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Interaction between prognostic factors and treatment
Interaction Between Prognostic Factors and Treatment Jonathan Shuster and Jan van Eys The Pediatric Oncology Group Statistical Office, Department of Statistics, University of Florida, Gainesville, Florida and the Pediatric Oncology Group, Scientific Support Chairman, Department of Pediatrics, University of Texas System Cancer Center, M.D. Anderson Hospital and Tumor Institute, Houston, Texas
ABSTRACT: In randomized trials, the better therapy may depend on values of a prognostic
factor. This article presents methodology for subdividing a target population into three subsets: one region of superiority of each treatment, and one region of uncertainty. The basis of such a subdivision is treatment by prognostic factor interaction. The importance of this analysis in group trial sequences is discussed briefly. KEY W O R D S :
interaction, prognostic factor, linear response
INTRODUCTION The therapeutic objective of randomized cancer trials is generally to determine which therapy (say A or B) is superior in terms of disease-free survival. To date, prognostic factor analyses have been used to identify high-risk, standard-risk, and low-risk patients. This is helpful in planning future trials, in that separate trials within risk groups can be considered. In "low-risk" groups, one considers less intensive therapy, while in "high-risk" groups, one might consider an aggressive therapy protocol. Examples of prognostic factors along these lines as deduced in group studies have been published [1, 2]. A second common use of prognostic factors is in analysis of covariance. This analysis retrospectively adjusts the treatment comparison for inbalances in prognostic factors. This is especially common in studies that utilize historical controls [3, 4]. Implicit in such analysis is the assumption that no treatment by prognostic factor interaction exists. That is, the relative risk at any instant for two patients with identical prognostic factors, but different treatments, is independent of the values of these prognostic factors. If one includes a treatment by prognostic factor interaction, one admits the possibility that the target population can be split into two subpopulations. In one subpopulation A is superior to B while in the other, B is superior to A. One of these subpopulations could be vacuous.
Address reprint requests to: Jonathan Shuster, Ph.D., University of Florida, 1105 W. University Avenue, Suite 200, GainesviUe, FL 32601. Received January 6, 1983; revised April 6, 1983. Controlled Clinical Trials 4:209-214 (1983) © Elsevier Science Publishing Co., Inc. 1983 52 Vanderbi]t Ave,, New York, New York 10017
209 0197-2456/83/$03.00
210
Jonathan Shuster and Jan van Eys Interaction is discussed briefly by Byar and co-workers [5] and Breslow [6], but no analytic methodology was provided. In this article, statistical methodology is developed for treatment by prognostic factor interaction. It must be noted that in general, prognostic factors are treatment artifacts. With improved therapy, the prognosis of one subset of patients may improve more than that of another. Hence, the dynamics of prognostic factors must be studied as a function of time and therapy. Treatment by prognostic factor interaction is a mechanism to accomplish this.
STATISTICAL METHODS
Consider a linear response function, L, for a target population: L = f3o + ~,X~ + f~2X2 + f3~X~X2,
where X~ = 1 for treatment A, = 0 for treatment B, X2 = Value of prognostic factor (covariate), and B0, [31, [32, [33 are parameters. Such response functions are used in analysis of hazards by Cox regression [7], logistic regression [8], and linear regression [9]. The difference between treatments for responses in the subpopulation with prognostic factor X2 = x2 is D = B1 + B3x2. If D > 0, then the subpopulation at x2 has a higher response on treatment A than on treatment B. Similarly, if D < 0, the opposite is true. It is therefore important to estimate the derived parameter Do = - [31/[33, the value of X2 where the response to A and B are identical. The theory of confidence intervals for ratios is well known from the theory of bioassays [10]. Whichever method is used for the problem at hand (Cox regression, logistic regression, or linear regression), one obtains asymptotically bivariate normal estimates of ([31,[33): ([31, [~3)with asymptotic covariance matrix
[Vll V13"~
V = \V~3 G d
By Fieller's Theorem, [10] an asymptotic 100(1 - 2~)% confidence interval for Do can be obtained as
I
-W2
-
X'/W~2 -
2W3
4WlW3.,
-
W 2 + X/W~2 -
2W3
4WlW3x I
J'
211
Treatment by Prognostic Factor Interaction where w , = ~? - z g v , ,
W2 = 2~1~3 - 2Z2V,3 w~ = ~
-
zgv~
and Z~ = upper (100c~) percent point of the standard (Z00s = 1.645, Zo02s = 1.96).
normal curve
I N T E R P R E T A T I O N OF THE CONFIDENCE INTERVAL FOR Do
The values of )(2 that lie in the confidence interval are those in which no significant difference (p > 2e0 exists between the treatments. The values outside the confidence interval are regions of superiority (p < 2c~) of the treatments. Since X2 is analytically treated as a continuous unbounded variable, the confidence interval for Do could contain exclusively impossible X2 values. In this situation, one of the treatments would be declared superior for all values of X2. (This could happen, for example, in a situation where X2 must be positive, but the confidence interval for Do contains only negative numbers.) Caution: The confidence interval is formed by solving the quadratic equation (for x):
(~,
+ x~3) 2 = z ~ ( v , ,
+ 2xv,3 + x2G3).
(1)
The roots can be imaginary. This is interpreted as no significant difference among the treatments for any value of X2. In such cases, the left-hand side of (1) would always be smaller than the right-hand side of (1). (The righthand side is positive definite while the left-hand side is zero at x =
-~,/&.)
If either W~ or W3 is greater than zero, then the solutions (1) are assured. Note that W~ (W3) greater than zero is equivalent to rejecting the hypothesis that/31 = 0 (/33 = 0) at p < 2a, two-sided. A S Y M P T O T I C RELATIVE EFFICIENCY W H E N 1~3 = 0
Although the methodology is designed for situations where/33 is non-zero, it is still effective when/33 = 0. Note that asymptotically: ~1 + x~3 is normally distributed, with mean/31 + x/33 = /31 and variance Vn + 2xV~3 + x2V33. For given )(2 = x, the asymptotic relative efficiency of classical analysis of covariance (which presumes/33 = 0) as compared to the above method is ARE(x) =
Wll -~ 2xV13 + x2V33 Vn - (~YV33)
212
Jonathan Shuster and Jan van Eys The above is the ratio of asymptotic variances of the two estimates of the treatment difference at X2 = x. Upon taking expectations over X2, the population m e a n relative efficiency is E(ARE(X2))
=
V11 q_ 2btV13 + (~j2 q_ O.2)V33 v.
-
(~dv.)
where bt = population m e a n of X2, and 0-2 = population variance of X2. EXTENSION TO MULTIPLE COVARIATES The technique is readily extended to several covariates. For two covariates, the response function is L = ~o + ftlx~ + B~x~ + f~x~ + ~ x , x ~ + ~ x l x ~
where X1 = 1
Treatment A,
= 0
Treatment B,
X2 = First prognostic factor, X3 --- Second prognostic factor. The analogue of the difference in response for the subpopulation at X2 = x2 a n d X 3 = x3is D = ~31 + [34x2 + ~35x3. A confidence region for the solutions to the equation D = 0 is the solution of the inequality (for x2,x3)
(10/
(f~, + ~4x2 + ~sx3)2 < Z2(1,0,O, x2,x3)V 0 ,
(2)
\::J where (~1. . . . ~s) are the parameter estimates and V = asymptotic covariance matrix of ~1,~2. . . . . Bs. Outside the region, are the domains of superiority (p < 2a) of the treatments. A l t h o u g h it is more difficult to visualize the confidence region in (2) than in the univariate case, one notes that for each fixed value of x2, one has a quadratic inequality in x3. For each fixed x2, there are three subsets corresponding to lack of treatment effect (between roots), a n d superiority of each treatment (two subsets--possibly vacuous outside of roots). The sign of ~1 + ~4x2 + f~sx3 determines where each regimen is declared superior,
213
Treatment by Prognostic Factor Interaction
This concept lends itself well to graphing the regions, as suggested by a referee.
NUMERICAL EXAMPLE
The following example is drawn from a trial of the Pediatric Oncology Group. Since the study is ongoing, it is premature to include the specifics. The dependent variable is disease-free survival. There were 336 patients in the randomized trial. The covariate of interest, a quantitative measurement taken at diagnosis, is expected to influence outcome. However, one of the two therapies has a much larger control over future values of this variable than does the other therapy. This suggests the possibility that the value of the covariate may have different prognostic significance within the two therapies. For the single prognostic factor (we call X2) of the statistical methodology section, the quantities were as follows, using Cox Regression. ~1 = 0.099,
f~2 = 0.00441,
~3 = -0.00763.
The asymptotic covariance matrix is V = 10-4
293.6 -3.085 1.6079 ~ -3.085 0.0954 - 0 . 0 4 3 3 | 1.6079 -0.0433 0.0431 ]
We use 2cx = 0.05 (95% confidence). An approximate 95% confidence interval for Do is (-27,92). Since low values of the response are favorable, in Cox Regression, we declare treatment A superior (p < 0.05, two-sided) for all X2 1-- 92. Since the range of X2 is X2 t> 0 there is no significant treatment effect for the remaining values of X2. It is estimated that treatment superiority can be declared for 29% (SE = 2%), since 98 of the 336 patients had X2 values above 92. Suppose we were to repeat the above study in a population where there was no treatment by prognostic factor interaction. Based on the sample mean and variance of the ongoing study (30.2 and 1981, respectively) we would estimate that classical analysis of covariance, which assumes [33 = 0, would have average asymptotic relative efficiency 1.99, compared to the methods of the article. However, it is of interest to note that in the ongoing study, the hypothesis that 63 = 0 can be rejected at p < 0.001. This fact renders the classical analysis of covariance invalid.
DISCUSSION The techniques above have two uses. First, by retrospective exploratory data analysis of large trials, one can generate interesting hypotheses for new prospective trials. Upon relating the interaction to treatment components one might build theories as to w h y patients react to therapy in a particular way. The second use deals with the actual prospective trial utilizing a planned
214
Jonathan Shuster and Jan van Eys interacting covariate. This interaction can be expected w h e n , as in the above example, the therapies t e n d to manipulate important prognostic factors differently. The structure of cooperative g r o u p protocol sequencing is highly suggestive of potential interaction. Stratification is based on extensive experience with the " s t a n d a r d " t h e r a p y a n d limited experience with the "experimental" therapy. O n e constructs strata, which yield vastly different prognosis on the s t a n d a r d therapy. O n e w o u l d expect (with no guarantees) that the prognostic i m p o r t a n c e of stratification t e n d s to be w e a k e r on the experimental therapy. ACKNOWLEDGMENTS This work was supported in part by the National Cancer Institute grants U10-CA-03713and U10CA-29139. The authors wish to thank the referees for their comments and suggestions.
REFERENCES 1. George SL, Fernbach DJ, Vietti TJ, Sullivan MP, Lane DM, Haggard ME, Berry DH, Lonsdale D, Komp D: Factors influencing survival in pediatric acute leukemia: The SWCCSG experience, 1958-1970. Cancer 32:1542-1553, 1973 2. Miller DR, Leikin S, Albo V, Vitale L, Sather H, Coccia P, Nesbit M, Karon M, Hammond D: The use of prognostic factors in improving the design and efficiency of clinical trials in childhood leukemia. Cancer Treat Rep 64:381-392, 1980 3. Gehan EA: Comparative clinical trials with historical controls: A statistician's view. Biomedicine 28:13-19, 1978 4. Gehan EA, Smith TL, Buzdar AV: Use of prognostic factors in analysis of historical control studies. Cancer Treat Rep 64:373-379, 1980 5. Byar DP, Simon RM, Friedewald WT, DeMets JJ, Ellenberg, JH, Gail MH, Ware JH: Randomized clinical trials: Perspectives on some recent ideas. New Engl J Med 295:74--80, 1976 6. Breslow N: Perspectives of the statistician's role in cooperative clinical research. Cancer 41, 326-332, 1978 7. Cox DR: Regression models and life-tables (with discussion). J R Stat Soc B 34:187-220, 1972 8. Cox DR: Analysis of Binary Data. London: Chapman and Hall, 1970 9. Draper NR, Smith H: Applied Regression Analysis. New York: Wiley, 1966 10. Finney DJ: Probit Analysis. London: Cambridge University Press, 1964
ABSTRACT: In randomized trials, the better therapy may depend on values of a prognostic
factor. This article presents methodology for subdividing a target population into three subsets: one region of superiority of each treatment, and one region of uncertainty. The basis of such a subdivision is treatment by prognostic factor interaction. The importance of this analysis in group trial sequences is discussed briefly. KEY W O R D S :
interaction, prognostic factor, linear response
INTRODUCTION The therapeutic objective of randomized cancer trials is generally to determine which therapy (say A or B) is superior in terms of disease-free survival. To date, prognostic factor analyses have been used to identify high-risk, standard-risk, and low-risk patients. This is helpful in planning future trials, in that separate trials within risk groups can be considered. In "low-risk" groups, one considers less intensive therapy, while in "high-risk" groups, one might consider an aggressive therapy protocol. Examples of prognostic factors along these lines as deduced in group studies have been published [1, 2]. A second common use of prognostic factors is in analysis of covariance. This analysis retrospectively adjusts the treatment comparison for inbalances in prognostic factors. This is especially common in studies that utilize historical controls [3, 4]. Implicit in such analysis is the assumption that no treatment by prognostic factor interaction exists. That is, the relative risk at any instant for two patients with identical prognostic factors, but different treatments, is independent of the values of these prognostic factors. If one includes a treatment by prognostic factor interaction, one admits the possibility that the target population can be split into two subpopulations. In one subpopulation A is superior to B while in the other, B is superior to A. One of these subpopulations could be vacuous.
Address reprint requests to: Jonathan Shuster, Ph.D., University of Florida, 1105 W. University Avenue, Suite 200, GainesviUe, FL 32601. Received January 6, 1983; revised April 6, 1983. Controlled Clinical Trials 4:209-214 (1983) © Elsevier Science Publishing Co., Inc. 1983 52 Vanderbi]t Ave,, New York, New York 10017
209 0197-2456/83/$03.00
210
Jonathan Shuster and Jan van Eys Interaction is discussed briefly by Byar and co-workers [5] and Breslow [6], but no analytic methodology was provided. In this article, statistical methodology is developed for treatment by prognostic factor interaction. It must be noted that in general, prognostic factors are treatment artifacts. With improved therapy, the prognosis of one subset of patients may improve more than that of another. Hence, the dynamics of prognostic factors must be studied as a function of time and therapy. Treatment by prognostic factor interaction is a mechanism to accomplish this.
STATISTICAL METHODS
Consider a linear response function, L, for a target population: L = f3o + ~,X~ + f~2X2 + f3~X~X2,
where X~ = 1 for treatment A, = 0 for treatment B, X2 = Value of prognostic factor (covariate), and B0, [31, [32, [33 are parameters. Such response functions are used in analysis of hazards by Cox regression [7], logistic regression [8], and linear regression [9]. The difference between treatments for responses in the subpopulation with prognostic factor X2 = x2 is D = B1 + B3x2. If D > 0, then the subpopulation at x2 has a higher response on treatment A than on treatment B. Similarly, if D < 0, the opposite is true. It is therefore important to estimate the derived parameter Do = - [31/[33, the value of X2 where the response to A and B are identical. The theory of confidence intervals for ratios is well known from the theory of bioassays [10]. Whichever method is used for the problem at hand (Cox regression, logistic regression, or linear regression), one obtains asymptotically bivariate normal estimates of ([31,[33): ([31, [~3)with asymptotic covariance matrix
[Vll V13"~
V = \V~3 G d
By Fieller's Theorem, [10] an asymptotic 100(1 - 2~)% confidence interval for Do can be obtained as
I
-W2
-
X'/W~2 -
2W3
4WlW3.,
-
W 2 + X/W~2 -
2W3
4WlW3x I
J'
211
Treatment by Prognostic Factor Interaction where w , = ~? - z g v , ,
W2 = 2~1~3 - 2Z2V,3 w~ = ~
-
zgv~
and Z~ = upper (100c~) percent point of the standard (Z00s = 1.645, Zo02s = 1.96).
normal curve
I N T E R P R E T A T I O N OF THE CONFIDENCE INTERVAL FOR Do
The values of )(2 that lie in the confidence interval are those in which no significant difference (p > 2e0 exists between the treatments. The values outside the confidence interval are regions of superiority (p < 2c~) of the treatments. Since X2 is analytically treated as a continuous unbounded variable, the confidence interval for Do could contain exclusively impossible X2 values. In this situation, one of the treatments would be declared superior for all values of X2. (This could happen, for example, in a situation where X2 must be positive, but the confidence interval for Do contains only negative numbers.) Caution: The confidence interval is formed by solving the quadratic equation (for x):
(~,
+ x~3) 2 = z ~ ( v , ,
+ 2xv,3 + x2G3).
(1)
The roots can be imaginary. This is interpreted as no significant difference among the treatments for any value of X2. In such cases, the left-hand side of (1) would always be smaller than the right-hand side of (1). (The righthand side is positive definite while the left-hand side is zero at x =
-~,/&.)
If either W~ or W3 is greater than zero, then the solutions (1) are assured. Note that W~ (W3) greater than zero is equivalent to rejecting the hypothesis that/31 = 0 (/33 = 0) at p < 2a, two-sided. A S Y M P T O T I C RELATIVE EFFICIENCY W H E N 1~3 = 0
Although the methodology is designed for situations where/33 is non-zero, it is still effective when/33 = 0. Note that asymptotically: ~1 + x~3 is normally distributed, with mean/31 + x/33 = /31 and variance Vn + 2xV~3 + x2V33. For given )(2 = x, the asymptotic relative efficiency of classical analysis of covariance (which presumes/33 = 0) as compared to the above method is ARE(x) =
Wll -~ 2xV13 + x2V33 Vn - (~YV33)
212
Jonathan Shuster and Jan van Eys The above is the ratio of asymptotic variances of the two estimates of the treatment difference at X2 = x. Upon taking expectations over X2, the population m e a n relative efficiency is E(ARE(X2))
=
V11 q_ 2btV13 + (~j2 q_ O.2)V33 v.
-
(~dv.)
where bt = population m e a n of X2, and 0-2 = population variance of X2. EXTENSION TO MULTIPLE COVARIATES The technique is readily extended to several covariates. For two covariates, the response function is L = ~o + ftlx~ + B~x~ + f~x~ + ~ x , x ~ + ~ x l x ~
where X1 = 1
Treatment A,
= 0
Treatment B,
X2 = First prognostic factor, X3 --- Second prognostic factor. The analogue of the difference in response for the subpopulation at X2 = x2 a n d X 3 = x3is D = ~31 + [34x2 + ~35x3. A confidence region for the solutions to the equation D = 0 is the solution of the inequality (for x2,x3)
(10/
(f~, + ~4x2 + ~sx3)2 < Z2(1,0,O, x2,x3)V 0 ,
(2)
\::J where (~1. . . . ~s) are the parameter estimates and V = asymptotic covariance matrix of ~1,~2. . . . . Bs. Outside the region, are the domains of superiority (p < 2a) of the treatments. A l t h o u g h it is more difficult to visualize the confidence region in (2) than in the univariate case, one notes that for each fixed value of x2, one has a quadratic inequality in x3. For each fixed x2, there are three subsets corresponding to lack of treatment effect (between roots), a n d superiority of each treatment (two subsets--possibly vacuous outside of roots). The sign of ~1 + ~4x2 + f~sx3 determines where each regimen is declared superior,
213
Treatment by Prognostic Factor Interaction
This concept lends itself well to graphing the regions, as suggested by a referee.
NUMERICAL EXAMPLE
The following example is drawn from a trial of the Pediatric Oncology Group. Since the study is ongoing, it is premature to include the specifics. The dependent variable is disease-free survival. There were 336 patients in the randomized trial. The covariate of interest, a quantitative measurement taken at diagnosis, is expected to influence outcome. However, one of the two therapies has a much larger control over future values of this variable than does the other therapy. This suggests the possibility that the value of the covariate may have different prognostic significance within the two therapies. For the single prognostic factor (we call X2) of the statistical methodology section, the quantities were as follows, using Cox Regression. ~1 = 0.099,
f~2 = 0.00441,
~3 = -0.00763.
The asymptotic covariance matrix is V = 10-4
293.6 -3.085 1.6079 ~ -3.085 0.0954 - 0 . 0 4 3 3 | 1.6079 -0.0433 0.0431 ]
We use 2cx = 0.05 (95% confidence). An approximate 95% confidence interval for Do is (-27,92). Since low values of the response are favorable, in Cox Regression, we declare treatment A superior (p < 0.05, two-sided) for all X2 1-- 92. Since the range of X2 is X2 t> 0 there is no significant treatment effect for the remaining values of X2. It is estimated that treatment superiority can be declared for 29% (SE = 2%), since 98 of the 336 patients had X2 values above 92. Suppose we were to repeat the above study in a population where there was no treatment by prognostic factor interaction. Based on the sample mean and variance of the ongoing study (30.2 and 1981, respectively) we would estimate that classical analysis of covariance, which assumes [33 = 0, would have average asymptotic relative efficiency 1.99, compared to the methods of the article. However, it is of interest to note that in the ongoing study, the hypothesis that 63 = 0 can be rejected at p < 0.001. This fact renders the classical analysis of covariance invalid.
DISCUSSION The techniques above have two uses. First, by retrospective exploratory data analysis of large trials, one can generate interesting hypotheses for new prospective trials. Upon relating the interaction to treatment components one might build theories as to w h y patients react to therapy in a particular way. The second use deals with the actual prospective trial utilizing a planned
214
Jonathan Shuster and Jan van Eys interacting covariate. This interaction can be expected w h e n , as in the above example, the therapies t e n d to manipulate important prognostic factors differently. The structure of cooperative g r o u p protocol sequencing is highly suggestive of potential interaction. Stratification is based on extensive experience with the " s t a n d a r d " t h e r a p y a n d limited experience with the "experimental" therapy. O n e constructs strata, which yield vastly different prognosis on the s t a n d a r d therapy. O n e w o u l d expect (with no guarantees) that the prognostic i m p o r t a n c e of stratification t e n d s to be w e a k e r on the experimental therapy. ACKNOWLEDGMENTS This work was supported in part by the National Cancer Institute grants U10-CA-03713and U10CA-29139. The authors wish to thank the referees for their comments and suggestions.
REFERENCES 1. George SL, Fernbach DJ, Vietti TJ, Sullivan MP, Lane DM, Haggard ME, Berry DH, Lonsdale D, Komp D: Factors influencing survival in pediatric acute leukemia: The SWCCSG experience, 1958-1970. Cancer 32:1542-1553, 1973 2. Miller DR, Leikin S, Albo V, Vitale L, Sather H, Coccia P, Nesbit M, Karon M, Hammond D: The use of prognostic factors in improving the design and efficiency of clinical trials in childhood leukemia. Cancer Treat Rep 64:381-392, 1980 3. Gehan EA: Comparative clinical trials with historical controls: A statistician's view. Biomedicine 28:13-19, 1978 4. Gehan EA, Smith TL, Buzdar AV: Use of prognostic factors in analysis of historical control studies. Cancer Treat Rep 64:373-379, 1980 5. Byar DP, Simon RM, Friedewald WT, DeMets JJ, Ellenberg, JH, Gail MH, Ware JH: Randomized clinical trials: Perspectives on some recent ideas. New Engl J Med 295:74--80, 1976 6. Breslow N: Perspectives of the statistician's role in cooperative clinical research. Cancer 41, 326-332, 1978 7. Cox DR: Regression models and life-tables (with discussion). J R Stat Soc B 34:187-220, 1972 8. Cox DR: Analysis of Binary Data. London: Chapman and Hall, 1970 9. Draper NR, Smith H: Applied Regression Analysis. New York: Wiley, 1966 10. Finney DJ: Probit Analysis. London: Cambridge University Press, 1964