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A program for balancing the allocation of subjects to treatment in a clinical trial
COMPUTERS
AND
BIOMEDICAL
A Program
RESEARCH
519-524 (1982)
15,
for Balancing the Allocation of Subjects Treatment in a Clinical Trial MIKEL
Sttrfistictrl
Consu/ting
Services,
517
to
AICKIN E. Lodge
Dr.,
Tempe,
Ari;onu
85283
Received March 10, 1982
Minimum likelihood allocation (MLA) is a new method for sequentially assigning subjects to treatment groups in a clinical trial. Its aim is to balance the trial, that is, to prevent various treatment groups from containing subjects who have much better or much worse prognoses than the average. The overall strategy of MLA is described, an illustrative example is provided, and the availability of the required computer program is announced. INTRODUCTION
A clinical trial is a controlled procedure for determining whether any of several treatments have beneficial effects, and for estimating the magnitudes of those effects. A crucial point in the design of a clinical trial comes when subjects are allocated to the treatment groups. The danger is that the allocation procedure may not be successful in balancing the treatment groups. If some groups contain a preponderance of subjects with favorable prognoses, and others contain a deficiency of such subjects, then it may be quite unconvincing, at the end of the trial, to argue that the observed differences in outcomes between the treatment groups are in fact due to the treatments, and not simply the result of differences on the prognostic factors. In some trials one may have a clear idea of which factors are of prognostic significance, while in others one may be reduced to consideration of the traditional epidemiologic trilogy-age, sex, and race. But it is a rare situation in which one would have no idea about prognosis; even a highly subjective “clinical impression” of an examining physician might be of great value in distinguishing patients. In some trials one has all the subjects available before beginning, so that at least in theory one could determine an adequately balanced allocation. However, due to the large number of possible ways of assigning subjects to treatments, this can be a formidable task from the standpoint of numerical computation. Moreover, it is far more common to see trials in which subjects are admitted over a certain period of time, so that one does not have a complete record of all subjects and their prognoses until the end of the trial. Here it is clearly necessary to have some adaptive procedure for allocating subjects as they are admitted. 519 OOlO-4809/82/060519-06$02.00/O Copyright 8 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
520
MIKEL
AICKIN
In this article a new method of allocation called Minimum Likelihood Allocation (MLA) is discussed. A computer program carries out MLA on each subject as he enters the trial, and adaptively balances prognostic factors identified by the investigators. MLA has desirable properties in large trials, which can be established mathematically, and has demonstrated its efficiency in simulated clinical trials, one of which is discussed here for the purpose of illustration.
THE STRATEGY
OF MLA
A precise description of the technical details of MLA would not contribute to an understanding of why it works (except, perhaps, to a mathematical statistician). But the basis on which MLA operates is quite practical, and based on commonsense. We assume that a number of prognostic factors have been identified, and that a suitable specification has been made of the “balance” desired in the trial. This will lead us into technical details that will be covered more fully in the next section, and so for now we simply assume that this first step has been taken. We imagine ourselves in midtrial, and consider the allocation of the current subject, who has just been admitted. We note the subject’s prognostic factors, and then provisionally allocate him to each of the possible treatments in turn. For each such provisional allocation, we compute a measure of the “imbalance” which would result if that were the actual allocation. The subject is either allocated to the ‘minimum imbalance” treatment, or a biased randomization is performed which makes the “balanced” treatments more likely than the “unbalanced” ones. At this level of generality MLA sounds much like imbalance minimization schemes which have been proposed before (I, 5, 6). There are several important conceptual differences between MLA and these other strategies, however, one of which hinges on the definition of “imbalance.” The previous allocation methods all define balance in terms of the distribution of treatment allocations within prognostic strata, while MLA defines balance in terms of the distribution of prognostic factors within treatment groups. It is, in fact, the definition used by MLA which has been the guideline for the assessment of balance adopted by trial investigators since the mid 1950s. Without going into too much detail, it is worthwhile to mention that the imbalance measure used by MLA is a defensive one. That is, MLA supposes that a critic of the trial might try to argue that treatment assignment and prognostic factors were related in the actual allocation, thus confounding their effects on the outcome. MLA mimics the actions of such a critic by statistically analysing the data at each allocation, using a log-linear model for the discrete treatment cells. MLA chooses the allocation which gives the critic the least statistical evidence of a prognostic factor-treatment allocation association. There are several other advantages of MLA over other allocation strategies, but they will not be discussed here. Perhaps the sole drawback of MLA is that it
BALANCING
requires a computer tive example.
TREATMENT
ALLOCATIONS
program for its implementation.
521
We now turn to an illustra-
EXAMPLEOF ACOMPLEX TRIAL Although MLA is suitable for attaining simple balance in a two-group trial, we will illustrate it in a more complex situation to indicate its flexibility and power. Let the primary treatment designation be L1 for active treatment, and L,, for placebo. We pose two auxiliary treatment factors E and R, each having two levels (E,,, E, and Ro, RI). Thus, there are eight possible treatment assignments, as shown at the top of Fig. 1. Examples of auxiliary factors might be (1) dose level of a drug (high vs low), (2) route of administration of a drug (oral vs intradermal), or (3) clinic at which the patient is treated. The subjects used for this illustration were chronic lung patients, and so the prognostic factors consisted of a continuous measure W of objective lung funcZ3
Treatment
"Target"
COnfigUratiOn
Frequencies 12 18 10
Attained
20
30
I--
Frequencies
E33 3
9
12
7
11
18
10
20
30
I-
w Means
r
I
I
FIG. 1. The configuration of a treatment involving three balance in terms of frequencies and W means was achieved displayed here.
factors, by MLA
E. L, and R. Attainment of for the margins of the table
522
MIKEL
AICKIN
tion, and a dichotomous variable D indicating a self-reported assessment of breathing impairment. We now turn to the specification of balance for this case. First, we imagine that the auxiliary factors E and R are noninteracting in their effect on the subject. This means that one can regard the trial as two different trials, one in which the joint effect of L and R is to be studied, and a second in which the joint effect of L and E is of interest. A consequence of this is that one does not require that all eight cells in Fig. 1 be balanced. It is only necessary that the marginal tables determined by the L, E and the L, R classifications be balanced simultaneously. That is, if we were to ignore the E factor, we would see balance in the four L, R cells, and similarly, if we were to ignore R we would see balance in the four L, E cells. With regard to the prognostic factors, we may specify balance either separately or jointly for D and IV. We would say that we had attained marginal balance for D and W separately if each of these factors were balanced when the other was ignored. Thus. we should see the same proportion of subjects with high D in the treatment cells, and the same means (for example) of W in the treatment cells. Joint balance, on the other hand, would require the same W means within both D groups, within each treatment cell. The choice between these two kinds of balance depends on whether one considers the D and W factors to have interactive effects on the outcome of the trial. If they interact. joint balance is required. One may also specify certain target frequencies in the various cells. In this case, one of the features of balance taken into account by MLA is how well the target frequencies are being produced. The target frequencies for our example appear in the second row of Fig. 1. Part of the results of allocating the first 100 subjects appears in the last 2 rows of Fig. 1. The attained frequencies are the actual numbers of subjects allocated to each cell. The L. E marginal target frequencies are 12. 18, 42, 28, while the attained frequencies are 12, 18, 43, 27, very close agreement. Similarly, the L. R frequencies (10, 20, 30, 40) are closely matched by 10, 20, 31, 39. Note that the frequencies appear in the margins of the 3-way table of Fig. 1. Note also, that the eight attained frequencies in the eight cells do not match the target frequencies quite so well. This is, of course, because MLA was not requested to balance the entire table, just the margins. We can see a similar pattern in the W means computed at the end of the allocation. The marginal means are all quite close, but there is considerable disparity in the eight cell means. Again, MLA was only requested to produce marginal balance, and would have produced balance in all eight cells if this had been the balance objective. Another view of the balancing process can be seen in Fig. 2. Here we have a record of the values of the F statistic for testing equality of W means in the four L. R treatment cells. The closer the value of F to zero, the more balance there is. A completely random allocation of subjects would produce F values in the neighborhood of 1.
BALANCING 1.75
TREATMENT
523
ALLOCATIONS
. . 1
1.40-
00 0 .
:*.
1.05-
.
.
00
. .
.
.70-
. b .
.35-
t St
.oo -
.
t--ttt@ 00
I
0
20
l
wttt?=tt tu*
40 Subject
l
60 Numbers
l ptttttu:t
80
100
FIG. 2. The F statistic for testing equality of W means across the .L. R margin, plotted against the current number of subjects in the trial. The sharp drop reflects the fact that balance was requested beginning with the 23rd subject.
What is especially interesting about Fig. 2 is that balance with regard to W was not specified until the 23rd subject was allocated. Before the 23rd subject, the F values behave as if the allocation were completely random, but afterwards they swing sharply towards balance. This also illustrates a feature of MLA that has practical importance; new prognostic factors can be added to the balance protocol in midtrial. It is also easy to omit factors which do not show the anticipated prognostic value. PROGRAM MANAGEMENT
Maintenance of a file of subjects is necessary for using MLA. The subjects must be listed in the order of their admission to the trial, and the prognostic measures must also be recorded. The file is updated to contain the allocations as they are made. Each new subject is added to the end of the file, and then MLA processes the entire file. The output identifies the minimum imbalance treatment assignment, and gives randomization probabilities (for all treatments) which favor the more balanced allocations (4). The randomization probabilities would only be used if there were concern about the possibility of investigator bias. It is also possible to allocate subjects in blocks, instead of one at a time. Specification of balance is made in a control block, described in detail by the user’s manual. Modification of the balance protocol is no more difhcult than the addition of a new subject.
524 A small FORTRAN author.
MIKEL
deck, listing,
AICKIN
and user’s manual are available
from the
CONCLUSION
The sequential allocation of subjects to treatments in a controlled clinical trial can be carried out efficiently by the MLA program. Complex treatments and prognostic factors, and complex relations among them can be accommodated by MLA. Although MLA requires a computer program because of the technical computations, it is in fact based on a common-sense strategy for attaining balance in the trial. Once installed on a computer, MLA is probably easier to use than previously published methods. REFERENCES I. BEGG, C. B., AND IGLEWICZ, B. A treatment allocation procedure for clinical trials. Biornerric~.\ 36, 81 (1980). 2. EFRON, B. Forcing a sequential experiment to be balanced. Biometrika 58, 403 (1971). 3. HARVILLE, D. A. Nearly optimal allocation of experimental units using observed covariate values. Technornefrics 16, 589 (1974). 4. KLOTZ, J. H. Maximum entropy constrained balance randomization for clinical trials. Biometrics 34, 283 (1978). 5. POCOCK, S. J., AND SIMON, R. Sequential treatment assignment with balancing for prognostic factors in the controlled ciinical trial. Biometrics 31, 103 (1975). 6. TAVES, D. R. Minimization: A new method of assigning patients to treatment and control groups. C/in. Pharmacol. Therapeut. 15, 443 (1974). 7. ZELEN, M. The randomization and stratification of patients to clinical trials. .J. Chrou. Dis. 27, 365 (1974).
AND
BIOMEDICAL
A Program
RESEARCH
519-524 (1982)
15,
for Balancing the Allocation of Subjects Treatment in a Clinical Trial MIKEL
Sttrfistictrl
Consu/ting
Services,
517
to
AICKIN E. Lodge
Dr.,
Tempe,
Ari;onu
85283
Received March 10, 1982
Minimum likelihood allocation (MLA) is a new method for sequentially assigning subjects to treatment groups in a clinical trial. Its aim is to balance the trial, that is, to prevent various treatment groups from containing subjects who have much better or much worse prognoses than the average. The overall strategy of MLA is described, an illustrative example is provided, and the availability of the required computer program is announced. INTRODUCTION
A clinical trial is a controlled procedure for determining whether any of several treatments have beneficial effects, and for estimating the magnitudes of those effects. A crucial point in the design of a clinical trial comes when subjects are allocated to the treatment groups. The danger is that the allocation procedure may not be successful in balancing the treatment groups. If some groups contain a preponderance of subjects with favorable prognoses, and others contain a deficiency of such subjects, then it may be quite unconvincing, at the end of the trial, to argue that the observed differences in outcomes between the treatment groups are in fact due to the treatments, and not simply the result of differences on the prognostic factors. In some trials one may have a clear idea of which factors are of prognostic significance, while in others one may be reduced to consideration of the traditional epidemiologic trilogy-age, sex, and race. But it is a rare situation in which one would have no idea about prognosis; even a highly subjective “clinical impression” of an examining physician might be of great value in distinguishing patients. In some trials one has all the subjects available before beginning, so that at least in theory one could determine an adequately balanced allocation. However, due to the large number of possible ways of assigning subjects to treatments, this can be a formidable task from the standpoint of numerical computation. Moreover, it is far more common to see trials in which subjects are admitted over a certain period of time, so that one does not have a complete record of all subjects and their prognoses until the end of the trial. Here it is clearly necessary to have some adaptive procedure for allocating subjects as they are admitted. 519 OOlO-4809/82/060519-06$02.00/O Copyright 8 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
520
MIKEL
AICKIN
In this article a new method of allocation called Minimum Likelihood Allocation (MLA) is discussed. A computer program carries out MLA on each subject as he enters the trial, and adaptively balances prognostic factors identified by the investigators. MLA has desirable properties in large trials, which can be established mathematically, and has demonstrated its efficiency in simulated clinical trials, one of which is discussed here for the purpose of illustration.
THE STRATEGY
OF MLA
A precise description of the technical details of MLA would not contribute to an understanding of why it works (except, perhaps, to a mathematical statistician). But the basis on which MLA operates is quite practical, and based on commonsense. We assume that a number of prognostic factors have been identified, and that a suitable specification has been made of the “balance” desired in the trial. This will lead us into technical details that will be covered more fully in the next section, and so for now we simply assume that this first step has been taken. We imagine ourselves in midtrial, and consider the allocation of the current subject, who has just been admitted. We note the subject’s prognostic factors, and then provisionally allocate him to each of the possible treatments in turn. For each such provisional allocation, we compute a measure of the “imbalance” which would result if that were the actual allocation. The subject is either allocated to the ‘minimum imbalance” treatment, or a biased randomization is performed which makes the “balanced” treatments more likely than the “unbalanced” ones. At this level of generality MLA sounds much like imbalance minimization schemes which have been proposed before (I, 5, 6). There are several important conceptual differences between MLA and these other strategies, however, one of which hinges on the definition of “imbalance.” The previous allocation methods all define balance in terms of the distribution of treatment allocations within prognostic strata, while MLA defines balance in terms of the distribution of prognostic factors within treatment groups. It is, in fact, the definition used by MLA which has been the guideline for the assessment of balance adopted by trial investigators since the mid 1950s. Without going into too much detail, it is worthwhile to mention that the imbalance measure used by MLA is a defensive one. That is, MLA supposes that a critic of the trial might try to argue that treatment assignment and prognostic factors were related in the actual allocation, thus confounding their effects on the outcome. MLA mimics the actions of such a critic by statistically analysing the data at each allocation, using a log-linear model for the discrete treatment cells. MLA chooses the allocation which gives the critic the least statistical evidence of a prognostic factor-treatment allocation association. There are several other advantages of MLA over other allocation strategies, but they will not be discussed here. Perhaps the sole drawback of MLA is that it
BALANCING
requires a computer tive example.
TREATMENT
ALLOCATIONS
program for its implementation.
521
We now turn to an illustra-
EXAMPLEOF ACOMPLEX TRIAL Although MLA is suitable for attaining simple balance in a two-group trial, we will illustrate it in a more complex situation to indicate its flexibility and power. Let the primary treatment designation be L1 for active treatment, and L,, for placebo. We pose two auxiliary treatment factors E and R, each having two levels (E,,, E, and Ro, RI). Thus, there are eight possible treatment assignments, as shown at the top of Fig. 1. Examples of auxiliary factors might be (1) dose level of a drug (high vs low), (2) route of administration of a drug (oral vs intradermal), or (3) clinic at which the patient is treated. The subjects used for this illustration were chronic lung patients, and so the prognostic factors consisted of a continuous measure W of objective lung funcZ3
Treatment
"Target"
COnfigUratiOn
Frequencies 12 18 10
Attained
20
30
I--
Frequencies
E33 3
9
12
7
11
18
10
20
30
I-
w Means
r
I
I
FIG. 1. The configuration of a treatment involving three balance in terms of frequencies and W means was achieved displayed here.
factors, by MLA
E. L, and R. Attainment of for the margins of the table
522
MIKEL
AICKIN
tion, and a dichotomous variable D indicating a self-reported assessment of breathing impairment. We now turn to the specification of balance for this case. First, we imagine that the auxiliary factors E and R are noninteracting in their effect on the subject. This means that one can regard the trial as two different trials, one in which the joint effect of L and R is to be studied, and a second in which the joint effect of L and E is of interest. A consequence of this is that one does not require that all eight cells in Fig. 1 be balanced. It is only necessary that the marginal tables determined by the L, E and the L, R classifications be balanced simultaneously. That is, if we were to ignore the E factor, we would see balance in the four L, R cells, and similarly, if we were to ignore R we would see balance in the four L, E cells. With regard to the prognostic factors, we may specify balance either separately or jointly for D and IV. We would say that we had attained marginal balance for D and W separately if each of these factors were balanced when the other was ignored. Thus. we should see the same proportion of subjects with high D in the treatment cells, and the same means (for example) of W in the treatment cells. Joint balance, on the other hand, would require the same W means within both D groups, within each treatment cell. The choice between these two kinds of balance depends on whether one considers the D and W factors to have interactive effects on the outcome of the trial. If they interact. joint balance is required. One may also specify certain target frequencies in the various cells. In this case, one of the features of balance taken into account by MLA is how well the target frequencies are being produced. The target frequencies for our example appear in the second row of Fig. 1. Part of the results of allocating the first 100 subjects appears in the last 2 rows of Fig. 1. The attained frequencies are the actual numbers of subjects allocated to each cell. The L. E marginal target frequencies are 12. 18, 42, 28, while the attained frequencies are 12, 18, 43, 27, very close agreement. Similarly, the L. R frequencies (10, 20, 30, 40) are closely matched by 10, 20, 31, 39. Note that the frequencies appear in the margins of the 3-way table of Fig. 1. Note also, that the eight attained frequencies in the eight cells do not match the target frequencies quite so well. This is, of course, because MLA was not requested to balance the entire table, just the margins. We can see a similar pattern in the W means computed at the end of the allocation. The marginal means are all quite close, but there is considerable disparity in the eight cell means. Again, MLA was only requested to produce marginal balance, and would have produced balance in all eight cells if this had been the balance objective. Another view of the balancing process can be seen in Fig. 2. Here we have a record of the values of the F statistic for testing equality of W means in the four L. R treatment cells. The closer the value of F to zero, the more balance there is. A completely random allocation of subjects would produce F values in the neighborhood of 1.
BALANCING 1.75
TREATMENT
523
ALLOCATIONS
. . 1
1.40-
00 0 .
:*.
1.05-
.
.
00
. .
.
.70-
. b .
.35-
t St
.oo -
.
t--ttt@ 00
I
0
20
l
wttt?=tt tu*
40 Subject
l
60 Numbers
l ptttttu:t
80
100
FIG. 2. The F statistic for testing equality of W means across the .L. R margin, plotted against the current number of subjects in the trial. The sharp drop reflects the fact that balance was requested beginning with the 23rd subject.
What is especially interesting about Fig. 2 is that balance with regard to W was not specified until the 23rd subject was allocated. Before the 23rd subject, the F values behave as if the allocation were completely random, but afterwards they swing sharply towards balance. This also illustrates a feature of MLA that has practical importance; new prognostic factors can be added to the balance protocol in midtrial. It is also easy to omit factors which do not show the anticipated prognostic value. PROGRAM MANAGEMENT
Maintenance of a file of subjects is necessary for using MLA. The subjects must be listed in the order of their admission to the trial, and the prognostic measures must also be recorded. The file is updated to contain the allocations as they are made. Each new subject is added to the end of the file, and then MLA processes the entire file. The output identifies the minimum imbalance treatment assignment, and gives randomization probabilities (for all treatments) which favor the more balanced allocations (4). The randomization probabilities would only be used if there were concern about the possibility of investigator bias. It is also possible to allocate subjects in blocks, instead of one at a time. Specification of balance is made in a control block, described in detail by the user’s manual. Modification of the balance protocol is no more difhcult than the addition of a new subject.
524 A small FORTRAN author.
MIKEL
deck, listing,
AICKIN
and user’s manual are available
from the
CONCLUSION
The sequential allocation of subjects to treatments in a controlled clinical trial can be carried out efficiently by the MLA program. Complex treatments and prognostic factors, and complex relations among them can be accommodated by MLA. Although MLA requires a computer program because of the technical computations, it is in fact based on a common-sense strategy for attaining balance in the trial. Once installed on a computer, MLA is probably easier to use than previously published methods. REFERENCES I. BEGG, C. B., AND IGLEWICZ, B. A treatment allocation procedure for clinical trials. Biornerric~.\ 36, 81 (1980). 2. EFRON, B. Forcing a sequential experiment to be balanced. Biometrika 58, 403 (1971). 3. HARVILLE, D. A. Nearly optimal allocation of experimental units using observed covariate values. Technornefrics 16, 589 (1974). 4. KLOTZ, J. H. Maximum entropy constrained balance randomization for clinical trials. Biometrics 34, 283 (1978). 5. POCOCK, S. J., AND SIMON, R. Sequential treatment assignment with balancing for prognostic factors in the controlled ciinical trial. Biometrics 31, 103 (1975). 6. TAVES, D. R. Minimization: A new method of assigning patients to treatment and control groups. C/in. Pharmacol. Therapeut. 15, 443 (1974). 7. ZELEN, M. The randomization and stratification of patients to clinical trials. .J. Chrou. Dis. 27, 365 (1974).