- Email: [email protected]
The bias coin design
The Bias Coin Design
Carol Sweeney, CNM, MS, and Marlene J. Egger,
PhD
ABSTRACT In clinical research with small sample size, problems can arise in allocating subjects into varin~~r nrolms when RW -__v_l.“ acrossinn& intn ctlv-k~ nuor .__ .__- treatment .__ _.... “... J.V’ r. . . . -.. thal ...vJ _.., ..w_ .. . .- R - .,...-J . ..“. ti,mp. The && Coin Design (BCD) was designed to overcome the problems of biased or unbalanced treatment groups, while ensuring the advantages of complete randomization, The BCD is discussed, and an example of its use in clinical research is given.
In recent years nurse-midwives have been encouraged to design and conduct clinical research. Clinical research by nurse-midwives both adds new knowledge to the profession and may validate current practice. The research conducted by nursemidwives should adhere to sound research designs. In experimental and quasiexperimental research this includes, among other factors, how the c,thiaz-tc au” ,TbLa
SVD v~nrlnmi~nrl UIG LUIIU”IIIIL~U
intr\ llll”
rl;4L&,1-
ferent treatment groups. One of the problems in clinical research is the acquisition of subjects into a study. In obstetrics, where cases of interest for research may infrequently occur, for example breech position of the fetus, the subjects are often accessioned into the studu over time. In other instances, the practice may be small, and sufficient numbers of all patients may take time to accumulate. A recent two-phased study provides an example of needing to
Address correspondence to: Carol Sweeney, CNM, MS, University of Utah College of Nursing, 25 South Medical Drive, Salt Lake City, UT 84112.
Journal of Nurse-Midwifery
??
acquire subjects over time. In “The Effects of a Nutrition Intervention Program During Pregnancy”’ a nutrition assessment and counseling program was to begin at 20 weeks’ gestation. Forty-seven subjects, who met the study criteria and consented to participate, were identified over a ten-week period, and one year later over an eight-week period. Each subject reached her 20th week of nncttltir\n rr+U = rli44nront yGJLuLr”rL UL UIII~ITIILtin-l” ULLIG. In the Nutrition Intervention Study, subjects accessioned were immediately assigned into the experimental or control groups as they reached the 20th week of gestation; however, the assignment of the total group of subjects occurred over a ten-week __.. oeriod ,__.~_ _. in ~~~the ~.~_ first phase of the study and over an eight-week pe-
nod in the second phase of the study one year later. To prevent biasing of results, subjects should be assigned immediately into the different treatment groups. Two common methods of assigning such subjects are by randomization, that is, “the independent flip of a fair coin,” or by systematic assignment. Both methods of treatment assignments have disadvan-
Vol. 30, No. 3, May/June
Copyright d 1985 by the American College of Nurse-Midwives
tages when there is a small sample size. Theoretically, the independent flip of a fair coin gives each subject an equal chance of being in either treatment group. This does not guarantee that at the end of randomization there will be an equal number of subjects in each group. In a small sample an imbalance in the numbers between treatment groups could be nnT\,,nh tn ctratictiCTJ.,OV,, 3.Z”GL.G r,,vuy,, L” r\v-o.ront i.JIT”CIII .ZLULI.7L1 cal analysis of the data. Another problem is that known confounding variables cannot be controlled, and resulting imbalance across treatment groups in a small sample may bias the results. In the Nutrition Intervention Study two confounding variables, low pregravid weight and low maternal weight gain, were of concern. Both of these variables have been correlated with low infant birth weight.2a3 The independent flip of a fair coin could not guarantee an equal number of subjects with low pregravid weight and low maternal weight gain between the experimental and control groups. The systematic assignment of subjects has been suggested to over-
175
1985 0091.2182’85,$03
30
come the problems of unequal numbers of subjects between the different treatment groups and imbalanced representation of confounding variables between the groups. Systematic assignment provides a balanced experiment by forcing subjects into a set pattern of allocations, for example, ECECEC or ECCEECCE. Some objections to this second approach have been raised, particularly on ethical grounds. For example, the researcher will know or can easily deduce the schedule of treatment assignments and may consciously or unconsciously schedule subjects to provide certain treatments to certain subjects. Although one would strive to remain objective in a research setting, such bias is extremely difficult to avoid. Competent researchers almost always have a subjective opinion regarding which treatment is better. The purpose of the clinical trial is to provide objective evidence regarding the relative merits of the treatments. During the course of a trial it is not uncommon for subjective opinion of the treatments, coupled with clinical concern for the welfare of each patient, to cause a conscious or unconscious ethical dilemma. The integrity of the clinical trial can be compromised severely if a treatment assignment procedure, such as systematic assignment with a randomized start, permits investiga-
Carol Sweeney
received
her BS in
nursing from Washington State University, and her MSN and CNM from the University
of Utah. She is current/y
clinical instructor graduate
of nursing
nurse-midwife
the Uniuersity
in the
y program
of Utah College
at
of
Nursing. Marlene
J. Egger
received
PhD in statistics from
University.
She is currently
Professor
in the Department
and Community Biostatistics, of Medicine.
176
her MS and
Stanford
Medicine,
Uniuersity
Assistant of Fami/y Division
of
of Utah School
tors to “get around” the treatment assignment schedule, for example, by shifting appointment dates to coincide with the preferred treatment. Additional discussion of ethics can be found elsewhere.4 If the clinicians who actually admit and observe subjects are kept “blind,” that is, they have no knowledge of which subjects receive which treatments or of the method of assignment, then the ethical objection to systematic assignment with a randomized start is removed. The Biased Coin Design (BCD) was proposed by Efron5 in 1971 to ensure both the benefits of a balanced experiment and the advantages of complete randomization. Within the Northern California Oncology Group’s (NCOG) randomized clinical trials in cancer,6,7 BCDs have been used routinely since 1977. Intuitively, the BCD allows the experimenter to “flip a biased coin” instead of a fair coin in the assignment of subjects to treatment groups. The first subject has an equal chance of being assigned to any treatment group. For subsequent subjects, “preference” scores are calculated to determine the best placement of the subject to balance the numbers of subjects assigned to each treatment within categories of potential confounding variables. Then a random number table is used so that the subject is assigned the “preferred” treatment 75% of the time and the “nonpreferred” treatment 25% of the time. It is as if a biased coin has been flipped so that the subject is randomly, not systematically, assigned a treatment. The coin is biased to achieve balanced numbers of subjects, and the researcher does not know which treatment will be assigned until after the coin flip. If more than two treatment groups are under study, the bias can be changed from 75% to other values. It should be stressed that “preference” scores are calculated based on the current imbalance in numbers of subjects in each treatment and confounding variable category, not on the re-
searcher’s preference for a particular treatment for an individual subject. Like complete randomization, the BCD still does not guarantee that at the end of randomization, there will be an equal number of subjects in each group. However, the probability of perfect balance is considerably higher than for complete randomization, and the probability of a severe imbalance is greatly reduced, even in small- and moderate-sized samples. This was known in a simulation study by Hannigan and Brown.6 Their technical report also describes the NCOG version of the BCD fully and demonstrates its simplicity of use. The BCD can provide for a balanced experiment of a small sample, even when the treatment groups are to be stratified. In the Nutrition Intervention Study, two confounding variables, low pregravid weight, and low maternal weight gain were identified before the different treatment programs were started; therefore, the goal was to obtain equal numbers of women in the control and the experimental groups with low pregravid weight and failure to gain at least 10 lb (4.5 kg) by 20 weeks of gestation. A protocol for use of the BCD in this specific study was designed and implemented. The BCD did achieve the stated goal, even though there was a small sample size of 47 subjects. There were 23 subjects in the experimental group and 24 subjects in the control group. Of the seven subjects with low pregravid weight, four were in the experimental group and three were in the control group. The 14 subjects with a failure to gain 10 lb (4.5 kg) by 20 weeks’ gestation were divided equally between the experimental and control groups. Pregravid weight and maternal weight gain have been observed to act independently of each other and are additive in their effects on infant birth weight2 This meant that the subjects could be further stratified into four categories in each treatment group: 1) normal or above pregravid
Journal of Nurse-Midwifery
??
Vol. 30, No. 3, May/June
1985
weight and high early weight gain, 2) normal or above pregravid weight and low early weight gain, 3) low pregravid weight and high early weight gain, and 4) low pregravid weight and low early weight gain. The BCD achieved a comparable number of subjects of each above category between the experimental and control group. Those subjects with both normal or above pregravid weight and a high early weight gain were divided into 12 in the experimental group and 15 in the control group. There were 13 subjects with a normal or above pregravid weight and a low early weight gain, seven in the experimental group and six in the control group. The subjects with low pregravid weight and high early weight gain were divided into four in the experimental group and two in the control group. The only subject with a low pregravid weight and a low early weight gain was in the control group. The BCD was implemented in the Nutrition Intervention Study without difficulty. A step-by-step approach of the protocol made learning easy. This protocol, available upon request from the authors, was designed spe-
Journal of Nurse-Midwifery
??
cifically for the Nutrition Intervention Study. A one-hour session, including explanation of the protocol and practice using the protocol, enabled the researchers to be proficient in its use. The researchers were able to randomize subjects into the different treatment groups, using the BCD protocol, within five minutes. In the Nutrition Intervention Study the protocol was done by hand; however, the protocol could easily be computerized or put in a small programmable calculator. To use the BCD in another study, a protocol would have to be tailored to reflect the specific treatments and potential confounding variables in that particular study. The BCD is an effective method of randomizing subjects into various treatment groups over time, by providing a balanced experiment and complete randomization of the subjects. The BCD can be adapted to meet the needs of different studies, such as multiple treatment groups and several stratification variables, and should be strongly considered whenever experimental research is being conducted with the subjects accessioned over time.
Vol. 30, No. 3, May/June
1985
REFERENCES 1. Sweeney
C,
Smith H, Foster JC, et
al: Effects of a nutrition intervention program during pregnancy: Maternal data phase 1. J Nurs-Midwif 30(3):149-158, 1985. 2. Eastman N, Jackson E: Weight relationships in pregnancy. Obstet Gynecol 23:1003-1025, 1968. 3. Higgins A: Nutritional Status and Outcome of Pregnancy. Paper presented at the Canadian Dietetic Association Annual Convention, Moneton, New Brunswick, Canada, 1975. 4. Tygstrup N, Lachin JM, Juhl E, (eds): The randomized clinical trial and therapeutic decisions. New York, Marcel Dekker, Inc., 1982, p 95. the
5. Efron B: Forcing a sequential experiment to be balanced. Biometrika 58:403-417, 1971. 6. Hannigan JF Jr, Brown BW Jr: Adaptive randomization biased coin design: Experience in a cooperative group clinical trial. Technical Report No. 74, Division of Biostatistics, Stanford University, Stanford, CA, 1982. 7. Efron B: Randomizing and balancing a complicated sequential experiment, in Miller RG Jr, Efron B, Brown BW Jr, and Moses LE (eds), Biostatistics case book. New York, Wiley, 1980, pp 19-30.
177
Carol Sweeney, CNM, MS, and Marlene J. Egger,
PhD
ABSTRACT In clinical research with small sample size, problems can arise in allocating subjects into varin~~r nrolms when RW -__v_l.“ acrossinn& intn ctlv-k~ nuor .__ .__- treatment .__ _.... “... J.V’ r. . . . -.. thal ...vJ _.., ..w_ .. . .- R - .,...-J . ..“. ti,mp. The && Coin Design (BCD) was designed to overcome the problems of biased or unbalanced treatment groups, while ensuring the advantages of complete randomization, The BCD is discussed, and an example of its use in clinical research is given.
In recent years nurse-midwives have been encouraged to design and conduct clinical research. Clinical research by nurse-midwives both adds new knowledge to the profession and may validate current practice. The research conducted by nursemidwives should adhere to sound research designs. In experimental and quasiexperimental research this includes, among other factors, how the c,thiaz-tc au” ,TbLa
SVD v~nrlnmi~nrl UIG LUIIU”IIIIL~U
intr\ llll”
rl;4L&,1-
ferent treatment groups. One of the problems in clinical research is the acquisition of subjects into a study. In obstetrics, where cases of interest for research may infrequently occur, for example breech position of the fetus, the subjects are often accessioned into the studu over time. In other instances, the practice may be small, and sufficient numbers of all patients may take time to accumulate. A recent two-phased study provides an example of needing to
Address correspondence to: Carol Sweeney, CNM, MS, University of Utah College of Nursing, 25 South Medical Drive, Salt Lake City, UT 84112.
Journal of Nurse-Midwifery
??
acquire subjects over time. In “The Effects of a Nutrition Intervention Program During Pregnancy”’ a nutrition assessment and counseling program was to begin at 20 weeks’ gestation. Forty-seven subjects, who met the study criteria and consented to participate, were identified over a ten-week period, and one year later over an eight-week period. Each subject reached her 20th week of nncttltir\n rr+U = rli44nront yGJLuLr”rL UL UIII~ITIILtin-l” ULLIG. In the Nutrition Intervention Study, subjects accessioned were immediately assigned into the experimental or control groups as they reached the 20th week of gestation; however, the assignment of the total group of subjects occurred over a ten-week __.. oeriod ,__.~_ _. in ~~~the ~.~_ first phase of the study and over an eight-week pe-
nod in the second phase of the study one year later. To prevent biasing of results, subjects should be assigned immediately into the different treatment groups. Two common methods of assigning such subjects are by randomization, that is, “the independent flip of a fair coin,” or by systematic assignment. Both methods of treatment assignments have disadvan-
Vol. 30, No. 3, May/June
Copyright d 1985 by the American College of Nurse-Midwives
tages when there is a small sample size. Theoretically, the independent flip of a fair coin gives each subject an equal chance of being in either treatment group. This does not guarantee that at the end of randomization there will be an equal number of subjects in each group. In a small sample an imbalance in the numbers between treatment groups could be nnT\,,nh tn ctratictiCTJ.,OV,, 3.Z”GL.G r,,vuy,, L” r\v-o.ront i.JIT”CIII .ZLULI.7L1 cal analysis of the data. Another problem is that known confounding variables cannot be controlled, and resulting imbalance across treatment groups in a small sample may bias the results. In the Nutrition Intervention Study two confounding variables, low pregravid weight and low maternal weight gain, were of concern. Both of these variables have been correlated with low infant birth weight.2a3 The independent flip of a fair coin could not guarantee an equal number of subjects with low pregravid weight and low maternal weight gain between the experimental and control groups. The systematic assignment of subjects has been suggested to over-
175
1985 0091.2182’85,$03
30
come the problems of unequal numbers of subjects between the different treatment groups and imbalanced representation of confounding variables between the groups. Systematic assignment provides a balanced experiment by forcing subjects into a set pattern of allocations, for example, ECECEC or ECCEECCE. Some objections to this second approach have been raised, particularly on ethical grounds. For example, the researcher will know or can easily deduce the schedule of treatment assignments and may consciously or unconsciously schedule subjects to provide certain treatments to certain subjects. Although one would strive to remain objective in a research setting, such bias is extremely difficult to avoid. Competent researchers almost always have a subjective opinion regarding which treatment is better. The purpose of the clinical trial is to provide objective evidence regarding the relative merits of the treatments. During the course of a trial it is not uncommon for subjective opinion of the treatments, coupled with clinical concern for the welfare of each patient, to cause a conscious or unconscious ethical dilemma. The integrity of the clinical trial can be compromised severely if a treatment assignment procedure, such as systematic assignment with a randomized start, permits investiga-
Carol Sweeney
received
her BS in
nursing from Washington State University, and her MSN and CNM from the University
of Utah. She is current/y
clinical instructor graduate
of nursing
nurse-midwife
the Uniuersity
in the
y program
of Utah College
at
of
Nursing. Marlene
J. Egger
received
PhD in statistics from
University.
She is currently
Professor
in the Department
and Community Biostatistics, of Medicine.
176
her MS and
Stanford
Medicine,
Uniuersity
Assistant of Fami/y Division
of
of Utah School
tors to “get around” the treatment assignment schedule, for example, by shifting appointment dates to coincide with the preferred treatment. Additional discussion of ethics can be found elsewhere.4 If the clinicians who actually admit and observe subjects are kept “blind,” that is, they have no knowledge of which subjects receive which treatments or of the method of assignment, then the ethical objection to systematic assignment with a randomized start is removed. The Biased Coin Design (BCD) was proposed by Efron5 in 1971 to ensure both the benefits of a balanced experiment and the advantages of complete randomization. Within the Northern California Oncology Group’s (NCOG) randomized clinical trials in cancer,6,7 BCDs have been used routinely since 1977. Intuitively, the BCD allows the experimenter to “flip a biased coin” instead of a fair coin in the assignment of subjects to treatment groups. The first subject has an equal chance of being assigned to any treatment group. For subsequent subjects, “preference” scores are calculated to determine the best placement of the subject to balance the numbers of subjects assigned to each treatment within categories of potential confounding variables. Then a random number table is used so that the subject is assigned the “preferred” treatment 75% of the time and the “nonpreferred” treatment 25% of the time. It is as if a biased coin has been flipped so that the subject is randomly, not systematically, assigned a treatment. The coin is biased to achieve balanced numbers of subjects, and the researcher does not know which treatment will be assigned until after the coin flip. If more than two treatment groups are under study, the bias can be changed from 75% to other values. It should be stressed that “preference” scores are calculated based on the current imbalance in numbers of subjects in each treatment and confounding variable category, not on the re-
searcher’s preference for a particular treatment for an individual subject. Like complete randomization, the BCD still does not guarantee that at the end of randomization, there will be an equal number of subjects in each group. However, the probability of perfect balance is considerably higher than for complete randomization, and the probability of a severe imbalance is greatly reduced, even in small- and moderate-sized samples. This was known in a simulation study by Hannigan and Brown.6 Their technical report also describes the NCOG version of the BCD fully and demonstrates its simplicity of use. The BCD can provide for a balanced experiment of a small sample, even when the treatment groups are to be stratified. In the Nutrition Intervention Study, two confounding variables, low pregravid weight, and low maternal weight gain were identified before the different treatment programs were started; therefore, the goal was to obtain equal numbers of women in the control and the experimental groups with low pregravid weight and failure to gain at least 10 lb (4.5 kg) by 20 weeks of gestation. A protocol for use of the BCD in this specific study was designed and implemented. The BCD did achieve the stated goal, even though there was a small sample size of 47 subjects. There were 23 subjects in the experimental group and 24 subjects in the control group. Of the seven subjects with low pregravid weight, four were in the experimental group and three were in the control group. The 14 subjects with a failure to gain 10 lb (4.5 kg) by 20 weeks’ gestation were divided equally between the experimental and control groups. Pregravid weight and maternal weight gain have been observed to act independently of each other and are additive in their effects on infant birth weight2 This meant that the subjects could be further stratified into four categories in each treatment group: 1) normal or above pregravid
Journal of Nurse-Midwifery
??
Vol. 30, No. 3, May/June
1985
weight and high early weight gain, 2) normal or above pregravid weight and low early weight gain, 3) low pregravid weight and high early weight gain, and 4) low pregravid weight and low early weight gain. The BCD achieved a comparable number of subjects of each above category between the experimental and control group. Those subjects with both normal or above pregravid weight and a high early weight gain were divided into 12 in the experimental group and 15 in the control group. There were 13 subjects with a normal or above pregravid weight and a low early weight gain, seven in the experimental group and six in the control group. The subjects with low pregravid weight and high early weight gain were divided into four in the experimental group and two in the control group. The only subject with a low pregravid weight and a low early weight gain was in the control group. The BCD was implemented in the Nutrition Intervention Study without difficulty. A step-by-step approach of the protocol made learning easy. This protocol, available upon request from the authors, was designed spe-
Journal of Nurse-Midwifery
??
cifically for the Nutrition Intervention Study. A one-hour session, including explanation of the protocol and practice using the protocol, enabled the researchers to be proficient in its use. The researchers were able to randomize subjects into the different treatment groups, using the BCD protocol, within five minutes. In the Nutrition Intervention Study the protocol was done by hand; however, the protocol could easily be computerized or put in a small programmable calculator. To use the BCD in another study, a protocol would have to be tailored to reflect the specific treatments and potential confounding variables in that particular study. The BCD is an effective method of randomizing subjects into various treatment groups over time, by providing a balanced experiment and complete randomization of the subjects. The BCD can be adapted to meet the needs of different studies, such as multiple treatment groups and several stratification variables, and should be strongly considered whenever experimental research is being conducted with the subjects accessioned over time.
Vol. 30, No. 3, May/June
1985
REFERENCES 1. Sweeney
C,
Smith H, Foster JC, et
al: Effects of a nutrition intervention program during pregnancy: Maternal data phase 1. J Nurs-Midwif 30(3):149-158, 1985. 2. Eastman N, Jackson E: Weight relationships in pregnancy. Obstet Gynecol 23:1003-1025, 1968. 3. Higgins A: Nutritional Status and Outcome of Pregnancy. Paper presented at the Canadian Dietetic Association Annual Convention, Moneton, New Brunswick, Canada, 1975. 4. Tygstrup N, Lachin JM, Juhl E, (eds): The randomized clinical trial and therapeutic decisions. New York, Marcel Dekker, Inc., 1982, p 95. the
5. Efron B: Forcing a sequential experiment to be balanced. Biometrika 58:403-417, 1971. 6. Hannigan JF Jr, Brown BW Jr: Adaptive randomization biased coin design: Experience in a cooperative group clinical trial. Technical Report No. 74, Division of Biostatistics, Stanford University, Stanford, CA, 1982. 7. Efron B: Randomizing and balancing a complicated sequential experiment, in Miller RG Jr, Efron B, Brown BW Jr, and Moses LE (eds), Biostatistics case book. New York, Wiley, 1980, pp 19-30.
177