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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“The analysis method ignoring blocks is more conservative regarding the statistical significance and thus less efficient and powerless (Matts and Lachin 1988).”
Method
Reverse total shoulder arthroplasty: a biomechanical evaluation of humeral and glenosphere hardware configuration
Tashjian, R. Z., Burks, R. T., Zhang, Y., & Henninger, H. B. (2014) · Journal of Shoulder and Elbow Surgery · doi:10.1016/j.jse.2014.08.017 · Cited by 55
Silymarin, Olibanum, and Nettle, A Mixed Herbal Formulation in the Treatment of Type II Diabetes: A Randomized, Double-Blind, Placebo-Controlled, Clinical Trial
Khalili, N., Fereydoonzadeh, R., Mohtashami, R., Mehrzadi, S., Heydari, M., & Huseini, H. F. (2017) · Journal of Evidence-Based Complementary & Alternative Medicine · doi:10.1177/2156587217696929 · Cited by 55
“The randomized list was generated using Microsoft Excel with a permuted balanced block randomization method, as previously described.(28) The physicians, researchers, and statisticians were blind to the allocation of patients.”
Method
Comparing the Effectiveness of Combined External Beam Radiation and Hyperthermia Versus External Beam Radiation Alone in Treating Patients With Painful Bony Metastases: A Phase 3 Prospective, Randomized, Controlled Trial
Chi, M., Yang, K.-L., Chang, Y.-C., Ko, H.-L., Lin, Y.-H., Huang, S.-C., Huang, Y.-Y., Liao, K., Kondo, M., & Chi, K.-H. (2017) · International Journal of Radiation Oncology*Biology*Physics · doi:10.1016/j.ijrobp.2017.09.030 · Cited by 55
“To include an equal number of patients in both treatment arms, a blocked randomization schedule was used for each stratum (16).”
Smoking cessation interventions with female smokers living with HIV/AIDS: a randomized pilot study of motivational interviewing
Manuel, J. K., Lum, P. J., Hengl, N. S., & Sorensen, J. L. (2013) · AIDS Care · doi:10.1080/09540121.2012.733331 · Cited by 52
“Allocation assignment was determined prior to the start of the study using permuted block randomization (Matts & Lachin, 1988).”
Method
An introduction to clinical trial design
Schultz, A., Saville, B. R., Marsh, J., & Snelling, T. (2019) · Paediatric Respiratory Reviews · doi:10.1016/j.prrv.2019.06.002 · Cited by 56
Effect of a clean stove intervention on inflammatory biomarkers in pregnant women in Ibadan, Nigeria: A randomized controlled study.
Olopade, C., Frank, E., Bartlett, E. C., Alexander, D., Dutta, A., Ibigbami, T., Adu, D., Olamijulo, J., Arinola, G., Karrison, T., et al. (2017) · Environment International · doi:10.1016/j.envint.2016.11.004 · Cited by 50
“N4 children) and presence/absence of diabetes using method of permuted blocks (Matts and Lachin, 1988) and MT64 random number generator in STATA (College Station, TX).”
Method
Automated phone call and text reminders for childhood immunisations (PRIMM): a randomised controlled trial in Nigeria
Ekhaguere, O. A., Oluwafemi, R. O., Badejoko, B. O., Oyeneyin, L. O., Butali, A., Lowenthal, E. D., & Steenhoff, A. P. (2019) · BMJ Global Health · doi:10.1136/bmjgh-2018-001232 · Cited by 49
Charting the Path Forward for Risk Prediction in Liver Transplant for Hepatocellular Carcinoma: International Validation of HALTHCC Among 4,089 Patients
Firl, D. J., Sasaki, K., Agopian, V. G., Gorgen, A., Kimura, S., Dumronggittigule, W., McVey, J. C., Iesari, S., Mennini, G., Vitale, A., et al. (2019) · Hepatology · doi:10.1002/hep.30838 · Cited by 49
“Block randomization was performed in a 3:1 fashion (with each center serving as a block) to maintain similar representation of every center.(21) Recalibration was performed on the training cohort (n = 1,021) using ordinal logistic modeling of a composite endpoint (1 point for either recurrence or mortality and 2 point…”
Method
A better alternative to stratified permuted block design for subject randomization in clinical trials
Zhao, W. (2014) · Statistics in Medicine · doi:10.1002/sim.6266 · Cited by 45
“Rosenberger and Lachin distinguish randomization procedures into four classes: complete randomization, restricted randomization, covariate-adaptive randomization, and response-adaptive randomization [3].”
MethodInfluential
The Efficacy of Screening for Common Dental Diseases by Hygiene-Therapists
Macey, R., Glenny, A., Walsh, T., Tickle, M., Worthington, H. V., Ashley, J., & Brocklehurst, P. R. (2015) · Journal of Dental Research · doi:10.1177/0022034514567335 · Cited by 46
Reproductive Health Choices for Young Adults With Sickle Cell Disease or Trait
Wilkie, D. J., Gallo, A. M., Yao, Y., Molokie, R. E., Stahl, C., Hershberger, P. E., Zhao, Z., Suarez, M. L., Labotka, R. J., Johnson, B., et al. (2013) · Nursing Research · doi:10.1097/nnr.0b013e3182a0316b · Cited by 44
“Stratified by SCD or SCT condition, we randomly assigned subjects in permuted blocks (Matts & Lachin, 1988) to e-Book (attention-control usual care) or CHOICES treatment groups using the random digit-based RandomAssignment software program (Pain & Symptom Management Research Group, Chicago, IL).”
Method
Effectiveness of airway clearance techniques in children hospitalized with acute bronchiolitis
Ginderdeuren, F. V., Vandenplas, Y., Deneyer, M., Vanlaethem, S., Buyl, R., & Kerckhofs, E. (2016) · Pediatric Pulmonology · doi:10.1002/ppul.23495 · Cited by 44
“com, 2011), using the method of randomly permuted blocks.(25) Each number was contained in a sealed opaque envelope opened by the physiotherapist after inclusion.”
Method
Kinesiotaping for the Rehabilitation of Rotator Cuff–Related Shoulder Pain: A Randomized Clinical Trial
de Oliveira, F. C. L., de Fontenay, B. P., Bouyer, L. J., Desmeules, F., & Roy, J. (2020) · Sports Health A Multidisciplinary Approach · doi:10.1177/1941738120944254 · Cited by 46
“Randomization was stratified by sex (as it was unclear whether sex could influence the KT responses), and a block design (block size of 4-6-8) was utilized.(27) The allocation was concealed in sequentially numbered sealed opaque envelopes that were opened by the treating physical therapist”
Method
Haemodynamic effects, safety, and tolerability of haemoglobin-based oxygen carrier-201 in patients undergoing PCI for CAD
Serruys, P. W. J. C., Vranckx, P., Slagboom, T., Regar, E. S., Meliga, E., de Winter, R. J., Heyndrickx, G. R., Schuler, G. C., van Remortel, E., Dubé, G. P., et al. (2008) · EuroIntervention · doi:10.4244/eijv3i5a108 · Cited by 44
The Integration of Virtual Simulation Gaming Into Undergraduate Nursing Resuscitation Education: A Pilot Randomised Controlled Trial
Keys, E., Luctkar‐Flude, M., Tyerman, J. J., Sears, K., & Woo, K. Y. (2021) · Clinical Simulation in Nursing · doi:10.1016/j.ecns.2021.01.013 · Cited by 43
Chronological Bias in Randomized Clinical Trials Arising from Different Types of Unobserved Time Trends
Hilgers, R.-D., & Tamm, M. (2014) · Methods of Information in Medicine · doi:10.3414/me14-01-0048 · Cited by 42
“This relation between blocked and unblocked analysis is well known from Matts and Lachin [22], who showed that for a positive intrablock correlation the analysis of variances ignoring the blocks is more conservative than the blocked analysis of variance.”
MethodInfluential
Randomization in substance abuse clinical trials
Hedden, S. L., Woolson, R. F., & Malcolm, R. J. (2006) · Substance Abuse Treatment Prevention and Policy · doi:10.1186/1747-597x-1-6 · Cited by 42
Environmental intervention for house dust mite control in childhood bronchial asthma
El‐Ghitany, E. M., & El-Salam, M. M. A. (2012) · Environmental Health and Preventive Medicine · doi:10.1007/s12199-011-0263-5 · Cited by 42
“Through block randomization [23] using random sequences of block sizes, the children were randomly allocated into one of four groups according to recommended HDM control measures:”
Method
Results and lessons learned from a prevention of weight gain program for low-income overweight and obese young mothers: Mothers In Motion
Chang, M., Brown, R., & Nitzke, S. A. (2017) · BMC Public Health · doi:10.1186/s12889-017-4109-y · Cited by 40
“Our randomization protocol utilized a variable permuted-block algorithm [22, 23].”
Method
Assessment of the safety and feasibility of administering antipyretic therapy in critically ill adults: A pilot randomized clinical trial
Niven, D. J., Stelfox, H. T., Léger, C., Kubes, P., & Laupland, K. B. (2012) · Journal of Critical Care · doi:10.1016/j.jcrc.2012.08.015 · Cited by 40
“The randomization scheme was developed using simple block randomization (n = 8), and allocation was concealed using the consecutively numbered, sealed opaque envelope technique [20].”
Method
Reproductive Health CHOICES for Young Adults with Sickle Cell Disease or Trait: Randomized Controlled Trial Outcomes over Two Years
Gallo, A. M., Wilkie, D. J., Yao, Y., Molokie, R. E., Stahl, C., Hershberger, P. E., Zhao, Z., Suarez, M. L., Johnson, B., Angulo, R., et al. (2015) · Journal of Genetic Counseling · doi:10.1007/s10897-015-9874-0 · Cited by 39
“Stratified by SCD or SCT status in permuted blocks (Matts and Lachin 1988), the RCT had baseline and multiple posttest measurements (immediately posttest, 6, 12, 18 and 24 months posttest).”
Method
Improving Breastfeeding by Empowering Mothers in Vietnam: A Randomised Controlled Trial of a Mobile App
Duong, D. T. T., Binns, C. W., Pham, N. M., Zhao, Y., Dinh, T. P. H., Ha, B. T. T., Tran, C. T., Nguyen, X. H., Giglia, R. C., Xu, F., et al. (2020) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph17155552 · Cited by 40
“≥2 children) to achieve the balance of the number of participants allocated to each treatment group [127,128].”
Background
Antenatal N-acetylcysteine to improve outcomes of premature infants with intra-amniotic infection and inflammation (Triple I): randomized clinical trial
Buhimschi, C. S., Bahtiyar, M. O., Zhao, G., Abdelghany, O., Schneider, L., Abdel‐Razeq, S. S., Dulay, A. T., Lipkind, H. S., Mieth, S., Rogers, L. K., et al. (2020) · Pediatric Research · doi:10.1038/s41390-020-01106-w · Cited by 39
Impact of receiving recorded mental health recovery narratives on quality of life in people experiencing psychosis, people experiencing other mental health problems and for informal carers: Narrative Experiences Online (NEON) study protocol for three randomised controlled trials
Rennick‐Egglestone, S., Elliott, R. A., Smuk, M., Robinson, C., Bailey, S., Smith, R., Keppens, J., Hussain, H., Pollock, K., Cuijpers, P. J., et al. (2020) · Trials · doi:10.1186/s13063-020-04428-6 · Cited by 40
“permuted block randomisation [103] with randomly”
Squill Oxymel, a traditional formulation from Drimia Maritima (L.) Stearn, as an add-on treatment in patients with moderate to severe persistent asthma: A pilot, triple-blind, randomized clinical trial
Nejatbakhsh, F., Karegar-Borzi, H., Amin, G., Eslaminejad, A., Hosseini, M., Bozorgi, M., & Gharabaghi, M. A. (2016) · Journal of Ethnopharmacology · doi:10.1016/j.jep.2016.12.032 · Cited by 35
“Subjects were allocated to 1 of these 3 treatment arms by randomized permuted blocks with a block size of four (Matts and Lachin, 1988) (27).”
Method
A randomized trial of motivational interviewing and facilitated contraceptive access to prevent rapid repeat pregnancy among adolescent mothers
Stevens, J. H., Lutz, R., Osuagwu, N., Rotz, D., & Goesling, B. (2017) · American Journal of Obstetrics and Gynecology · doi:10.1016/j.ajog.2017.06.010 · Cited by 37
“A permuted block design was used to seek an even balance of participants across the intervention and control groups throughout the study period.(16) We specified a variable block size of up to 6 characters and a 1:1 allocation of participants across the 2 study”
Method
Robust Tests for Treatment Effect in Survival Analysis under Covariate-Adaptive Randomization
Ye, T., & Shao, J. (2020) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/rssb.12392 · Cited by 35
“…permuted block randomization is very easy to implement, but sometimes its deterministic aspect makes the future sequence predictable if the investigator is unblinded to the previous treatment assignments, which may result in an increased risk of investigator selection bias (Matts and Lachin, 1988).”
Background
Short-Term Impacts of Pulse: An App-Based Teen Pregnancy Prevention Program for Black and Latinx Women
Manlove, J. S., Cook, E. L., Whitfield, B., Johnson, M., Martínez-García, G., & Garrido, M. (2019) · Journal of Adolescent Health · doi:10.1016/j.jadohealth.2019.08.017 · Cited by 35
Biomechanical comparison of reverse total shoulder arthroplasty systems in soft tissue–constrained shoulders
Henninger, H. B., King, F. K., Tashjian, R. Z., & Burks, R. T. (2013) · Journal of Shoulder and Elbow Surgery · doi:10.1016/j.jse.2013.08.008 · Cited by 34
Early Removal of Urethral Catheter with Suprapubic Tube Drainage Versus Urethral Catheter Drainage Alone after Robot-Assisted Laparoscopic Radical Prostatectomy
Prasad, S. M., Large, M. C., Patel, A. R., Famakinwa, O. J., Galocy, R. M., Karrison, T., Shalhav, A. L., & Zagaja, G. P. (2014) · The Journal of Urology · doi:10.1016/j.juro.2014.01.004 · Cited by 33
“Interestingly, all 202 patients elected removal of the UC rather than the SPT. Chicago patients and the 202 Detroit patients (reference 12 in article)?”
BackgroundInfluential
Can standardized diagnostic assessment be a useful adjunct to clinical assessment in child mental health services? A randomized controlled trial of disclosure of the Development and Well-Being Assessment to practitioners
Ford, T. J., Last, A., Henley, W., Norman, S., Guglani, S., Kelesidi, K., Martin, A., Moran, P., Latham-Cork, H., & Goodman, R. (2012) · Social Psychiatry and Psychiatric Epidemiology · doi:10.1007/s00127-012-0564-z · Cited by 32
“Randomisation was carried out by an independent clinical trials unit; it used permuted block randomization [20], stratifying by clinical team to ensure balance for the different length of waiting list among the three clinical teams.”
Method
Efficacy of Addition of Transcutaneous Electrical Nerve Stimulation to Standardized Physical Therapy in Subacute Spinal Spasticity: A Randomized Controlled Trial
Oo, W. M. (2014) · Archives of Physical Medicine and Rehabilitation · doi:10.1016/j.apmr.2014.06.001 · Cited by 32
Clinical Trials in Orthopaedics Research. Part III. Overcoming Operational Challenges in the Design and Conduct of Randomized Clinical Trials in Orthopaedic Surgery
Losina, E., Wright, J. G., & Katz, J. N. (2012) · Journal of Bone and Joint Surgery · doi:10.2106/jbjs.k.00009 · Cited by 34
Allocation of Patients to Conditions in Headache Clinical Trials: Randomization, Stratification, and Treatment Matching
Lipchik, G. L., Nicholson, R. A., & Penzien, D. B. (2005) · Headache The Journal of Head and Face Pain · doi:10.1111/j.1526-4610.2005.05093.x · Cited by 29
Empiric antibiotics for an elevated prostate‐specific antigen ( PSA ) level: a randomised, prospective, controlled multi‐institutional trial
Eggener, S. E., Large, M. C., Gerber, G. S., Pettus, J. A., Yossepowitch, O., Smith, N. D., KUNDU, S. D., Kunnavakkam, R., Zorn, K. C., & Raman, J. D. (2013) · British Journal of Urology · doi:10.1111/bju.12241 · Cited by 28
Effects of unstratified and centre‐stratified randomization in multi‐centre clinical trials
Anisimov, V. V. (2010) · Pharmaceutical Statistics · doi:10.1002/pst.412 · Cited by 26
Dynamic randomization and a randomization model for clinical trials data
Kaiser, L. D. (2012) · Statistics in Medicine · doi:10.1002/sim.5448 · Cited by 26
“This parallels the situation with PBR where the variance estimate ignoring the blocking in time will generally be greater than the estimate accounting for this blocking when there is low-frequency variation or a smooth time trend in the unit effect residuals, as quantified by the intrablock correlation [1].”
BackgroundMethodInfluential
Racial Differences in the Effectiveness of Internet-Delivered Mental Health Care
Jonassaint, C. R., Belnap, B. H., Huang, Y., Karp, J. F., Abebe, K. Z., & Rollman, B. L. (2019) · Journal of General Internal Medicine · doi:10.1007/s11606-019-05542-1 · Cited by 24

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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