Incomplete (Stepped Wedge) Designs with SteppedPower

Philipp Mildenberger1

Institute of Medical Biostatistics, Epidemiology and Informatics (IMBEI, Mainz)

Federico Marini2

Institute of Medical Biostatistics, Epidemiology and Informatics (IMBEI, Mainz)

2023-09-12

1 Incomplete Designs

In general, a study design is referred to as incomplete if not all clusters are observed at every time period (Hemming et al. 2015).

Suppose you do not plan to observe all clusters over the whole study period. Rather, clusters that switch early to the intervention are not observed until the end. Analogous, observation starts later in clusters that switch towards the end of the study.

2 Incomplete Designs in SteppedPower

There are essentially three ways to define cluster periods without observation.

  • The first - and generally preferred - option is to use the incomplete argument. Input can be either a scalar or a matrix of dimension clusters\(\cdot\)timepoints or sequences\(\cdot\)timepoints:
    • A scalar is interpreted as the number of observed periods before and after the switch from control to intervention in each cluster
    • A matrix must contain 1s for cluster cells that are observed and 0 or NAs for cluster cells that are not observed.
  • Insert NAs into an explicitly defined treatment matrix, easiest done with the argument trtmatrix=.
  • Insert NAs into the vector for delayed treatment start trtDelay=.

glsPower() calls the function construct_DesMat() to construct the design matrix with the relevant arguments. All the above options can be used in the main wrapper function, but the examples below focus on construct_DesMat() directly.

SteppedPower stores information about (un)observed cluster cells separately from the treatment allocation. This is done for more consistency in the code as the indices in the covariance and design matrices is

3 Examples

3.1 1

If for example the a stepped wedge study consists of eight clusters in four sequences (i.e. five timepoints), and we observe two timepoints before and after the switch, then we receive

Dsn1.1 <- construct_DesMat(Cl=rep(2,4), incomplete=2)

A slightly more tedious, but more flexible way is to define a matrix where each row corresponds to either a cluster or a wave of clusters and each column corresponds to a timepoint. If a cluster is not observed at a specific timepoint, set the value in the corresponding cell to 0. For the example above, such a matrix would look like this:

TM  <- toeplitz(c(1,1,0,0))
incompleteMat1 <- cbind(TM[,1:2],rep(1,4),TM[,3:4])
incompleteMat2 <- incompleteMat1[rep(1:4,each=2),]

A matrix where each row represents a wave of clusters

1 1 1 0 0
1 1 1 1 0
0 1 1 1 1
0 0 1 1 1

or each row represents a cluster

1 1 1 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 0
0 1 1 1 1
0 1 1 1 1
0 0 1 1 1
0 0 1 1 1

Now all that’s left to do is to plug that into the function and we receive the same design matrix

Dsn1.2 <- construct_DesMat(Cl=rep(2,4), incomplete=incompleteMat1)
Dsn1.3 <- construct_DesMat(Cl=rep(2,4), incomplete=incompleteMat2)

all.equal(Dsn1.1,Dsn1.2)
## [1] "Component \"incompMat\": 'is.NA' value mismatch: 0 in current 12 in target"
all.equal(Dsn1.1,Dsn1.3)
## [1] "Component \"incompMat\": 'is.NA' value mismatch: 0 in current 12 in target"

The argument incomplete with matrix input works also for other design types, but makes (supposedly) most sense in the context of stepped wedge designs

3.2 2

Now suppose we want to use a SWD to investigate the intervention effects after at least one month,
i.e. cluster periods directly after the switch to intervention conditions are not observed. That leads to an incomplete design that is easiest modelled with trtDelay=

Dsn2 <- construct_DesMat(Cl=rep(2,4), trtDelay = c(NA) )
Dsn2
## Timepoints                         = 5
## Number of clusters per seqence     = 2, 2, 2, 2
## Design type                        = stepped wedge
## Time adjustment                    = factor
## Dimension of design matrix         = 40 x 6
## 
## Treatment status (clusters x timepoints):
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0   NA    1    1    1
## [2,]    0   NA    1    1    1
## [3,]    0    0   NA    1    1
## [4,]    0    0   NA    1    1
## [5,]    0    0    0   NA    1
## [6,]    0    0    0   NA    1
## [7,]    0    0    0    0   NA
## [8,]    0    0    0    0   NA

3.3 3

The above arguments can also be combined, e.g.

Dsn3 <- construct_DesMat(Cl=rep(2,4), incomplete=2, trtDelay=c(NA) )
Dsn3
## Timepoints                         = 5
## Number of clusters per seqence     = 2, 2, 2, 2
## Design type                        = stepped wedge
## Time adjustment                    = factor
## Dimension of design matrix         = 40 x 6
## 
## Treatment status (clusters x timepoints):
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0   NA    1   NA   NA
## [2,]    0   NA    1   NA   NA
## [3,]    0    0   NA    1   NA
## [4,]    0    0   NA    1   NA
## [5,]   NA    0    0   NA    1
## [6,]   NA    0    0   NA    1
## [7,]   NA   NA    0    0   NA
## [8,]   NA   NA    0    0   NA
Hemming, Karla, Terry P Haines, Peter J Chilton, Alan J Girling, and Richard J Lilford. 2015. “The Stepped Wedge Cluster Randomised Trial: Rationale, Design, Analysis, and Reporting.” Bmj 350.