis, is given by Gooley and colleagues.4
The methodology of competing risks is standard in many medical areas. The article by Gaynor and coworkers5 discusses an oncology example with three competing risks; a standard error formula is given, but it is more difficult to use than the one in the article by Andersen and associates.3 The article by Klein6 is also in the oncology area; it discusses handling of covariates.
In all of the aforementioned references the probability of a particular event being the first event observed is computed as a function of time; the term "cumulative incidence" is used to denote the graph showing these probabilities. (Note that the term "cumulative hazard" is often used for a different concept; the two should not be confused.)
In the valve examples studied by Blackstone and Kirklin7 and by Grunkemeier and colleagues,8 there will be some true mathematical distribution for valve failure, and there will be another true mathematical distribution for death. In the actuarial analysis of valve failure, death is merely a circumstance that prevents one from observing failure events. The end of follow-up in a clinical trial also prevents observation of failure. Both circumstances have the same meaning to the statistician, albeit not to the patient; in either case, data are censored at the last time that the valve was known to be good. The Kaplan–Meier algorithm is well suited for estimating both the valve failure and death distributions, but there are circumstances in which a parametric model is preferable.
There will also be some true mathematical distribution for the random variable valve failure observed before death. This distribution could be computed if the other two distributions were known; if one has a series of patient data, the distribution must be estimated. In this situation a patient death directly affects the cumulative incidence estimate, because this represents a case in which the first event will not be valve failure. The end of follow-up still causes censoring, because in such a case one does not know which event would ultimately occur first. The distribution to be estimated here is mathematically different from the distribution of valve failure, and the algorithms are accordingly different. Competing risks analysis is not implemented as a standard part of SAS (SAS Institute, Inc, Cary, NC); however, the SAS code, including the standard error estimates of reference 3, has been published by Anderson.9
As an example of correct use of competing risks analysis, the valve studies7,8 considered advising a patient as to his or her own future risks. Other situations would include an insurance company that might want to estimate the potential cost of replacement surgery or a valve manufacturer that might want to estimate future sales of replacement valves. The Kaplan–Meier analysis would not be appropriate in any of these cases; meaningful results cannot be obtained without use of competing risks analysis.
Another example comes from left ventricular assist devices. There are current clinical studies involving a combined end point of death, device failure, and stroke. Whichever event comes first defines the end point for the particular patient; censoring is almost exclusively the result of a patient being alive and event-free at the end of follow-up. In planning such a trial, it is vital for a manufacturer to use competing risks analysis to study the cumulative incidence of the various components of the combined end point.
Finally, there are the numerous oncology studies, including two aforementioned articles5,6; the article by Freidlin and Korn10 also includes a simulation comparing different analysis methods.
If one is going to present competing risks analysis, in the above or other circumstances in which the analysis is appropriate, the question remains as to how the results might be presented. I have two suggestions.
First, I suggest that it may be useful to present on one set of axes a graph with three (or more) curves. There will be one curve showing the cumulative incidence of each risk. There will also be one curve showing the freedom from all events. In these graphs the various cumulative incidence curves increase, and the freedom from all events decreases; at all times the values represented by the curves add to 1. There is no place in this graph for the Kaplan–Meier actuarial curves showing freedom from individual events; using the vocabulary of Blackstone and Bodnar, the competing risks curves present the apples, and the actuarial curves are the oranges.
An example of such a graph is shown in Figure 1; the analysis is of explant resulting from structural valve deterioration for a hypothetical bioprosthetic valve. Literature examples of such graphs include Figure 1 in Banbury and colleagues,11 Figure 3 in Blackstone and Lytle,12 and Figure 3 in Kojori and colleagues.13
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The influence of age on valve failure presents an important issue, and it is natural to include age as a covariate in actuarial analysis. In this context, actuarial analysis is generally performed with the Cox proportional hazards algorithm. The results of such an analysis are widely known: valves last somewhat longer in older patients. If different valve models are being compared, the valve model effect can be separated from the age effect by proportional hazards.
Alternatively, patients can be stratified into age groups, and then the Kaplan–Meier algorithm can be used within each age group; slight differences in age distribution will not have much impact on the final results. Countless valve series have been presented using various age stratifications, and valuable comparisons can be made from published data.
The situation is considerably different with competing risks analysis. Age has a dramatic effect on patient survival, and this translates directly into a dramatic effect on cumulative incidence of valve failure. Inasmuch as this effect goes in the same direction as the known effect of age on valve failure itself, the combination of the effects makes it virtually impossible to validly compare cumulative incidence curves from different valve series. I agree with the recommendation of Blackstone and Bodnar that this comparison should never be done. I would go further and suggest that cumulative incidence curves from different series should never be presented together in the same graph; the temptation to do an invalid comparison is simply too strong.
If one had complete outcome and covariate information for all patients, a proportional hazards version of the competing risks analysis could in principle be performed.2,6,14 No single analyst would generally have such data for valves from different manufacturers. Even if such data were available, I suggest that it would be much safer to use actuarial analysis to compare valves; the latter removes one source of noise, and it does not seem that the competing risks analysis adds anything valuable to the comparison.
Footnotes
* From 1994 through 2001 the author was a full time employee of Edwards Lifesciences, LLC. Since that time he has been a consultant to various cardiovascular device manufacturers, including Edwards. 
References
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