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**Unequal Randomisation**

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Background Most RCTs randomised participants into equally sized groups. Equal randomisation 1:1 ratio is statistically the most EFFICIENT method. For any given TOTAL sample size the most power to detect a difference occurs with equal group sizes. Statisticians usually recommend 1:1 as this satisfies their desire for POWER.

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**Allocation Ratios and Power**

Although a ratio of 1:1 does produce the most power ratios of 2:1 or 3:2 do not substantially reduce power. 2:1 for example moves a study’s power from 80% to 75%. Therefore, quite LARGE imbalances in sample sizes have little effect on power.

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**Why unequal allocation?**

Sometimes it is better to put more participants into one group than another. Reasons are as follows: Practical Learning curve Cost Statistical

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**Practical/Administrative**

For some treatments such as group therapy sessions there might be a minimum number of participants needed to make the group sessions viable. For example, Hundley et al allocated more women in a mid-wives trial to the new intervention in order to keep the ward full.

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Learning Curve A new technique, e.g. surgery, may require some learning. More participants allocated to the new treatment can allow a more precise estimate of any learning effects. For example Garry et al, (BMJ 2004,328,129) used unequal allocation in favour of a laparoscopic surgery so surgons had more people to practice on. Interestingly, they did not look at the effects of learning in their analysis

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Treatment experience As well as learning curve we might be interested in the side-effect profile of a new treatment. For standard therapy side-effects will be well established, but for the new treatment there are more unknowns. Therefore, we might have more people so that we have more power to pick up any unknown side-effects.

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Ethics Some people advocate unequal allocation to minimise exposure to either the control treatment or new, hazardous treatment. This suggests, to me, that there is a strong belief in one treatment, which would question the necessity of the trial. I would not use unbalanced allocation for ethical reasons.

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**Cost An important reason, commonly overlooked, is due to cost.**

One treatment may be much more expensive than the alternative and the trial can be made much cheaper if more people are allocated to the cheaper treatment. Indeed this could make a trial more powerful.

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**Cost and power ALL trials have a limited budget.**

We want to get MOST power from this money. The idea behind putting more people onto the cheaper treatment is that it the savings released can be used to put MORE people into the trial.

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Cost savings Trial cost efficiency may be improved by allocating more participants to the less expensive treatment and more to the cheaper treatment. Statistical power can be maintained by increasing the sample size. OR power can even be increased by recruiting MORE participants.

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**Optimum Randomisation Ratio**

The most efficient allocation ratio is calculated by the square root of the cost ratio of two treatments. If treatment A costs 4 X as much as treatment B then the optimum allocation ratio is 2 or 9 x as much then the ratio is 3.

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Example MRC Taxol trial for ovarian cancer. The new drug, taxol, extremely expensive about £10,000 per patient. In order to reduce costs the trialists allocated twice as many women to the control group (2:1) than in the treatment arm.

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**Cost Savings of Unequal allocation**

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Cost Savings By using an unequal allocation ratio the trial saved about £1 million. Many studies do not have as dramatic cost difference but important savings can still be made.

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Hip Protector Trial In the hip protector trial a key additional cost was the cost of the hip protectors (about £80 per person for 3 pairs including postage). The cost of controls, after recruitment costs, was mailing out follow-up q’naires.

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Estimating the ratio Initially we thought we would recruit 10% of women we approached. The cost of the mailout was about £1 a person. To recruit 100 women would cost £1,000 (£10 per person). To follow-up the women would be another £5 in postage after randomisation (£15 in total). The intervention group would cost an additiona £80 (£95 in total) 95/15 = 6.3, square root = 2.51.

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**Ratio We, therefore adopted an allocation ratio of 2:1.**

BUT recruitment costs went up to £20 per woman therefore the ratio of costs were 4.2, square root is about 2. Therefore, our optimum ratio still remained about 2.

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Cost savings We estimated to have saved 10% of our research budget by using unequal allocation, which allowed us to mail out to more participants (to compensate for the unexpected shortfall in recruitment) and follow up participants for longer.

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**Recruitment with fixed budget**

Increase allocation to control using saved money to increase mail out. Disadvantages will increase workload for local trial co-ordinators in terms of data entry and data management.

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**Allocation Ratio - lessons**

A fixed allocation ratio is unlikely to be correct through the lifetime of a trial. Should plan for an adaptive allocation and change ratio during recruitment if cost ratio changes. Budget planning should probably start with an ‘inefficiently’ high allocation (e.g. 3:2 or 1:1) ratio and adapt downwards (e.g. 2:1) as trial proceeds if necessary.

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**Allocation ratios: a review**

What is done in actual practice? To find out we are undertaking a review of trials using unequal allocation ratios to see why. We searched electronic databases using unequal allocation, unbalanced randomisation etc, plus personal knowledge. We couldn’t find many trials. This confirms that it is not used widely (unfortunately).

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**Reasons for unequal allocation**

Cost 4 (7%) Drop-outs 5 (9%) Pt acceptability Ethics 3 (5%) Experience 8 (14%) Other Not stated 30 (52%) Dumville et al, 2005.

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‘Other’ reasons Expected variability differs in trial arms. Can increase power if more patients are allocated to group with larger SD as central limit theorum helps improve normality. Comparison of two treatment arms vs a control treatment (larger numbers in treatment arms to increase power of treatment vs treatment comparison).

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**Comparison of treatments**

We might have 3 arms: control; dose 1; dose 2. To compared dose 1 and 2 we would expect a muted treatment response, and therefore, we would need larger sample sizes to observe a treatment effect.

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A Digression Unequal allocation, if undertaken randomly, STILL results in equivalent groups in terms of equal distribution of confounders. It does NOT lead to BIASED allocation.

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**Analysis of unequal allocation**

This is exactly the same as a trial that uses equal allocation EXCEPT if the allocation ratio changes part of the way through the study. If the allocation does change this needs to be taken into account in the analysis.

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Summary Most trials have unequal costs and probably could benefit from unequal randomisation. Most trials use even allocation. WARNING - many grant referees do NOT understand unequal allocation and some see it as UNSCIENTIFIC.

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