Properties of Simple Randomization in Clinical Trials John M. Lachin, ScD The George Washington University, Department of Statistics~Computer and Information Systems, The Biostatistics Center, Rockville, Maryland ABSTRACT: This article presents the properties of complete randomization (e.g., coin toss) and of the random allocation rule (random permutation of n/2 of n elements). The latter is principally used in cases where the total sample size n is known exactly a priori. The likelihood of treatment imbalances is readily computed and is shown to be negligible for large trials (n > 200), regardless of whether a stratified randomization is used. It is shown that substantial treatment imbalances are extremely unlikely in large trials, and therefore there is likely to be no substantial effect on power. The large-sample permutational distribution of the family of linear rank tests is presented for complete randomization unconditionally and conditionally, and for the random allocation rule. Asymptotically the three are equivalent to the distribution of these tests under a sampling-based population model. Permutation tests are also presented for a stratified analysis within one or more subgroups of patients defined post hoc on the basis of a covariate. This provides a basis for analysis when some patients' responses are assumed to be missing-at-random. Using the Blackwell-Hodges model, it is shown that complete randomization elim- inates the potential for selection bias, but that the random allocation rule yields a substantial potential for selection bias in an unmasked trial. Finally, the Efron model for accidentalbias is used to assess the potential for bias in the estimate of treatment effect due to covariate imbalance. Asymptotically, this probability approaches zero for complete randomization and for the random allocation rule. However, for finite n, complete randomization minimizes the probability of accidental bias, whereas this probability is slightly higher with a random allocation rule. It is concluded that complete randomization has merit in large clinical trials. KEY WORDS: Randomization, complete binomial, random allocation rule, permutation tests, stratified analysis, missing data, selection bias, accidental bias INTRODUCTION Many procedures have been proposed for the random assignment of treat- ments to patients in clinical trials [1-4]. In the simplest cases, no restrictions are placed on the nature of the randomization sequence, apart perhaps from prespecification of the total sample size and the sample size in each group, Address reprint requests to: John M. Lachin, ScD, The George Washington University, Department of Statistics~Computer and Information Systems, The Biostatistics Center, 6110 Executive Boulevard, Suite 750, Rockville, MD 20852. Received April 29, 1987; revised June 13, 1988. 312 Controlled Clinical Trials 9:312-326 (1988) 0197-2456/1988/$3.50 © Elsevier Science Publishing Co., Inc. 1988 655 Avenue of the Americas, New York, New York 10010 Simple Randomization 313 where feasible. Such procedures for unrestricted randomization are com- monly referred to as simple randomization. Lachin [1] describes some of the properties by which randomization pro- cedures have been evaluated. In this article, the properties of simple ran- domization are described as a yardstick against which the properties of re- stricted randomization procedures can be compared. Likewise, Matts and Lachin [5] describe the properties of permuted-block randomization, and Wei and Lachin [6] describe those of urn adaptive biased-coin randomization. The factors considered are the probabilities of treatment imbalances and their potential effects on power, the permutational distribution of linear rank tests, the potential for selection bias, and the potential for accidental bias. Lachin [1] describes the models by which these properties are assessed. Only the case of two treatment groups (a and b) is considered, where approximately equal sample sizes are desired. SIMPLE RANDOMIZATION In order to describe "simple" randomization, two different cases must be distinguished. In the first case, the total sample size n and the sample sizes in each group are prespecified exactly and are under the direct control of the investigator. In this case the usual simple randomization procedure is the random allocation rule [7, 8], whereby a randomly choosen subset of n/2 out of n is assigned to group a, the remainder to group b. Thus, the sample sizes in each group, say na and r/b, each equal to 1/2. In the second case, which is most common in large clinical trials, a target sample size is established but the final sample size is not known with certainty. In this case, the usual simple randomization procedure is complete randomi- zation [7], analogous to tossing a fair coin. Here the sample sizes in each group are binomially distributed random variables. For complete randomization, the marginal and conditional probability of assignment is a constant 1/2 for all assignments. That is, pj = Pr(a) = 1/2 marginally for the jth assignment, and the conditional probabilities are such that pj~, = 1/2 for all j' < j, even if all prior assignments were to one of the two groups. For the random allocation rule, the marginal probability of assignment is also 1/2 for all assignments. However, the conditional probability of assign- ment at the jth step given prior assignments is not always 1/2. Specifically, let na = nb = m = n/2 and let n~j and nbj denote the numbers assigned to a and b after j assignments. For the random allocation rule, pjv, = (m - 1)/(n - 1) if j and j' have the same assignment, pj~, = m/(n- 1) otherwise. Also, PJ~'<i = (m - n~d_l)/(n - j + 1). Clearly pj~,<j = 1/2 only if the numbers of prior assignments n~,j-1 and nb,j-1 are equal. For example, suppose n = 10 (n~ = nb= m = 5) and the first two assignments are a and a. Then, the unconditional (marginal) probability of an a on the third assignment is I/2 over the set of all possible permutations, but within the particular permutation chosen, the conditional probability of an a on the third assignment given the past assignments is p3~'<3 = 3/8. 314 j.M. Lachin TREATMENT IMBALANCES With a random allocation rule na = m provided that n patients are ran- domized. Ifj < n patients are randomized, then the probability of an imbalance of na/to nb/, or worse, can be obtained from Fisher's exact test for the resulting 2 x 2 table with cells naj, rlbj, m -- na/, and m - nbj. Asymptotically, this probability can be assessed by the ordinary chi-square test for the 2 x 2 table. With complete randomization, after n (or j) assignments, n~ (or n~j) follows the binomial distribution with p = 1/2 and sample size n (or j). The specific probability of a treatment imbalance of na to nb can readily be calculated using the large sample approximation to the binomial, where na is approximately normally distributed with mean (0.5)n = m and variance (0.5)2n -- n/4. If we denote the larger of the two sample fractions as qu = max(n~, nb)/n, then Pr[qu > r] ~ 2~(- ZF), where ~(.) is the standard normal cumulative distribution [e.g., ci)(1.645) = 0.95] and Zr = 2(r - 0.5)V~n. Therefore, the probability of a stated treatment imbalance r decreases as the sample size increases. This property is displayed in Figures 1 and 2. Figure 1 presents the prob- ability of obtaining a treatment imbalance (larger sample fraction) of q~ --- r for r = 0.55, 0.60, and 0.70. Figure 1 shows that there is a probability of less than 0.05 of an imbalance of 0.70 or more for a sample size of 30 or more; of an imbalance of 0.60 or more for a sample size of 100 or more; and of an imbalance of 0.55 or more for a sample size of 400 or more. Figure 2 shows those degrees of imbalance q~ --- r that would occur with probabilities 0.01, 0.05, or 0.10 as sample size increases. This shows that severe imbalances become extremely unlikely with increasing sample sizes. Now consider the case of a stratified randomization with K strata, each of size nk (1 -----k --< K). By the normal approximation to the binomial, the number assigned to treatment a in the kth strata, nak, is asympotically normally dis- tributed with mean P-k = nd2 and variance o~k = nd4. Therefore, in aggregate overall K strata, n~ = Y.n~k is also asympotically normally distributed with mean p, = Xp,k = n/2 and variance oa = Y.(r2 = n/4. Therefore the probability of an aggregate treatment imbalance in a stratified trial is the same as that in an unstratified trial of the same total sample size. The principal concern is that a treatment imbalance may affect the power of a statistical test. It has been shown that power is nontrivially reduced only if the treatment imbalance is on the order of qu = 0.70 or greater [1]. However, from Figures I and 2 it is clear that even for a small-sized study, the probability of an imbalance this great is extremely remote. Thus, there is little support for the widespread concern that complete randomization is likely to result in treatment imbalances that could have a substantial effect on statistical power. PERMUTATION TESTS As described in ref. 1, there is a fundamental difference between the use of a population model and a randomization model as a basis for statistical tests. Permutation tests make no assumptions regarding the origin of the patient samples or the distribution of the measurements. However, the vari- ance of permutation tests depends on the precise nature of the randomization Probability of Imbalance b0 b0 .o o ~0 0 0 ~ 0 0"1 0 O~ 0 -[~ I I I I I /f f 0 f IV ~ 0 0 b~ 0 0 9P~ 7 // /// j::LO // O / / 3 / p~ / mg / O / / I O I O / O / / I ?, ?, o p o O O 0 0 ¢.~ ex ¢D0 N O" ¢3 O ~[~ uo.q~z!tuopu~I oldtu!S 0.75-T O~ I I J 0,70 - Probability = 0.01 0.05 i a) 0 t- 0.10 O "~ 0.65- E C 0 o b-E i ~- o.go- E 0.55 - 0.50 Jl-- -- I ......