Cited by 322Statistical Methods in Clinical TrialsOptimal Experimental Design MethodsAdvanced Causal Inference Techniques

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Properties of permuted-block randomization in clinical trials

Matts, John P., Lachin, John M.·1988·Controlled Clinical Trials

DOI 10.1016/0197-2456(88)90047-5

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Abstract

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)

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Method
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“Besonders in einer nicht-doppelblind angelegten Studie entsteht hier durch das Erraten der Zuteilungen (vor allem gegen Ende der Studie) Spielraum für das Auftreten von Selektionsbias, was aber natürlich nicht auf die Ebene der permutierten Blockrandomisierung mit kleineren Blockgrößen zutrifft [24,25].”
Background
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Method
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The Randomization Process
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Background
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Background
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“In the present paper, we continue the approach described by Matts and Lachin (1) by quantifying predictability with certainty in situations other than two-arm balanced blocks: trials with three or more arms, and trials with unbalanced treatment allocation.”
BackgroundMethodInfluential
Best practices for design and implementation of human clinical trials studying dietary oils
MacKay, D., Jew, S., & Jones, P. J. H. (2016) · Progress in Lipid Research · doi:10.1016/j.plipres.2016.10.003 · Cited by 10
Attention-Deficit/Hyperactivity Disorder
(2012) · Springer eBooks · doi:10.1007/springerreference_179716 · Cited by 10
Continuous Risk Score Predicts Waitlist and Post-transplant Outcomes in Hepatocellular Carcinoma Despite Exception Changes
Akabane, M., McVey, J. C., Firl, D. J., Kwong, A. J., Melcher, M. L., Kim, W. R., & Sasaki, K. (2024) · Clinical Gastroenterology and Hepatology · doi:10.1016/j.cgh.2024.05.046 · Cited by 9
Text Prediction on Structured Data Entry in Healthcare
Hua, L., Wang, S., & Gong, Y. (2014) · Applied Clinical Informatics · doi:10.4338/aci-2013-11-ra-0095 · Cited by 9
“Experimental design With a permuted-block algorithm [28], the 52 participants were randomly assigned to two groups.”
Method
Minimal sufficient balance randomization for sequential randomized controlled trial designs: results from the ESCAPE trial
Sajobi, T. T., Singh, G., Lowerison, M. W., Engbers, J. D. T., Menon, B. K., Demchuk, A. M., Goyal, M., & Hill, M. D. (2017) · Trials · doi:10.1186/s13063-017-2264-1 · Cited by 9
Replacing permuted block design with big stick design in stratified randomization
Cai, H., & Zhou, X. (2021) · Contemporary Clinical Trials · doi:10.1016/j.cct.2021.106450 · Cited by 9
Regulatory Guidance on Randomization and the Use of Randomization Tests in Clinical Trials: A Systematic Review
Carter, K., Scheffold, A., Renteria, J., Berger, V. W., Luo, Y., Chipman, J., & Sverdlov, O. (2023) · Statistics in Biopharmaceutical Research · doi:10.1080/19466315.2023.2239521 · Cited by 10
“A good summary of statistical properties of PBD is given by Matts and Lachin (1988), and practical implementation aspects of PBD are discussed by McEntegart (2014).”
A random allocation system with the minimization method for multi-institutional clinical trials
Akazawa, K., Odaka, T., Sakamoto, M., Ohtsuki, S., Shimada, M., Kamakura, T., & Nose, Y. (1991) · Journal of Medical Systems · doi:10.1007/bf00999168 · Cited by 9
Leveraging user's performance in reporting patient safety events by utilizing text prediction in narrative data entry
Gong, Y., Hua, L., & Wang, S. (2016) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2016.03.031 · Cited by 9
“With a permuted-block algorithm and random block [30], fiftytwo participants were randomly assigned to two groups.”
Improving Diabetes Management in Emerging Adulthood: An Intervention Development Study Using the Multiphase Optimization Strategy
Carcone, A. I., Ellis, D. A., Eggly, S. S., MacDonell, K. K., Ghosh, S., Buggs‐Saxton, C., & Ondersma, S. J. (2020) · JMIR Research Protocols · doi:10.2196/20191 · Cited by 7
“Permuted blocks have the advantage of ensuring balance between treatment arms for important prognostic variables without unmasking the next treatment allocation [51].”
Background
Psychological interventions for parents of children with intellectual disabilities to enhance child behavioral outcomes or parental well-being: A systematic review, content analysis and effects
Ranta, K., Saarimäki, H., Gummerus, J. K., Virtanen, J., Peltomäki, S., & Kontu, E. K. (2024) · Journal of Intellectual Disabilities · doi:10.1177/17446295241302857 · Cited by 10
“Permuted-block designs were considered to cause potential for selection bias (Matts and Lachin, 1988).”
Marine ω-3 PUFA Supplementation Enhances FFAR4 Activation and Reduces Inflammatory Markers in PBMC of Subjects with Obesity: A Randomized Controlled Trial (EPICO)
Reyes-Pérez, S. D., Cambrón-Mora, D., Torres-Vanegas, J., Mojica-Zamudio, K. L., Olaez-Ramos, C. E., Lauriano-Rivera, R. G., Rivera-Valdes, J. J., Torres‐Castillo, N., Valencia-Sosa, E., Rodriguez‐Echevarría, R., et al. (2025) · Nutrients · doi:10.3390/nu17233630 · Cited by 9
“The randomization of the groups was carried out by the permuted blocks technique [19,20].”
Balance of group sizes in randomized controlled trials published in American Psychological Association journals.
Cañedo-Ayala, M., Rice, D. B., Levis, A. W., Chiovitti, M. J., & Thombs, B. D. (2020) · Health Psychology · doi:10.1037/hea0001020 · Cited by 8
Efficacy and safety of oral Nigella sativa oil for symptomatic treatment of knee osteoarthritis: A double-blind, randomized, placebo-controlled clinical trial
Huseini, H. F., Mohtashami, R., Sadeghzadeh, E., Shadmanfar, S., Hashem-Dabaghian, F., & Kianbakht, S. (2022) · Complementary Therapies in Clinical Practice · doi:10.1016/j.ctcp.2022.101666 · Cited by 8
Which Randomization Methods Are Used Most Frequently in Clinical Trials? Results of a Survey by the Randomization Working Group
Sverdlov, O., Carter, K., Hilgers, R.-D., Everett, C., Berger, V. W., Luo, Y., Chipman, J., Ryeznik, Y. V., Ross, J., Knight, R., et al. (2023) · Statistics in Biopharmaceutical Research · doi:10.1080/19466315.2023.2225451 · Cited by 8
“Matts and Lachin (1988) Stratified permuted block randomization Trial participants are grouped according to covariate values prior to randomization, and they are sequentially randomized to treatment or control using permuted block randomization within strata.”
Properties of randomization in clinical trials: Foreword
Lachin, J. M. (1988) · Controlled Clinical Trials · doi:10.1016/0197-2456(88)90044-x · Cited by 8
Self-citation
Effects of Two Exercise Regimes on Patients with Chiari Malformation Type 1: a Randomized Controlled Trial
Türkmen, C., Köse, N., Bal, E., Bilgin, S., Çetin, H., Zengin, H. Y., Gümeler, E., & Mut, M. (2022) · The Cerebellum · doi:10.1007/s12311-022-01397-1 · Cited by 9
“1 3 All patients who met the inclusion criteria were randomly allocated to either the TEP-CM (n = 8) or CSSE (n = 8) group using a computer-generated permuted block randomization scheme with blocks of four [14].”
Method
How important is exact balance in treatment and control sample sizes to evaluations?
Elliott, M. N., McCaffrey, D. F., & Lockwood, J. R., III (2007) · Journal of Substance Abuse Treatment · doi:10.1016/j.jsat.2006.12.007 · Cited by 8
“The results we obtain are an extension of work by Matts and Lachin (1988) and Wei and Lachin (1988).”
Result
Guided Participation Support of Coparenting an Infant With Complex Congenital Heart Disease: A Randomized Pilot Feasibility Study
Pridham, K. F., Melby, J. N., Harrison, T. M., Brown, R., & Mussatto, K. A. (2022) · Research and theory for nursing practice · doi:10.1891/rtnp-2021-0096 · Cited by 7
Inflammatory markers changes following acceptance-based behavioral psychotherapy in generalized anxiety disorder patients: Evidence from a randomized controlled trial
Gandarela, L., de A. Sampaio, T. P., Marçal, L., Burdmann, E. A., Neto, F. L., & Bernik, M. A. (2024) · Brain, Behavior, & Immunity - Health · doi:10.1016/j.bbih.2024.100779 · Cited by 6
“A random number generator (http://www.randomization.com) was used to generate a random number for each participant and randomly allocate each enrolled participant to one of the two treatment groups, in a 1:1 ratio, following the method of randomly permuted blocks (Matts and Lachin, 1988).”
Method
Choosing and evaluating randomisation methods in clinical trials: a qualitative study
Bruce, C. L., Iflaifel, M., Montgomery, A. A., Ogollah, R. O., Sprange, K., & Partlett, C. (2024) · Trials · doi:10.1186/s13063-024-08005-z · Cited by 6
“(Statistician 6).”
Method
Statistical Modeling to Adjust for Time Trends in Adaptive Platform Trials Utilizing Non‐Concurrent Controls
Krotka, P., Posch, M., Gewily, M., Höglinger, G. U., & Roig, M. B. (2025) · Biometrical Journal · doi:10.1002/bimj.70059 · Cited by 6
“This leads to deflation in the type I error rate, which gets stronger the more extreme the time trend is (Matts and Lachin 1988).”
Home-delivered meals for people with dementia: Which model delays nursing home placement? - Protocol for a feasibility pilot
Thomas, K. S., Dosa, D. M., Fisher, A., Gadbois, E. A., Harrison, J., Hilgeman, M. M., Largent, E. A., Lima, J. C., McAuliff, K., McCreedy, E. M., et al. (2022) · Contemporary Clinical Trials · doi:10.1016/j.cct.2022.106897 · Cited by 6
Impacts of healthy marriage and relationship education for expectant and new mothers
Patnaik, A., Wood, R. G., & Gonzalez, K. E. (2022) · Family Relations · doi:10.1111/fare.12818 · Cited by 6
Evaluation of Imbalance in Stratified Blocked Randomization
Glass, A., & Kundt, G. (2011) · Methods of Information in Medicine · doi:10.3414/me10-01-0078 · Cited by 6
“permuted-block randomization [2] or the biased coin design [3] with a suitable choice of values of parameters [4] and different balance effectiveness [5].”
Background
Cardiovascular Disease Prevention Education Using a Virtual Environment in Sexual-Minority Men of Color With HIV: Protocol for a Sequential, Mixed Method, Waitlist Randomized Controlled Trial
Ramos, S. R., Johnson, C., Melkus, G. D., Kershaw, T., Gwadz, M. V., Reynolds, H. R., & Vorderstrasse, A. A. (2022) · JMIR Research Protocols · doi:10.2196/38348 · Cited by 6
“Permuted blocks randomization with varying block sizes will be used to ensure allocation concealment and balanced allocation to the intervention and control groups over the study period [68].”
Protocol for diagnostic test accuracy study: the efficacy of screening for common dental diseases by Dental Care Professionals
Macey, R., Walsh, T., Glenny, A., Worthington, H. V., Tickle, M., Ashley, J., & Brocklehurst, P. R. (2013) · BMC Oral Health · doi:10.1186/1472-6831-13-45 · Cited by 6
“To reduce order effects each practice will be randomly allocated using a permuted block sampling [21] system to either DCP first or GDP first.”
Method
A Comparison of Randomization Methods for Multi-Arm Clinical Trials
Azher, R., Wason, J. M. S., & Grayling, M. J. (2023) · Statistics in Biopharmaceutical Research · doi:10.1080/19466315.2023.2238645 · Cited by 7
“Wei (1977) proposed the UD to force a small trial to be better-balanced in terms of group size, but to provide behavior like SR as the size of the trial increases (Matts and Lachin 1988).”
BackgroundMethod
Impacts of healthy marriage and relationship education with integrated economic stability services
Moore, Q., Wood, R., & Wu, A. (2023) · Family Relations · doi:10.1111/fare.12877 · Cited by 5
Design of the CHARGE study: A randomized control trial evaluating a novel treatment for Veterans with binge eating disorder and overweight and obesity
Boutelle, K. N., Afari, N., Obayashi, S., Eichen, D. M., Strong, D. R., & Peterson, C. B. (2023) · Contemporary Clinical Trials · doi:10.1016/j.cct.2023.107234 · Cited by 5
Impact of Stratified Randomization in Clinical Trials
Anisimov, V. V. (2010) · Contributions to statistics · doi:10.1007/978-3-7908-2410-0_1 · Cited by 5
“The properties of various types of randomization schemes are studied in the papers Hallstrom and Davis (1988), Lachin (1988), Matts and Lachin (1988) , and books by Pocock (1983), Rosenberger and Lachin (2002).”
Background
Properties of the Random Block Design for Clinical Trials
Shao, H., & Rosenberger, W. F. (2016) · Contributions to statistics · doi:10.1007/978-3-319-31266-8_26 · Cited by 5
Implementing online interventions in ICare: A biostatistical perspective
Görlich, D., & Faldum, A. (2019) · Internet Interventions · doi:10.1016/j.invent.2018.12.004 · Cited by 5
“prioritized the beneficial balancing properties (cf. Lachin et al., 1988; Matts and Lachin, 1988).”
Background
Regulation of Cues vs Cognitive Behavioral Therapy for Binge Eating and Weight Loss Among Veterans
Boutelle, K. N., Afari, N., Obayashi, S., Eichen, D. M., Strong, D. R., Pasquale, E. K., & Peterson, C. B. (2025) · JAMA Network Open · doi:10.1001/jamanetworkopen.2025.25064 · Cited by 5
Factorial Trial to Optimize an Internet-Delivered Intervention for Sexual Health After Breast Cancer: Protocol for the WF-2202 Sexual Health and Intimacy Enhancement (SHINE) Trial
Shaffer, K. M., Reese, J. B., Dressler, E. V. M., Glazer, J. V., Cohn, W. F., Showalter, S. L., Clayton, A. H., Danhauer, S. C., Loch, M. M., Kadi, M. S., et al. (2024) · JMIR Research Protocols · doi:10.2196/57781 · Cited by 4
“Block randomization will be used to help maintain balance across conditions, with varying block sizes of 16, 32, and 48 [44,45].”
Method
Misconduct in research integrity: Assessment the quality of systematic reviews in Cochrane urological cancer review group
Salehi‐Pourmehr, H., Naseri, A., Mostafaei, A., Vahedi, L., Sajjadi, S., Tayebi, S., Mostafaei, H., & Hajebrahimi, S. (2021) · Urology Research and Practice · doi:10.5152/tud.2021.21038 · Cited by 6

Authors: Matts, J. P., & Lachin, J. M.
Affiliations: University of Minnesota (US)
George Washington University (US)
Corresponding author: Matts, John P.
ORCID: Lachin, John M.
Venue: Controlled Clinical Trials
Publisher: Elsevier BV
Published: 1988-12
Volume: 9
Issue: 4
Pages: 327-344
ISSN: 0197-2456 (ISSN-L)
DOI: 10.1016/0197-2456(88)90047-5
OpenAlex: W2005415125
Cited by: 321
References: 18
Citations by year: 2026: 14 · 2025: 13 · 2024: 23 · 2023: 22 · 2022: 19 · 2021: 18 · 2020: 33 · 2019: 14 · 2018: 6 · 2017: 15 · 2016: 13 · 2015: 18 · 2014: 16 · 2013: 12 · 2012: 15
FWCI: 4.39
Type: journal article
Language: en
Open access: closed
Licence: www.elsevier.com/tdm/userlicense/1.0/ (tdm)
Locations: Controlled Clinical Trials (publishedVersion)
PubMed (publishedVersion)
Funders: U.S. Public Health Service (R01-DI-39826)
National Heart, Lung, and Blood Institute (HL15265)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Clinical Trials
Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
Keywords: permuted block randomization, permutation tests, nonparametric rank tests, stratified analysis, selection bias, covariate imbalance, treatment effect estimation
MeSH terms: Analysis of Variance, Clinical Trials as Topic, Humans, Random Allocation, Research Design, Models, Statistical

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Properties of permuted-block randomization in clinical trials

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