Ann Thorac Surg 2001;71:1885-1887
© 2001 The Society of Thoracic Surgeons
Original article: cardiovascular
Actuarial and actual analysis of surgical results: empirical validation
Gary L. Grunkemeier, PhDa,
Richard P. Anderson, MDc,
Albert Starr, MDb
a Medical Data Research Center, Providence Health System, Portland, Oregon, USA
b Providence Heart Institute, Providence Health System, Portland, Oregon, USA
c Department of Surgery, Virginia Mason Clinic, Seattle, Washington, USA
Accepted for publication February 12, 2001.
Address reprint requests to Dr Grunkemeier, 9155 SW Barnes Rd, Suite #33, Portland, OR 97225
e-mail: ggrunkemeier{at}providence.org
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Abstract
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This report validates the use of the Kaplan-Meier (actuarial) method of computing survival curves by comparing 12-year estimates published in 1978 with current assessments. It also contrasts cumulative incidence curves, referred to as "actual" analysis in the cardiac-related literature with Kaplan-Meier curves for thromboembolism and demonstrates that with the former estimate the percentage of events that will actually occur.
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Introduction
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In 1974, we [1] published an expository report recommending use of the actuarial method for the analysis of time-related postoperative events. This method estimates survival and event-free curves for an ongoing cohort out to the time of the longest-surviving patient, even though the observation times for most of the patients are much shorter. We now have the opportunity to confirm the validity of this technique by comparing survival estimates from an early series [2] published in 1978, when relatively few of the patients had died, to the current estimates, when the majority of patients have died.
In the last few years, the use of cumulative incidence analysis, a method referred to as "actual" analysis, has become popular for reporting nonfatal events [3] because it estimates the percentage of patients who will actually have an event. Our mature series also provides an opportunity to demonstrate the difference between actuarial and "actual" analysis, as most of the patients from the 1978 series [2] have died.
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Statistical methods
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Actuarial analysis
To estimate the survival curve for a group of patients before they have all died, there are two nonparametric (ie, distribution-free) methods in use: the life-table method [4] and the product-limit method of Kaplan and Meier [5]. They differ by whether or not the data are grouped before analysis. Some use the word actuarial to refer only to the life-table method, but we generalize that usage to include the Kaplan-Meier (K-M) method.
Both methods give estimates of the percentage of patients estimated to be alive at a given time, say 20 years, even though most patients may still be alive with follow-up much shorter than 20 years. The life-table method groups the data, often into yearly intervals; survival percentages are plotted for each year and are usually connected by a straight line. The K-M method plots a point wherever an event occurs; the points are connected by a step function. Before the widespread use of computers and statistical packages, the life-table method was more popular, because the computations could be done using a hand calculator or could be programmed into a spreadsheet. Now, the K-M method is used almost universally.
"Actual" analysis
The K-M method is also used to produce event-free curves for events other than death, such as thromboembolism (TE) and structural valvular deterioration. For this exposition, we present TE more directly as event percentages, rather than event-free percentages, by subtracting the K-M estimates from 100%. The K-M curves thus estimate the percentage of patients in whom TE would occur by, say, 20 years if every patient lived and was at risk for such an event for at least 20 years. This estimate is greater than the percentage of patients who really will sustain a thromboembolic event by that period because many patients will die TE free before 20 years, thus reducing the number of events that will actually occur. Also, some valve explantation will be performed to preclude a subsequent thromboembolic event.
The estimate of the percentage of patients who actually will experience TE is given by the cumulative incidence function [3], often referred to as "actual" analysis in cardiac-related reports [6–10]. Because the "actual" method is still an estimate of events yet to happen, at least until the series has been completely observed, we often put it in quotes.
Clinical material
In 1978, we [2] reported the results for isolated Starr-Edwards Silastic ball valves implanted from 1965 through 1977. We published K-M survival and TE-free curves to 12 years for the 249 operative survivors who underwent aortic valve replacement (models 1200 and 1260) and the operative survivors of mitral valve replacement (model 6120). At that time, the maximum follow-up was 12 years, with a mean follow-up of 4 years for the aortic position and 5 years for the mitral position (Table 1). We have continued to prospectively follow these patients and recently identified them in our database. The maximum follow-up was then 32 years, with means of 12 years and 11 years for the aortic and mitral positions, respectively.
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Table 1. Survival Information for Operative Survivors of Isolated Starr-Edwards Silastic Ball Valve Implantation From 1965 Through 1977a
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Results
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In 1978, only 19% of the patients having aortic valve replacement and 28% of the group having mitral valve replacement had died; the rest were censored (still at risk) before 12 years (see Table 1). In 1998, 74% of the patients with aortic valve replacement and 67% of the patients who had mitral valve replacement had died, and only 8% and 16%, respectively, were censored before 12 years (see Table 1). The current survival estimates are very similar to those computed in 1978 (Fig 1; see Table 1). The 1978 K-M results for TE are not directly comparable because our definitions have changed since then. Originally we excluded cerebrovascular events in patients with proven cerebrovascular disease, but since publication of the guidelines for reporting [11] in 1988, we have included them.
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Fig 1. Survival curve for patients having mitral valve replacement as reported in 1978, shown as filled circles, compared with 1998 estimates, shown as solid line. Numbers of patients at risk (alive) at 10, 20, and 30 years are shown above the horizontal axis. (SE = standard error.)
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For TE, the current "actual" estimates are much lower than the K-M estimates and only slightly higher than the percentage of patients who actually have experienced TE to date (Table 2). The 30-year K-M estimates of TE for the aortic series is 63%, whereas the "actual" estimate is only 34% (Fig 2). Only 80 patients who underwent aortic valve replacement sustained a thromboembolic event to date, for a raw percentage of 33%. This agrees well with the "actual" estimate for this early series, in which few additional thromboembolic events would be expected to occur before 30 years, as only 24 patients, at a mean follow-up of 17 years, are still at risk for such an event in the future (see Table 2). For the mitral valve series, the 30-year actuarial and "actual" TE estimates are 58% and 42%, respectively. The percentage of patients who have sustained at least one thromboembolic event to date is 41% (47 patients), and only a few more events would be expected to occur before 30 years, as only 14 patients, at a mean follow-up of 18 years, are still at risk for a first TE (see Table 2).
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Table 2. Current Thromboembolism Information for Operative Survivors of Isolated Starr-Edwards Silastic Ball-Valve Implantation From 1965 Through 1977a
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Fig 2. Thromboembolism (TE) curves for patients having aortic valve replacement. Kaplan-Meier (KM) estimates are shown by the upper line and cumulative incidence or "actual" estimates, the lower line. Numbers of patients at risk (alive and TE free) at 10, 20, and 30 years are shown above the horizontal axis.
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Comment
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The report has two primary goals. The first is to show that K-M survival estimates made in 1978, when most patients had not completed long-term (ie, 12-year) follow-up, are remarkably consistent with the findings when these patients have all had the opportunity to complete 12 years. Thus, the K-M estimates have been validated by the empirical survival curves. The second goal is to contrast the actuarial (K-M) and "actual" methods for analyzing events and to demonstrate that the actuarial method overestimates the percentages of nonfatal events that will really occur.
If the death times of all the patients in a series are known, it is a simple matter to cumulate them across time to get the cumulative death curve or, subtracting that from 100%, the survival curve. This is the "empirical" or observed, as opposed to the estimated, survival curve. However, in the usual ongoing series, we know the death times of only some patients because most are still alive. These are called "censored" patients, which means that they have not yet experienced the event in question, in this case, death. (Anyone reading this sentence has a survival time that is censored at his or her current age.) The K-M method allows us to estimate the final death or survival curve as we would expect it to look after all the patients have completed their lifetimes.
The fundamental assumption of the K-M method is that those who are censored (ie, have not yet died) at any postoperative time have the same risk of future death as those who have already died at longer postoperative times. That is, they will eventually contribute a death to the series, and the distribution of those future deaths will be the same as for the deaths that have already happened. The close agreement between our recent review and the 1978 report confirms that this was the case in this series.
This can be framed as a problem of competing risks [12]; being censored (alive) is a competing risk for death. The K-M curve estimates what the survival curve would look like if "alive" were eliminated as a competing risk and all patients had been allowed to continue on to death. When this method is applied to TE, being censored means not having had a thromboembolic event (yet), and there are (at least) two categories of TE-free patients: those who have died TE free and those who are still alive. In this setting, the K-M curve estimates what the TE curve would look like if all the censored (alive and dead) patients were allowed to continue on to sustain a thromboembolic event. This results in a nonrealistic curve that includes a percentage of future TEs that will never happen, namely, those attributed to the currently dead patients. The cumulative incidence curve provides the realistic estimate by making the dead (but not the alive) patients ineligible to contribute future thromboembolic events to the series. This, of course, results in a lower estimate of patients with TE, or with any other nonfatal event. Figure 2 shows how dramatic this difference can be.
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References
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Abstract
Introduction
