Life Tables for Clinical Scientists1 John G. Ferguson, MD Index terms: Radiology and radiologists, research Statistical analysis
JVIR 1992; 3:607-615 Abbreviation: PTA = percutaneous transluminal angioplasty
The life-table, or Cutler-Ederer,method of survival analysis is a simple and efficient means of estimating the probability that the first instance of an event will occur in a given period of time in studies complicatedby incomplete patient follow-up.This discussion is designed to acquaint the nonstatistician with the general concepts, assumptions, advantages, and disadvantages of life-table analysis. The arcane nature of the calculations frustrates attempts at simplification. A glossary of statistical terms and SaInple calculations are provided for interested readers.
life table, or Cutler-Ederer method (I),is one of several forms of survival analysis. Survival analysis is the preferred statistical method when the time elapsed prior to an event, and not just the occurrence of the event, is important. The life-table method permits valid statistical inference when patients enter a study serially and some are withdrawn prior to occurrence of the event of interest (Fig 1).Statistics such as the crude mortality rate, mortality rate per person-years of observation, mean duration of survival, and median survival time do not permit this (2). The event of interest need not be survival. However, it must be the first occurrence of a dichotomous outcome, such as an outcome that is either present or absent. For example, time to first restenosis and time elapsed prior to first amputation after peripheral percutaneous transluminal angioplasty (PTA) are equally appropriate. Analysis restricted to the first occurrence of an event precludes counting a patient more than once; that is, one patient undergoing two PTA procedures is not statistically equivalent to two patients undergoing one PTA each. T H E
From 3529 Lily Lane, Memphis, TN 38111, ReceivedJuly 29, 1992; accepted A U ~ U 2. S ~Address reprint requests to the author. SCVIR, 1992
STUDY DESIGN Life-table analysis is appropriate in a properly executed cohort study or clinical trial. A properly executed cohort study/clinical trial is one that begins with an inception cohort and involves precise and accurate measurements, a high rate of follow-up, and a design and execution that effectively minimizes selection bias, information bias, and confounding. Ideally, the study should involve random sampling or random allocation to avoid violation of statistical assumptions.
DATA REQUIRED FOR CALCULATIONS Data requirements for calculations include the following: ( a )a unique patient identifier, ( b ) date of inception (Inception is defined by a patient characteristic, such as time of diagnosis or start of therapy, which serves as a clear, objective, and collective starting point for follow-up, eg, date of PTA), ( c ) a clear, objective outcome (eg, restenosis after PTA),
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( d l date a t time of last follow-up, ( e ) event status a t the time of last follow-up (eg, either the patient experienced an event or helshe was free of the event of interest a t the time of the last follow-up), and ( f ) group membership-if the aim is to compare groups. (For example, groups can be defined by gender or treatment.)
Figure 1. Life-table analysis with serial entry of patients and incomplete follow-up.
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R = restenosls W = wthdrawal
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CALCULATION OF THE SURVIVAL CURVE 2
Data format and preparation varies with the statistical analysis package used and is explained in software manuals. The following procedure pertains to calculations done by hand (Appendix A). The follow-up period is divided into consecutive time intervals. The intervals can be of different lengths. However, the length is chosen to minimize the number of censored observations in any given interval. An observation is censored if follow-up ends before the subject under observation develo~sthe outcome of interest (eg, due to termination of the study, loss to follow-up, or death resulting from a competing event). In some cases, the choice of intervals may be guided by a priori knowledge of the pathophysiologic process under study. The unit of time selected for grouping must exceed that used for measuring survival; for example, if event-free survival is measured in days, the unit used for grouping can be weeks, months, years, etc. Next, the data are sorted by increasing length of follow-up and are arranged in a format that conforms to and simplifies the calculations. Once tabulated, the data are utilized to estimate three probabilities. The first is the probability of experiencing the event of interest-for example, restenosis after PTA-in each time interval, given event-free survival in all preceding intervals (qi). For any given interval, this probability is defined as the number of subjects who experience the event of interest during the interval, divided by the number of subjects who are at
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risk. The number of patients a t risk is the number of patients who are alive and event free a t the beginning of the interval, modified by a n adjustment for censoring. Adjustment for censoring is required because censored patients are not, for purposes of observation, at risk during the entire interval. Censoring thereby effectively reduces the number of patients at risk for the event of interest. The second estimate is the probability of event-free survival for each time interval, given event-free survival of all preceding time intervals (pi).This is equal the compliment of qi and equals 1 - q,. The third estimate is the probability of surviving multiple, contiguous time intervals, without experiencing the event of interest. This is the cumulative probability of event-free survival to a given time. The calculation is based on the multi~licative law of probability. For eximple, the probability of remaining free of restenosis for 2 months after PTA is equal to the product of the probability of remaining free of restenosis during the first postangioplasty month multiplied by the probability of remaining free of restenosis during the second postangioplasty month, given the absence of restenosis in the first month after PTA. The results are plotted against time to generate the empirical (observed) survival distribution. If the study was properly designed and executed, the empirical
survival distribution is an unbiased estimator of the true survival distribution in the target population and can be used for making inferences about the population.
ESTIMATION AND HYPOTHESIS TESTING BASED ON LIFE TABLES Uncertainty is associated with the life-table calculations because they are derived in samples and because of assumptions that relate to grouping; for example, it is assumed that censoring occurs randomly within an interval. Therefore, a truly informative survival estimate requires calculation of the confidence intervals associated with the life table. The confidence interval is a range of values for survival that includes the true chance of survival, with a probability that is determined by the level of statistical significance, which is chosen by the investigators in the design stage. Confidence limits can be derived by using an approximation called Greenwood's formula (3). The approximation is valid if the sample size is large enough and censoring is minimal. A statistician should be consulted to determine if the approximation is justifiable. The confidence bands delimit a plausible range within which the true survival is likely to be, and underscore the uncertainty inherent in survival estimates. For example, inspec-
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2. 3. Figures 2,3. (2) Precision of estimate decreases with increasing duration of follow-up. (3) When survival curves cross, point comparisons may be misleading.
tion of the survival curve and its confidence bands demonstrates the dwindling precision of the life-table estimates with increasing length of follow-up (Fig 2). This is due to a decrease in the number of subjects at risk for the event of interest with time and because the reliability of the survival estimates is inversely proportional to the number of subjects available for calculation. While the information stemming from confidence bands is useful, most investigators wish to compare the survival experience of different groups and to make inferences about the target populations. For example, it may be of interest to know if the time to restenosis after peripheral PTA varies with gender. Hypothesis testing is used to determine if survival differences noted in the sample could be attributable to chance alone. In general, comparing event-free survival at individual points in time is not recommended (4-6). First, this form of comparison is statistically inefficient; it wastes information and has less power to detect real differences in event-free survival than methods that use the data fully. Second, such comparisons may be biased if they are made after inspection of the data; post-hoc comparisons based on inspection of the data will tend to focus on large differences. This practice inflates the alpha error beyond
the nominalp value ( p value obtained from statistical tables) to a degree that is difficult to estimate. Third, point-in-time comparisons can be erroneous if the form of the survival curve is complex (Fig 3). The method of choice compares the entire survival experience of the groups in question by means of a statistic like the log-rank test or the Gehan (generalized Wilcoxon) test. Details concerning the generalized Wilcoxon test can be found elsewhere (7). Suffice it to say that this form of the Wilcoxon test emphasizes early losses and it should be used when the assumption of a constant hazard ratio is inappropriate (see Assumptions). The log-rank test is a widely accepted, efficient, and unbiased estimator of the true survival difference between two or more target populations. It has several forms, which the clinical investigator may consider to be equivalent. The test compares the number of events in each group, over the entire course of the study, with the number of events that would be expected if the event-free survival of the two groups was, in reality, the same. The resulting statistic has an approximate X 2 distribution with k - 1 degrees of freedom, where k is the number of groups being compared (not the number of patients). The significance of the sample X 2 is deter-
mined by reference to a table of critical values of the X 2 distribution with k - 1degrees of freedom. Tables of critical values for x2 can be found at the end of most biostatistical textbooks (2).A value of the X 2 that exceeds the critical value implies that the observed difference is unlikely to be due to chance alone. When this is the case, an alternative explanation should be considered. In so doing, it is best to remember that this difference may be due to several things other than chance, only one of which is a bona fide treatment effect (eg, selection bias, detection bias, confounding, etc).
ASSUMPTIONS Assumptions underlie every statistical test. Violation of these assumptions may preclude valid inference. Preventing losses to follow-up is a crucial and conceptually straightforward method for curtailing this problem. However, because exhaustive follow-up is rarely possible, an attempt should be made, with the guidance of a statistician, to obtain supplementary information about lost patients. If available, these ancillary data may suggest violation of key assumptions and provide evidence to support the use of an alternative statistical test. In performing a life-table
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analysis the following assumptions are made: 1.There are no secular trends in survival during the period of observation; event-free survival is independent of the time of entry into the study. This assumption is necessary because grouping may combine data from several different calendar periods into the same interval. For example, in a 5-year study with a grouping interval equal to 1month, patients with event-free survival of 1month or less may have been observed during the 1st year, the 5th year or any year in between. If the period of observation is long enough, the process under observation may have changed over time: For example, in a 5-year study of restenosis after PTA, a drug may be found in the 4th year that markedly reduced the rate of restenosis. 2. Censoring is unrelated to the subject's true time to response. That is to say, censored patients have the same probability of experiencing the event of interest after they depart the study as patients who remain in the study. When treatments are compared by using the life-table method, this implies that censoring is independent of treatment. If not, estimates of survival and comparisons based on the life table will be biased. The practical significance of this assumptions varies directly with the number of patients lost to follow-up. 3. Censoring occurs randomly and uniformly over the interval in which it occurs. 4. Censoring that results from death or that is caused by something other than the event of interest is unrelated to the event of interest. 5. Censored subjects are assumed to have been followed up for half the interval during which they were censored. 6. The risk of experiencing the event of interest is uniform within a given interval. The shorter the interval, the less likely violation of this assumption is to occur. If risk is not constant, the problem can sometimes be resolved by choosing shorter intervals. As in assumption number 2, if
the number of patients lost to follow-up is small, this assumption matters little, re-emphasizing the importance of complete patient follow-up. 7. With use of the log-rank test, it is assumed that the ratio of the mortality rates in the groups being compared is constant over time (8). 8. The probability of event-free survival in any given interval is independent of the probability of eventfree survival in all other intervals. 9. The probability that a subject remains event free, is independent of the probability that all other subjects remain event free.
ADVANTAGES
However, the potential to do so is limited compared with other techniques, for example, regression based on the proportional hazards model.
DISADVANTAGES AND PITFALLS IN INTERPRETATION 1. Life-table analysis shows cumulative event-free survival and, therefore, does not directly show changes in the event rate. 2. It does not take into account the exact ordering of deaths and losses within each interval. 3. It is an approximation to the Kaplan-Meier product-limit survival estimate and owes its popularity, in some measure, to the relative ease of calculations performed in large data sets. However, this is no longer an important consideration given the ubiquity of computers. 4. Survival estimates are somewhat arbitrary in that they depend on the time intervals chosen. This is not a problem with the Kaplan-Meier product-limit method of survival analysis. 5. Large steps and long flat regions in the survival curve can result from small numbers of ~ a t i e n t as t risk. These distortions i r e not biologically sienificant. 6. The most recent observations are the least reliable because of the decreasing number of patients a t risk for the event of interest. 7. The log-rank test is relatively insensitive to early differences in event-free survival. Consequently, it is less likely than the Gehan test to detect these. v
1. Life-table analysis uses the data of all patients. 2. Calculations are simple enough to be performed by a non-statistician and by hand. 3. I t does not require knowledge of exact follow-up times. 4. Hypothesis testing is based on distribution-free methods. That is to say, no assumption is required regarding the distribution of survival times and, therefore, the form of the survival distribution need not be known. This is particularly important, since in general it is not known. 5. Results can be represented graphically, promoting comprehension and simplifying communication of results. This is particularly important when the target audience consists of non-statisticians. 6. Life tables can suggest which factors are important for establishing a prognosis, for example, by comparing life tables in separate patient categories based on treatment, gender, etc. CONCLUSIONS 7. The log-rank test is optimal for assessing the equality of life tables In general, the life-table method is when the event rate in one group simple, accurate, and efficient relaconsistently exceeds that of another. tive to other methods of survival 8. The log-rank tests can be used to analysis. The calculations can be percompare more than two groups. formed by a non-statistician. How9. It is possible to adjust survival ever, a statistician should be concomparisons for the confounding efsulted to minimize the risk that violations of assumptions will invalifects of prognostic variables by means of a stratified log-rank test (5). date inferences and to ensure that u
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Table A1 Summary of Observations Date of Patient No. Entry 8810934 12108190 8815097 02116190 05/26/90 8903457 8829910 11130190 8828956 03/21/90 8801244 06/15/91 8978920 01130190 06102190 8801234
the study design has been implemented well enough to justify statistical analysis.
APPENDIX Sample Calculations This appendix is intended for those who need a deeper understanding of the mechanics of life-table analysis. The calculations are best done by computer, with programs like BMDP (9). However, the ease with which computer programs produce answers belies the complexity of the underlying procedures. Working through the calculations will help crystallize this impression. To promote understanding, an attempt has been made to minimize the complexity of the computational formulas. Those seeking to go beyond the level presented here should refer to textbooks and seminal publications (10,ll).An example based on a small number of patients was chosen for illustrative simplicity. Be forewarned that the inferential process does not lend itself reliably to samples of such a small size.
Data Preparation Consider a hypothetical study involving eight patients who underwent peripheral PTA for claudication during a study that lasted a total of approximately 2 years. Half the patients were male. We wish to compare the pattern of event-free survival, in male and female patients, where the
Date of Last Follow-up 12129190 03112190 07101190 02/03/91 08/14/90 12/09/91 07130190 12109190
Survival (d) 21 24 36 65 146 177 181 190
event of interest is restenosis during the first 7 months after PTA. First, divide the follow-up period into consecutive and mutually exclusive time intervals. The units of time used for grouping must be larger than the units of time used to measure event-free survival. The time intervals can be of different lengths. However, the length of an interval should be chosen to minimize the number of censored observations and may be guided by a priori knowledge of the pathophysiologic process being evaluated. An observation is censored if follow-up ends before the subject under observation develops the outcome of interest (eg, loss to follow-up, termination of the study, etc). Second, list the individual observations (Table A l ) in order of increasing trial time (5) (eg, duration of follow-up), together with survival status and, if comparison of survival is desired, group membership. In this case group 1consisted of the male patients and group 2, the female patients. Survival status at last followup has two possible states: restenosis and no restenosis. For estimating event-free survival, all patients without the event of interest a t the end of follow-up are censored without distinction. However, when evaluating statistical assumptions, judging the validity of the study, and interpreting results, patients who are lost to follow-up must be distinguished from those who are withdrawn bv virtue of study termination (see Assumptions). Patients who are lost to follow-up are
Group 1 1 2 2 1
2 2 1
Survival Status Alive Restenosis Alive Alive Restenosis Restenosis Alive Alive
more likely to be inappropriately censored. Third, for each group, tabulate by sequential intervals (i) (a)the number of patients (n;) who were alive and had not had a documented restenosis by the beginning of the interval, (b) the number of subjects who experienced restenosis during the interval (di),and ( c ) the number of subjects who had no documented restenosis, and were withdrawn or lost to follow-up during the interval (wi). Allow three additional columns for the results of calculations (Tables A2, A3).
Calculations Probability of the event of interest in each interval.-First, calculate the probability of experiencing the event of interest (q, (eg, restenosis after PTA), in each time interval for each group as follows:
For any given time interval, this probability is estimated by the number of subjects who experience the event of interest during the interval, divided by the number of subjects who are at risk (alive and event-free) a t the beginning of the interval, modified by an adjustment for censoring. An observation is censored if follow-up ends before the subject under observation develops the outcome of interest. Adjustment for censoring is
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Table A2 Life Table for Group 1 Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
n, 4 2 2 2 2 1 1
d, No. of Events 1 0 0 0 1 0 0
W,
1 0 0 0 0 0 1
q,= d, l[n,- (w,/2)1 Chance of Event in Interval x 11(4 - 0.5)= 0.286 0 0 0 1/(2- 0.0)= 0.500 0 0
PL = 1 - 4, Chance of No Event in Interval x 0.714 1 1 1 0.500 1 1
P, = P, . ? , - I . . . P I Cumulative Chance of No Event 0.714 0.714 0.714 0.714 0.357 0.357 0.357
q1= d, l[n,- (~,12)1 Chance of Event in Interval x 0 0 0 0 0 11(2 - 0.0) 0.500 0
PL = 1 - 9, Chance of No Event in Interval x 1 1 1 1 1 0.500 1
P,= P ; ? , - I . . . P I Cumulative Chance of No Event 1 1 1 1 1 0.500 0.500
Table A3 Life Table for Group 2 Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
n, 4 4 3 2 2 2 1
dl No. of Events 0 0 0 0 0 1 0
required because censored patients are not, for purposes of observation, a t risk for the event of interest during the entire interval. Censoring, thereby, effectively reduces the number of patients a t risk for the outcome of interest. The life-table method assumes that censoring occurs randomly and uniformly during the interval in question. It follows that, on average, a censored subject is observed for half the interval in question. Therefore, the number of patients at risk a t the beginning of the interval must be reduced by '/z the number of subject who are censored (wi1. This is accounted for in Equation (Al),by dividing wi by two and subtracting the quotient from ni. Probability of event-free survival in each interval.--Second, calculate the probability of event-free survival for each time interval (pi 1, given eventfree survival of all preceding time intervals. This is equal to 1 - qi :
W,
0 1 1 0 0 0 1
2
Cumulative probability of eventfree survival.-Third, calculate the probability of surviving multiple, contiguous time intervals (Tables A2, A3). The cumulative probability of event-free survival to time i is:
The calculation of P, is based on the multiplicative law for probabilities: For two events A and B, the probability that both A and B occur is the probability that A occurs, multiplied by the probability that B occurs, given that A has already taken place. This identity can be summarized as P(A and B ) = P(A) x P(B/A). Thus, the probability of event-free survival to the end of time interval i is the probability of surviving interval i, multiplied by the probability of surviving each of the preceding time intervals. For example, the probability of remaining free of restenosis for 2 months after PTA, is equal to the product of the probability of remaining free of restenosis during the 1st
postangioplasty month, multiplied by the probability of remaining free of restenosis during the 2nd postangioplasty month, given that restenosis did not occur in the 1st postangioplasty month.
Plot of the Survival Curve Plotted against time, this string of conditional probabilities is the empirical survival distribution. Reports of survival data should include graphic or tabular presentation of the life tables with details concerning how many patients were at risk a t the different follow-up times (FigA). If the study was properly designed and executed, the empirical survival distributions are unbiased estimators of the true survival distributions in the target populations, and these can be used for making inferences about the populations. Estimation and Hypothesis Testing Based on Life Tables Uncertainty is associated with the life-table calculations because they
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Table A4 Log-RankTest Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
Alive and Event Free at Start of Interval G1 G2 4 4 2 4 2 3 2 2 2 2 1 2 1 1
T 8 6 5 4 4 3 2
G1 1 0 0 0 1 0 0
Total G1 = group 1,G2 = group 2, T = total.
2
5 2
Months since first PTA
are derived in samples and because of assumptions that relate to grouping (eg, censoring occurs randomly within an interval). Therefore, a truly informative survival estimate requires calculation of the confidence intervals associated with the life table. The confidence interval is a range of values for survival that includes the true chance of survival, with a probability that is determined by the level of statistical significance chosen. Confidence limits can be derived by using Greenwood's formula if the sample size is large and censoring is minimal. The confidence limits depict a plausible range within which the true survival is likely to be and underscore the uncertainty inherent in the survival estimates (Fig 2). Greenwood's formula.-Greenwood's formula for calculation of the
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No. of Events (0,) G2 0 0 0 0 0 1 0 1
Figure A. Survival curve for data in Table Al.
standard error SE(Pi)of the estimates of the cumulative probability of survival is defined below:
When SE(Pi)/Pi 5 ?hand di r 10, SE(Pi)can be used to obtain approximate 95% confidence limits on Pi. The upper and lower confidence limits are calculated for each estimate of cumulative, event-free survival by means of the following formula: The confidence limits are then plotted on the same graph as the estimates of cumulative event-free sur-
T 1 0 0 0 1 1 0
G1 0.5 0 0 0 0.5 0.3 0
3
1.3
Expected No. of Events (e, G2 0.5 0 0 0 0.5 0.7 0
1.7
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vival. The terms 21.96 . SE(Pi)are critical values that cut off the upper and lower 2.5% of the standardized normal (Gaussian) distribution. The points, comprising a given set of limits, are linked by straight lines to yield confidence bands. Caveats pertaining to SE(Pi) and worked examples can be found elsewhere (2,11,12). While the information stemminn " from confidence bands is useful, most investigators wish to compare the survival experience of different groups to make inferences about the corresponding target populations. For example, it may be of interest to know if the time to restenosis after peripheral PTA varies with gender. Hypothesis testing is used to determine if survival differences, observed in the sample, are likely to be due to chance or to a real difference in the corresponding target populations. As indicated earlier, a comparison of survival at individual points in time, for example, by using the formulas based on SE(P,), is not recommended. The method of choice compares the entire survival experience of the groups in question by means of a statistic like the log-rank test. The log-rank test is a widely accepted, easily calculated, efficient, and unbiased estimator of the true survival difference in the target populations. The test compares the number of events in each group over the entire course of the study with the number
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of events that would be expected if the survival of the two groups was, in reality, the same. Calculation of the log-rank statistic.-For each time interval, list the number of patients in each group who are alive and free of the event of interest at the beginning of the interval (Table A4). Adjacent to these, list the number of patients who experienced the event of interest in each group during the interval (0~1, where interval = i and group number = j. Calculation of the log-rank statistic requires an introduction to hypothesis testing. Hypothesis testing is a statistical techniaue used to analvze a postulate, k n o h as the null h i ~othesis.that there is no survival difference in the target populations. Calculation of the ex~ectednumber of events is based o n i h e null hypothesis. When the null hwothesis is true, that is, the eventrate in the groups is equal, the expected number of events in a group ( e V )in a given interval is equal. If group size is unequal, then e, is directly proportional to the size of the group in question. Thus, the expected number of events in a group, for a given interval, is the product of the total number of events in the interval, multiplied by the number of patients a t risk in the group, divided by the total number of patients a t risk in the interval. For example, in the interval 0-1 month in Table A4, the expected number of events in group 1is:
The ex~ectednumber of events is then summed across the i intervals. This is done separately for each of the j groups and yields the expected number of events over the entire study for that group (Ej). The expected number of events over the entire study in each group is then subtracted from the total number of events in the same group (0;). The result of this subtraction is then squared and divided by the expected number of events over the entire study for that group (Ej). The terms,
difference-for example, in restenosis after PTA-when in reality there is none. bias: any tendency for a value to consistently deviate in one direction from the actual value. censored observation: a study subject in whom the event of interest has not been observed a t the end of histher trial time. cohort: a group of people who have something in common when they are The statistical significance of the X 2 is first assembled, and who are then determined by reference to a table of observed for a period of time to see critical values of the X 2 distribution what happens to them (13). with k - 1degrees of freedom. The cohort study: a study in which a larger the x2, the less likely the differ- cohort is assembled. none of whom ence observed in the sample is due to have experienced the outcome of inchance alone. For the example in Taterest. Baseline characteristics. including putative risk factors for the ble A4: outcome are measured upon entry to the study. These subjects are then followed over time to characterize the relationship between baseline characteristics and the outcome. confounding: an apparent association between a risk factor and outBecause the table of critical values of come that is due to a second factor, the X 2 distribution is a partial listing, which is extraneous to the question small values such as X 2 = 0.60 are under study. not tabulated. Computer software, dichotomous variable: a variable like the Electronic Tables for Statisti- measured on a scale that has only cians (Hawkeye, Softworks, Iowa two possible states, for example, City, Iowa) can be used to pinpoint younglold, yesino, maleifemale. the P value when this is the case. The euent-free suruiual: in this article, P value corresponding to a X 2 of 0.60, event-free survival and survival are synonymous. The term event-free with (2- 1)= 1degree of freedom, is survival is used to reinforce the con0.44. Therefore, a difference in survival of the observed magnitude cept that the event of interest need would be expected, by chance alone, not be death, as is implied by the unin 44% of studies of the same size. As qualified use of the term survival. a result, the null hypothesis cannot event o f interest: the outcome under study. In this case, restenosis. be rejected at the .05 level of statistical significance; that is, it would be hypothesis test: also known as a inappropriate to conclude that there significance test. A rule for deciding is a statistically significant difference whether a n assumption (hypothesis) in survival between the two groups. about the distribution of a random (Note: failing to reject the null hyvariable should be rejected, with use pothesis is not synonymous with acof a sample from the distribution. cepting it, for reasons that are beThe hypothesis is called the null hyyond the scope of this discussion.) pothesis, written Ho, and it is tested against some alternative hypothesis, H I . A statistic is computed from the sample data. If it falls within the critGLOSSARY OF TERMS ical region, where the value of the staalpha error (type I error): the prob- tistic is significantly different from that ability of concluding that there is a expected under Ho,Ho is rejected (15). so derived for each group, are then added together. If the expected number of events in any given interval is less than lo%, the sum has an approximate X 2 distribution with k - 1 degrees of freedom, where k represents the number of groups being compared
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inception: inception is defined by a patient characteristic, such as time of diagnosis or start of therapy, which serves as a clear, well-defined, and collective starting point for follow-up, for example, date of PTA. inference (statistical): the logical basis by which conclusions regarding populations are drawn from results obtained in a sample (15). life table: a method for analyzing survival times that have been grouped into arbitrary intervals, when observation of some patients is incomplete in that they have not experienced the event of interest when the trial is stopped. mutually exclusive time intervals: time intervals that are non-overlapping, such that the assignment of a trial time to an interval is unique. null hypothesis: see hypothesis test-
ing. outcome: see event o f interest. suruiual, event free: see event-free suruival. table of critical values: a subset of the possible values of a test statistic, each of which corresponds to the probability of observing a value of the test statistic, that is at least as extreme as the one in question, by chance alone.
target population: the entire collection of patients, about whom the investigator wishes to draw conclusions based on observations made in a sample. trial time: the time from inception to date of loss to follow-up, to the trial stopping date, or occurrence of the event of interest. References 1. Cutler J, Ederer F. Maximum utilization of the life-table method in analyzing survival. J Chron Dis 1958; 8:699-713. 2. Colton T. Statistics in medicine. Boston: Little Brown, 1974; 237250. 3. Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects 33. London: H.M. Stationary Office, 1926. 4. Peto R, Pike M, Armitage P, et al. Design and analysis of randomized controlled clinical trials requiring prolonged observation of each patient. I. Introduction and design. Br J Cancer 1976; 34:585-612. 5. Peto R, Pike M, Armitage P, et al. Design and analysis of randomized controlled clinical trials requiring prolonged observation of each patient. 11. Analysis and examples. Br J Cancer 1977; 35:l-39.
Dawson-Saunders B, Trapp RG. Basic and clinical biostatistics 1990. Norwalk, Conn: Appleton Lange, 1990; 166-206. Gehan EA. A generalized Wilcoxon test for comparing arbitrarily singlycensored samples. Biometrika 1965; 52:15-21. Matthews DE, Farewell VT. Using and understanding medical statistics. New York: Karger, 1988: 78. BMDP Statistical software manual. Dixon WJ, Brown MB, Engleman, L, et al. Berkeley, Calif: University of California Press, 1985; 557-575. Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. Am Stat Assoc J 1958; 53:457-481. Kalbfleisch JD, Prentice RL. The statistical analysis of failure time data. New York: Wiley, 1980; 10-19. Kahn HA. An introduction to epidemiologic methods. Monographs in epidemiology and biostatistics. v5. New York: Oxford University Press, 1983: 121-144. Fletcher RH, Fletcher SW, Wagner EH. Clinical epidemiology: the essentials. Baltimore: Williams & Wilkins, 1982; 96-97. Gibson C. Dictionary of mathematics. Berkshire: Intercontinental Book Productions, 1981; 81. Colton T. Statistics in medicine. Boston: Little Brown, 1974; 3.
JVIR 1992; 3:607-615 Abbreviation: PTA = percutaneous transluminal angioplasty
The life-table, or Cutler-Ederer,method of survival analysis is a simple and efficient means of estimating the probability that the first instance of an event will occur in a given period of time in studies complicatedby incomplete patient follow-up.This discussion is designed to acquaint the nonstatistician with the general concepts, assumptions, advantages, and disadvantages of life-table analysis. The arcane nature of the calculations frustrates attempts at simplification. A glossary of statistical terms and SaInple calculations are provided for interested readers.
life table, or Cutler-Ederer method (I),is one of several forms of survival analysis. Survival analysis is the preferred statistical method when the time elapsed prior to an event, and not just the occurrence of the event, is important. The life-table method permits valid statistical inference when patients enter a study serially and some are withdrawn prior to occurrence of the event of interest (Fig 1).Statistics such as the crude mortality rate, mortality rate per person-years of observation, mean duration of survival, and median survival time do not permit this (2). The event of interest need not be survival. However, it must be the first occurrence of a dichotomous outcome, such as an outcome that is either present or absent. For example, time to first restenosis and time elapsed prior to first amputation after peripheral percutaneous transluminal angioplasty (PTA) are equally appropriate. Analysis restricted to the first occurrence of an event precludes counting a patient more than once; that is, one patient undergoing two PTA procedures is not statistically equivalent to two patients undergoing one PTA each. T H E
From 3529 Lily Lane, Memphis, TN 38111, ReceivedJuly 29, 1992; accepted A U ~ U 2. S ~Address reprint requests to the author. SCVIR, 1992
STUDY DESIGN Life-table analysis is appropriate in a properly executed cohort study or clinical trial. A properly executed cohort study/clinical trial is one that begins with an inception cohort and involves precise and accurate measurements, a high rate of follow-up, and a design and execution that effectively minimizes selection bias, information bias, and confounding. Ideally, the study should involve random sampling or random allocation to avoid violation of statistical assumptions.
DATA REQUIRED FOR CALCULATIONS Data requirements for calculations include the following: ( a )a unique patient identifier, ( b ) date of inception (Inception is defined by a patient characteristic, such as time of diagnosis or start of therapy, which serves as a clear, objective, and collective starting point for follow-up, eg, date of PTA), ( c ) a clear, objective outcome (eg, restenosis after PTA),
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( d l date a t time of last follow-up, ( e ) event status a t the time of last follow-up (eg, either the patient experienced an event or helshe was free of the event of interest a t the time of the last follow-up), and ( f ) group membership-if the aim is to compare groups. (For example, groups can be defined by gender or treatment.)
Figure 1. Life-table analysis with serial entry of patients and incomplete follow-up.
6
5
* F Z ," >
R = restenosls W = wthdrawal
-
PTA
4 -
3
PTA - -
2
PTA
W Z -
CALCULATION OF THE SURVIVAL CURVE 2
Data format and preparation varies with the statistical analysis package used and is explained in software manuals. The following procedure pertains to calculations done by hand (Appendix A). The follow-up period is divided into consecutive time intervals. The intervals can be of different lengths. However, the length is chosen to minimize the number of censored observations in any given interval. An observation is censored if follow-up ends before the subject under observation develo~sthe outcome of interest (eg, due to termination of the study, loss to follow-up, or death resulting from a competing event). In some cases, the choice of intervals may be guided by a priori knowledge of the pathophysiologic process under study. The unit of time selected for grouping must exceed that used for measuring survival; for example, if event-free survival is measured in days, the unit used for grouping can be weeks, months, years, etc. Next, the data are sorted by increasing length of follow-up and are arranged in a format that conforms to and simplifies the calculations. Once tabulated, the data are utilized to estimate three probabilities. The first is the probability of experiencing the event of interest-for example, restenosis after PTA-in each time interval, given event-free survival in all preceding intervals (qi). For any given interval, this probability is defined as the number of subjects who experience the event of interest during the interval, divided by the number of subjects who are at
1
I
I
3
4
5
(months)
TIME
risk. The number of patients a t risk is the number of patients who are alive and event free a t the beginning of the interval, modified by a n adjustment for censoring. Adjustment for censoring is required because censored patients are not, for purposes of observation, at risk during the entire interval. Censoring thereby effectively reduces the number of patients at risk for the event of interest. The second estimate is the probability of event-free survival for each time interval, given event-free survival of all preceding time intervals (pi).This is equal the compliment of qi and equals 1 - q,. The third estimate is the probability of surviving multiple, contiguous time intervals, without experiencing the event of interest. This is the cumulative probability of event-free survival to a given time. The calculation is based on the multi~licative law of probability. For eximple, the probability of remaining free of restenosis for 2 months after PTA is equal to the product of the probability of remaining free of restenosis during the first postangioplasty month multiplied by the probability of remaining free of restenosis during the second postangioplasty month, given the absence of restenosis in the first month after PTA. The results are plotted against time to generate the empirical (observed) survival distribution. If the study was properly designed and executed, the empirical
survival distribution is an unbiased estimator of the true survival distribution in the target population and can be used for making inferences about the population.
ESTIMATION AND HYPOTHESIS TESTING BASED ON LIFE TABLES Uncertainty is associated with the life-table calculations because they are derived in samples and because of assumptions that relate to grouping; for example, it is assumed that censoring occurs randomly within an interval. Therefore, a truly informative survival estimate requires calculation of the confidence intervals associated with the life table. The confidence interval is a range of values for survival that includes the true chance of survival, with a probability that is determined by the level of statistical significance, which is chosen by the investigators in the design stage. Confidence limits can be derived by using an approximation called Greenwood's formula (3). The approximation is valid if the sample size is large enough and censoring is minimal. A statistician should be consulted to determine if the approximation is justifiable. The confidence bands delimit a plausible range within which the true survival is likely to be, and underscore the uncertainty inherent in survival estimates. For example, inspec-
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2. 3. Figures 2,3. (2) Precision of estimate decreases with increasing duration of follow-up. (3) When survival curves cross, point comparisons may be misleading.
tion of the survival curve and its confidence bands demonstrates the dwindling precision of the life-table estimates with increasing length of follow-up (Fig 2). This is due to a decrease in the number of subjects at risk for the event of interest with time and because the reliability of the survival estimates is inversely proportional to the number of subjects available for calculation. While the information stemming from confidence bands is useful, most investigators wish to compare the survival experience of different groups and to make inferences about the target populations. For example, it may be of interest to know if the time to restenosis after peripheral PTA varies with gender. Hypothesis testing is used to determine if survival differences noted in the sample could be attributable to chance alone. In general, comparing event-free survival at individual points in time is not recommended (4-6). First, this form of comparison is statistically inefficient; it wastes information and has less power to detect real differences in event-free survival than methods that use the data fully. Second, such comparisons may be biased if they are made after inspection of the data; post-hoc comparisons based on inspection of the data will tend to focus on large differences. This practice inflates the alpha error beyond
the nominalp value ( p value obtained from statistical tables) to a degree that is difficult to estimate. Third, point-in-time comparisons can be erroneous if the form of the survival curve is complex (Fig 3). The method of choice compares the entire survival experience of the groups in question by means of a statistic like the log-rank test or the Gehan (generalized Wilcoxon) test. Details concerning the generalized Wilcoxon test can be found elsewhere (7). Suffice it to say that this form of the Wilcoxon test emphasizes early losses and it should be used when the assumption of a constant hazard ratio is inappropriate (see Assumptions). The log-rank test is a widely accepted, efficient, and unbiased estimator of the true survival difference between two or more target populations. It has several forms, which the clinical investigator may consider to be equivalent. The test compares the number of events in each group, over the entire course of the study, with the number of events that would be expected if the event-free survival of the two groups was, in reality, the same. The resulting statistic has an approximate X 2 distribution with k - 1 degrees of freedom, where k is the number of groups being compared (not the number of patients). The significance of the sample X 2 is deter-
mined by reference to a table of critical values of the X 2 distribution with k - 1degrees of freedom. Tables of critical values for x2 can be found at the end of most biostatistical textbooks (2).A value of the X 2 that exceeds the critical value implies that the observed difference is unlikely to be due to chance alone. When this is the case, an alternative explanation should be considered. In so doing, it is best to remember that this difference may be due to several things other than chance, only one of which is a bona fide treatment effect (eg, selection bias, detection bias, confounding, etc).
ASSUMPTIONS Assumptions underlie every statistical test. Violation of these assumptions may preclude valid inference. Preventing losses to follow-up is a crucial and conceptually straightforward method for curtailing this problem. However, because exhaustive follow-up is rarely possible, an attempt should be made, with the guidance of a statistician, to obtain supplementary information about lost patients. If available, these ancillary data may suggest violation of key assumptions and provide evidence to support the use of an alternative statistical test. In performing a life-table
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analysis the following assumptions are made: 1.There are no secular trends in survival during the period of observation; event-free survival is independent of the time of entry into the study. This assumption is necessary because grouping may combine data from several different calendar periods into the same interval. For example, in a 5-year study with a grouping interval equal to 1month, patients with event-free survival of 1month or less may have been observed during the 1st year, the 5th year or any year in between. If the period of observation is long enough, the process under observation may have changed over time: For example, in a 5-year study of restenosis after PTA, a drug may be found in the 4th year that markedly reduced the rate of restenosis. 2. Censoring is unrelated to the subject's true time to response. That is to say, censored patients have the same probability of experiencing the event of interest after they depart the study as patients who remain in the study. When treatments are compared by using the life-table method, this implies that censoring is independent of treatment. If not, estimates of survival and comparisons based on the life table will be biased. The practical significance of this assumptions varies directly with the number of patients lost to follow-up. 3. Censoring occurs randomly and uniformly over the interval in which it occurs. 4. Censoring that results from death or that is caused by something other than the event of interest is unrelated to the event of interest. 5. Censored subjects are assumed to have been followed up for half the interval during which they were censored. 6. The risk of experiencing the event of interest is uniform within a given interval. The shorter the interval, the less likely violation of this assumption is to occur. If risk is not constant, the problem can sometimes be resolved by choosing shorter intervals. As in assumption number 2, if
the number of patients lost to follow-up is small, this assumption matters little, re-emphasizing the importance of complete patient follow-up. 7. With use of the log-rank test, it is assumed that the ratio of the mortality rates in the groups being compared is constant over time (8). 8. The probability of event-free survival in any given interval is independent of the probability of eventfree survival in all other intervals. 9. The probability that a subject remains event free, is independent of the probability that all other subjects remain event free.
ADVANTAGES
However, the potential to do so is limited compared with other techniques, for example, regression based on the proportional hazards model.
DISADVANTAGES AND PITFALLS IN INTERPRETATION 1. Life-table analysis shows cumulative event-free survival and, therefore, does not directly show changes in the event rate. 2. It does not take into account the exact ordering of deaths and losses within each interval. 3. It is an approximation to the Kaplan-Meier product-limit survival estimate and owes its popularity, in some measure, to the relative ease of calculations performed in large data sets. However, this is no longer an important consideration given the ubiquity of computers. 4. Survival estimates are somewhat arbitrary in that they depend on the time intervals chosen. This is not a problem with the Kaplan-Meier product-limit method of survival analysis. 5. Large steps and long flat regions in the survival curve can result from small numbers of ~ a t i e n t as t risk. These distortions i r e not biologically sienificant. 6. The most recent observations are the least reliable because of the decreasing number of patients a t risk for the event of interest. 7. The log-rank test is relatively insensitive to early differences in event-free survival. Consequently, it is less likely than the Gehan test to detect these. v
1. Life-table analysis uses the data of all patients. 2. Calculations are simple enough to be performed by a non-statistician and by hand. 3. I t does not require knowledge of exact follow-up times. 4. Hypothesis testing is based on distribution-free methods. That is to say, no assumption is required regarding the distribution of survival times and, therefore, the form of the survival distribution need not be known. This is particularly important, since in general it is not known. 5. Results can be represented graphically, promoting comprehension and simplifying communication of results. This is particularly important when the target audience consists of non-statisticians. 6. Life tables can suggest which factors are important for establishing a prognosis, for example, by comparing life tables in separate patient categories based on treatment, gender, etc. CONCLUSIONS 7. The log-rank test is optimal for assessing the equality of life tables In general, the life-table method is when the event rate in one group simple, accurate, and efficient relaconsistently exceeds that of another. tive to other methods of survival 8. The log-rank tests can be used to analysis. The calculations can be percompare more than two groups. formed by a non-statistician. How9. It is possible to adjust survival ever, a statistician should be concomparisons for the confounding efsulted to minimize the risk that violations of assumptions will invalifects of prognostic variables by means of a stratified log-rank test (5). date inferences and to ensure that u
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Table A1 Summary of Observations Date of Patient No. Entry 8810934 12108190 8815097 02116190 05/26/90 8903457 8829910 11130190 8828956 03/21/90 8801244 06/15/91 8978920 01130190 06102190 8801234
the study design has been implemented well enough to justify statistical analysis.
APPENDIX Sample Calculations This appendix is intended for those who need a deeper understanding of the mechanics of life-table analysis. The calculations are best done by computer, with programs like BMDP (9). However, the ease with which computer programs produce answers belies the complexity of the underlying procedures. Working through the calculations will help crystallize this impression. To promote understanding, an attempt has been made to minimize the complexity of the computational formulas. Those seeking to go beyond the level presented here should refer to textbooks and seminal publications (10,ll).An example based on a small number of patients was chosen for illustrative simplicity. Be forewarned that the inferential process does not lend itself reliably to samples of such a small size.
Data Preparation Consider a hypothetical study involving eight patients who underwent peripheral PTA for claudication during a study that lasted a total of approximately 2 years. Half the patients were male. We wish to compare the pattern of event-free survival, in male and female patients, where the
Date of Last Follow-up 12129190 03112190 07101190 02/03/91 08/14/90 12/09/91 07130190 12109190
Survival (d) 21 24 36 65 146 177 181 190
event of interest is restenosis during the first 7 months after PTA. First, divide the follow-up period into consecutive and mutually exclusive time intervals. The units of time used for grouping must be larger than the units of time used to measure event-free survival. The time intervals can be of different lengths. However, the length of an interval should be chosen to minimize the number of censored observations and may be guided by a priori knowledge of the pathophysiologic process being evaluated. An observation is censored if follow-up ends before the subject under observation develops the outcome of interest (eg, loss to follow-up, termination of the study, etc). Second, list the individual observations (Table A l ) in order of increasing trial time (5) (eg, duration of follow-up), together with survival status and, if comparison of survival is desired, group membership. In this case group 1consisted of the male patients and group 2, the female patients. Survival status at last followup has two possible states: restenosis and no restenosis. For estimating event-free survival, all patients without the event of interest a t the end of follow-up are censored without distinction. However, when evaluating statistical assumptions, judging the validity of the study, and interpreting results, patients who are lost to follow-up must be distinguished from those who are withdrawn bv virtue of study termination (see Assumptions). Patients who are lost to follow-up are
Group 1 1 2 2 1
2 2 1
Survival Status Alive Restenosis Alive Alive Restenosis Restenosis Alive Alive
more likely to be inappropriately censored. Third, for each group, tabulate by sequential intervals (i) (a)the number of patients (n;) who were alive and had not had a documented restenosis by the beginning of the interval, (b) the number of subjects who experienced restenosis during the interval (di),and ( c ) the number of subjects who had no documented restenosis, and were withdrawn or lost to follow-up during the interval (wi). Allow three additional columns for the results of calculations (Tables A2, A3).
Calculations Probability of the event of interest in each interval.-First, calculate the probability of experiencing the event of interest (q, (eg, restenosis after PTA), in each time interval for each group as follows:
For any given time interval, this probability is estimated by the number of subjects who experience the event of interest during the interval, divided by the number of subjects who are at risk (alive and event-free) a t the beginning of the interval, modified by an adjustment for censoring. An observation is censored if follow-up ends before the subject under observation develops the outcome of interest. Adjustment for censoring is
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Table A2 Life Table for Group 1 Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
n, 4 2 2 2 2 1 1
d, No. of Events 1 0 0 0 1 0 0
W,
1 0 0 0 0 0 1
q,= d, l[n,- (w,/2)1 Chance of Event in Interval x 11(4 - 0.5)= 0.286 0 0 0 1/(2- 0.0)= 0.500 0 0
PL = 1 - 4, Chance of No Event in Interval x 0.714 1 1 1 0.500 1 1
P, = P, . ? , - I . . . P I Cumulative Chance of No Event 0.714 0.714 0.714 0.714 0.357 0.357 0.357
q1= d, l[n,- (~,12)1 Chance of Event in Interval x 0 0 0 0 0 11(2 - 0.0) 0.500 0
PL = 1 - 9, Chance of No Event in Interval x 1 1 1 1 1 0.500 1
P,= P ; ? , - I . . . P I Cumulative Chance of No Event 1 1 1 1 1 0.500 0.500
Table A3 Life Table for Group 2 Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
n, 4 4 3 2 2 2 1
dl No. of Events 0 0 0 0 0 1 0
required because censored patients are not, for purposes of observation, a t risk for the event of interest during the entire interval. Censoring, thereby, effectively reduces the number of patients a t risk for the outcome of interest. The life-table method assumes that censoring occurs randomly and uniformly during the interval in question. It follows that, on average, a censored subject is observed for half the interval in question. Therefore, the number of patients at risk a t the beginning of the interval must be reduced by '/z the number of subject who are censored (wi1. This is accounted for in Equation (Al),by dividing wi by two and subtracting the quotient from ni. Probability of event-free survival in each interval.--Second, calculate the probability of event-free survival for each time interval (pi 1, given eventfree survival of all preceding time intervals. This is equal to 1 - qi :
W,
0 1 1 0 0 0 1
2
Cumulative probability of eventfree survival.-Third, calculate the probability of surviving multiple, contiguous time intervals (Tables A2, A3). The cumulative probability of event-free survival to time i is:
The calculation of P, is based on the multiplicative law for probabilities: For two events A and B, the probability that both A and B occur is the probability that A occurs, multiplied by the probability that B occurs, given that A has already taken place. This identity can be summarized as P(A and B ) = P(A) x P(B/A). Thus, the probability of event-free survival to the end of time interval i is the probability of surviving interval i, multiplied by the probability of surviving each of the preceding time intervals. For example, the probability of remaining free of restenosis for 2 months after PTA, is equal to the product of the probability of remaining free of restenosis during the 1st
postangioplasty month, multiplied by the probability of remaining free of restenosis during the 2nd postangioplasty month, given that restenosis did not occur in the 1st postangioplasty month.
Plot of the Survival Curve Plotted against time, this string of conditional probabilities is the empirical survival distribution. Reports of survival data should include graphic or tabular presentation of the life tables with details concerning how many patients were at risk a t the different follow-up times (FigA). If the study was properly designed and executed, the empirical survival distributions are unbiased estimators of the true survival distributions in the target populations, and these can be used for making inferences about the populations. Estimation and Hypothesis Testing Based on Life Tables Uncertainty is associated with the life-table calculations because they
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Table A4 Log-RankTest Time from PTA (mo) 0-1 1-2 2-3 3-4 4-5 5-6 6-7
Alive and Event Free at Start of Interval G1 G2 4 4 2 4 2 3 2 2 2 2 1 2 1 1
T 8 6 5 4 4 3 2
G1 1 0 0 0 1 0 0
Total G1 = group 1,G2 = group 2, T = total.
2
5 2
Months since first PTA
are derived in samples and because of assumptions that relate to grouping (eg, censoring occurs randomly within an interval). Therefore, a truly informative survival estimate requires calculation of the confidence intervals associated with the life table. The confidence interval is a range of values for survival that includes the true chance of survival, with a probability that is determined by the level of statistical significance chosen. Confidence limits can be derived by using Greenwood's formula if the sample size is large and censoring is minimal. The confidence limits depict a plausible range within which the true survival is likely to be and underscore the uncertainty inherent in the survival estimates (Fig 2). Greenwood's formula.-Greenwood's formula for calculation of the
6
7 0
No. of Events (0,) G2 0 0 0 0 0 1 0 1
Figure A. Survival curve for data in Table Al.
standard error SE(Pi)of the estimates of the cumulative probability of survival is defined below:
When SE(Pi)/Pi 5 ?hand di r 10, SE(Pi)can be used to obtain approximate 95% confidence limits on Pi. The upper and lower confidence limits are calculated for each estimate of cumulative, event-free survival by means of the following formula: The confidence limits are then plotted on the same graph as the estimates of cumulative event-free sur-
T 1 0 0 0 1 1 0
G1 0.5 0 0 0 0.5 0.3 0
3
1.3
Expected No. of Events (e, G2 0.5 0 0 0 0.5 0.7 0
1.7
T 1 0 0 0 1 1 0 3
vival. The terms 21.96 . SE(Pi)are critical values that cut off the upper and lower 2.5% of the standardized normal (Gaussian) distribution. The points, comprising a given set of limits, are linked by straight lines to yield confidence bands. Caveats pertaining to SE(Pi) and worked examples can be found elsewhere (2,11,12). While the information stemminn " from confidence bands is useful, most investigators wish to compare the survival experience of different groups to make inferences about the corresponding target populations. For example, it may be of interest to know if the time to restenosis after peripheral PTA varies with gender. Hypothesis testing is used to determine if survival differences, observed in the sample, are likely to be due to chance or to a real difference in the corresponding target populations. As indicated earlier, a comparison of survival at individual points in time, for example, by using the formulas based on SE(P,), is not recommended. The method of choice compares the entire survival experience of the groups in question by means of a statistic like the log-rank test. The log-rank test is a widely accepted, easily calculated, efficient, and unbiased estimator of the true survival difference in the target populations. The test compares the number of events in each group over the entire course of the study with the number
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of events that would be expected if the survival of the two groups was, in reality, the same. Calculation of the log-rank statistic.-For each time interval, list the number of patients in each group who are alive and free of the event of interest at the beginning of the interval (Table A4). Adjacent to these, list the number of patients who experienced the event of interest in each group during the interval (0~1, where interval = i and group number = j. Calculation of the log-rank statistic requires an introduction to hypothesis testing. Hypothesis testing is a statistical techniaue used to analvze a postulate, k n o h as the null h i ~othesis.that there is no survival difference in the target populations. Calculation of the ex~ectednumber of events is based o n i h e null hypothesis. When the null hwothesis is true, that is, the eventrate in the groups is equal, the expected number of events in a group ( e V )in a given interval is equal. If group size is unequal, then e, is directly proportional to the size of the group in question. Thus, the expected number of events in a group, for a given interval, is the product of the total number of events in the interval, multiplied by the number of patients a t risk in the group, divided by the total number of patients a t risk in the interval. For example, in the interval 0-1 month in Table A4, the expected number of events in group 1is:
The ex~ectednumber of events is then summed across the i intervals. This is done separately for each of the j groups and yields the expected number of events over the entire study for that group (Ej). The expected number of events over the entire study in each group is then subtracted from the total number of events in the same group (0;). The result of this subtraction is then squared and divided by the expected number of events over the entire study for that group (Ej). The terms,
difference-for example, in restenosis after PTA-when in reality there is none. bias: any tendency for a value to consistently deviate in one direction from the actual value. censored observation: a study subject in whom the event of interest has not been observed a t the end of histher trial time. cohort: a group of people who have something in common when they are The statistical significance of the X 2 is first assembled, and who are then determined by reference to a table of observed for a period of time to see critical values of the X 2 distribution what happens to them (13). with k - 1degrees of freedom. The cohort study: a study in which a larger the x2, the less likely the differ- cohort is assembled. none of whom ence observed in the sample is due to have experienced the outcome of inchance alone. For the example in Taterest. Baseline characteristics. including putative risk factors for the ble A4: outcome are measured upon entry to the study. These subjects are then followed over time to characterize the relationship between baseline characteristics and the outcome. confounding: an apparent association between a risk factor and outBecause the table of critical values of come that is due to a second factor, the X 2 distribution is a partial listing, which is extraneous to the question small values such as X 2 = 0.60 are under study. not tabulated. Computer software, dichotomous variable: a variable like the Electronic Tables for Statisti- measured on a scale that has only cians (Hawkeye, Softworks, Iowa two possible states, for example, City, Iowa) can be used to pinpoint younglold, yesino, maleifemale. the P value when this is the case. The euent-free suruiual: in this article, P value corresponding to a X 2 of 0.60, event-free survival and survival are synonymous. The term event-free with (2- 1)= 1degree of freedom, is survival is used to reinforce the con0.44. Therefore, a difference in survival of the observed magnitude cept that the event of interest need would be expected, by chance alone, not be death, as is implied by the unin 44% of studies of the same size. As qualified use of the term survival. a result, the null hypothesis cannot event o f interest: the outcome under study. In this case, restenosis. be rejected at the .05 level of statistical significance; that is, it would be hypothesis test: also known as a inappropriate to conclude that there significance test. A rule for deciding is a statistically significant difference whether a n assumption (hypothesis) in survival between the two groups. about the distribution of a random (Note: failing to reject the null hyvariable should be rejected, with use pothesis is not synonymous with acof a sample from the distribution. cepting it, for reasons that are beThe hypothesis is called the null hyyond the scope of this discussion.) pothesis, written Ho, and it is tested against some alternative hypothesis, H I . A statistic is computed from the sample data. If it falls within the critGLOSSARY OF TERMS ical region, where the value of the staalpha error (type I error): the prob- tistic is significantly different from that ability of concluding that there is a expected under Ho,Ho is rejected (15). so derived for each group, are then added together. If the expected number of events in any given interval is less than lo%, the sum has an approximate X 2 distribution with k - 1 degrees of freedom, where k represents the number of groups being compared
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inception: inception is defined by a patient characteristic, such as time of diagnosis or start of therapy, which serves as a clear, well-defined, and collective starting point for follow-up, for example, date of PTA. inference (statistical): the logical basis by which conclusions regarding populations are drawn from results obtained in a sample (15). life table: a method for analyzing survival times that have been grouped into arbitrary intervals, when observation of some patients is incomplete in that they have not experienced the event of interest when the trial is stopped. mutually exclusive time intervals: time intervals that are non-overlapping, such that the assignment of a trial time to an interval is unique. null hypothesis: see hypothesis test-
ing. outcome: see event o f interest. suruiual, event free: see event-free suruival. table of critical values: a subset of the possible values of a test statistic, each of which corresponds to the probability of observing a value of the test statistic, that is at least as extreme as the one in question, by chance alone.
target population: the entire collection of patients, about whom the investigator wishes to draw conclusions based on observations made in a sample. trial time: the time from inception to date of loss to follow-up, to the trial stopping date, or occurrence of the event of interest. References 1. Cutler J, Ederer F. Maximum utilization of the life-table method in analyzing survival. J Chron Dis 1958; 8:699-713. 2. Colton T. Statistics in medicine. Boston: Little Brown, 1974; 237250. 3. Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects 33. London: H.M. Stationary Office, 1926. 4. Peto R, Pike M, Armitage P, et al. Design and analysis of randomized controlled clinical trials requiring prolonged observation of each patient. I. Introduction and design. Br J Cancer 1976; 34:585-612. 5. Peto R, Pike M, Armitage P, et al. Design and analysis of randomized controlled clinical trials requiring prolonged observation of each patient. 11. Analysis and examples. Br J Cancer 1977; 35:l-39.
Dawson-Saunders B, Trapp RG. Basic and clinical biostatistics 1990. Norwalk, Conn: Appleton Lange, 1990; 166-206. Gehan EA. A generalized Wilcoxon test for comparing arbitrarily singlycensored samples. Biometrika 1965; 52:15-21. Matthews DE, Farewell VT. Using and understanding medical statistics. New York: Karger, 1988: 78. BMDP Statistical software manual. Dixon WJ, Brown MB, Engleman, L, et al. Berkeley, Calif: University of California Press, 1985; 557-575. Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. Am Stat Assoc J 1958; 53:457-481. Kalbfleisch JD, Prentice RL. The statistical analysis of failure time data. New York: Wiley, 1980; 10-19. Kahn HA. An introduction to epidemiologic methods. Monographs in epidemiology and biostatistics. v5. New York: Oxford University Press, 1983: 121-144. Fletcher RH, Fletcher SW, Wagner EH. Clinical epidemiology: the essentials. Baltimore: Williams & Wilkins, 1982; 96-97. Gibson C. Dictionary of mathematics. Berkshire: Intercontinental Book Productions, 1981; 81. Colton T. Statistics in medicine. Boston: Little Brown, 1974; 3.
