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COMM E N T A R Y Open Access
Sample size calculations for cluster randomised
controlled trials with a fixed number of clusters
Karla Hemming
*
, Alan J Girling, Alice J Sitch, Jennifer Marsh and Richard J Lilford
Abstract
Background: Cluster randomised controlled trials (CRCTs) are frequently used in health service evaluation.
Assuming an average cluster size, required sample sizes are readily computed for both binary and continuous
outcomes, by estimating a design effect or inflation factor. However, where the number of clusters are fixed in
advance, but where it is possible to increase the number of individuals within each cluster, as is frequently the
case in health service evaluation, sample size formulae have been less well studied.
Methods: We systematically outline sample size formulae (including required number of randomisation units,
detectable difference and power) for CRCTs with a fixed number of clusters, to provide a concise summary for
both binary and continuous outcomes. Extensions to the case of unequal cluster sizes are provided.
Results: For trials with a fixed number of equal sized clusters (k), the trial will be feasible provided the number of
clusters is greater than the product of the number of individuals required under individual randomisation (n
I
) and
the estimated intra-cluster correlation (r). So, a simple rule is that the number of clusters (k) will be sufficient
provided:
k>nI×
ρ.
Where this is not the case, investigators can determine the maximum available power to detect the pre-specified
difference, or the minimum detectable difference under the pre-specified value for power.
Conclusions: Designing a CRCT with a fixed number of clusters might mean that the study will not be feasible,
leading to the notion of a minimum detectable difference (or a maximum achievable power), irrespective of how
many individuals are included within each cluster.
Introduction
Cluster randomised controlled trials (CRCTs), in which
clusters of individuals are randomised to intervention
groups, are frequently used in the evaluation of service
delivery interventions, primarily to avoid contamination
but also for logistic and economic reasons [1-3]. Whilst
a well conducted individually Randomised Controlled
Trial (RCT) is the gold standard for assessing the effec-
tiveness of pharmacological treatments, the evaluation
of many health care service delivery interventions is dif-
ficult or impossible without recourse to cluster trials.
Standard sample size formulae for CRCTs require the
investigator to pre-specify an average cluster size, to
determine the number of clusters required. In so doing,
these sample size formulae implicitly assume that the
number of clusters can be increased as required [1,3-5].
However, when evaluating health care service delivery
interventions the number of clusters might be limited to
a fixed number even though the sample size within each
cluster can be increased. In a real example, evaluating
lay pregnancy support workers, clusters consisted of
groups of pregnant women under the care of different
midwifery teams [6,7]. The available number of clusters
was restricted to the midwifery teams within a particular
geographical region. Yet within each midwifery team it
was possible to recruit any reasonable number of indivi-
duals by extending the recruitment period. In another
real example, a CRCT to evaluate the effectiveness of a
combined polypill (statin, aspirin and blood pressure
lowering drugs) in Iran was limited to a fixed number of
* Correspondence: k.hemming@bham.ac.uk
Department of Public Health, Epidemiology and Biostatistics, University of
Birmingham, UK
Hemming et al.BMC Medical Research Methodology 2011, 11:102
http://www.biomedcentral.com/1471-2288/11/102
© 2011 Hemming et al; licensee BioMed Centra l Ltd. This is an Open Access a rticle distributed under the terms of the Creative
Commons Attri bution License (http://creativecommons.org/licenses/by/2.0), which permits u nrestricted use, distri bution, and
reproductio n in any medium, provided the original work is pro perly cited.
villages participating in an existing cohort study [8].
Other such examples of designs in which a limited num-
ber of clusters were available include trials of commu-
nity based diabetes educational programs [9] and
general practice based interventions to reduce primary
care prescribing errors [10], both of which were limited
to the number of general practices which agreed to
participate.
Theexistingliteratureonsamplesizeformulaefor
CRCTs focuses largely on the case where there is no
limit on the number of available clusters [3-5,11,12].
Whilst it is well known that the statistical power that can
be achieved by additional recruitment within clusters is
limited, and that this depends on the intra-cluster corre-
lation [11-13], little attention has been paid to the limita-
tions imposed when the number of clusters is fixed in
advance. This paper aims to fill this gap by exploring the
range of effect sizes, and differences between proportions,
that can be detected when the number of clusters is fixed.
We describe a simple check to determine whether it is
feasible to detect a specified effect size (or difference
between proportions) when the number of clusters are
fixed in advance; and for those cases in which it is infeasi-
ble, we determine the minimum detectable difference
possible under the required power and the maximum
achievable power to detect the required difference. We
illustrate these ideas by considering the design of a
CRCT to detect an increase in breastfeeding rates where
the number of clusters are fixed.
For completeness we outline formulae for simpler
designs for which the sample size formulae are relatively
well known, or easily derived, as an important prelude.
In so doing, the simple relationships between the formu-
lae are clear and this allows progressive development to
the less simple situation (that of binary detectable differ-
ence or power). It is hoped that by developing the for-
mulae in this way the material will be accessible to
applied statisticians and more mathematically minded
health care researchers. We also provide a set of guide-
lines useful for investigators when designing trials of
this nature.
Background
Generally, suppose a trial is to be designed to test the
null hypothesis H
0
:μ
0
=μ
1
where μ
0
and μ
1
represent
the means of some variable in the control and interven-
tion arms respectively; and where it is assumed that var
(μ
0
)=var(μ
1
)=s
2
. Suppose further that there are an
equal number of individuals to be randomised to both
arms, letting ndenote the number of individuals per
arm and letting ddenote the difference to be detected
such that d=μ
0
-μ
1
,1-bdenotes the power and a
the significance level. We limit our consideration to
trials with two equal sized parallel arms, with common
standard deviation, two-sided test, and assume normality
of outcomes and approximate the variance of the differ-
ence of two proportions. The sub-script, I (for Indivi-
dual randomisation), is used throughout to highlight any
quantities which are specific to individual randomisa-
tion; and likewise the sub-script, C (for Cluster rando-
misation), is used throughout to highlight any quantities
which are specific to cluster randomisation. No sub-
scripts are used to distinguish cluster from individual
randomisation for variables which are pre-specified by
the user.
RCT: sample size formulae under individual
randomisation
Following standard formulae, for a trial using individual
randomisation[14], for fixed power (1 - b)andfixed
sample size (n) per arm, the detectable difference, d
I
,
with variance var(d
I
)=2s
2
/n
I
is:
dI=
2σ2
n(zα/2 +zβ
)
(1)
where z
a/2
denotes the upper 100a/2 standard normal
centile.
For a trial with nindividuals per arm, the power to
detect a pre-specified difference of d,is1-b
I
, such that:
z
βI=
n
2
d
σ−zα/
2
or equivalently:
βI=
n
2
d
σ−zα/2
(2)
where Fis the cumulative standardised Normal
distribution.
And,finallytherequiredsamplesizeperarmfora
trial at pre-specified power 1 - bto detect a pre-speci-
fied difference of d,isn
I
, where:
n
I=2σ2
(zα/2 +zβ)2
d2
(3)
Using Normal approximations, the above formulae can
be used for binary outcomes, by approximating the var-
iance (s
2
) of the proportions π
1
and π
2
, by:
σ2≈1/2[π1
(
1−π1
)
+π2
(
1−π2
)]
(4)
for testing the two sided hypothesis H
0
:π
1
=π
2
.
CRCTs: standard sample size formulae under cluster
randomisation
Suppose, instead of randomising over individuals, the
trial will randomise the intervention over kclusters per
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arm each of size m, to provide a total of n
C
=mk indivi-
duals per arm. Then, by standard results [1], the var-
ianceofthedifferencetobedetectedd
C
is inflated by
the Variance Inflation Factor (VIF):
V
IF =[1+
(
m−1
)
ρ](5)
where ris the Intra-Cluster Correlation (ICC) coeffi-
cient, which represents how strongly individuals within
clusters are related to each other. Where the cluster
sizes are unequal this variance inflation factor can be
approximated by:
V
IF =[1+
((
cv2+1
)
¯
m−1
)
ρ
]
(6)
where cv represents the coefficient of variation of the
cluster sizes and ¯
m
is the average cluster size [15]. Thus,
the variance of d
C
(for fixed cluster sizes) becomes:
var(dC)= 2σ
2
[1+(m−1)ρ]
mk
(7)
and this is simply extended for varying cluster sizes
using equation 6. To determine the required sample size
for a CRCT with a pre-specified power 1 - b,todetect
the pre-specified difference d, and where there are m
individuals within each cluster, then the required sample
size n
C
=km per arm, follows straightforwardly from
equations 3 and 5 and is:
nC=2σ2
(zα/2 +zβ)2[1 + (m−1)ρ]
d2
=nI[1 + (m−1)ρ]
=n
I
×VIF
(8)
where n
I
istherequiredsamplesizeperarmusinga
trial with individual randomization to detect a difference
d, and VIF can be modified to allow for variation in
cluster sizes (equation 6). This is the standard result,
thattherequiredsamplesizeforaCRCTisthat
required under individual randomisation, inflated by the
variance inflation factor [1]. The number of clusters
required per arm is then:
k=
nI(1+(m−1)ρ)
m
(9)
assuming equal cluster sizes. This slight modification
of the common formula for the number of required
clusters (over that say presented in [2]), has rounded up
the total sample size to a multiple of the cluster size
(using the ceiling function). For, unequal cluster sizes
(using the VIF at equation 6) this becomes:
k=
nI[1 + ((cv2+1)¯
m−1)ρ]
¯
m
(10)
again with rounding up to the average cluster size.
CRCTs of fixed size: fixed number of clusters each
of fixed size
Where a CRCT is to be designed with a completely
fixed size, that is with a fixed number of clusters, each
of a fixed size (although this size may vary between
clusters), then it is possible to evaluate both the detect-
able difference and the power, as would be the case in a
design using individual randomisation. CRCTs of fixed
size might not be the commonest of designs, but formu-
lae presented below: are an important prelude to later
formulae, might be useful for retrospectively computing
power once a trial has commenced (and thus the size
has been determined), and will also be useful in those
limited number of studies for which the trial sample
size is indeed completely fixed (for example within a
cohort study) [9,10].
CRCT of fixed size: detectable difference
For a CRCT with a fixed number of clusters kper arm,
with a fixed number of individuals per cluster mand
with power 1 -b, then the detectable difference, d
C
,fol-
lows straightforwardly from equation 1:
dC=2σ2[1 + (m−1)ρ]
mk (zα/2 +zβ
)
=dI[1 + (m−1)ρ]
=d
I
√VIF
(11)
where d
I
is the detectable difference using individual
randomisation and VIF might be either of those pre-
sented at equations 5 and 6. So the detectable difference
in a CRCT can be thought of as the detectable differ-
ence in a trial using individual randomisation, inflated
by the square-root of the variance inflation factor.
CRCTs of fixed size: power
The power 1 - b
C
of a trial designed to detect a differ-
ence of dwith fixed sample size n
C
=mk per arm, fol-
lowing equation 2, is:
z
βC=mk
2
d
σ
√
VIF −zα/
2
or equivalently, that:
βC=
mk
2
d
σ√VIF −zα/2
(12)
where again, VIF might be either of those presented at
equations 5 and 6. So, power in a CRCT can be thought
of as the power available under individual randomisation
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for a standardised effect size which is deflated by the
square-root of the variance inflation factor.
CRCTs with fixed number of clusters but flexible
cluster size
Standard sample size formulae for CRCTs, by assuming
knowledge of the cluster size (m) and determining the
required number of clusters (k), implicitly assume that
the number of clusters can be increased as required.
However, in the design of health service interventions, it
is often the case that the number of clusters will be lim-
ited by the number of cluster units willing or able to
participate. So for example, in two general practice
based CRCTs (one to evaluate lay education in diabetes
and the other to evaluate a general practice-based inter-
vention to reduce primary care prescribing errors), the
number of clusters was limited to the number of pri-
mary care practices that agreed to participate in the
study. From an estimate of the number of clusters avail-
able, it is relatively straightforward to determine the
required cluster size for each of the clusters. However,
due to the limited increase in precision available by
increasing cluster sizes, it might not always be feasible
to detect the required difference at required power
under a design with a fixed number of kclusters. These
issues are explored below.
CRCTs with a fixed number of clusters: sample size per
cluster
ThestandardsamplesizeformulaeforCRCTsassumes
knowledge of cluster size (m) and consequently deter-
mines the number of clusters (k) required. For a pre-
specified available number of clusters (k), investigators
need instead to determine the required cluster size (m).
Whilst this sample size formula is not commonly pre-
sented in the literature, it consists of a simple re-
arrangement of the above formulae presented at equa-
tion 8 [2]. So, for a trial with a fixed number of equal
sized clusters (k) the required sample size per arm for a
trial with pre-specified power 1 - b, to detect a differ-
ence of d,isn
C
, such that:
nC=nI[1+(m−1)ρ]
=nI1+nC
k−1ρ
=nIk[1 −ρ]
[
k−nIρ
]
(13)
where n
I
is the sample size required under individual
randomisation. This increase in sample size, over that
required under individual randomisation, is no longer a
simple inflation, as the inflation required is now depen-
dent on the sample size required under individual
randomisation.
The corresponding number of individuals in each of
the kequally sized clusters is:
m=
nI(1 −ρ)
(
k−nIρ
)
(14)
this time rounding up the total sample size to a multi-
ple of the number of clusters (k) available (using the
ceiling function).
For unequal cluster sizes, using the VIF from equation
6, the required sample size is:
nC=nIk[1 −ρ]
[k−nI
(
cv2+1
)
ρ](15)
and the average number of individuals per cluster
becomes:
¯
m=
nI(1 −ρ)
k−nI
(
cv2+1
)
(16)
again rounding up to a multiple of the number of
clusters (k) available.
CRCT with a fixed number of clusters: feasibility check
When designing a CRCT with a fixed number of clus-
ters, because of the diminishing returns that sets in
when the sample size of each cluster is increased, it may
not be possible to detect the required difference at pre-
specified power [2]. In a CRCT with a fixed number of
individuals per cluster, but no limit on the number of
clusters, no such limit will exist. This limit on the differ-
ence detectable (or alternatively available power) stems
from the maximum precision available within a CRCT
with a limited number of clusters. Recall that the preci-
sion of the estimate of the difference is:
var−1(dC)= mk
2σ2VIF =mk
2σ2[1 +
(
m−1
)
ρ](17)
As the cluster size (m) becomes large, this precision
reaches a theoretical limit:
lim
m→∞
var
−1
(d
C
) = lim
m→∞
mk
2σ
2
[1 +
(
m−1
)
ρ]=k
2σ
2
ρ(18)
This limit therefore provides an upper bound on the
precision of an estimate from a CRCT. If the CRCT is
to achieve the same or greater power as a corresponding
individually randomised design, it is required that:
k
2σ2
ρ
>
nI
2σ2(19)
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for equal cluster sizes; and:
k
2σ2ρ
(
cv2+1
)
>
nI
2σ2(20)
for unequal cluster sizes. A simple feasibility check, to
determine whether a fixed number of available clusters
will enable a trial to detect a required difference at
required power, therefore consists of evaluating whether
the following inequality holds:
k>nI
ρ
(21)
for equal cluster sizes [2], and
k>nIρ
(
cv2+1
)
(22)
for unequal cluster sizes. Here, n
I
is the required sam-
plesizeunderindividual randomisation, kis the avail-
able number of clusters, ris the estimated intra-cluster
correlation coefficient, and cv represents the coefficient
of variation of cluster sizes. When this inequality does
not hold, it will be necessary to re-evaluate the specifica-
tions of this sample size calculation. This might consist
of a re-evaluation of the power and significance level of
the trial, or it might consist of a re-evaluation of the
detectable difference. Bounds, imposed as a result of the
limited precision, on the detectable difference and
power are derived below.
CRCT with a fixed number of clusters: minimum
detectable difference
For a trial with a fixed number of clusters (k), and
power 1 - b, the theoretical Minimum Detectable Differ-
ence (MDD) for an infinite cluster size is d
MDD
, where:
dMDD =limm→∞2σ2[1 + (m−1)ρ]
mk (zα/2 +zβ
)
=
2σ2ρ
k(zα/2 +zβ)
(23)
which follows naturally from the formula for detect-
able difference (equation 1) and the bound on precision
(equation 18). This therefore gives a bound on the
detectable difference achievable in a trial with a fixed
number of clusters.
For the case of two binary outcomes, where π
1
is fixed
(and π
2
>π
1
), then the minimum detectable difference
for a fixed number of clusters per arm k,isd
C
=π
2
-π
1
such that:
d2
MDD =ρ(zα/2 +zβ)
2
[π1(1 −π1)+π2(1 −π2)].
k
(24)
Re-arranging this as a function of π
2
is:
0=aπ2
2
+bπ2+
c
(25)
where a=-(1+w), b=2π
1
+w,
c=wπ1(1 −π1)−π
2
1
,andw=r(z
a/2
+z
b
)
2
/k.Solving
this quadratic gives:
π2=−b±(b2−4ac)
2
a.
(26)
Each of these two solutions to this quadratic will pro-
vide the limit on π
2
for two sided tests.
CRCT with a fixed number of clusters: maximum
achievable power
For a trial again with a fixed number of clusters (k), the
theoretical Maximum Achievable Power (MAP) to
detect a difference dis 1 - b
MAP
where:
lim
m
→∞
zβMAP = lim
m→∞ km
2(1+(m−1)ρ)
d
σ−zα/
2
=
k
2ρ
d
σ−zα/2
(27)
which again follows from the formula for power
(equation 2) and the bound on precision (equation 18).
So the maximum achievable power is 1 - b
MAP
where:
β
MAP =
k
2ρ
d
σ−zα/2
.
(28)
This therefore provides an upper limit on the power
available under a design with a fixed number of clus-
ters k.
CRCT with a fixed number of clusters: practical advice
When designing a CRCT with a fixed number of clus-
ters, researchers should be aware that such trials will
have a limited available power, even when it is possible
to increase the number of individuals per cluster. In
such circumstances, it will be necessary to:
(a) Determine the required number of individuals
per arm in a trial using individual randomisation
(n
I
).
(b) Determine whether a sufficient number of clus-
ters are available. For equal sized clusters, this will
occur when:
k>nI
ρ
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where n
I
is the sample size required under individual
randomisation, ris the intra-cluster correlation coef-
ficient, and kis the number of clusters available in
each arm. For unequal sized clusters:
k>nIρ
(
cv2+1
)
where cv is the coefficient of variation of cluster
sizes.
(c) Where the design is not feasible and cluster sizes
are unequal, determine whether the design becomes
feasible with equal cluster sizes (i.e. if k>n
I
r).
(d) Where the design is still not feasible:
(i) Either: the power must be reset at a value
lower than the maximum available power (equa-
tion 28),
(ii) Or: the detectable difference must be set
greater than the minimum detectable difference
(equations 23 (continuous outcomes) and 26
(binary outcomes)),
(iii) Or: both power and detectable difference are
adjusted in combination.
(e) Once a feasible design is found, determine the
required number of individuals per cluster from
equations 14 (for equal cluster sizes) and 16 (for
varying cluster sizes).
General examples
Maximum achievable power for cluster designs with 10,
20, 30, 50 or 100 clusters per arm are presented in Fig-
ure 1 for standardised effect sizes ranging from 0.05 to
0.30 and for ICCs in the range 0 to 0.1 (which are com-
mon ICCs in the medical literature [16]). As expected,
achievable power increases with increasing numbers of
clusters and increasing effect size. For the smallest effect
size considered, 0.05, even 100 clusters per arm is not
sufficient to obtain anywhere near an acceptable power
level for ICCs above about 0.02. For less extreme effect
sizes, such as 0.2 when there are 50 or 100 clusters
available per arm, for ICCs less than about 0.1 power in
the level of 80% will be obtainable; yet where there are
just 10 or 20 clusters available, 80% power will only be
attainable for ICCs less than about 0.06. Figure 2 shows
similar estimates of maximum achievable power for bin-
ary comparisons at baseline proportions ranging from
0.05 to 0.5 to detect increases of 0.1 (i.e. 10 percentage
points on a percentage scale).
Minimal detectable differences are also presented for
both standardised effect sizes (Figure 3) and proportions
(Figure 4) for 80% power. As expected, increasing the
number of clusters reduces the minimum detectable dif-
ference. Therefore with a large number of clusters avail-
able and sufficient numbers of individuals per cluster,
trials are possible to detect small changes in proportions
and standardised effect sizes. On the other hand, for
trials with few clusters (say 10 or 20 per arm), minimum
detectable differences become large. So, for example for
continuous outcomes, with say 10 clusters per arm and
an ICC in the region of 0.02, then the MDD is in the
region of 0.2 standardised effect sizes (Figure 3). For
binary outcomes (Figure 4) with 10 clusters per arm and
ICC in the region of 0.02 the minimum detectable dif-
ference is in the region of about a 10 percentage point
change (i.e. from about 15% to 25%).
Example
In a real example, a CRCT is to be designed to evaluate
the effectiveness of lay support workers to promote
breastfeeding initiation and sustainability until 6 weeks
postpartum. Due to fears of contamination, whereby
new mothers indivertibly gain access and support from
the lay workers, the intervention is to be randomised
over cluster units. Cluster randomisation will also
ensure that the trial is logistically simpler to run, as ran-
domisation will be carried out at a single point in time,
and midwives will have the benefit of remaining in
either the intervention or control arm for the duration
of the trial. The cluster units to be used are midwifery
teams, which are teams of midwives who visit a set
number of primary care general practices to deliver
antenatal and postnatal care. The trial is to be carried
out within a single primary care trust within the West
Midlands. The nature of this design therefore means
that the number of clusters available is fixed at the
number of midwifery teams delivering care within the
region.
Atthetimeofdesigningthetrial,currentbreastfeed-
ing rates, at 6 weeks postpartum, in the region were
around 40%. National targets had been set to encourage
all regions to increase rates to around 50%. It was
known that 40 clusters are available (i.e. there are 40
midwifery teams within the region), so that the number
of clusters per arm was fixed to k= 20. Estimates of
ICC range from 0.005 to 0.07 in similar trials [6,7].
Firstly, the feasibility check is implemented to deter-
mine whether the 20 available clusters per arm are suffi-
cient to detect the 10 percentage point change assuming
the lower estimated ICC (0.005). Where the power is set
at 80%, the required sample size per arm to detect an
increase in percentages from 40% to 50%, under indivi-
dual randomisation, is n
I
= 385 (Table 1). When multi-
plied by the ICC this gives 385 × 0.005 = 1.925 which is
less than k= 20. This therefore means that 20 clusters
per arm will be sufficient for this design (provided an
adequate number of individuals are recruited in each
cluster). A similar design with 90% power would require
519 individuals per arm using individual randomisation.
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Again, because 515 × 0.005 = 2.57 < 20, this also means
that 20 clusters per arm will be sufficient to detect an
increase from 40% to 50% with 90% power (again pro-
vided an adequate number of individuals are recruited
in each cluster). Equation 14 shows that under the
assumption that r= 0.005, either 22 or 30 individuals
will be required per cluster (for 80% and 90% power
respectively).
Secondly, the feasibility check is evaluated to deter-
mine whether the 20 available clusters per arm is suffi-
cient to detect the 10 percentage point change assuming
the higher estimated ICC (0.07). However, in this case
as 385 × 0.07 = 26.95 >20, so the condition is not met
at the 80% power level (and so neither at the 90%
power level). Therefore, 20 clusters per arm is not a suf-
ficient number of clusters, however many individuals are
included within each cluster, to detect the required
effect size at the pre-specified power and significance.
Since this latter design is not feasible, formulae at
equation 25 allow determination of the minimum
detectable difference (or maximum achievable power
from equation 27). For a cluster trial with 80% power,
and assuming a baseline event rate of π
1
= 0.40, the
minimum detectable difference is 0.12 (to 2 d.p.). That
is, a change from 40% to 52%. To detect a change from
40% to 52% with 80% power, 189 individuals would be
required per cluster. For a trial with 90% power, the
minimum detectable difference is 0.14 (i.e. a change
from 40% to 54%). To detect a change from 40% to 54%
with 90% power, 146 individuals would be required per
cluster.
Discussion
In health care service evaluation cluster RCTs, pre-spe-
cifying the numbers of clusters available, are frequently
used. That is, trials are designed based on a limited
number of cluster units (e.g. GP practices) willing or
able to participate [6,7,9,10]. In contrast, sample size
methods are almost exclusively based on pre-specified
average cluster sizes, as opposed to number of clusters
available [1,4]. Whilst mapping sample size formulae
from one method to the other is straightforward, a limit
Figure 1 Maximum achievable power for various different standardised effect sizes: limiting values as the cluster size approaches
infinity.
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on the precision of estimates in such designs leads to a
maximum available power (that is, a limit on the power
available irrespective of how large the clusters are) and
minimum detectable differences (that is, a limit on the
difference detectable irrespective of how large the clus-
ters are).
For example, with just 15 clusters available per arm
and an ICC of 0.05, power achievable for a trial aiming
to detect an increase in percentage change from 40% to
50% is limited to about 62%, irrespective of how large
the clusters are made. Cluster trials with just 15 clusters
available per arm are not uncommon and a 10 percen-
tage point change not an unrealistic goal in many set-
tings. However, power levels as low as 60% are clearly
sub-optimal, and might not be regarded as sufficiently
high to warrant the costs of a clinical trial. Formulae
provided here for minimum detectable differences show
that to retain a power level in the region of 80%, trial-
lists would have to be content with detecting a differ-
ence above a twelve percentage point change. Re-
formulation of the problem in terms of minimum
detectable difference can thus be used to compare the
difference which is statistically detectable (at acceptable
power levels) to that which is clinically, or managerially,
important.
Should the situation arise in which the postulated ICC
suggests that it is not possible to detect the required dif-
ference (at pre-specified power), it might be tempting to
lower the estimated ICC. Such an approach should be
strongly discouraged, since loss of power will most likely
result, potentially leading to a non-significant finding
[12]. Rather, formulae here allow sensitivity of the
design to be explored in light of possible variations in
the ICC. However, other avenues to increase available
power might reasonably be considered. For example, it
may be plausible to consider relaxing alpha and even to
set alpha and beta equivalent [17]. Or alternatively,
incorporating prior information in a Bayesian framework
may lead to increases in power. It might further be
argued that studies of limited power are of importance
as they contribute to the evidence framework by ulti-
mately becoming part of future systematic reviews [18],
Figure 2 Maximum achievable power to detect increases in 10 percentage points for various different baseline proportions (π
1
):
limiting values as the cluster size approaches infinity.
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and the methods presented here thus allow for the
achievable power to be computed. Before-and-after type
studies offer a further avenue of exploration, as by their
very nature induce smaller intra-cluster correlations.
Methodological limitations of the work presented here
include the assumption of equal sized arms; equal stan-
dard deviations; Normality assumptions (which might
not be tenable for small numbers of clusters as well as
small numbers of individuals); and lack of continuity
correction for binary variables. Furthermore, CRCTs
with a small number of clusters are controversial, pri-
marily because the small number of units randomised
open results to the possibility of bias and approxima-
tions to Normality become questionable. However,
despite this, CRCTs with a small number of clusters are
frequently reported. The Medical Research Council, for
instance, has issued guidelines that cluster trials with
fewer than 5 clusters per arm are inadvisable [19].
Figure 3 Minimum detectable difference (effect size) at 80%
power for continuous outcomes: limiting values as the cluster
size approaches infinity.
Figure 4 Minimum detectable difference (π
2
) at 80% power various different baseline proportions (π
1
): limiting values as the cluster
size approaches infinity.
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Others have considered some of the issues involved in
community based intervention trials with a small num-
ber of clusters, but have focused on issues of restricted
randomisation and whether the analysis should be at the
individual or cluster level [20].
Conclusions
Evaluations of health service interventions using CRCTs,
are frequently designed with a limited available number
of clusters. Sample size formulae for CRCTs, are almost
exclusively evaluated as a function of the average cluster
size. Where no formal limits exist on the number of
individuals enrolled within each cluster, increasing the
numbers of individuals leads to a limited increase in the
study power. This in turn means that for a trial with a
fixed number of clusters, some designs will not be feasi-
ble, and we have provided simple guidelines to evaluate
feasibility. A simple rule is that the number of clusters
(k) will be sufficient provided:
k>nI×
ρ.
For infeasible designs to retain acceptable levels of
power, detectable difference might not be as small as
desired, leading to the notion of a minimum detectable
difference. Useful aidese memoires are that the detect-
able difference in a CRCT is that of an individual RCT
inflated by the square root of the variance inflation fac-
tor; and the power is that under individual randomisa-
tion with the standardised effect size deflated by the
square root of the variance inflation factor. A STATA
function, clusterSampleSize.ado, allows practical imple-
mentation of all formulae discussed here and is available
from the author.
Acknowledgements
K. Hemming, R. J. Lilford and A. Girling were funded by the Engineering and
Physical Sciences Research Council of the UK through the MATCH
programme (grant GR/S29874/01) and by a National Institute of Health
Research grant for Collaborations for Leadership in Applied Health Research
and Care (CLAHRC), for the duration of this work. The views expressed in
this publication are not necessarily those of the NIHR or the Department of
Health. The authors would like to express their gratitude to Monica Taljaard
and Sandra Eldridge for review comments which helped to develop the
material.
Authors’contributions
KH, JM and AS conceived the idea. KH wrote the first and subsequent drafts.
AG and RJL helped developed the ideas. All authors read and approved the
final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 15 February 2011 Accepted: 30 June 2011
Published: 30 June 2011
References
1. Murray DM: The Design and Analysis of Group-Randomized Trials.
London: Oxford, University Press; 1998.
2. Donner A, Klar N: Design and Analysis of Cluster Randomised Trials in
Health Research. London: Arnold 2000.
3. Campbell MK, Thomson S, Ramsay CR, MacLennan GS, Grimshaw JM:
Sample size calculator for cluster randomised trials. Computers in Biology
and Medicine 2004, 34:113-125.
4. Donner A, Birkett N, Buck C: Randomization by cluster: sample size
requirements and analysis. American Journal of Epidemiology 1981,
114:906-914.
5. Kerry SM, Bland MJ: Sample size in cluster randomisation. BMJ 1998,
316:549.
6. MacArthur C, Win ter HR, Bick DE, Lilford RJ, La ncashire RJ, Knowles H,
et al:Redesigning postnatal care: a randomised controlled trial of
protocol-based midwifery-led care focused on individual womens
physical and psychological health needs. Health Technology Assessment
2003, 7:37.
7. MacArthur C, K KJ, Ingram L, Freemantle N, Dennis CL, Hamburger R, et al:
Antenatal peer support workers and breastfeeding initiation: a cluster
randomised controlled trial. BMJ 2009, 338.
8. Pourshams A, Khademi H, Malekshah AF, Islami F, Nouraei M, Sadjadi AR,
et al:Cohort profile: The Golestan Cohort Study - a prospective study of
oesophagael cancer in northern Iran. International Journal of Epidemiology
2009, 39:52-59.
9. Davies MJ, Heller S, Skinner TC, Campbell MJ, Carey ME, Cradock S, et al:
Effectiveness of the diabetes education and self management for
ongoing and newly diagnosed (DESMOND) programme for people with
newly diagnosed type 2 diabetes: cluster randomised controlled trial.
BMJ 2008, 336:491-495.
10. Avery AJ, Rodgers S, Cantrill JA, Armstrong S, Elliott R, Howard R, et al:
Protocol for the PINCER trial: a cluster randomised trial comparing the
effectiveness of a pharmacist-led IT-based intervention with simple
feedback in reducing rates of clinically important errors in medicines
management in general practices. Trials 2009, 10:28.
11. Feng Z, Diehr P, Peterson A, McLerran D: Selected issues in group
randomized trials. Annual Review of Public Health 2001, 22:167-87.
12. Guittet L, Giraudea B, Ravaud P: A priori postulated and real power in
cluster randomised triasl: mind the gap. BMC Medical Research
Methodology 2005, 5:25.
13. Brown C, Hofer T, Johal A, Thomson R, Nicholl J, Franklin BD, et al:An
epistemology of patient safety research: a framework for study design
and interpretation. Part 2. Study design. Qual Saf Health Care 2008,
17:163-169.
14. Armitage P, Berry G, Matthews JNS: Statistical methods in medical
research. London: Blackwell Publishing; 2002.
15. Kerry SM, Bland MJ: Sample size in cluster randomised trials: effect of
coefficient of variation of cluster size and cluster analysis method.
International Journal of Epidemiology 2006, 35:1292-1300.
16. Campbell MK, Fayers PM, Grimshaw JM: Determinants of the intracluster
correlation coefficient in cluster randomized trials: the case of
implementation research. Clinical Trials 2005, 2:99-107.
17. Lilford RJ, Johnson N: The alpha and beta errors in randomized trials.
New England Journal of Medicine 1990, 322:780-781.
18. Edwards SJ, Braunholtz D, Jackson J: Why underpowered trials are not
necessarily unethical. Lancet 2001, 350:804-807.
Table 1 Estimates of the Minimum Detectable Difference
(MDD) for trial with 20 clusters per arm, to detect an
increase in an event rate from 40%
Power = 80% Power = 90%
40% vs 50% MDD 40% vs 50% MDD
ICC = 0.005 n
I
= 385 N/A n
I
= 515 N/A
n
C
= 440 n
C
= 600
m = 22 m = 30
ICC = 0.07 MDD = 12% MDD = 14%
n
I
= 267 n
I
= 262
n
C
= 3,780 n
C
= 2,920
m = 189 m = 146
Hemming et al.BMC Medical Research Methodology 2011, 11:102
http://www.biomedcentral.com/1471-2288/11/102
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19. Medical Research Council: Cluster randomsied trials: methodological and
ethical considerations; 2002.[http://www.mrc.ac.uk].
20. Yudkin PL, Moher M: Putting theory into practice: a cluster randomised
trial with a small number of clusters. Statistics in Medicine 2001,
20:341-349.
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Cite this article as: Hemming et al.: Sample size calculations for cluster
randomised controlled trials with a fixed number of clusters. BMC
Medical Research Methodology 2011 11:102.
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