Unequal allocation ratios

Unequal Randomisation
• 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.
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.
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
• 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.
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
Treatment experience
• As well as learning curve we might be
interested in the side-effect profile of a
new treatment. For standard therapy sideeffects 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.
• 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
• 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
• Indeed this could make a trial more
Cost and power
• ALL trials have a limited budget.
• We want to get MOST power from this
• 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.
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.
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.
• 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
Cost Savings of Unequal
S a m p le E q u a l a llo c a tio n
C o st
£ 1 0 ,0 0 0
£ 1 ,0 0 0
£ 4 ,0 7 0 ,0 0 0
£ 4 0 7 ,0 0 0
T o ta l
C o st
T o ta l
4 ,4 7 7 ,0 0 0
£ 2 ,6 7 0 ,0 0 0
£ 8 5 0 ,0 0 0
£ 3 ,5 2 0 ,0 0 0
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.
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
• The cost of controls, after recruitment
costs, was mailing out follow-up q’naires.
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 followup the women would be another £5 in postage
after randomisation (£15 in total). The
intervention group would cost an additional £80
(£95 in total) 95/15 = 6.3, square root = 2.51.
• 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.
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.
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.
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
• 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.
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).
Reasons for unequal allocation
Pt acceptability
Not stated
N = 58
4 (7%)
5 (9%)
4 (7%)
3 (5%)
8 (14%)
4 (7%)
30 (52%)
Dumville et al, Contemporary Clin Trials 2005;27:1-12.
‘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 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
A Digression
• Unequal allocation, if undertaken
randomly, STILL results in equivalent
groups in terms of equal distribution of
• It does NOT lead to BIASED allocation.
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.
Unequal allocation and power
• Equal allocation is nearly always most powerful
given a FIXED sample size. If the budget is
fixed this is no longer true and unequal
allocation is more powerful.
• For example, in a criminal justice trial 150 crime
hot spots were identified, but there were only
enough police resources to patrol 55 – what did
the authors do?
• They randomised 110, 55 in each group – what
a waste they should have randomised 95 to the
control and 55 to the intervention.
• Most trials have unequal costs and
probably could benefit from unequal
• Most trials use even allocation.
• WARNING - many grant referees do NOT
understand unequal allocation and some

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