STATISTICS IN MEDICINE, VOL. 11,975-977 (1992)
LETTERS TO THE EDITOR THE ANALYSIS OF FAILURE TIME DATA I N CROSSOVER STUDIES by L. A. France, J. A. Lewis and R. Kay, Statistics in Medicine, 10, 1099-1113 (1991)
From: David R. Bristol Janssen Research Foundation Inc. Piscataway. NJ 08855, U.S.A.
France, Lewis, and Kay presented several techniques for analysis of ‘time to event’ data in a two-by-two crossover design with emphasis on an angina pectoris trial for the comparison of atenolol and the combination of atenolol and nifedipine. They presented a brief review of several common approaches that do not incorporate the censoring. Several of their criticisms of these techniques are similar to the criticisms presented by Bristol and Castellana,’ who considered ‘time to angina’ in parallel-group studies. Furthermore, they proposed several techniques to incorporate the censoring, including Cox’s proportional hazards regression model.’ They proposed this technique ‘to use the full power and complexity of Cox’s proportional hazards regression model’. However, they emphasize use of this model to categorize the data into ‘preference’ (that is, one treatment is better than the other for each patient). Here a direct approach using this model is used to analyse the data given in their example. The variable of interest in France et al. is the time to a 1 mm ST-segment depression on an exercise stress test using a standard Bruce exercise protocol. This variable is associated with the presence of angina. Categorizing the data into ‘preference’ is inappropriate since the observed difference is important. Small differences, such as less than 15 seconds, may be within the range of variability and are not indicative of a clinically significant treatment difference. For example, at least two patients in their study had very small non-zero differences. patient 13 in sequence 2 had 210 seconds on the combination and 207 seconds on atenolol and patient 33 in sequence 2 had 248 seconds on the combination and 250 seconds on atenolol. It is not reasonable to conclude that the combination was better for the former patient and atenolol was better for the latter. The model of France et al. is l i ( t ;xl, x2) = l o i ( t )exp(fllxl flzx2),where
+
x1 = 0 for treatment B 1 for treatment A, x2 = 0 for period 1 1 for period 2, and l i ( t ;x,, x2) is the hazard function for patient i with this pair of values of xI and xz. Note that this model does not include a sequence (carryover) effect. This model can be easily revised to Ai(t;x1?x29x3)=
exdblxl + b Z X Z + f13x3),
where x3 = 0 for sequence 1 (AB) 1 for sequence 2 (BA) and l o ( [is) an unspecified function oft, which is the same for all patients. Using PROC PHGLM of SAS,3 anabsis of the data results in estimates of the regression coefficients (standard error) of fll = - 0.21 (0.15), f12 = - 0.20 (0.16),and b3 = - 004 (0.15). The corresponding pvalues are 0.177, 0.193, and 0.806, respectively. As the sequence effect is clearly not significant, one might analyse this data without the se_quenceeffect in the Fodel. This analysis results in estimates of the regression coefficients (standard error) of b1 = 0.21 (0.15)and f12 = - 0.21 (016). The corresponding p-values are 0.174 and 0.189, respectively. Analysis based on the Kaplan-Meier4 estimate of the survival function using PROC LIFETEST of SASS to further summarize the data yields the following results.
LETTERS TO THE EDITOR
976
25th percentile Median 75th percentile Mean (SE) Per cent censored P-value (logrank)
Atenolol
Combination
240 329 432 347.8 ( 1 3.4) 16.0
260 360 460 378.3 (15.2) 22% 0.163
Although the values in the previous table are useful for summary, they do not incorporate the effects in the crossover design. They are quite different from the estimates given in France et a/., which were obtained using a modification of Cox’s model to incorporate the crossover design. As noted in Bristol and Castellana,’ ‘time to event’ data from angina pectoris trials, such as total walking time and time to 1 mm ST-segment depression, can be analysed using Cox’s proportional hazards regression model. Using an appropriate model, a two-by-two crossover design can be used to incorporate period, treatment, and sequence effects, as well as the censoring due to termination of the exercise stress test due to fatigue or other ‘non-anginal’ reasons. REFERENCES
1. Bristol, D. R. and Castellana, J. V. ‘Survival analysis techniques in angina pectoris trials’, Statistics in Medicine, 9, 293-299 (1990). 2. Cox, D. R. ‘Regression models and life tables’ (with discussion), Journal ofthe Royal Statistical Society, Series B, 34, 187-220 (1972). 3. SAS Institute, Inc. SAS Supplemental Library User’s Guide, SAS Institute, Inc., Cary, NC, 1980. 4. Kaplan, E. L. and Meier, P. ‘Nonparametric estimation from incomplete observations’, Journal qf the American Statistical Association, 53, 45748 1 (1958). 5. SAS Institute, Inc. SAS User’s Guide: Statistics, Version 5 Edition, SAS Institute, Inc., Cary, NC, 1985.
AUTHORS’ REPLY We have read the letter from David Bristol and his associated paper, Bristol and Castellana.’ Our views are that their arguments are not appropriate for the crossover setting. The model on which our development was based allowed a patient-specific underlying hazard function. Using likelihood methods, this led naturally to a within-patient assessment of treatment effect which was based on a within-patient ranking of the survival times, as would be expected from the class of proportional hazards methods. Our use of the term ‘preference’ in this context simply reflected the statistical procedure of ranking each patient’s exercise times, and we did not expect readers to infer from this term a real clinical preference, as Bristol appears to have done. We did, of course, realize the importance of interpreting the results in meaningful clinical terms, and our paper covered that issue thoroughly. By contrast to our approach, the model used by Bristol and Castellana is based on a common underlying hazard for all patients, and a multiplicative constant for the sequence effect. This produces a measure of the treatment difference which is a combination of within- and between-patient effects. We feel that this is inappropriate. One well accepted feature of crossover trials is that inferences regarding treatment should be based on within-patient differences. As a minimum, we would have expected a multiplicative term for ‘patient’ in Bristol’s model, although the subsequent fitting of such a model might have posed computational problems. To illustrate our point further, Bristol goes on to fit the model without the sequence effect. This corresponds to treating the data as if it were a collection of 2n (n = number of patients) independent observations, which is clearly inappropriate in the crossover context and leads to a confusion of the within-
LETTERS TO THE EDITOR
977
and between-patient elements of variation. What is more, Bristol’s approach reduces the sensitivity of the treatment comparison, and hence it is no surprise that the statistical significance of the treatment effect is greatly reduced, opening the door for a type I1 error.
JOHN A. LEWIS Institute of Mathematics and Statistics Cornwallis Building University of Kent Canterbury, Kent CT2 7NF, U . K . RICHARD KAY Shefield Statistical Services Shefield Science Park Shefield S l 2 N S , U.K. REFERENCE
1. Bristol D. R. and Castellana, J. V. ‘Survival analysis techniques in angina pectoris trials’, Statistics in Medicine, 9, 293-299 (1 990).
LETTERS TO THE EDITOR THE ANALYSIS OF FAILURE TIME DATA I N CROSSOVER STUDIES by L. A. France, J. A. Lewis and R. Kay, Statistics in Medicine, 10, 1099-1113 (1991)
From: David R. Bristol Janssen Research Foundation Inc. Piscataway. NJ 08855, U.S.A.
France, Lewis, and Kay presented several techniques for analysis of ‘time to event’ data in a two-by-two crossover design with emphasis on an angina pectoris trial for the comparison of atenolol and the combination of atenolol and nifedipine. They presented a brief review of several common approaches that do not incorporate the censoring. Several of their criticisms of these techniques are similar to the criticisms presented by Bristol and Castellana,’ who considered ‘time to angina’ in parallel-group studies. Furthermore, they proposed several techniques to incorporate the censoring, including Cox’s proportional hazards regression model.’ They proposed this technique ‘to use the full power and complexity of Cox’s proportional hazards regression model’. However, they emphasize use of this model to categorize the data into ‘preference’ (that is, one treatment is better than the other for each patient). Here a direct approach using this model is used to analyse the data given in their example. The variable of interest in France et al. is the time to a 1 mm ST-segment depression on an exercise stress test using a standard Bruce exercise protocol. This variable is associated with the presence of angina. Categorizing the data into ‘preference’ is inappropriate since the observed difference is important. Small differences, such as less than 15 seconds, may be within the range of variability and are not indicative of a clinically significant treatment difference. For example, at least two patients in their study had very small non-zero differences. patient 13 in sequence 2 had 210 seconds on the combination and 207 seconds on atenolol and patient 33 in sequence 2 had 248 seconds on the combination and 250 seconds on atenolol. It is not reasonable to conclude that the combination was better for the former patient and atenolol was better for the latter. The model of France et al. is l i ( t ;xl, x2) = l o i ( t )exp(fllxl flzx2),where
+
x1 = 0 for treatment B 1 for treatment A, x2 = 0 for period 1 1 for period 2, and l i ( t ;x,, x2) is the hazard function for patient i with this pair of values of xI and xz. Note that this model does not include a sequence (carryover) effect. This model can be easily revised to Ai(t;x1?x29x3)=
exdblxl + b Z X Z + f13x3),
where x3 = 0 for sequence 1 (AB) 1 for sequence 2 (BA) and l o ( [is) an unspecified function oft, which is the same for all patients. Using PROC PHGLM of SAS,3 anabsis of the data results in estimates of the regression coefficients (standard error) of fll = - 0.21 (0.15), f12 = - 0.20 (0.16),and b3 = - 004 (0.15). The corresponding pvalues are 0.177, 0.193, and 0.806, respectively. As the sequence effect is clearly not significant, one might analyse this data without the se_quenceeffect in the Fodel. This analysis results in estimates of the regression coefficients (standard error) of b1 = 0.21 (0.15)and f12 = - 0.21 (016). The corresponding p-values are 0.174 and 0.189, respectively. Analysis based on the Kaplan-Meier4 estimate of the survival function using PROC LIFETEST of SASS to further summarize the data yields the following results.
LETTERS TO THE EDITOR
976
25th percentile Median 75th percentile Mean (SE) Per cent censored P-value (logrank)
Atenolol
Combination
240 329 432 347.8 ( 1 3.4) 16.0
260 360 460 378.3 (15.2) 22% 0.163
Although the values in the previous table are useful for summary, they do not incorporate the effects in the crossover design. They are quite different from the estimates given in France et a/., which were obtained using a modification of Cox’s model to incorporate the crossover design. As noted in Bristol and Castellana,’ ‘time to event’ data from angina pectoris trials, such as total walking time and time to 1 mm ST-segment depression, can be analysed using Cox’s proportional hazards regression model. Using an appropriate model, a two-by-two crossover design can be used to incorporate period, treatment, and sequence effects, as well as the censoring due to termination of the exercise stress test due to fatigue or other ‘non-anginal’ reasons. REFERENCES
1. Bristol, D. R. and Castellana, J. V. ‘Survival analysis techniques in angina pectoris trials’, Statistics in Medicine, 9, 293-299 (1990). 2. Cox, D. R. ‘Regression models and life tables’ (with discussion), Journal ofthe Royal Statistical Society, Series B, 34, 187-220 (1972). 3. SAS Institute, Inc. SAS Supplemental Library User’s Guide, SAS Institute, Inc., Cary, NC, 1980. 4. Kaplan, E. L. and Meier, P. ‘Nonparametric estimation from incomplete observations’, Journal qf the American Statistical Association, 53, 45748 1 (1958). 5. SAS Institute, Inc. SAS User’s Guide: Statistics, Version 5 Edition, SAS Institute, Inc., Cary, NC, 1985.
AUTHORS’ REPLY We have read the letter from David Bristol and his associated paper, Bristol and Castellana.’ Our views are that their arguments are not appropriate for the crossover setting. The model on which our development was based allowed a patient-specific underlying hazard function. Using likelihood methods, this led naturally to a within-patient assessment of treatment effect which was based on a within-patient ranking of the survival times, as would be expected from the class of proportional hazards methods. Our use of the term ‘preference’ in this context simply reflected the statistical procedure of ranking each patient’s exercise times, and we did not expect readers to infer from this term a real clinical preference, as Bristol appears to have done. We did, of course, realize the importance of interpreting the results in meaningful clinical terms, and our paper covered that issue thoroughly. By contrast to our approach, the model used by Bristol and Castellana is based on a common underlying hazard for all patients, and a multiplicative constant for the sequence effect. This produces a measure of the treatment difference which is a combination of within- and between-patient effects. We feel that this is inappropriate. One well accepted feature of crossover trials is that inferences regarding treatment should be based on within-patient differences. As a minimum, we would have expected a multiplicative term for ‘patient’ in Bristol’s model, although the subsequent fitting of such a model might have posed computational problems. To illustrate our point further, Bristol goes on to fit the model without the sequence effect. This corresponds to treating the data as if it were a collection of 2n (n = number of patients) independent observations, which is clearly inappropriate in the crossover context and leads to a confusion of the within-
LETTERS TO THE EDITOR
977
and between-patient elements of variation. What is more, Bristol’s approach reduces the sensitivity of the treatment comparison, and hence it is no surprise that the statistical significance of the treatment effect is greatly reduced, opening the door for a type I1 error.
JOHN A. LEWIS Institute of Mathematics and Statistics Cornwallis Building University of Kent Canterbury, Kent CT2 7NF, U . K . RICHARD KAY Shefield Statistical Services Shefield Science Park Shefield S l 2 N S , U.K. REFERENCE
1. Bristol D. R. and Castellana, J. V. ‘Survival analysis techniques in angina pectoris trials’, Statistics in Medicine, 9, 293-299 (1 990).
