1. Alan Girling University of Birmingham, UK A.J.Girling@bham.ac.uk
An algorithm to optimise the sampling scheme within the clusters of a stepped-wedge design
Alan Girling
University of Birmingham, UK
Funding support (AG) from the NIHR through:
The NIHR Collaborations for Leadership in Applied Health Research and Care for West Midlands (CLAHRC WM).
The HiSLAC study (NIHR ref 12/128/07)
London November 2018

2. Scope Cross-sectional cluster designs:
with (possibly) time-varying cluster-level effects,
and uni-directional switching between treatment regimes (as in Stepped-Wedge)
Equal numbers of observations in each cluster
Freedom to choose the timing of observations within each cluster
*Other constraints are available! *

3. Treatment Effect Estimate: 𝜃 = 𝑘,𝑡 𝑎 𝑘𝑡 𝑦 𝑘𝑡
SW4: 20 observations per cluster at each time-point ICC = ; fixed time effects (Hussey & Hughes model)
Cell Mean
Treatment Effect Estimate: 𝜃 = 𝑘,𝑡 𝑎 𝑘𝑡 𝑦 𝑘𝑡
Design Layout
No.s of Observations
𝑚 𝑘𝑡
Total
(𝑀)
Clusters (k) 20
100
Time (t)
Coefficients:
𝑎 𝑘𝑡 (×100)
7.5 30
17.5 5
2.5
15
22.5 10
2.5
10
22.5 15
7.5 5
17.5
30
Precision = var 𝜃 −1  0.400

4. Proposal: modify the 𝑚 𝑘𝑡 s to make 𝑚 𝑘𝑡 ∗ ∝ 𝑎 𝑘𝑡 within each row
𝑎 𝑘𝑡 (×100)
∑|𝑎|
7.5 30
17.5 5
67.5
2.5
15
22.5 10
52.5
2.5
10
22.5 15
7.5 5
17.5
30
𝜃 = 𝑘,𝑡 𝑎 𝑘𝑡 𝑦 𝑘𝑡
Some observations have greater influence on the estimate than others (unlike many classical designs)
?Layout might be improved by moving observations from low-influence to high-influence cells within the same cluster
(For equal influence, need 𝑎 𝑘𝑡 𝑚 𝑘𝑡 to be the same in each cell)
Proposal: modify the 𝑚 𝑘𝑡 s to make 𝑚 𝑘𝑡 ∗ ∝ 𝑎 𝑘𝑡 within each row
𝑚 𝑘𝑡 ∗ =𝑀 𝑎 𝑘𝑡 𝑠=1 𝑇 𝑎 𝑘𝑠

5. Revised Layout: 𝑚 𝑘𝑡 ∗ =𝑀 𝑎 𝑘𝑡 𝑠=1 𝑇 𝑎 𝑘𝑠 .
Also
Update Treatment Estimate
No.s of Observations ( 𝑚 𝑘𝑡 ∗ )
Total
(𝑀)
Clusters (k)
11.1
44.4
25.9
7.4
100
4.8
28.6
42.9
19.0
Time (t)
New Coefficients
𝑎 𝑘𝑡 ∗ (×100)
∑ 𝑎 ∗
3.9
30.1
11.6
1.6
51.1
0.7
20.4
28.1
8.1
58.0
0.7
8.1
28.1
20.4
3.9
1.6
11.6
30.1
Precision  0.624
Precision has improved from to 0.624
But, 𝑎 𝑘𝑡 ∗ 𝑚 𝑘𝑡 ∗ ≠𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, still, within each row

6. No.s of Observations ( 𝑚 𝑘𝑡 (∞) )
After repeated iteration the process converges (to a ‘Staircase’ Design)
𝜃 = 𝑘,𝑡 𝑎 𝑘𝑡 ∞ 𝑦 𝑘𝑡
No.s of Observations ( 𝑚 𝑘𝑡 (∞) )
Total
Clusters (k)
100 50
Time (t)
𝑎 𝑘𝑡 ∞ ×100
Clusters
33.3
33.3
Time
Precision  0.750
𝑎 𝑘𝑡 ∞ 𝑚 𝑘𝑡 ∞ =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 within rows, at least for occupied cells

7. The Algorithm
For the current allocation 𝑚 𝑘𝑡 𝑛 compute the coefficients 𝑎 𝑘𝑡 𝑛 of the best estimate 𝜃. 𝜃 𝑛 = 𝑘,𝑡 𝑎 𝑘𝑡 𝑛 𝑦 𝑘𝑡
Update the allocation to make 𝑚 𝑘𝑡 𝑛+1 ∝ 𝑎 𝑘𝑡 𝑛 within each cluster using 𝑚 𝑘𝑡 𝑛+1 =𝑀 𝑎 𝑘𝑡 𝑛 𝑠=1 𝑇 𝑎 𝑘𝑠 𝑛 (𝑀 is the total number in each cluster, assumed fixed.)
Repeat ad lib.

8. Model 𝑦 𝑘𝑡𝑖 = 𝛽 𝑡 + 𝜃 𝑋 𝑘𝑡 + 𝛾 𝑘𝑡 + 𝜀 𝑘𝑡𝑖
… for Observation 𝑖 in Cluster 𝑘 at time 𝑡.
𝑦 𝑘𝑡𝑖 = 𝛽 𝑡 + 𝜃 𝑋 𝑘𝑡 𝛾 𝑘𝑡 𝜀 𝑘𝑡𝑖
Time Treatment Cluster x Time Residual
var 𝛾 𝑘𝑡 = 𝜏 2 , var 𝜀 𝑘𝑡𝑖 = 𝜎 2 , corr 𝛾 𝑘𝑠 , 𝛾 𝑘𝑡 = Γ 𝑠𝑡
(Hussey & Hughes) Γ 𝑠𝑡 ≡1
(Exchangeable) Γ 𝑠𝑡 =𝜋+ 1−𝜋 𝛿 𝑠𝑡
(Exponential) Γ 𝑠𝑡 = 𝑟 𝑠−𝑡
Fixed Effects
Random Effects
𝜃 = 𝑘,𝑡 𝑎 𝑘𝑡 𝑦 𝑘𝑡 is the weighted least squares estimator ( BLUE)

9. Properties of the Algorithm
Improvement happens at every step: I.e. var 𝜃 𝑛+1 ≤ var 𝜃 𝑛 with equality only if 𝑚 𝑘𝑡 𝑛 ∝ 𝑎 𝑘𝑡 𝑛 within each cluster.
Convergence to a stable point is guaranteed
This is usually the optimal allocation
Any stable point is a ‘best’ allocation among all allocations with that support (i.e. collection of non-zero cells)
But, if an empty cell appears at any step i.e. 𝑎 𝑘𝑡 𝑛 =0 that cell remains empty at every subsequent step.
(In principle the best allocation could be missed.)
On the other hand, this property allows us to obtain improved/optimal designs in situations where sampling in some cells is prohibited
Behaviour depends on 𝜎 2 , 𝜏 2 and M only through 𝑅= 𝑀 𝜏 2 𝑀 𝜏 2 + 𝜎 2
𝑅 is related to the Cluster-Mean Correlation 𝐶𝑀𝐶 1−𝐶𝑀𝐶 = 1 ′ Γ1 𝑇 2 ⋅ 𝑅 1−𝑅

10. Examples: 1) Hussey & Hughes model
Initial Allocation:
Equal % of observations at each time-point.
(Row-totals = 100%) 14
When 𝑅< 1 2 solution is NOT Unique
𝑅=0.25
Unique solution when 𝑅≥ 1 2 , apart from trades between end columns
𝑅=0.75 79 18 3 50 39 9 2 8 42 40 1 17 67 47 53 49 51
Efficiency improves
from (initially) to
Efficiency improves
from (initially) to
(Efficiency computed relative to a Cluster Cross-Over design with same number of observations.)

11. Efficiency of Optimised Allocation: Hussey & Hughes Model

12. Examples: 2) Exponential model: r = 0.9
Initial Allocation:
Equal % of observations at each time-point.
(Row-totals = 100%) 14
Exact General Behaviour unknown
𝑅=0.25
𝑅=0.75 72 3 4 8 14 47 53 49 51 14 73 46 54 49 51
Efficiency improves
from (initially) to
Efficiency improves
from (initially) to
(Efficiency relative to an “Ideal” Cluster Cross-Over design with same number of observations.)

13. Efficiency of Optimised Allocation: Exponential Model with r = 0.9

14. Example with prohibited cells: ‘Transition’/ ‘Washout’ periods (under H&H model)
Initial Allocation:
Equal % of observations at each permissible time-point.
(Row-totals = 100%) 14
𝑅=0.25
𝑅=0.75 78 16 5 72 25 2 50 44 6 12 67 9 8 13 4 50
Efficiency improves
from (initially) to
Efficiency improves
from (initially) to

15. Why it works
For any linear estimate 𝜃 =∑ 𝑎 𝑘𝑡 𝑦 𝑘𝑡 , var 𝜃 =𝑉(𝑎,𝑚)= 𝜏 2 𝑘=1 𝐾 𝑎 𝑘 ′ Γ 𝑎 𝑘 + 𝜎 2 𝑘=1 𝐾 𝑡=1 𝑇 𝑎 𝑘𝑡 2 𝑚 𝑘𝑡 It is always true that 𝑡=1 𝑇 𝑎 𝑘𝑡 2 𝑚 𝑘𝑡 ∗ ≤ 𝑡=1 𝑇 𝑎 𝑘𝑡 2 𝑚 𝑘𝑡 where 𝑚 𝑘𝑡 ∗ =𝑀 𝑎 𝑘𝑡 𝑠=1 𝑇 𝑎 𝑘𝑠 So the variance of the estimate is reduced by the reallocation of observations.
Now apply this argument to the BLUE of 𝜃 under allocation 𝑚 𝑘𝑡 .
It follows that the BLUE of 𝜃 under allocation ( 𝑚 𝑘𝑡 ∗ ) has
smaller variance than the BLUE under 𝑚 𝑘𝑡 .

16. Spin-off: An Objective Function
The best allocation corresponds to a stable point of the algorithm.
At any stable point (i.e. where 𝑚 𝑘𝑡 ∝| 𝑎 𝑘𝑡 |within each cluster):
𝑉 𝑎,𝑚 ∝Ψ 𝑎 =𝑅 𝑘=1 𝐾 𝑎 𝑘 ′ Γ 𝑎 𝑘 +(1−𝑅) 𝑘=1 𝐾 𝑡=1 𝑇 𝑎 𝑘𝑡 2
Any optimal design corresponds to a (constrained) minimum value of Ψ.
Ψ 𝑎 = min 𝑎 Ψ(𝑎) (subject to unbiasedness constraints on the 𝑎 𝑘𝑡 s, and 𝑎 𝑘𝑡 =0 in any prohibited cells)
…with cell numbers given by:
𝑚 𝑘𝑡 =𝑀 𝑎 𝑘𝑡 𝑠=1 𝑇 𝑎 𝑘𝑠
Ψ 𝑎 is not a smooth function, but it is convex.

17. Potential for Exact results using Ψ 𝑎
Eg. For the Hussey and Hughes Model (Γ 𝑠𝑡 ≡1)
Ψ 𝑎 =𝑅 𝑘=1 𝐾 𝑡=1 𝑇 𝑎 𝑘𝑡 −𝑅 𝑘=1 𝐾 𝑡=1 𝑇 𝑎 𝑘𝑡 2

18. (Exact) Optimal Design under HH: R≥ 1 2
The matrix of 𝑎 𝑘𝑡 s has an “Anchored Staircase” form:
q0/2 q1
q1 q2
q2 q3
q3 q4
q4 q5
+q6/2
q5
𝑞 𝑘 ∝ coth 𝜙 2 ∙ sinh 𝐾𝜙 2 − cosh 𝑘− 𝐾 2 𝜙
cosh 𝜙= 2𝑅−1 −1
E.g. 𝑅=0.75; Efficiency ≈0.76
Efficiency=1− 1 𝐾 2− tanh 𝜙 2 tanh 𝐾𝜙 2
=1− 1 𝐾 2− 1−𝑅 𝑅 +𝑂 1 𝐾 2 17 67 47 53 49 51
(Rel. to CXO)

19. Optimal Design under HH: R< 1 2
One possible matrix of 𝑎 𝑘𝑡 s is:
x+y y x x y
xy
𝑥= 𝐾−2𝑅 −1 ,
𝑦= 1 2 −𝑅 ⋅ 𝐾−2𝑅 −1
Eg. 𝑅=0.25 Efficiency = ≈0.92
Efficiency=1− 2𝑅 𝐾 83 17 50

20. An alternative solution was given earlier:
𝑅=0.25; Efficiency = 0.92 79 18 3 50 39 9 2 8 42 40 1
…and there are many others.

21. Summary
Flexible approach to improving design, often leading to substantial improvements in precision
Works for sparse layouts and designs with prohibited cells
Where the solution is a staircase-type design the experiment may take longer Partly this is a consequence of improved precision
A fair comparison is between designs with the same precision (i.e. SW vs an optimised design with fewer total observations).
The objective function Ψ provides an alternative approach via convex optimisation methods, and a tool for finding exact results

22. Further developments
Optimal allocation of clusters to (optimised) sequences
Readily accomplished by adding an extra computation to the algorithm
Little advantage for precision, it seems, but there may be scope for alternative near-optimal designs
Alternative constraints
Fixed total size of study
Constraints over specific time-periods
Unequal clusters
Explore optimal designs with prohibited cells
Eg. the Washout example
Or to seek more compact designs