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Doi:10.1016/j.cct.2005.08.003

Contemporary Clinical Trials 27 (2006) 1 – 12 The use of unequal randomisation ratios in clinical trials: A review J.C. Dumville a,*, S. Hahn b, J.N.V. Miles a, D.J. Torgerson a a Area 4, York Trials Unit, Department of Health Sciences, University of York, York, YO10 5DD, United Kingdom b Medical Research Collaborating Center, Seoul National University College of Medicine, Seoul National University Hospital, Seoul 110-744, Republic of Korea Received 28 September 2004; accepted 11 August 2005 Objective: To examine reasons given for the use of unequal randomisation in randomised controlled trials (RCTs).
Main Measures: Setting of the trial; intervention being tested; randomisation ratio; sample size calculation; reason given forrandomisation.
Methods: Review of trials using unequal randomisation.
Databases and sources: Cochrane library, Medline, Pub Med and Science Citation Index.
Results: A total of 65 trials were identified; 56 were two-armed trials and nine trials had more than two arms. Of the two-arm trials,50 trials recruited patients in favour of the experimental group. Various reasons for the use of unequal randomisation were given.
Six studies stated that they used unequal randomisation to reduce the cost of the trial, with one screening trial limited by theavailability of the intervention. Other reasons for using unequal allocation were: avoiding loss of power from drop-out or cross-over, ethics and the gaining of additional information on the treatment. Thirty seven trials papers (57%) did not state why they hadused unequal randomisation and only 14 trials (22%) appeared to have taken the unequal randomisation into account in theirsample size calculation.
Conclusion: Although unequal randomisation offers a number of advantages to trials the method is rarely used and is especiallyunder-utilised to reduce trial costs. Unequal randomisation should be considered more in trial design especially where there arelarge differences between treatment costs.
D 2005 Elsevier Inc. All rights reserved.
Keywords: Unequal randomisation; Randomised controlled trials Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Learning curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Search strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Corresponding author. Tel.: +441904321325; fax: +441904 321383.
E-mail address: [email protected] (J.C. Dumville).
1551-7144/$ - see front matter D 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.cct.2005.08.003 J.C. Dumville et al. / Contemporary Clinical Trials 27 (2006) 1–12 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reasons for unequal randomisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sample size calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maximise available participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differential loss to follow-up and cross over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Treatment experience. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Unequal allocation in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reasons for limited use of unequal allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 In randomised controlled trials (RCTs) the allocation of participants to experimental and control groups is usually balanced leading to approximately equal group sizes. However, whilst balanced group sizes will maximise a studyTsstatistical power, the use of unequal randomisation ratios will only significantly reduce the power of a study if theratio is 3 : 1 or more There are a number of situations, discussed below, where unequal group sizes in a trial maybe beneficial. Indeed, it has been argued that the use of unequal randomisation could and should be substantiallyincreased In most RCTs there is probably an unequal distribution of either research or treatment costs between the trial arms.
When this is the case, economic efficiency can potentially be improved using unequal randomisation. For a fixed trialbudget, using unequal randomisation to favour the least expensive trial arm allows more participants to be recruitedthus increasing the power of the trial Where there is a fixed total sample size available for recruitment, unequalrandomisation can confer large financial savings to a trial with limited impact on statistical power Furthermore,when there is a fixed quantity of treatments available for one trial arm, but more than twice the number of trialparticipants are available, trial efficiency can be improved using unequal allocation. Allocating more participants tothe treatment with no restrictions on availability will increase the statistical power of the trial by utilising all theavailable participants.
There are situations where unequal randomisation that favours the experimental group is useful. In evaluations of a new technology that has a dlearningT element, for example a new surgical procedure, more participants might beallocated to the intervention group to reduce the impact of the dlearning curveT on the results of the trial Allocatingmore participants to the experimental treatment allows the clinician to climb the learning curve more rapidly andallows for an analysis that compares both treatments when at the top of this curve.
Some have advocated unequal allocation in order to expose fewer participants to the hazards of a treatment in a trial This view pre-supposes that there is a high degree of certainty regarding the risks and benefits accruing to thetreatment under evaluation. Yet, as many RCTs are conducted when there is equipoise, this view cannot generally beused as justification for unequal allocation in trials. However, there may be cases, for example in clinical drug trials,where the beneficial nature of a treatment, known to have unpleasant side-effects, is still being investigated. Hereunequal allocation could, potentially, be justified on ethical grounds.
J.C. Dumville et al. / Contemporary Clinical Trials 27 (2006) 1–12 Unequal allocation can be useful in studies that compare a group with an individual treatment (e.g., group versus individual cognitive behavioural therapy) where it helps to obtain enough participants for the group therapy. Usingunequal allocation to increase the number of groups will increase the study’s power for a fixed total sample size.
Additionally, a larger number of participants may be allocated to one group when a higher drop-out rate is anticipated, as this will enhance statistical power for a dper protocolT analysis.
Whilst the literature cites a number of reasons why unequal allocation may be beneficial in RCTs, to date there has been no review undertaken that assesses why this method has been used in practice. We undertook a review of trialsthat employed unequal randomisation to investigate why it had been used and to discuss these applications further.
The Cochrane Library, Medline, Pub Med and the Science Citation Index were searched for all paper abstracts that contained words and phrases related to unequal randomisation. These were: bunequal or unbalanced, randomis(z)ation, allocation or ratioQ.
Papers from this literature search were also supplemented by the authorsT detailed personal knowledge of trials that had unequal group sizes and from the bibliographies of papers discussing unequal randomisation. Trials that used acluster design were excluded from this review.
Double data extraction was carried out by JD and SH. Data was extracted on: the setting of the trial (primary care, secondary care or university); the intervention being tested; the randomisation ratio; the sample size calculation usedand the reason given for unequal randomisation. A sample size calculation was recorded as being outlined when somedetails of the calculation were provided. We recorded that unequal randomisation had been taken into account in thesample size calculation if the authors explicitly stated they had accounted for unequal randomisation or we were able torepeat and confirm the sample size calculation for the proposed ratio. Conversely, unequal randomisation was recordedas having not been taken into account if the authors’ suggested sample size calculation was for equal group sizes; if wecould not follow the sample size calculation from the details supplied we recorded the calculation as being unclear.
If the reason for unequal randomisation was not given in the paper, where possible, authors were e-mailed to gain further information about why they used unequal randomisation.
We identified 65 trials that used unequal randomisation . One further paper was identified but was excluded from the study as, although it used unequal randomisation (2 : 1 : 2 : 2 : 1 : 1), of the six arms, only three arms were compared inthe study and these were balanced. Most included trials (~80%) were identified from personal knowledge andbibliographies rather than from databases. Details of the 65 included trials, in terms of settings, interventions, allocationratios, sample size calculation, and reasons for use of unequal allocation, are shown in Appendix 1. summarisesthe trial characteristics. Many trials papers (n = 37; 57%) did not state why they used unequal randomisation in theirdesign; after e-mailing authors 15 replies clarifying the reason for use of unequal randomisation were received.
Of the trials, 56 (86%) were two armed and nine trials had more than two arms. Unequal randomisation favouring the experimental group was used most frequently with the allocation favouring the control group(s) used only seventimes ).
The reasons for unequal randomisation given in each study are detailed in the Appendix. Of the seven studies where allocation favoured the control group, four trials used unequal allocation to reduce financial costs as the J.C. Dumville et al. / Contemporary Clinical Trials 27 (2006) 1–12 a In 2 of these trials unequal randomisation was performed in only some of the centres.
experimental treatment or intervention was more expensive than the control . One further trial was alarge screening programme, where the authors state there were a limited number of screenings available: thusthey wanted to maximise the power of the study for this finite number of examinations A second screeningtrial did not give a clear reason for the use of unequal randomisation that favoured the control group but it islikely to be for similar reasons as the first . The remaining trial that allocated in favour of the control groupdid so for reasons related to statistical considerations, however, this reasoning did not make the use of unequalrandomisation clear Finally, one of the two trials that compared analogous treatments also used unequalrandomisation to reduce cost by minimising the number of participants in the most expensive treatment group The remaining studies mostly had more participants in their treatment group(s) than their control group. Five trials used unequal allocation as they were concerned about drop-out or transfer from the treatment group Six trials used unequal randomisation to increase patient acceptability of the trial and therefore recruitment ratesand three trials stated ethical reasons for maximising participants exposure to the treatment. Individual trials also gave other specific reasons for using unequal randomisation to maximise allocationto the experimental group. These were: the late starting of one arm of the trial limited variability of the controlgroup more participants required in the treatment group for the next phase of the study ; to give increasedpower for a secondary analysis increased availability of the intervention service compared to the controlservice and to ensure maximum use of the available counselling intervention Only one trial clearly attributed the used of an unbalanced allocation to overcome a learning curve and to gain more experience of a treatment However, gaining more experience of a treatment, including increasedsafety and toxicity data, were given as reasons for the use of unequal randomisation in thirteen further trials J.C. Dumville et al. / Contemporary Clinical Trials 27 (2006) 1–12 Of the 65 trials, 40 (62%) were recorded as having outlined a sample size calculation. Fourteen of these 40 studies appeared to have taken the unequal randomisation into account. The remaining 26 studies that outlined a sample sizecalculation either did not take unequal randomisation into account (n = 5) or did not supply enough detail to establishwhether unequal randomisation had been accounted for (n = 21).
We found few RCTs that employed unequal randomisation. Because unequal randomisation was rarely described in paper abstracts, most papers for this review were identified from extensive combined personalknowledge of clinical trials. This non-systematic approach could be viewed as a possible limitation of thisstudy. Of those trials identified as having unbalanced groups, many gave no apparent reason for using thismethod. The use of unequal randomisation is an important methodological feature of a trial and studies shouldgive a reason for its use. Trials that did discuss their use of unequal randomisation describe doing so for avariety of reasons.
Although the use of unequal allocation can lead to considerable cost savings, economic efficiency was one of the least frequently cited for using unequal allocation. Those studies that did use unequal randomisation to confer savingsappeared to do so successfully. For example, in a screening trial where screening cost o70 per person, 195,000participants were randomised with unbalanced allocation leading to savings of more than o2 million with very littleloss in power Similarly, unequal allocation in a trial of hip protectors reduced the overall cost of the trial by10% (Torgerson, unpublished observation).
Three studies, in which the availability of one treatment was not restricted used unequal allocation to improve statistical power. For example, had Turnbull et al. used equal allocation rather than unequalallocation they would have recruited over 100 less patients into their intervention group and thus into theirtrial.
Unequal randomisation can improve power in three armed trials when a dose comparison between two drug arms is being carried out as the differences between drug dosages would be expected to be more muted compared with theplacebo comparison. In this review no trial gave this as a reason for using unequal randomisation although one three-armed trial used this design 4.3. Differential loss to follow-up and cross over Unequal allocation was used because of anticipated differences in drop-out or treatment cross-over rates between groups in five studies . The advantage of unequal randomisation in this situation is that it willpreserve statistical power for an don treatmentT or dper protocolT analysis. However, it is important to note that whenlarge drop-out or cross-over rates are anticipated, unequal randomisation will not reduce the risk of bias throughdrop-out nor will it prevent ddilutionT effects of participants crossing over into the other treatment arm. Intention totreat analysis, that is analysing all participants in their originally allocated groups, must still be carried out as thisgives a conservative but unbiased estimate of treatment effects. The five studies identified here did undertake anintention to treat analysis.
Thirteen studies exposed increased numbers of participants to the experimental treatment or procedures to gain more experience in using the treatment. This included the collection of safety and toxicity data including adverse J.C. Dumville et al. / Contemporary Clinical Trials 27 (2006) 1–12 events. Additionally, further studies that did not state a reason for the use of unequal randomisation were phase I–III-type trials that tested drugs, radiological or surgical treatments and often reported safety and adverse event data. Wepostulate that these studies used unequal randomisation to increase the amount of information on the new treatment.
This application of unequal randomisation has been only briefly discussed in the literature and has been acknowl-edged as being potentially controversial Only one study used unequal allocation specifically because of learningcurve effects . However, this study did not appear to examine whether there was any interaction betweentreatment effects and patient recruitment. If learning curve effects are a reason for unequal allocation, it would belogical to look for any effect of learning and, if observed, present an analysis of participants recruited don the topT ofthe learning curve.
Practically, carrying out unequal allocation adds little complexity when compared to using balanced allocation.
The process of randomisation must simply ensure that, in the case of a 2 : 1 ratio, double the number of peoplerandomly enter one group compared to the other.
It is important when using unequal allocation to ensure that a correct sample size calculation is performed, and ideally a statistician should be consulted. In a majority of the studies we examined it did not appear that unequalrandomisation had been taken into account for the sample size calculation. This is particularly serious wherestudies have used randomisation ratios of 3 : 1 or more where the power of the study will have been significantlyreduced.
Input from a statistician is also useful in the analysis of data from trials using unequal randomisation. The statistical tests commonly used to analyse data from trials, such as the t-test, ANOVA, ANCOVA and regression analysis, makean assumption about the homogeneity of the variance of the residuals across the levels of the independent variables. Ifthe variances are not equal, but groups are equal this does not seriously affect the type 1 error rate. However, wherethe groups are not equally allocated this does not always hold. Thus the decision to use unequal randomisation musttake into account the potential for unequal variance.
4.6. Reasons for limited use of unequal allocation There may be several reasons why trials rarely employ unbalanced allocation. Some might, incorrectly, consider unequal groups sizes as somehow being dunscientificT. Indeed, when advocating unequal randomisation, one of theauthors (DT) has had such feedback on two different occasions from grant-giving bodies. Also, the need to maximisethe power of the study might be the over-riding concern among some trial statisticians and for a given total samplesize, balanced allocation, usually, gives the most statistical power. However, for a fixed sample size, unequalrandomisation of 2 : 1 can be used without a serious loss of power . One reason why unequal allocation is notused in factorial trials may be the view that this makes the analysis more complicated. When computations of trialswere done dby handT completely balanced factorial designs were easier to calculate; however, with the use of moderncomputers this justification no longer holds.
Resistance to unequal randomisation may stem from concern that its use may require the recruitment of an increased number of participants, since recruitment is often a limiting factor in trials. Indeed, where there are a smallnumber patients that can be recruited unequal randomisation may not be advantageous. However, as previously noted,where there is a fixed budget, savings introduced by use of unequal randomisation may allow more sites to berecruited to the trial — thus allowing increased recruitment, which can increase the power of a study Furthermore,for a fixed sample size potential cost savings from using unequal allocation should be balanced against the loss ofpower that unequal randomisation will cause.
Unequal randomisation is not commonly undertaken. Although by no means suitable for every trial, there are a number of situations where unequal randomisation can be useful, especially by conferring financialsavings on trials — yet unequal randomisation is rarely used for this reason and could be implemented moreoften.
Effect of direct payment on questionnaire Drop out (due to transfer) expected intreatment group Drop out (due to transfer) expected intreatment group Prevention of otitis media with inactivated Lamivudine in children with chronic hepatitis Effectiveness of influenza vaccine in healthy, Cognitive–behavioural therapy in the treatment Comparison of zolmitriptan and sumatriptan Discontinuing drug therapy in HIV-infected Peginterferon-a2a and ribavirin combination exposure of difficult to treat patientsto the experimental treatment — not Adefovir dipivoxil for the treatment of chronic Prophylactic implantation of a defibrillator Effects of antenatal day care for medical Evidence of real world effectiveness of a Laparoscopic versus abdominal hysterectomy Dermatan sulphate for the prevention of deep Pirenzepine ophthalmic gel in myopic children therapy with losartan and hydrochlorothiazide Insulin vs. human insulin in type 1 diabetes Effects of bicarbonate/lactate-based dialysis Endoscopic versus open carpel tunnel release: Uterine artery embolization vs. abdominal Pantoprazole in the prevention of peptic ulcers Large-scale prostrate cancer screening trials Zidovudine and Zalcitabine in HIV patients Neutron radiation therapy for head and neck Effects of Alendronate and risedronate on bone Effect of sibutramine in weight maintenance Nicardipine therapy in ambulatory elderly Sustained-release formulation of ibuprofen Trandolapril a new angiotensin converting Hypocaloric high protein diet spares lean body Radiotherapy in the treatment of lung cancer Cost effectiveness of transplantation types for Cost-effectiveness analysis of improved blood pressure control in hypertensive patients a Primary care (PC), Secondary care (SC), University (U).
b Reason for unequal randomisation not stated in paper but given on e-mail.
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