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On Numerators, Denominators, and Virgules
O n Numerators, Denominators, and Virgules
T
o express the accomplishmentsof therapy, physicians are accustomed to reporting an outcome rate. The denominator in such a rate consists of the people exposed to the treatment. The numerator consists of the number of those people who experienced the outcome event, which may be death, life, or some other entity. When the numerator and denominator numbers are divided to form the rate, the quotient is written in the form of a fraction, such as ah,in which the virgule (“slash mark”) also has a distinct role. It represents the serial time, beginning after onset of therapy, during which the people in the denominator have been followed for observation of the outcome event noted in the numerator. This chronological role of the virgule is acknowledged in the names that are given to these rates. We talk about such entities asjve-year survival rates, one-year mortality rates, or postoperative complication rates. The virgule in an outcome rate has another role beyond that of merely marking time. It also denotes the idea thatall the people in the denominator have been observed or followed for the cited duration. Thus, in reporting a five-year survival rate with such numbers as 23/68 (34%),we imply that all 68 people in the denominator have been followed for at least five years. This aspect of the virgule creates a major problem in many analyses of therapy. When an investigator assembles his data to report the results of therapy, not all the patients under study will have been followed for identical lengths of serial time. Of the total population of patients who have been followed after treatment, some will have been observed for the maximum serial time cited in an outcome rate, but many others will have had a “shortened term” of observation. These shortened terms can occur in two different ways for patients who were still alive when last examined. Some patients, who are still being actively followed, may have entered observation at different dates on the calendar. For example, in 1975, we can report five-year survival for a patient who was first treated in 1969,but not for a patient first treated in 1972. Other patients may have “dropped out” after their last visit. Thus, a patient whom we began to follow in 1969 may have become “lost to follow-up” in 1973. We can report such a patient’s status at four years, when he was last examined, but we know nothing about him thereafter. In the “direct method” of reporting outcome rates, the denominator at each serial time is restricted to the patients who were followed for the cited length of time. In a group that contains a total of 118 patients, the investigator can cite a one-year survival rate for the 96 patients who have been followed for one year, a two-year survival rate for the 73 patients followed for two years, and a three-year survival rate for the 26 patients who have been followed for three years. The direct method is a quite satisfactory procedure for citing outcome rates. The deSupported by US. Public Health Service Grant no. HS 00408 from the National Center for Health Services Research and Development.
364
T H E ANNALS OF THORACIC SURGERY
Editorial nominators and numerators consist of real patients, followed and counted for real lengths of time. In recent years, however, an actuarial (or “life-table”)form of analysis has become popular for dealing with the problems of unequal lengths of follow-up. The main difference between direct and actuarial analysis is in the statistical fate of the shortened-term patients. With the direct procedure, a patient whose follow-up ends between the second and third time interval is counted as one unit in the denominator of the second group but not at all in the third. In the actuarial procedure, such a patient is still counted as one unit in the second time interval but is counted as one-half of a unit in the third. Because of the prestige derived from its association with life insurance work and demography, and because biostatisticians seem to like it, the actuarial procedure is now regularly used for reporting the outcome rates of therapeutic studies. Regardless of whether the denominator of an outcome rate is determined by direct or actuarial methods, however, a major problem can still occur in the numerator. What do we do if the outcome events include something besides life or death? An answer to this question is proposed in this issue of The Annals (p 289) by Drs. Grunkemeier, Lambert, Bonchek, and Starr. The technique they offer has several components. First, they define an outcome event as either death or a specified complication. With this definition, a numerator can be determined for patients who have been “event free.” We can therefore contemplate not merely patients who have survived, but particularly patients who have survived without any complications. What can then be calculated, as shown in the investigators’ Figure 1, is two curves and three rates: the rate of mortality, the rate of complication-accompanied survival, and the rate of complication-free survival. This technique of reporting results is clear, straightforward, and an obviously useful method for denoting posttherapeutic accomplishment. The technique is also applicable in many fields other than cardiac surgery. Not content with this contribution, however, the authors propose two other relative innovations for analyzing the denominators of outcome rates. One idea is to convert the denominator from numbers of patients to numbers of something else, such as prosthetic valves. Instead of reporting the death rates of patients, we could then report the death rate for prosthetic valves. (A valve “dies”either when it is replaced or when the patient dies.) This procedure seems reasonable for assessing the performance of different valves, although the result obviously does not indicate what happened to individual patients. The procedure can thus be useful as an auxiliary set of data to supplement direct information about patients. The other new tactic proposed by the authors is to calculate an “effective sample size.” Instead of citing outcome rates at different serial times for different numbers of patients in the appropriate denominators, this new technique would agglomerate the individual temporal rates into a single value for each treated group. Although the process would permit single, direct comparisons among different treated groups, the technique removes the dynamic aspect of a sequence of serial rates and converts everything into a single time point, which happens, in this case, to be a mean of the temporal results. The effective sample size procedure VOL. 20, NO. 3, SEPTEMBER, 1975
365
Editorial may not receive an enthusiastic reception from many readers since it seems arbitrary, excessively statisticated,and difficult to understand. Many investigative clinicians may prefer to continue comparing one-year rates versus one-year rates and five-year rates versus five-year rates rather than against a mingled temporal collection. Nevertheless, the possible rejection of this proposal by Dr. Grunkemeier and his colleagues should not keep clinicians from recognizing the potential virtues of the other two procedures. An actuarial analysis with multiple numerator events offers a useful description of complex phenomena. An actuarial analysis that enters prostheses, rather than people, in the denominator offers an interesting adjunct to conventional tabulations. Both procedures could be applied, of course, in direct rather than actuarial analyses, but the tide of statistical fashion is currently flowing in the direction of the actuarial technique. For most readers of clinical literature, the evaluation of therapy will continue to require answers to such questions as: (1) Can I understand what the author is doing and what he found?; (2) Does it make sense?; and (3)Do I believe him? An author who wants to convince his readers will also want to display his statistical data in a manner that provides a comprehensible documentation of the results. N o matter how the data are arranged for analysis, the main goal is to give a clear, complete account of the basic information needed for a thoughtful reader to determine what happened and to draw his own conclusions. ALVAN R. FEINSTEIN, M.D.
Department of Internal Medicine Yale University School .fMedicine New Haven, Conn. 06510
366
THE ANNALS OF THORACIC SURGERY
T
o express the accomplishmentsof therapy, physicians are accustomed to reporting an outcome rate. The denominator in such a rate consists of the people exposed to the treatment. The numerator consists of the number of those people who experienced the outcome event, which may be death, life, or some other entity. When the numerator and denominator numbers are divided to form the rate, the quotient is written in the form of a fraction, such as ah,in which the virgule (“slash mark”) also has a distinct role. It represents the serial time, beginning after onset of therapy, during which the people in the denominator have been followed for observation of the outcome event noted in the numerator. This chronological role of the virgule is acknowledged in the names that are given to these rates. We talk about such entities asjve-year survival rates, one-year mortality rates, or postoperative complication rates. The virgule in an outcome rate has another role beyond that of merely marking time. It also denotes the idea thatall the people in the denominator have been observed or followed for the cited duration. Thus, in reporting a five-year survival rate with such numbers as 23/68 (34%),we imply that all 68 people in the denominator have been followed for at least five years. This aspect of the virgule creates a major problem in many analyses of therapy. When an investigator assembles his data to report the results of therapy, not all the patients under study will have been followed for identical lengths of serial time. Of the total population of patients who have been followed after treatment, some will have been observed for the maximum serial time cited in an outcome rate, but many others will have had a “shortened term” of observation. These shortened terms can occur in two different ways for patients who were still alive when last examined. Some patients, who are still being actively followed, may have entered observation at different dates on the calendar. For example, in 1975, we can report five-year survival for a patient who was first treated in 1969,but not for a patient first treated in 1972. Other patients may have “dropped out” after their last visit. Thus, a patient whom we began to follow in 1969 may have become “lost to follow-up” in 1973. We can report such a patient’s status at four years, when he was last examined, but we know nothing about him thereafter. In the “direct method” of reporting outcome rates, the denominator at each serial time is restricted to the patients who were followed for the cited length of time. In a group that contains a total of 118 patients, the investigator can cite a one-year survival rate for the 96 patients who have been followed for one year, a two-year survival rate for the 73 patients followed for two years, and a three-year survival rate for the 26 patients who have been followed for three years. The direct method is a quite satisfactory procedure for citing outcome rates. The deSupported by US. Public Health Service Grant no. HS 00408 from the National Center for Health Services Research and Development.
364
T H E ANNALS OF THORACIC SURGERY
Editorial nominators and numerators consist of real patients, followed and counted for real lengths of time. In recent years, however, an actuarial (or “life-table”)form of analysis has become popular for dealing with the problems of unequal lengths of follow-up. The main difference between direct and actuarial analysis is in the statistical fate of the shortened-term patients. With the direct procedure, a patient whose follow-up ends between the second and third time interval is counted as one unit in the denominator of the second group but not at all in the third. In the actuarial procedure, such a patient is still counted as one unit in the second time interval but is counted as one-half of a unit in the third. Because of the prestige derived from its association with life insurance work and demography, and because biostatisticians seem to like it, the actuarial procedure is now regularly used for reporting the outcome rates of therapeutic studies. Regardless of whether the denominator of an outcome rate is determined by direct or actuarial methods, however, a major problem can still occur in the numerator. What do we do if the outcome events include something besides life or death? An answer to this question is proposed in this issue of The Annals (p 289) by Drs. Grunkemeier, Lambert, Bonchek, and Starr. The technique they offer has several components. First, they define an outcome event as either death or a specified complication. With this definition, a numerator can be determined for patients who have been “event free.” We can therefore contemplate not merely patients who have survived, but particularly patients who have survived without any complications. What can then be calculated, as shown in the investigators’ Figure 1, is two curves and three rates: the rate of mortality, the rate of complication-accompanied survival, and the rate of complication-free survival. This technique of reporting results is clear, straightforward, and an obviously useful method for denoting posttherapeutic accomplishment. The technique is also applicable in many fields other than cardiac surgery. Not content with this contribution, however, the authors propose two other relative innovations for analyzing the denominators of outcome rates. One idea is to convert the denominator from numbers of patients to numbers of something else, such as prosthetic valves. Instead of reporting the death rates of patients, we could then report the death rate for prosthetic valves. (A valve “dies”either when it is replaced or when the patient dies.) This procedure seems reasonable for assessing the performance of different valves, although the result obviously does not indicate what happened to individual patients. The procedure can thus be useful as an auxiliary set of data to supplement direct information about patients. The other new tactic proposed by the authors is to calculate an “effective sample size.” Instead of citing outcome rates at different serial times for different numbers of patients in the appropriate denominators, this new technique would agglomerate the individual temporal rates into a single value for each treated group. Although the process would permit single, direct comparisons among different treated groups, the technique removes the dynamic aspect of a sequence of serial rates and converts everything into a single time point, which happens, in this case, to be a mean of the temporal results. The effective sample size procedure VOL. 20, NO. 3, SEPTEMBER, 1975
365
Editorial may not receive an enthusiastic reception from many readers since it seems arbitrary, excessively statisticated,and difficult to understand. Many investigative clinicians may prefer to continue comparing one-year rates versus one-year rates and five-year rates versus five-year rates rather than against a mingled temporal collection. Nevertheless, the possible rejection of this proposal by Dr. Grunkemeier and his colleagues should not keep clinicians from recognizing the potential virtues of the other two procedures. An actuarial analysis with multiple numerator events offers a useful description of complex phenomena. An actuarial analysis that enters prostheses, rather than people, in the denominator offers an interesting adjunct to conventional tabulations. Both procedures could be applied, of course, in direct rather than actuarial analyses, but the tide of statistical fashion is currently flowing in the direction of the actuarial technique. For most readers of clinical literature, the evaluation of therapy will continue to require answers to such questions as: (1) Can I understand what the author is doing and what he found?; (2) Does it make sense?; and (3)Do I believe him? An author who wants to convince his readers will also want to display his statistical data in a manner that provides a comprehensible documentation of the results. N o matter how the data are arranged for analysis, the main goal is to give a clear, complete account of the basic information needed for a thoughtful reader to determine what happened and to draw his own conclusions. ALVAN R. FEINSTEIN, M.D.
Department of Internal Medicine Yale University School .fMedicine New Haven, Conn. 06510
366
THE ANNALS OF THORACIC SURGERY