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• W2893331902 abstract "•Simple randomization leads most of the time to unequal group sizes.•Simple randomization is suggested for large samples, and block randomization is suggested for smaller samples.•Sample size influences the power more than the group imbalance. It is virtually impossible to control all the factors that may confound results in research. Factors unknown to the researcher can create differences between groups that are not related to the primary factor under analysis. Randomization is perhaps one of the best ways to try to overcome such unknown factors.1Pandis N. Polychronopoulou A. Eliades T. Randomization in clinical trials in orthodontics: its significance in research design and methods to achieve it.Eur J Orthod. 2011; 33: 684-690Crossref PubMed Scopus (15) Google Scholar It homogenizes groups so that each specimen, participant, or intervention has the same chance of being allocated to each experimental group. Therefore, it is important to include randomization in experimental designs of orthodontic studies when possible. Imagine a research project where 2 adhesive systems from 2 manufacturers are being compared for shear bond strength. The researcher may collect data of the adhesive system from manufacturer A in the morning and from manufacturer B in the afternoon. It is possible that the researcher might be tired toward the end of the day and therefore could, unconsciously, be less precise when collecting data. Thus, if differences were found between the groups, would those be attributed only to the adhesive system or to the researcher's fatigue as well? Similarly, imagine a clinical study aiming to compare the bracket debonding rates using 2 composites. To compare them in the same patient in a split-mouth design, the researcher could bond the right-side brackets with composite A and the left-side ones with composite B. Because the orthodontist might have a different perspective between the patient's right and left sides, the bonding quality could be slightly different between the sides, and this may result in differences that are not really attributed to the composites. This could undermine the results because it is expected that differences between 2 interventions are explained only by the factors being evaluated. In these 2 examples, one might think that the biases mentioned could be controlled if the adhesive systems (in the in-vitro test) or composite types (in the in-vivo test) were evaluated or applied in an alternate manner. Even though this idea may seem interesting, it could also lead to selection bias: eg, if the operator knows which adhesive is being evaluated, a systematic bias might be introduced. Besides eliminating biases and giving all specimens the same chance of allocation to any of the groups, randomization is a fundamental premise that justifies most of the statistical procedures used in data analysis.2Hahn G.J. Meeker W.Q. Assumptions for statistical inference.Am Stat. 1993; 47: 1-11Google Scholar Thus, any attempt at randomization that has a logical pattern of allocation and that deviates from pure chance (eg, chart identification numbers, day of the week, date of birth) can possibly, even if unconsciously, introduce unknown factors into the groups being studied. This could confound the intervention investigated and compromise the research results. In orthodontics, these problems are usually circumvented by generating random numbers that do not follow any pattern by a method called simple randomization. Specific softwares or functions, such as RANDBETWEEN (Excel; Microsoft, Redmond, Wash), are normally used to produce a list of random numbers that leave all expected or unforeseen biases to chance alone. Nevertheless, simple randomization has a significant problem in studies with small sample sizes, because there is a high probability that the groups will be in a state of imbalance3Lachin J.M. Properties of simple randomization in clinical trials.Control Clin Trials. 1988; 9: 312-326Abstract Full Text PDF PubMed Scopus (110) Google Scholar; in most of the current orthodontic literature, sample sizes are usually small. Orthodontic randomized controlled trials published from 1992 through 2012 were shown to have a median sample size of 46 to obtain adequate power, and a median sample size of 60 was necessary to produce results.4Koletsi D. Pandis N. Fleming P.S. Sample size in orthodontic randomized controlled trials: are numbers justified?.Eur J Orthod. 2014; 36: 67-73Crossref PubMed Scopus (20) Google Scholar However, the percentages of these studies with groups in a state of imbalance, or whether they used any methods to restrict randomization, decreasing the chance of producing groups in a state of imbalance, are unknown. This imbalance can produce differences in distribution and variances, decreasing the statistical power.5Chen Z. Zhao Y. Cui Y. Kowalski J. Methodology and application of adaptive and sequential approaches in contemporary clinical trials.J Probab Stat. 2012; 2012: 1-20Crossref Scopus (11) Google Scholar, 6Suresh K. An overview of randomization techniques: an unbiased assessment of outcome in clinical research.J Hum Reprod Sci. 2011; 4: 8-11Crossref PubMed Scopus (624) Google Scholar One way to overcome this problem is to use block randomization, which produces equal numbers of specimens in each group. In such a method, the specimens are distributed into blocks of multiple numbers related to the number of groups under study, containing all possible combinations of allocation but maintaining a 1:1 balance. Thus, in the aforementioned clinical research example, when determining which composite, A or B, will be tested on the right or left side of 24 patients in a split-mouth design, 4 allocations are arranged into 6 blocks (AABB, BBAA, BABA, ABAB, ABBA, and BAAB). These blocks will then undergo simple randomization to determine the sequence in which they will be applied until all patients are bonded. The investigator must be careful, though, when using blocks of the same size, which are easier to manage, because that could lead to a prediction of which treatment will be allocated next. Different block sizes can be used to overcome that issue. Therefore, the aim of this article was to determine when block or simple randomization is necessary, based on the probability of imbalance between groups and on the influence of the statistical power. Four hypothetical research designs were analyzed, varying the numbers of subjects (20, 30, 60, and 90) allocated into 2 groups, using an independent 2-tailed t test, with α = 0.05, by simple randomization using the RANDBETWEEN function of the Excel 2011 software. A total of 100 allocation simulations were made for each research design to describe the differences between specimens in the groups, their frequencies, and the balance ratios. Statistical power was also calculated for each balance ratio using the G*Power software (http://www.gpower.hhu.de/en.html).7Faul F. Erdfelder E. Lang A.G. Buchner A. G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences.Behav Res Methods. 2007; 39: 175-191Crossref PubMed Scopus (30383) Google Scholar The effect sizes were varied from small and medium to large using Cohen's d (d = 0.2, d = 0.5, and d = 0.8, respectively)8Cohen J. A power primer.Psychol Bull. 1992; 112: 155-159Crossref PubMed Scopus (28369) Google Scholar and the aforementioned parameters. The values obtained with block randomization simulations, with a fixed equilibrium ratio of 1:1, were compared with those using simple randomization. When simulating with 20 specimens, 17% of the simulations showed a ratio of 1:1 between the samples. On the other hand, 70% of the simulations showed imbalances from 1.2:1 to 1.9:1, which caused a maximum reduction of 10% of the test power to evaluate large effects, but less than 1% when medium or small effects were evaluated. The remaining 13% of the simulations had larger imbalances (from 2.3:1 to 5.7:1), with gradually decreasing test power. This decrease was marked especially when a large effect was used in the simulation, causing a 17% decrease (Table I). The decrease in test power with imbalanced samples occurred in all effect sizes, with greater reductions for larger effect sizes. In block randomization, all groups have a 1:1 equilibrium rate; thus, the test power will be 40% with large effect sizes, 19% with medium effect sizes, and 7% with small effect sizes.Table IResults from 100 simulations of the allocation of 20 patients into 2 groups by simple randomizationG1G2ImbalanceImbalance ratioFrequencyPower (%)d = 0.8d = 0.5d = 0.210101:117%401971190.81.2:125%391871280.71.5:125%381871370.51.9:120%371771460.42.3:16%341671550.33:14%311571640.34:12%271461730.25.7:11%23126G, Group. Open table in a new tab G, Group. Only 8% of the simple randomization simulations with 30 specimens showed balanced groups (1:1). In the remaining simulations, imbalance ranged from 1.1:1 to 2.3:1, with maximum decreases of 7% in the test power with a large effect size, 3% with a medium effect size, and less than 1% with a small effect size. In all effect sizes, the test power dropped, but the drop was more pronounced when a large effect size was used. As in the previous simulation, block randomization groups had a 1:1 equilibrium, but the test power was higher for these comparisons. Test powers were 56%, 26%, and 8% for large, medium, and small effect sizes, respectively (Table II).Table IIResults from 100 simulations of the allocation of 30 patients into 2 groups by simple randomizationG1G2ImbalanceImbalance ratioFrequencyPower (%)d = 0.8d = 0.5d = 0.215151:18%5626816140.91.1:130%5626817130.81.3:118%5526818120.71.5:119%5425819110.61.7:112%5325820100.52:19%512482190.42.3:14%49238G, Group. Open table in a new tab G, Group. With 60 specimens, only 10% of the simulations had balanced groups, which resulted in an 86% power in a test with a large effect size. In the remaining simulations of large effect size, test power was above 80% even when the imbalance was 2.2:1. When a medium effect size was used, the maximum decrease of the test power was 5% (from 1:1 to 2.2:1); in small effect sizes, the test power decreased by only 1%. In the block randomization, the groups had a 1:1 equilibrium rate, and the test power values were 86%, 48%, and 12% with large, medium, and small effect sizes, respectively (Table III).Table IIIResults from 100 simulations of the allocation of 60 patients into 2 groups by simple randomizationG1G2ImbalanceImbalance ratioFrequencyPower (%)d = 0.8d = 0.5d = 0.230301:110%86481231290.91.1:114%86481233270.81.2:118%86471234260.81.3:123%86471235250.71.4:17%85471236240.71.5:17%85461237230.61.6:13%84461139210.51.9:14%83441141190.52.2:11%814311G, Group. Open table in a new tab G, Group. When the sample size was 90, 4% of the simulations produced balanced groups, whereas 15% of the groups were similar (1:1) “on average.” In the other 77% of the simulations, where imbalances ranged from 1.1:1 to 1.6:1, the test power (96%) did not change for the large effect size, but it fell by 2% and 1% for the medium and small effect sizes, respectively. The largest imbalance (1.8:1) had a frequency of 3% and was responsible for decreasing the test power by 1%, 4%, and 1% in large, medium, and small effect sizes, respectively. In block randomization, the groups had a 1:1 equilibrium, and the test power values were 96%, 65%, and 16% for the large, medium, and small effect sizes, respectively (Table IV).Table IVResults from 100 simulations of the allocation of 90 patients into 2 groups by simple randomizationG1G2ImbalanceImbalance ratioFrequencyPower (%)d = 0.8d = 0.5d = 0.245451:14%96651646441.01:115%96651647430.91.1:116%96651648420.91.1:113%96651549410.81.2:111%96651550400.81.3:111%96641551390.81.3:111%96641552380.71.4:13%96641553370.71.4:17%96641554360.71.5:12%96631555350.61.6:13%96631556340.61.6:11%95621558320.61.8:13%956115G, Group. Open table in a new tab G, Group. The decreases in test power became less as the sample size increased, and the effect sizes were larger. The probability of groups being balanced after simple randomization is small, regardless of the sample size. This does not happen with block randomization, where the groups are balanced, as has already been shown in the literature.9Lachin J.M. Matts J.P. Wei L.J. Randomization in clinical trials: conclusions and recommendations.Control Clin Trials. 1988; 9: 365-374Abstract Full Text PDF PubMed Scopus (196) Google Scholar The problem with group imbalance is that the power of the test decreases, increasing the likelihood of a type II error, which is what our results showed.10Lachin J.M. Statistical properties of randomization in clinical trials.Control Clin Trials. 1988; 9: 289-311Abstract Full Text PDF PubMed Scopus (129) Google Scholar As the imbalance increases, regardless of sample size, the test power decreases. This may lead to the need of having groups balanced to maintain a high power in the statistical tests. Thus, researchers may use a block type of randomization or erroneously try to balance the groups by improperly interfering with the simple randomization process. This improper balancing may occur due to a simple randomization imbalance or even by losing some specimens of 1 group. Let us analyze the following example: in a sample size calculation for an independent 2-tailed t test, with a large effect size (0.8), with the type I error probability set to 5% and type II error set to 80%, the result will be a sample of 26 participants in each group. If 3 patients drop out of 1 group and 3 are removed from the other group (in an attempt to recover a 1:1 balance), 2 groups of 23 patients will be left. This would decrease the power of the test by 4.4%, to 75.6%. On the other hand, if the patient drop is not compensated for and the imbalance is maintained, resulting in 23 patients in 1 group and 26 in the other, the test power would only drop by 2%, to 78%, with a smaller decrease than when trying to balance the groups. The innocent attempt to remove specimens to balance groups will reduce the test power because it is more sensitive to the overall sample size than to the balance between the groups (Fig). Although this procedure is not considered common, it happens more often than expected. A search conducted in the PubMed database on July 17, 2017, using the terms orthodontic [All Fields] AND (Controlled Clinical Trial[ptyp] AND (2016/07/17[PDAT]: 2017/07/17[PDAT]) AND humans[MeSH Terms] AND English [lang]) returned 30 studies. A more in-depth analysis of their methodologies showed that 15 articles applied the simple randomization method, and surprisingly, 10 of them obtained identical groups. As the sample size increases, larger imbalance ratios are less frequent, and as mentioned, these large ratios will significantly affect the test power.10Lachin J.M. Statistical properties of randomization in clinical trials.Control Clin Trials. 1988; 9: 289-311Abstract Full Text PDF PubMed Scopus (129) Google Scholar In sample sizes larger than 60, the highest observed imbalance ratio was 2.2:1, reducing the test power by 5%, regardless of the effect size. If 80% test power is considered to be adequate for an orthodontic research, simple randomization can be used safely in samples larger than 60 if the effect size is large, because the test power will not be lower than 80%. In small samples (n = 30) or experiments that intend to demonstrate small differences between groups, block randomization is suggested as a safe way to maintain balance between groups and to preserve the test power. If a simple randomization is used instead, not only would it produce a small test power (based on the experimental design itself), but it could also decrease the test power even further due to an almost certain imbalance between groups. Even though a sample of 46 has been shown to be adequate to obtain enough power to produce results in the orthodontic literature,4Koletsi D. Pandis N. Fleming P.S. Sample size in orthodontic randomized controlled trials: are numbers justified?.Eur J Orthod. 2014; 36: 67-73Crossref PubMed Scopus (20) Google Scholar it is always wise to restrict randomization, decreasing the chance of producing imbalanced groups, because this sample size is considered too small to produce balanced results. In addition to block randomization, there are other methods to overcome some of the problems of simple random allocation: eg, stratification and minimization. They can be used when known factors could confound the results and need to be controlled, especially in small sample sizes. A good example may be the influence of sex on compliance. If women comply more in using their retainers, a study comparing relapse between 2 appliances may need to have a balance between the sexes within each group; otherwise, the outcome can be confounded.11Pratt M.C. Kluemper G.T. Lindstrom A.F. Patient compliance with orthodontic retainers in the postretention phase.Am J Orthod Dentofacial Orthop. 2011; 140: 196-201Abstract Full Text Full Text PDF PubMed Scopus (56) Google Scholar The first method, stratification, involves dividing the groups studied into subgroups, called strata, where known factors that could confound the results can be controlled. Then the features of block randomization are used in each stratum to balance them; therefore, it is a combination of both methods. It is good for small sample sizes when imbalances of prognostic predictors are more likely to occur.1Pandis N. Polychronopoulou A. Eliades T. Randomization in clinical trials in orthodontics: its significance in research design and methods to achieve it.Eur J Orthod. 2011; 33: 684-690Crossref PubMed Scopus (15) Google Scholar However, when there are too many prognostic factors that could confound the results, the division of the sample into many strata will result in a large number of subgroups with fewer specimens, which could imbalance the treatment allocation. The alternative method for the problem of having several prognostic factors in small samples is minimization.12Altman D.G. Bland J.M. Treatment allocation by minimisation.BMJ. 2005; 330: 843Crossref PubMed Scopus (264) Google Scholar Minimization works by randomizing the sample into strata dynamically during the experiment. In contrast to stratified randomization, where the specimens are allocated before the experiment, in minimization, the treatment allocated to the next participant enrolled in the trial depends on the characteristics of participants already enrolled.12Altman D.G. Bland J.M. Treatment allocation by minimisation.BMJ. 2005; 330: 843Crossref PubMed Scopus (264) Google Scholar Besides being slightly more complicated to use, it is best performed with the help of software; the main issue with minimization is that it may be possible to predict the next allocation by looking at the past assignment. Another issue with small sample sizes that should be discussed is the situation where the covariates may remain imbalanced even when the groups are numerically balanced. Normally, this will not occur in large samples where randomization was performed correctly, but it can affect the results of small samples, particularly with fewer than 50 specimens.13Grizzle J.E. A note on stratifying versus complete random assignment in clinical trials.Control Clin Trials. 1982; 3: 365-368Abstract Full Text PDF PubMed Scopus (49) Google Scholar When baseline covariates are imbalanced, a multiple linear regression, where the baseline variable of the dependent variable is also included in the model (analysis of covariance), should be used.14Kirkwood B.R. Sterne J.A. Malden M.A. Essential medical statistics.2nd ed. Blackwell, Oxford, United Kigdom2003Google Scholar This will account for possible bias because of the baseline values of covariates between groups.15Pandis N. Analysis of covariance.Am J Orthod Dentofacial Orthop. 2016; 150: 200-201Abstract Full Text Full Text PDF PubMed Scopus (3) Google Scholar However, the interpretation of this postadjustment approach is often difficult because the imbalance of covariates frequently leads to unanticipated interaction effects, such as unequal slopes among subgroups of covariates.16Kang M Ragan BG Park JH Issues in outcomes research: an overview of randomization techniques for clinical trials.J Athl Train. 2008; 43: 215-221Crossref PubMed Scopus (225) Google Scholar For independent parallel group studies, the following conclusions were made.1.Simple randomization almost always produces imbalances between groups, which is a problem in studies with fewer than 60 specimens. Therefore, simple randomization can be used for studies with samples larger than 60.2.Block randomization or another method that restricts randomization and manages imbalance, such as stratified randomization or minimization, should be used in studies with less than 60 samples.3.The sample size influences the test power more than the imbalance rate." @default.
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• W2893331902 title "Simple randomization may lead to unequal group sizes. Is that a problem?" @default.
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