Imbalance in treatment assignments in stratified blocked randomization

Selecting a randomization method for a multi-center clinical trial with stochastic recruitment considerations

BACKGROUND

The design of a multi-center randomized controlled trial (RCT) involves multiple considerations, such as the choice of the sample size, the number of centers and their geographic location, the strategy for recruitment of study participants, amongst others. There are plenty of methods to sequentially randomize patients in a multi-center RCT, with or without considering stratification factors. The goal of this paper is to perform a systematic assessment of such randomization methods for a multi-center 1:1 RCT assuming a competitive policy for the patient recruitment process.

METHODS

We considered a Poisson-gamma model for the patient recruitment process with a uniform distribution of center activation times. We investigated 16 randomization methods (4 unstratified, 4 region-stratified, 4 center-stratified, 3 dynamic balancing randomization (DBR), and a complete randomization design) to sequentially randomize n = 500 patients. Statistical properties of the recruitment process and the randomization procedures were assessed using Monte Carlo simulations. The operating characteristics included time to complete recruitment, number of centers that recruited a given number of patients, several measures of treatment imbalance and estimation efficiency under a linear model for the response, the expected proportions of correct guesses under two different guessing strategies, and the expected proportion of deterministic assignments in the allocation sequence.

RESULTS

Maximum tolerated imbalance (MTI) randomization methods such as big stick design, Ehrenfest urn design, and block urn design result in a better balance-randomness tradeoff than the conventional permuted block design (PBD) with or without stratification. Unstratified randomization, region-stratified randomization, and center-stratified randomization provide control of imbalance at a chosen level (trial, region, or center) but may fail to achieve balance at the other two levels. By contrast, DBR does a very good job controlling imbalance at all 3 levels while maintaining the randomized nature of treatment allocation. Adding more centers into the study helps accelerate the recruitment process but at the expense of increasing the number of centers that recruit very few (or no) patients-which may increase center-level imbalances for center-stratified and DBR procedures. Increasing the block size or the MTI threshold(s) may help obtain designs with improved randomness-balance tradeoff.

CONCLUSIONS

The choice of a randomization method is an important component of planning a multi-center RCT. Dynamic balancing randomization with carefully chosen MTI thresholds could be a very good strategy for trials with the competitive policy for patient recruitment.

Minimisation for the design of parallel cluster-randomised trials: An evaluation of balance in cluster-level covariates and numbers of clusters allocated to each arm

BACKGROUND

Cluster-randomised trials often use some form of restricted randomisation, such as stratified- or covariate-constrained randomisation. Minimisation has the potential to balance on more covariates than blocked stratification and can be implemented sequentially unlike covariate-constrained randomisation. Yet, unlike stratification, minimisation has no inbuilt guard to maintain close to a 1:1 allocation. A departure from a 1:1 allocation can be unappealing in a setting with a small number of allocation units such as cluster randomisation which typically include about 30 clusters.

METHODS

Using simulation (10,000 per scenario), we evaluate the performance of a range of minimisation procedures on the likelihood of a 1:1 allocation of clusters (10-80 clusters) to treatment arms, along with its performance on covariate imbalance. The range of minimisation procedures includes varying: the proportion of clusters allocated to the least imbalanced arm (known as the stochastic element) - between 0.7 and 1, percentage of first clusters allocated completely at random (known as the bed-in period) - between 0% and 20% and adding 'number of clusters allocated to each arm' as a covariate in the minimisation algorithm. We additionally include a comparison of stratifying and then minimising within key strata (such as country within a multi country cluster trial) as a potential aid to increasing balance.

RESULTS

Minimisation is unlikely to result in an exact 1:1 allocation unless the stochastic element is set higher than 0.9. For example, with 20 clusters, 2 binary covariates and setting the stochastic element to 0.7: only 41% of the possible randomisations over the 10,000 simulations achieved a 1:1 allocation. While typical sizes of imbalance were small (a difference of two clusters per arm), allocations as extreme as of 10:10 were observed. Adding the 'number of clusters' into the minimisation algorithm reduces this risk slightly, but covariate imbalance increases slightly. Stratifying and then minimising within key strata improve balance within strata but increase imbalance across all clusters, both on the number of clusters and covariate imbalance.

CONCLUSION

In cluster trials, where there are typically about 30 allocation units, when using minimisation, unless the stochastic element is set very high, there is a high risk of not achieving a 1:1 allocation, and a small but nonetheless real risk of an extreme departure from a 1:1 allocation. Stratification with minimisation within key strata (such as country) improves the balance within strata although compromises overall balance.

Risk of selection bias assessment in the NINDS rt-PA stroke study

OBJECTIVES

The NINDS rt-PA Stroke Study is frequently cited in support of alteplase for acute ischemic stroke within 3 h of symptom onset. Multiple post-hoc reanalyses of this trial have been published to adjust for a baseline imbalance in stroke severity. We performed a risk of selection bias assessment and reanalyzed trial data to determine if the etiology of this baseline imbalance was more likely due to random chance or randomization errors.

METHODS

A risk of selection bias assessment was conducted using signaling questions from the Cochrane Risk of Bias 2 (ROB 2) tool. Four sensitivity analyses were conducted on the trial data based on the randomization process: assessment of imbalances in allocation in unique strata; adherence to a pre-specified restriction on randomization between time strata at each randomization center; assessment of differences in baseline computed tomography (CT) results in unique strata; and comparison of baseline characteristics between allocation groups within each time strata. A multivariable logistic regression model was used to compare reported treatment effects with revised treatment effects after adjustment of baseline imbalances identified in the sensitivity analyses.

RESULTS

Based on criteria from the ROB 2 tool, the risk of bias arising from the randomization process was high. Sensitivity analyses found 11 of 16 unique strata deviated from the expected 1:1 allocation ratio. Three randomization centers violated an apriori rule regarding a maximum difference in allocation between the time strata. Three unique strata had imbalances in baseline CT results that prognostically favored alteplase. Four imbalances in baseline characteristics were identified in the 91-180-min time stratum that all prognostically favored alteplase and were consistent with a larger alteplase treatment effect size compared to the 0-90-min time stratum. After adjustments for baseline imbalances, all reported treatment effects were reduced. Three out of seven originally positive reported results were revised to non-significant.

CONCLUSION

This risk of selection bias assessment revealed a high risk of selection bias in the NINDS rt-PA Stroke Study. Sensitivity analyses conducted based on the randomization process supported this assessment. Baseline imbalances in the trial were more likely due to randomization errors than random chance. Adjusted analyses accounting for baseline imbalances revealed a reduction in reported treatment effects supporting the presence of selection bias in the trial. Treatment decisions and guideline recommendations based on the original treatment effect reported in the NINDS rt-PA Stroke Study should be done cautiously.

Best Practices for the Design of Clinical Trials Related to the Visual System

Clinical trials for conditions affecting the visual system need to not only conform to the guidelines for all clinical trials, but also accommodate the possibility of both eyes of a single patient qualifying for the trial. In this review, I present the interplay of the key components in the design of a clinical trial, along with the modifications or options that may be available for trials addressing ocular conditions. Examples drawn from published reports of the design and results of clinical trials of ocular conditions are provided to illustrate application of the design principles. Current approaches to data analysis and reporting of trials are outlined, and the oversight and regulatory procedures to protect participants in clinical trials are discussed. Expected final online publication date for the Annual Review of Vision Science, Volume 7 is September 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Short-Duration Prednisolone in Children with Nephrotic Syndrome Relapse: A Noninferiority Randomized Controlled Trial

BACKGROUND AND OBJECTIVES

In children with nephrotic syndrome, steroids are the cornerstone of therapy for relapse. The adequate duration and dosage of steroids, however, have not been an active area of research, especially in children with infrequently relapsing nephrotic syndrome. This study investigated the efficacy of an abbreviated regimen for treatment of a relapse in this population.

DESIGN, SETTING, PARTICIPANTS, & MEASUREMENTS

In a single-center, open-label, randomized controlled trial, we evaluated the efficacy of prednisolone as a "short regimen" (40 mg/m² on alternate days for 2 weeks) compared with "standard regimen" (40 mg/m² on alternate days for 4 weeks) for children aged 1-16 years who achieved remission of a relapse. The primary outcome was the proportion of children developing frequent relapses or steroid dependence at 12 months.

RESULTS

A total of 117 patients were enrolled and randomized to short (55) or standard (62) regimen. Fourteen (24%) patients in standard regimen and 12 (23%) in short regimen developed frequent relapses or steroid dependence over a period of 1 year (risk difference, -1%; 95% confidence interval, -15 to 16; P=0.90). A large 95% confidence interval crossed the proposed noninferiority margin. In a time to event analysis, there was no significant difference in the proportion of children developing frequent relapses or steroid dependence and time to outcome between the two groups (hazard ratio, 1.01; 95% confidence interval, 0.83 to 1.23; P=0.98). Time to relapse, relapse rate, and steroid-related adverse events were similar in both groups. Cumulative steroid exposure was significantly lower in the short regimen (risk difference, -541 mg/m²; 95% confidence interval, -917 to -164 mg/m²; P<0.001).

CONCLUSIONS

In children with infrequently relapsing nephrotic syndrome, a short steroid treatment for relapse resulted in a similar proportion of patients developing frequent relapses or steroid dependence; however, noninferiority of a short regimen was not established.

CLINICAL TRIAL REGISTRY NAME AND REGISTRATION NUMBER

CTRI/2015/11/006345.

Imbalance properties of centre-stratified permuted-block and complete randomisation for several treatments in a clinical trial

Randomisation schemes are rules that assign patients to treatments in a clinical trial. Many of these schemes have the common aim of maintaining balance in the numbers of patients across treatment groups. The properties of imbalance that have been investigated in the literature are based on two treatment groups. In this paper, their properties for K > 2 treatments are studied for two randomisation schemes: centre-stratified permuted-block and complete randomisation. For both randomisation schemes, analytical approaches are investigated assuming that the patient recruitment process follows a Poisson-gamma model. When the number of centres involved in a trial is large, the imbalance for both schemes is approximated by a multivariate normal distribution. The accuracy of the approximations is assessed by simulation. A test for treatment differences is also considered for normal responses, and numerical values for its power are presented for centre-stratified permuted-block randomisation. To speed up the calculations, a combined analytical/approximate approach is used. Copyright © 2016 John Wiley & Sons, Ltd.

Generalized multidimensional dynamic allocation method

Dynamic allocation has received considerable attention since it was first proposed in the 1970s as an alternative means of allocating treatments in clinical trials which helps to secure the balance of prognostic factors across treatment groups. The purpose of this paper is to present a generalized multidimensional dynamic allocation method that simultaneously balances treatment assignments at three key levels: within the overall study, within each level of each prognostic factor, and within each stratum, that is, combination of levels of different factors Further it offers capabilities for unbalanced and adaptive designs for trials. The treatment balancing performance of the proposed method is investigated through simulations which compare multidimensional dynamic allocation with traditional stratified block randomization and the Pocock-Simon method. On the basis of these results, we conclude that this generalized multidimensional dynamic allocation method is an improvement over conventional dynamic allocation methods and is flexible enough to be applied for most trial settings including Phases I, II and III trials.

How many strata in an RCT? A flexible approach

Background:

The need to allow for prognostic factors when designing and analysing cancer trials is well recognised, but the possibility of overstratification should be avoided by restricting the number of strata. The proposed method improves on existing guidance by being based on explicit principles and being more adaptable to circumstances, and should be of particular use to clinicians when designing a trial.

Methods:

Given a proposed sample size, a minimum allowable number in a stratum and an acceptable risk of observing fewer than this minimum, the number of strata can then be obtained by assuming a Poisson distribution for the number of observations per stratum. This can easily be programmed into Excel.

Results:

An example is given for a hypothetical typical trial of 250 patients, which for 80% power and 5% two-sided significance would correspond to a Cohen's effect size of 0.355 (about halfway between the ‘small’ and ‘moderate’ thresholds). To have a <1% risk of fewer than 10 patients in a stratum, no >13 strata should be considered. For a survival analysis with the same overall sample size but 170 deaths, no >9 strata would be prudent. In the context of a cancer trial this could easily be met by only two prognostic variables.

Conclusion:

The method proposed is flexible and based on explicit principles and may be applied in the design or analysis of both clinical trials and epidemiological studies.

The benefit of stratification in clinical trials revisited

Stratification is common in clinical trials because it can reduce the variance of the estimated treatment effect. The traditional demonstration of variance reduction relies on the assumption of stratum sizes being fixed quantities. However, in practice, to speed up enrollment, and to obtain a study population with a similar distribution as the overall population, the stratum sizes are allowed to vary. Under the condition that the total sample size is fixed and that the stratum sizes have a multinomial distribution, the criterion changes for achieving a reduction in variance. The relationship between the stratified and unstratified variances is established and shown to be approximately the same for prestratified and post-stratified trials. It is demonstrated why stratification may actually increase the variance compared with no stratification even when the mean square error is reduced on account of stratification. Data from a real clinical trial will be used for illustration. The benefit attributed to stratification needs to be re-examined in light of the findings presented, particularly given its widespread use.

Evaluation of imbalance in stratified blocked randomization: some remarks on the range of validity of the model by Hallstrom and Davis

OBJECTIVES

If in a clinical trial prognostic factors are known in advance, it is often recommended that randomization of patients should be stratified. The best-known method is permuted-block randomization within strata. But it suffers from the disadvantage that imbalance still occurs in the trial as a whole if there are a large number of strata, or/and the block sizes are too large for the number of patients. The results of Hallstrom and Davis are appropriate for evaluating the risk of such a troubled situation by using two special cases of their general variance formula. But it is merely generally argued for whichever practical situations these special cases are valid. Consequently, additional investigations are required to reveal the conditions for correct application.

METHODS

We investigated the range of validity of special cases by performing computer simulations, varying a number of trial characteristics, and discuss the application of results for practical situations.

RESULTS

The validity of special cases is not given in each situation. Depending on block size, a binomial distribution model is valid for a permitted average maximum number of patients per stratum between 36% and 57% of considered block sizes, whereas the uniform distribution model works adequately from at least 70%. In an intermediate range of invalidity, implementation of a simulation study is necessary to compute the probability distribution of differences.

CONCLUSIONS

Our results are important if choosing the stratified permuted-block randomization to estimate the risk for an intolerable overall imbalance when planning a trial.

Effects of unstratified and centre-stratified randomization in multi-centre clinical trials

This paper deals with the analysis of randomization effects in multi-centre clinical trials. The two randomization schemes most often used in clinical trials are considered: unstratified and centre-stratified block-permuted randomization. The prediction of the number of patients randomized to different treatment arms in different regions during the recruitment period accounting for the stochastic nature of the recruitment and effects of multiple centres is investigated. A new analytic approach using a Poisson-gamma patient recruitment model (patients arrive at different centres according to Poisson processes with rates sampled from a gamma distributed population) and its further extensions is proposed. Closed-form expressions for corresponding distributions of the predicted number of the patients randomized in different regions are derived. In the case of two treatments, the properties of the total imbalance in the number of patients on treatment arms caused by using centre-stratified randomization are investigated and for a large number of centres a normal approximation of imbalance is proved. The impact of imbalance on the power of the study is considered. It is shown that the loss of statistical power is practically negligible and can be compensated by a minor increase in sample size. The influence of patient dropout is also investigated. The impact of randomization on predicted drug supply overage is discussed.

Some practical problems in implementing randomization

BACKGROUND

While often theoretically simple, implementing randomization to treatment in a masked, but confirmable, fashion can prove difficult in practice.

PURPOSE

At least three categories of problems occur in randomization: (1) bad judgment in the choice of method, (2) design and programming errors in implementing the method, and (3) human error during the conduct of the trial. This article focuses on these latter two types of errors, dealing operationally with what can go wrong after trial designers have selected the allocation method.

RESULTS

We offer several case studies and corresponding recommendations for lessening the frequency of problems in allocating treatment or for mitigating the consequences of errors. Recommendations include: (1) reviewing the randomization schedule before starting a trial, (2) being especially cautious of systems that use on-demand random number generators, (3) drafting unambiguous randomization specifications, (4) performing thorough testing before entering a randomization system into production, (5) maintaining a dataset that captures the values investigators used to randomize participants, thereby allowing the process of treatment allocation to be reproduced and verified, (6) resisting the urge to correct errors that occur in individual treatment assignments, (7) preventing inadvertent unmasking to treatment assignments in kit allocations, and (8) checking a sample of study drug kits to allow detection of errors in drug packaging and labeling.

LIMITATIONS

Although we performed a literature search of documented randomization errors, the examples that we provide and the resultant recommendations are based largely on our own experience in industry-sponsored clinical trials. We do not know how representative our experience is or how common errors of the type we have seen occur.

CONCLUSIONS

Our experience underscores the importance of verifying the integrity of the treatment allocation process before and during a trial. Clinical Trials 2010; 7: 235-245. http://ctj.sagepub.com.

Low patient enrollment sites in multicenter randomized clinical trials of cerebrovascular diseases: associated factors and impact on trial outcomes

Wide variability in patient enrollment among participating sites is a common phenomenon in multicenter trials. We examined stroke trial-related factors associated with the proportion of sites with low patient enrollment and the effect of these low-enrollment sites on trial outcome. We identified efficacy clinical trials enrolling patients with cerebrovascular diseases between 1980 and 2008 using an electronic database. The trials included in our analyses were multicenter randomized controlled trials (RCTs) comparing efficacy endpoints between two or more treatment groups and having >5 sites. Sites enrolling <10 patients or <2% of total trial patients were defined as low- enrollment sites. Trials were classified into tertiles based on the proportion of low-enrollment sites. Factors associated with trials that could be ascertained through a systematic review of published data were identified and examined. The association between low enrollment and a conclusive trial designation (defined by the ability to reject the primary null hypothesis either at or before target enrollment or demonstrate equivalence/noninferiority with adequate statistical power, depending on the initial design) was assessed using a multivariate logistic regression model. We identified 51 trials that met the inclusion criteria and provided information regarding patients enrolled per center. A total of 3059 participating centers enrolled a total of 53,742 trial participants; 78% of the participating sites enrolled <2% of trial participants. Trials enrolling acute stroke patients (within 24 hours of symptom onset) or those evaluating endovascular/surgical intervention had a higher proportion of low-enrollment sites (<10 patients per site). Studies with a higher proportion of low-enrollment sites were more likely to target acute stroke patients and less likely to randomize ≥1000 patients, use general efficacy endpoints, and stratify by site. There was no association between the studies with a higher proportion of low-enrollment sites and designation as a conclusive trial. A better understanding of factors associated with low-enrollment sites in clinical trials and the impact on a trial's ability to demonstrate conclusive outcomes may lead to strategies to make trial enrollments more efficient and cost-effective.

Randomization in substance abuse clinical trials

BACKGROUND

A well designed randomized clinical trial rates as the highest level of evidence for a particular intervention's efficacy. Randomization, a fundamental feature of clinical trials design, is a process invoking the use of probability to assign treatment interventions to patients. In general, randomization techniques pursue the goal of providing objectivity to the assignment of treatments, while at the same time balancing for treatment assignment totals and covariate distributions. Numerous randomization techniques, each with varying properties of randomness and balance, are suggested in the statistical literature. This paper reviews common randomization techniques often used in substance abuse research and an application from a National Institute on Drug Abuse (NIDA)-funded clinical trial in substance abuse is used to illustrate several choices an investigator faces when designing a clinical trial.

RESULTS

Comparisons and contrasts of randomization schemes are provided with respect to deterministic and balancing properties. Specifically, Monte Carlo simulation is used to explore the balancing nature of randomization techniques for moderately sized clinical trials. Results demonstrate large treatment imbalance for complete randomization with less imbalance for the urn or adaptive scheme. The urn and adaptive randomization methods display smaller treatment imbalance as demonstrated by the low variability of treatment allocation imbalance. For all randomization schemes, covariate imbalance between treatment arms was small with little variation between adaptive schemes, stratified schemes and unstratified schemes given that sample sizes were moderate to large.

CONCLUSION

We develop this paper with the goal of reminding substance abuse researchers of the broad array of randomization options available for clinical trial designs. There may be too quick a tendency for substance abuse researchers to implement the fashionable urn randomization schemes and other highly adaptive designs. In many instances, simple or blocked randomization with stratification on a major covariate or two will accomplish the same objectives as an urn or adaptive design, and it can do so with more simply implemented schedules and without the dangers of overmatching. Furthermore, the proper analysis, fully accounting for the stratified design, can be conducted.

Ensuring the comparability of comparison groups: is randomization enough?

BACKGROUND

It is widely believed that baseline imbalances in randomized trials must necessarily be random. In fact, there is a type of selection bias that can cause substantial, systematic and reproducible baseline imbalances of prognostic covariates even in properly randomized trials. It is possible, given complete data, to quantify both the susceptibility of a given trial to this type of selection bias and the extent to which selection bias appears to have caused either observable or unobservable baseline imbalances. Yet, in articles reporting on randomized trials, it is uncommon to find either these assessments or the information that would enable a reader to conduct them. Nevertheless, there have been a few published reports that contain descriptions of either this type of selection bias or indicators that it may have occurred.

OBJECTIVE

To document that the same type of selection bias has been described in numerous randomized trials and therefore that it represents a problem deserving of greater attention.

STUDY SELECTION

Computerized searches were not useful in locating trials with one or more elements that contribute to or are indicative of selection bias in randomized trials. We limit our treatment to trials that were previously questioned for susceptibility to selection bias or for large baseline imbalances.

RESULTS

We found 14 randomized trials that appear to be suspicious for selection bias. This may represent only the tip of the iceberg, because the status of other trials is inconclusive.

CONCLUSIONS

Authors of clinical trial reports should be required to disclose sufficient details to allow for an assessment of both allocation concealment and selection bias. The extent to which a randomized study was susceptible to selection bias should be considered in determining the relative contribution it makes to any subsequent meta-analysis, policy or decision.

Stratified randomization for clinical trials

Trialists argue about the usefulness of stratified randomization. For investigators designing trials and readers who use them, the argument has created uncertainty regarding the importance of stratification. In this paper, we review stratified randomization to summarize its purpose, indications, accomplishments, and alternatives. In order to identify research papers, we performed a Medline search for 1966-1997. The search yielded 33 articles that included original research on stratification or included stratification as the major focus. Additional resources included textbooks. Stratified randomization prevents imbalance between treatment groups for known factors that influence prognosis or treatment responsiveness. As a result, stratification may prevent type I error and improve power for small trials (<400 patients), but only when the stratification factors have a large effect on prognosis. Stratification has an important effect on sample size for active control equivalence trials, but not for superiority trials. Theoretical benefits include facilitation of subgroup analysis and interim analysis. The maximum desirable number of strata is unknown, but experts argue for keeping it small. Stratified randomization is important only for small trials in which treatment outcome may be affected by known clinical factors that have a large effect on prognosis, large trials when interim analyses are planned with small numbers of patients, and trials designed to show the equivalence of two therapies. Once the decision to stratify is made, investigators need to chose factors carefully and account for them in the analysis.

How many stratification factors are "too many" to use in a randomization plan?

The issue of stratification and its role in patient assignment has generated much discussion, mostly focused on its importance to a study or lack thereof. This report focuses on a much narrower problem: assuming that stratified assignment is desired, how many factors can be accommodated? This problem is investigated for two methods of balanced patient assignments; the first is based on the minimization method of Taves and the second on the commonly used method of stratified assignment. Simulation results show that the former method can accommodate a large number of factors (10-20) without difficulty but that the latter begins to fail if the total number of distinct combination of factor levels is greater than approximately n/2. The two methods are related to a linear discriminant model, which helps to explain the results.

A random allocation system with the minimization method for multi-institutional clinical trials

This paper describes the random allocation system used to perform precise and rapid treatment assignments in multi-institutional clinical trials. This system is based on sophisticated randomization procedures, according to Pocock and Simon's minimization method and Zelen's method for institution balancing. The major advantage of randomized treatment assignments with this system is to balance treatment numbers for each level of various prognostic factors over the entire trial and at the same time balance the allocation of treatments within an institution. Therefore, the randomized treatment assignments by this system can prevent degrading of the statistical power of a particular treatment factor. This system is designed to run on a small-sized notebook computer and therefore can be set up beside a telephone for registration, without occupying a large space. At present, this system is conveniently being used in two clinical trials.

Properties of permuted-block randomization in clinical trials

This article describes some of the important statistical properties of the commonly used permuted-block design, also known simply as blocked-randomization. Under a permutation model for statistical tests, proper analyses should employ tests that incorporate the blocking used in the randomization. These include the block-stratified Mantel-Haenszel chi-square test for binary data, the blocked analysis of variance F test, and the blocked nonparametric linear rank test. It is common, however, to ignore the blocking in the analysis. For these tests, it is shown that the size of a test obtained from an analysis incorporating the blocking (say T), versus an analysis ignoring the blocking (say TI), is related to the intrablock correlation coefficient (R) as TI = T(1-R). For blocks of common length 2m, the range of R is from -1/(2m-1) to 1. Thus, if there is a positive intrablock correlation, which is more likely than not for m greater than 1, an analysis ignoring blocking will be unduly conservative. Permutation tests are also presented for the case of stratified analyses within one or more subgroups of patients defined post hoc on the basis of a covariate. This provides a basis for the analysis when responses from some patients are assumed to be missing-at-random. An alternative strategy that requires no assumptions is to perform the analysis using only the subset of complete blocks in which no observations are missing. The Blackwell-Hodges model is used to assess the potential for selection bias induced by investigator attempts to guess which treatment is more likely to be assigned to each incoming patient. In an unmasked trial, the permuted-block design provides substantial potential for selection bias in the comparison of treatments due to the predictability of the assignments that is induced by the requirement of balance within blocks. Further, this bias is not eliminated by the use of random block sizes. We also modify the Blackwell-Hodges model to allow for selection bias only when the investigator is able to discern the next assignment with certainty. This type of bias is reduced by the use of random block sizes and is eliminated only if the possible block sizes are unknown to the investigators. Finally, the Efron model for accidental bias is used to assess the potential for bias in the estimation of treatment effects due to covariate imbalances. For the permuted-block design, the variance of this bias approaches that of complete randomization as the half-block length m----infinity.(ABSTRACT TRUNCATED AT 400 WORDS)