1.
Design Choices for Cluster-
Randomised Trials
Alan Girling
University of Birmingham, UK
A.J.Girling@bham.ac.uk
CLAHRC Scientific Advisory
Group Birmingham, June 2015

2.
… a Statistical viewpoint
• Ethical and Logistical concerns are suspended
• RCTs and Parallel Cluster trials are well-
understood; ‘Stepped Designs’ less so.
Two questions:
1. How does the Stepped Wedge Design perform?
…but The Genie is out of the bottle!
2. What about alternative stepped designs?
Does it have to be a “Wedge”?

3.
Some Alternative Designs:
8 months 4 groups of Clusters (“Arms”)
Arms
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Months 
Arms
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Months 

4.
Assumptions
• Study of fixed duration (8 months)
• Constant recruitment rate in each cluster
• Continuous Outcome
– Additive treatment effect
– Cross-sectional observations with constant ICC = 
• Cross-over in one direction only (i.e. Treated to Control prohibited)
• The analysis allows for a secular trend (“time effect”)
– This has been questioned; but if time effects are ignored, the ‘best’
statistical design involves simple before-and-after studies in each cluster
(i.e. not good at all!)
Goal: To compare statistical performance of different designs
under these assumptions, especially the Stepped-Wedge

5.
1. ‘Precision Factor Plots’ for
Comparing Designs

6.
Clusters
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Months 
1. Simple Parallel Design
Two groups of clusters.
Treatment implemented in
one group only, in month 1.
Treatment Effect Estimate =
(Mean difference between two groups over all months)
Two Simple Candidate Designs

7.
Clusters
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Months 
1. Parallel Study with (multiple) baseline controls
“Controlled Before-and-After Design”
Two groups of clusters.
Treatment implemented in
one group only, in month 5.
Treatment Effect Estimate =
(Mean difference between two groups in months 5 – 8)
minus
r x (Mean difference between two groups in months 1 – 4)
(r is a correlation coefficient “derived from the ICC”)

8.
Performance measured by Precision of the effect estimate:
Precision = 1/(Sampling Variance) = 1/(Standard Error)2
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
 













R
N
4
1
2
1
14 2
  
 R
N







1
14 2

Precision =
Where R is the “Cluster-Mean Correlation”
… and is the same for both designs  

11 

m
m
R
(m = number of observations per cluster,  = ICC)
CBA Parallel

9.
Relative Efficiency of designs – by comparing straight
lines on a Precision-Factor plot
CBA design is better (“more efficient”) if R > 2/3.
Otherwise the Parallel design is better.
Parallel
CBA

10.
2. The Cluster-Mean Correlation
(CMC)

11.
Cluster-Mean Correlation (CMC)
• Relative efficiency of different designs depends on
cluster-size (m) and ICC (), but only through the CMC (R)
• The CMC (R) =
proportion of the variance of the average
observation in a cluster that is attributable to
differences between clusters
• (The ICC () =
proportion of variance of a single observation
attributable to differences between clusters)
• CMC can be large (close to 1) even if ICC is small
  



m
m
m
m
R




111

12.
Cluster-Mean Correlation: relation with cluster size
• CMC can be large (close to 1) even if ICC is small
• So CBA can be more efficient than a parallel design for
reasonable values of the ICC, if the clusters are large enough




 11
m
R
R

13.
3. Some other designs

14.
1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Controlled Before & After Before & After + Parallel
R
4
1
2
1
 R
2
1
4
3

0 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 1
R
32
9
16
9

0 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1
0 0 0 0 0 1 1 1
0 0 0 0 0 0 0 1
R
16
5
8
5

+ Baseline & Full Implementation Stepped Wedge
Precision
Factors

15.
• Stepped-Wedge is best when the CMC is close to 1
• Parallel Design is best when the CMC is close to 0 – but risky if ICC
is uncertain
• Before & After/Parallel mixture is a possible compromise
Parallel
Stepped-Wedge
B & A/Parallel

16.
4. Is there a ‘Best’ Design?

17.
• If Cross-over from Treatment to Control is permitted,
the Bi-directional Cross-Over (BCO) design has Precision
Factor = 1 at every R, better than any other design.
0 0 0 0 1 1 1 1
0 0 0 0 1 1 1 1
1 1 1 1 0 0 0 0
1 1 1 1 0 0 0 0
R01
Precision Factor =
• But this design is usually not feasible!
• If ‘reverse cross-over’ is disallowed, the Precision Factor
cannot exceed 2
3
1
1 RR 

18.
1 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 0
• Available design performance limited by choosing only 4 groups of
clusters
Some (nearly) ‘Best’ designs with 4 groups of clusters
Stepped-
Wedge
Parallel
Region
Prohibited by
Irreversibility
of Intervention

19.
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1
0 0 0 0 0 1 1 1
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Some (nearly) ‘Best’ designs with 8 groups of clusters
Stepped-
Wedge
Parallel

20.
‘Best’ Design for large studies: This is a mixture
of Parallel and Stepped-Wedge clusters.
100R%
Stepped-Wedge
Clusters
100(1 – R)%
Parallel
Clusters
• Includes Parallel and Stepped-Wedge designs as special cases
• When R is close to 1 the Stepped-Wedge is the best possible
design

21.
5. Conclusions
• Efficiency of different cluster designs depends on the
ICC () and the Cluster-size (m) but only through the
CMC (R).
• Precision Factor plots are useful for comparing designs
– Comparisons are linear in R
• The Stepped-Wedge Design is most advantageous
when R is close to 1.
– In studies with large clusters this can arise even if the ICC is
relatively small.
• The theoretically ‘best’ design choice is sensitive to R,
and combines Parallel with Stepped-Wedge clusters

22.
Limitations
• Continuous data, simple mixed model
– Natural starting point – exact results are possible
– ?Applies to Binary observations through large-
sample approximations
• Extension to cohort designs through nested
subject effects is straightforward and gives
essentially the same answers