Abstract
The randomized controlled clinical trial is the standard by which all trials are judged since other designs have certain undesirable features. In the simplest case, randomization is a process by which each participant has the same chance of being assigned to either intervention or control. An example would be the toss of a coin, in which heads indicates intervention group and tails indicates control group. Even in the more complex randomization strategies, the element of chance underlies the allocation process. Of course, neither trial participant nor investigator should know what the assignment will be before the participant’s decision to enter the study. Otherwise, the benefits of randomization can be lost. The role that randomization plays in clinical trials has been discussed in Chap. 5 as well as by numerous authors [1–12]. While not all accept that randomization is essential [11, 12], most agree it is the best method for achieving comparability between study groups and is the basis for statistical inference [2, 3].
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References
Hill AB. The clinical trial. Br Med Bull 1951;7:278–282.
Armitage P. The role of randomization in clinical trials. Stat Med 1982;1:345–352.
Byar DP, Simon RM, Friedewald WT, et al. Randomized clinical trials: Perspectives on some recent ideas. N Engl J Med 1976;295:74–80.
Zelen M. The randomization and stratification of patients to clinical trials. J Chronic Dis 1974;27:365–375.
Pocock SJ. Allocation of patients to treatment in clinical trials. Biometrics 1979;35:183–197.
Peto R. Clinical trial methodology. Biomedicine 1978;28(special issue):24–36.
Peto R, Pike MC, Armitage P, et al. Design and analysis of randomised clinical trials requiring prolonged observation of each patient. 1. Introduction and design. Br J Cancer 1976;34: 585–612.
Brown BW. Statistical controversies in the design of clinical trials – some personal views. Control Clin Trials 1980;1:13–27.
Lachin JM. Statistical properties of randomization in clinical trials. Control Clin Trials 1988;9:289–311.
Lachin JM, Matts JP, Wei LJ. Randomization in clinical trials: Conclusions and recommendations. Control Clin Trials 1988;9:365–374.
Royall RM. Ethics and statistics in randomized clinical trials. Stat Sci 1991;6(1):52–88.
Weinstein MC. Allocation of subjects in medical experiments. N Engl J Med 1974;291: 1278–1285.
Bather JA. On the allocation of treatments in sequential medical trials. Int Stat Rev 1985;53:1–13.
Kalish LA, Begg CB. Treatment allocation methods in clinical trials: A review. Stat Med 1985;4:129–144.
Stigler SM. The use of random allocation for the control of selection bias. Biometrika 1969;56:553–560.
Wei LJ. On the random allocation design for the control of selection bias in sequential experiments. Biometrika 1978;65:79–84.
Altman D, Dore CJ. Randomization and baseline comparisons in clinical trials. Lancet 1990;335:149–155.
Williams DS, Davis CE. Reporting of assignment methods in clinical trials. Control Clin Trials 1994;15:294–298.
Moher D, Schulz KF, Altman DG, CONSORT Group (Consolidated Standards of Reporting Trials). The CONSORT statement: revised recommendations for improving the quality of reports of parallel-group randomized trials. Ann Intern Med 2001;134:657-662.
Mills EJ, Wu P, Gagnier J, Devereaux PJ. The quality of randomized trial reporting in leading medical journals since the revised CONSORT statement. Contemp Clin Trials 2005;26: 480–487.
Brittain E, Schlesselman JJ. Optimal allocation for the comparison of proportions. Biometrics 1982;38:1003–1009.
Lachin JM. Properties of simple randomization in clinical trials. Control Clin Trials 1988;9:312–326.
Louis TA. Optimal allocation in sequential tests comparing the means of two Gaussian populations. Biometrika 1975;62:359–369.
Louis TA. Sequential allocation in clinical trials comparing two exponential survival curves. Biometrics 1977;33:627–634.
Kalish LA, Harrington DP. Efficiency of balanced treatment allocation for survival analysis. Biometrics 1988;44:815–821.
Matts JP, Lachin JM. Properties of permutated-block randomization in clinical trials. Control Clin Trials 1988;9:327–344.
Kalish LA, Begg CB. The impact of treatment allocation procedures on nominal significance levels and bias. Control Clin Trials 1987;8:121–135.
Smythe RT, Wei LJ. Significance tests with restricted randomization design. Biometrika 1983;70:496–500.
Steele JM. Efron’s conjecture on vulnerability to bias in a method for balancing sequential trials. Biometrika 1980;67:503–504.
Titterington DM. On constrained balance randomization for clinical trials. Biometrics 1983;39:1083–1086.
Matts JP, McHugh RB. Analysis of accrual randomized clinical trials with balanced groups in strata. J Chronic Dis 1978;31:725–740.
Zelen M. Aspects of the planning and analysis of clinical trials in cancer. In Srivastava JN (ed.). A Survey of Statistical Design and Linear Models. Amsterdam: North-Holland, 1975.
Coronary Drug Project Research Group. Factors influencing long term prognosis after recovery from myocardial infarction – Three year findings of the Coronary Drug Project. J Chronic Dis 1974;27:267–285.
Pocock SJ, Simon R. Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics 1975;31:103–115.
Green SB, Byar DP. The effect of stratified randomization on size and power of statistical tests in clinical trials. J Chronic Dis 1978;31:445–454.
Ducimetiere P. Stratification. In Boissel JP, Klimt CR (eds.). Multi-center Controlled Trials: Principals and Problems. Paris: INSERM, 1979.
Simon R. Restricted randomization designs in clinical trials. Biometrics 1979;35:503–512.
Meier P. Stratification in the design of a clinical trial. Control Clin Trials 1981;1:355–361.
Grizzle JE. A note on stratifying versus complete random assignment in clinical trials. Control Clin Trials 1982;3:365–368.
McHugh R, Matts J. Post-stratification in the randomized clinical trial. Biometrics 1983;39:217–225.
Fleiss JL. Multicentre clinical trials: Bradford Hill’s contributions and some subsequent developments. Stat Med 1982;1:353–359.
Feinstein AR, Landis JR. The role of prognostic stratification in preventing the bias permitted by random allocation of treatment. J Chronic Dis 1976;29:277–284.
Mantel N. Pre-stratification or post-stratification (Letter). Biometrics 1984;40:256–258.
Palta M. Investigating maximum power losses in survival studies with nonstratified randomization. Biometrics 1985;41:497–504.
Palta M, Amini SB. Magnitude and likelihood of loss resulting from non-stratified randomization. Stat Med 1982;1:267–275.
Aspirin Myocardial Infarction Study Research Group. A randomized controlled trial of aspirin in persons recovered from myocardial infarction. JAMA 1980;243:661–669.
Efron B. Forcing a sequential experiment to be balanced. Biometrika 1971;58:403–417.
Freedman LS, White SJ. On the use of Pocock and Simon’s method for balancing treatment numbers over prognostic factors in the controlled clinical trial. Biometrics 1976;32:691–694.
Begg CD, Iglewicz B. A treatment allocation procedure for sequential clinical trials. Biometrics 1980;36:81–90.
Atkinson AC. Optimum biased coin designs for sequential clinical trials with prognostic factors. Biometrika 1982;69:61–67.
Taves DR. Minimization: A new method of assigning patients to treatment and control groups. Clin Pharmacol Ther 1974;15:443–453.
White SJ, Freedman LS. Allocation of patients to treatment groups in a controlled clinical study. Br J Cancer 1978;37:849–857.
Forsythe AB, Stitt FW. Randomization or minimization in the treatment assignment of patient trials: validity and power of tests. Technical Report No. 28, Health Science Computer Facility, University of California, Los Angeles, 1977.
Begg CB. On inferences from Wei’s biased coin design for clinical trials. Biometrika 1990;77:467–484.
Efron B. Randomizing and balancing a complicated sequential experiment. In Miller RG Jr. Efron B, Brown BW Jr, Moses LE (eds.). Biometrics Casebook. New York: Wiley, 1980, pp. 19–30.
Halpern J, Brown BW Jr. Sequential treatment allocation procedures in clinical trials – with particular attention to the analysis of results for the biased coin design. Stat Med 1986;5: 211–229.
Hannigan JR Jr, Brown BW Jr. Adaptive randomization based coin-design: Experience in a cooperative group clinical trial. Technical Report 74, Division of Biostatistics, Stanford University, Stanford, California, 1982.
Klotz JH. Maximum entropy constrained balance randomization for clinical trials. Biometrics 1978;34:283–287.
Raghavaro D. Use of distance function in sequential treatment assignment for prognostic factors in the controlled clinical trial. Calcutta Stat Assoc Bull 1980;29:99–102.
Smith RL. Sequential treatment allocation using biased coin designs. J R Stat Soc Series B Stat Methodol 1984;46:519–543.
Soares JF, Wu CFJ. Some restricted randomization rules in sequential designs. Commun Stat Theory Methods A 1983;12:2017–2034.
Wei LJ. The adaptive biased coin design for sequential experiments. Ann Stat 1978;6: 92–100.
Wei LJ. A class of designs for sequential clinical trials. J Am Stat Assoc 1977;72:382–386.
Wei LJ. A class of treatment assignment rules for sequential experiments. Commun Stat Theory Methods A 1978;7:285–295.
Wei LJ, Lachin JM. Properties of the urn randomization in clinical trials. Control Clin Trials 1988;9:345–364.
Wei LJ, Smythe RT, Lin DY, Park TS. Statistical inferences with data-dependent treatment allocation rules. J Am Stat Assoc 1990;85:156–162.
Wei LJ, Smythe RT, Smith RL. K-treatment comparisons with restricted randomization rules in clinical trials. Ann Stat 1986;14:265–274.
Wei LJ. An application of an urn model to the design of sequential controlled clinical trials. J Am Stat Assoc 1978;73:559–563.
The DCCT Research Group. Diabetes Control and Complications Trial (DCCT): Design and methodologic considerations for the feasibility phase. Diabetes 1986;35:530–545.
Begg CB, Kalish LA. Treatment allocation for nonlinear models in clinical trials: The logistic model. Biometrics 1984;40:409–420.
Begg CB, Kalish LA. Treatment allocation in sequential clinical trials: Nonlinear models. Proc Stat Comput Sect, Am Stat Assoc 1982:57–60.
Gail MH, Wieand S, Piantadosi S. Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 1984;71:431–444.
Birkett JJ. Adaptive allocation in randomized controlled trials. Control Clin Trials 1985;6:146–155.
Russell M, Fleg JL, Galloway J, et al. Examination of lower targets for low-intensity lipoprotein cholesterol and blood pressure in diabetes—the Stop Atherosclerosis in Native Diabetics Study (SANDS). Am Heart J 2006:152;867–875.
Howard BV, Roman MJ, Devereux RB, et al. Effect of lower targets for blood pressure and LDL cholesterol on atherosclerosis in diabetes: The SANDS randomized trial. JAMA 2008:299;1678–1689.
Zelen M. Play-the-winner rule and the controlled clinical trial. J Am Stat Assoc 1969;64: 131–146.
Robbins H. Some aspects of the sequential design of experiments. Bull Am Math Soc 1952;58:527–535.
Bailar JC. Patient assignment algorithms: An overview. In Proceedings of the 9th International Biometric Conference, Raleigh, NC: The Biometric Society, 1976; Vol I, pp. 189–206.
Simon R. Adaptive treatment assignment methods and clinical trials. Biometrics 1977;33: 743–749.
Armitage P. The search for optimality in clinical trials. Int Stat Rev 1985;53:15–24.
Nordbrock E. An improved play-the-winner sampling procedure for selecting the better of two binomial populations. J Am Stat Assoc 1976;71:137–139.
Wei LJ. Exact two-sample permutation tests based on the randomized play-the-winner rule. Biometrika 1988;75:603–606.
Bartlett RH, Roloff DW, Cornell RG, et al. Extracorporeal circulation in neonatal respiratory failure: A prospective randomized study. Pediatrics 1985;76:479–487.
O’Rourke PP, Crone RK, Vacanti JP, et al. Extracorporeal membrane oxygenation and conventional medical therapy in neonates with persistent pulmonary hypertension of the newborn: A prospective randomized study. Pediatrics 1989;84:957–963.
Simon R, Weiss GH, Hoel DG. Sequential analysis of binomial clinical trials. Biometrika 1975;62:195–200.
Simon R, Hoel DG, Weiss GH. The use of covariate information in the sequential analysis of dichotomous response experiments. Commun Stat Theory Methods 1977;8:777–788.
Paneth N, Wallenstein S. Extracorporeal membrane oxygenation and the play the winner rule. Pediatrics 1985;76:622–623.
Ware JH. Investigating therapies of potentially great benefit: ECMO. Stat Sci 1989;4:298–340.
Ware JH, Epstein MF. Extracorporeal circulation in neonatal respiratory failure: A prospective randomized study. Pediatrics 1985;76:849–851.
Pocock SJ, Lagakos SW. Practical experience of randomization in cancer trials: An international survey. Br J Cancer 1982;46:368–375.
Chalmers TC, Celano P, Sacks HS, et al. Bias in treatment assignment in controlled clinical trials. N Engl J Med 1983;309:1358–1361.
Beta-Blocker Heart Attack Trial Research Group. A randomized trial of propranolol in patients with acute myocardial infarction. I. Mortality results. JAMA 1982;247:1707–1714.
Hypertension Detection and Follow-up Program Cooperative Group. Five-year findings of the Hypertension Detection and Follow-up Program. Reduction in mortality of persons with high blood pressure, including mild hypertension. JAMA 1979;242:2562–2571.
Multiple Risk Factor Intervention Trial Research Group. Multiple Risk Factor Interventional Trial. Risk factor changes and mortality results. JAMA 1982;248:1465–1477.
CASS Principal Investigators and Their Associates. Coronary Artery Surgery Study (CASS): A randomized trial of coronary artery bypass surgery, survival data. Circulation 1983;68:939–950.
Collaborative Group on Antenatal Steroid Therapy. Effect of antenatal dexamethasone administration on the prevention of respiratory distress syndrome. Am J Obstet Gynecol 1981;141:276–287.
Krischer J, Hurley C, Pillamarri M, et al. An automated patient registration and treatment randomization system for multicenter clinical trials. Control Clin Trials 1991;12:367–377.
Kjekshus J, Apetrei E, Barrios V, et al. Rosuvastatin in older patients with systolic heart failure. N Engl J Med 2007;357:2248–2261.
SPORTIF Executive Steering Committee for the SPORTIF-V Investigators. Ximelagatran vs Warfarin for stroke prevention in patients with nonvalvular atrial fibrillation. A randomized trial. JAMA 2005;293:690–698.
Ahlmark G, Saetre H. Long-term treatment with β-blockers after myocardial infarction. Eur J Clin Pharmacol 1976;10:77–83.
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Appendix
Appendix
Adaptive Randomization Algorithm
Adaptive randomization can be used for more than two intervention groups, but for the sake of simplicity only two will be used here. To describe this procedure in more detail, a minimum amount of notation needs to be defined. First, let
And define
The \( {x}_{ik}^{t}\) represents the change in balance of allocation if the new participant is assigned intervention t. Finally, let
Many possible definitions of B(t) can be identified. As an illustrative example, let
where w i = the relative importance of factor i to the other factors and the range is the absolute difference between the largest and smallest values of \( {x}_{i1}^{t}\ and \ {x}_{i2}^{t}\).
The value of B(t) is determined for each intervention (t = 1 and t = 2). The intervention with the smaller B(t) is preferred, because allocation of the participant to that intervention will cause the least imbalance. The participant is assigned, with probability p > 1/2, to the intervention with the smaller score, B(1) or B(2). The participant is assigned, with probability (1−p), to the intervention with the larger score. These probabilities introduce the random component into the allocation scheme. Note that if p = 1 and, therefore, 1−p = 0, the allocation procedure is deterministic (no chance or random aspect) and has been referred to by the term “minimization” [51, 53].
As a simple example of the adaptive stratification method, suppose there are two groups and two prognostic factors to control. The first factor has two levels and the second factor has three levels. Assume that 50 participants have already been randomized and the following table summarizes the results (Table 6.2).
In addition, the function B(t) as defined above will be used with the range of the x1 ik s as the measure of imbalance, where w 1 = 3 and w 2 = 2; that is, the first factor is 1.5 times as important as the second as a prognostic factor. Finally, suppose p = 2/3 and 1−p = 1/3.
If the next participant to be randomized has the first level of the first factor and the third level of the second factor, then this corresponds to the first and fifth columns in the table. The task is to determine B(1) and B(2) for this participant as shown below.
-
(a)
Determine B(1)
-
Factor 1, Level 1
K
x 1k
\( {{x}^{1}}_{1k}\)
Range (\( {x^{1}}_{11},{x^{1}}_{12}\))
Group
1
16
17
|17–14| = 3
2
14
14
-
Factor 2, Level 3
K
x 2k
\( {x}_{2k}^{1}\) x 1 2k
Range (\( {x^{1}}_{21},{x^{1}}_{22}\))
Group
1
4
5
|5–6| = 1
2
6
6
Using the formula given, B(1) is computed as 3 × 3 + 2 × 1 = 11.
-
-
(b)
Determine B(2)
-
Factor 1, Level 1
K
x 1k
\( {x}_{1k}^{2}\)
Range (\( {x^{2}}_{11}, {x^{2}}_{12}\))
Group
1
16
16
|16–15| = 1
2
14
15
-
Factor 2, Level 3
K
x 2k
\( {x^{1}}_{1k}\)
Range (\( {x^{2}}_{21},{x^{2}}_{22}\))
Group
1
4
4
|4–7| = 3
2
6
7
Then B(2) is computed as 3 × 1 + 2 × 3 = 9.
-
-
(c)
Now rank B(1) and B(2) from smaller to larger and assign with probability p the group with the smaller B(t).
t
B(t)
Probability of assigning t
2
B(2) = 9
p = 2/3
1
B(1) = 11
1−p = 1/3
Thus, this participant is randomized to Group 2 with probability 2/3 and to Group 1 with probability 1/3. Note that if minimization were used (p = 1), the assignment would be Group 2.
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Friedman, L.M., Furberg, C.D., DeMets, D.L. (2010). The Randomization Process. In: Fundamentals of Clinical Trials. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1586-3_6
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