3  Detailed statistical information

This chapter offers additional statistical information.

  1. Steps within one simulation experiment
  2. How to conduct design matrices
  3. How to determine fidelity patterns and corresponding design matrices

3.1 Explanation of simulation experiment and provided corresponding function

For a simulation experiment it is not necessary to understand the whole statistical modeling background, which is figured out here.

Each simulation step includes three necessary steps:

  • Determining the design matrix regarding the specified design

  • Sampling data

  • Effect estimation from the data

and will be explained with more detail in the following subsections.

For the first two steps, the R-package {samplingDataCRT} v1.0 (Trutschel and Treutler 2017) is needed, for the last {lme4} (Bates et al. 2015). The first step is done once and the latter two are used repeatedly through one simulation experiment within each scenario.

3.1.1 Determining the design matrix regarding the chosen design

To specify a cluster randomized study the following parameter has to be determined:

  • Number of clusters (e.g. hospitals, nursing homes, …) obtained through the study
  • Number of time points the clusters are followed
  • Cluster size refers to the number of individuals obtained within one cluster
  • Study design: parallel design or Stepped wedge design (cross-over as well possible)
  • Study type: cross-sectional if individuals could be different within the cluster between time points or longitudinal if individuals are followed over time

Based on the example in the main article, we specify a hypothetical example including parameter settings for a reference setup of the simulation experiment: a cross-sectional stepped wedge cluster randomized trial with 6 nursing homes, 7 time points and 10 individuals within each nursing home and time point (see main article). All parameters can be adapted to an own practical example. The resulting reference design matrix can be create manually (by create a corresponding matrix) or by designMatrix().

######################################################
#using the parameter setting of Table 1 in the article
######################################################

## Design ##
############
K<-7 #number of time points
I<-6 #number of cluster
Sw<-1 #number of cluster switches per time point can be manually set
J<-10 #Subjects =  Number of individuals per cluster

#design matrix of "SWD" with given setting
(X<-samplingDataCRT::designMatrix(nC=I, nT=K, nSw=Sw))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    1    1    1    1    1    1
[2,]    0    0    1    1    1    1    1
[3,]    0    0    0    1    1    1    1
[4,]    0    0    0    0    1    1    1
[5,]    0    0    0    0    0    1    1
[6,]    0    0    0    0    0    0    1

3.1.2 Sampling data of a cluster randomized trial with a given design

To sample data for a cluster randomized trial using sampling from a multivariate normal distribution provided by the package {samplingDataCRT}, several model parameter has also be specified:

  • Baseline mean of the outcome of interest
  • Intervention effect (change/difference in mean outcome) by implementing the intervention)
  • Time trend effects
  • Variance within the multilevel data: between clusters, between individuals, within individuals which provide an estimate of the intra-cluster correlation coefficient (ICC) as an measure of dependencies within clusters.

In the example here we set the mean outcome to 10 (e.g. the mean population value of measured quality of life), the intervention effect which aimed to change quality of life of 1, and no time trends. With a given between cluster variance of sigma.3 and a error variance of sigma.1, the resulting ICC is ICC.

## Model parameter ##
####################
mu.0<- 10           # Baseline mean of the outcome of interest
theta <- 1          # intervention effect
betas<-rep(0, K-1)  # no Time trend, but could be included

# variability within or error variance (H&H sigma)
sigma.1<-2    
# variability within clusters, if longitudinal data
sigma.2<-NULL
# between clusters variability (H&H tau)
sigma.3<-sigma.1*sqrt(0.001/(1-0.001))    

#resulting ICC
(ICC<-sigma.3^2/(sigma.3^2+sigma.1^2))
[1] 0.001

To note, by choosing longitudinal or cross-sectional design, we need to specify in the first case 3, in the second only two variances regarding using a three- instead of a two-level hierarchical experiment with the following meaning:

  • three-level design (longitudinal data):
    • between clusters variability
    • within cluster (or between individuals) variability
    • within individuals (or error) variability
  • two-level design (cross-sectional data):

A complete data set for a special design with a given setup can be sampled by the function sampleData(). Therefore,the complete data design matrix and the covariance-variance matrix for the data given the design are also needed to be specified with completeDataDesignMatrix() and CovMat.Design(). The complete data design matrix has the size of (Number of cluster x cluster size x Number of time points) rows and (Number of model parameters) columns. To sample the data from for a cluster randomized trial, it has additionally be specified, if the individuals are followed over time (longitudinal design) or not (cross-sectional design).

#complete data design matrix
D<-samplingDataCRT::completeDataDesignMatrix(J, X)
(dim(D))
[1] 420   8
#covariance-variance matrix for the data given the design
V<-samplingDataCRT::CovMat.Design(K, J, I, sigma.1=sigma.1, sigma.3=sigma.3)
dim(V)
[1] 420 420
#corresponding fixed effects in linear mixed model
parameters<-c(mu.0, betas, theta)

#sample complete data given the setup
# study design type = cross-sectional
type<-"cross-sec" 
sample.data<-samplingDataCRT::sampleData(type = type, K=K,J=J,I=I, 
                                         D=D, V=V, parameters=parameters)

To validate the number of observations provided by the sampling method a summary of the data can be conducted.

dim(sample.data)
[1] 420   5
#show the number of observations within the SWD
xtabs(~cluster+measurement, data=sample.data)
       measurement
cluster  1  2  3  4  5  6  7
      1 10 10 10 10 10 10 10
      2 10 10 10 10 10 10 10
      3 10 10 10 10 10 10 10
      4 10 10 10 10 10 10 10
      5 10 10 10 10 10 10 10
      6 10 10 10 10 10 10 10

3.1.3 Effect estimation from the data

The sampled data can then be analyzed by a linear mixed model with lme4::lmer(), where the parameter of the model will be estimated.

#analysis of the two-level data by a linear mixe model
lme4::lmer(val~intervention+measurement + (1|cluster), data=sample.data)
Linear mixed model fit by REML ['lmerMod']
Formula: val ~ intervention + measurement + (1 | cluster)
   Data: sample.data
REML criterion at convergence: 1489.207
Random effects:
 Groups   Name        Std.Dev.
 cluster  (Intercept) 0.2053  
 Residual             1.4098  
Number of obs: 420, groups:  cluster, 6
Fixed Effects:
 (Intercept)  intervention  measurement2  measurement3  measurement4  
    10.17891       0.82068       0.21980       0.42806       0.15609  
measurement5  measurement6  measurement7  
     0.02882       0.07451      -0.03113

This process of sampling data and effect estimation from the data will be used repeatedly through the simulation experiment within each scenario setting.

3.2 Determine differrent design matrices

The function designMatrix() of the R-package {samplingDataCRT} provides the design matrix of cluster randomized trials with several study designs, given the number of clusters, time points and cluster size assumed for the trial. We show here for two examples:

  • parallel
  • SWD

3.2.1 Parallel cluster randomized trials

Within the function the design argument is set to ‘parallel’,
and the parameter nSW indicates the number of clusters that are being control group.

## Design paramter ##
######################
I <- 6 # number of clusters
J <- 10 # number of individuals per cluster (used later in simulation-step)
K <- 7 # number of time points
Sw <- round(I / 2) # number of cluster within the control group

# design matrix for parallel design
(designMat_prll <- samplingDataCRT::designMatrix(nC = I, nT = K, nSw = round(I / 2), 
                                                 design = "parallel"))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    0    0    0    0    0    0
[2,]    0    0    0    0    0    0    0
[3,]    0    0    0    0    0    0    0
[4,]    1    1    1    1    1    1    1
[5,]    1    1    1    1    1    1    1
[6,]    1    1    1    1    1    1    1

3.2.2 Stepped wedge cluster randomized trial

Whereas the design parameters can be the same as in the parallel study the design matrix differs accordingly when changing the argument design = "SWD" (which is also set by default, so it has not to be specified). The parameter ‘nSW’ indicates here the number of cluster switches per time point from control to intervention group.

## Design matrix ##

I <- 6 # number of clusters
J <- 10 # number of individuals per cluster (used later in simulation-step)
K <- 7 # number of time points
Sw <- 1 # number of cluster switches per time point can be manually set

# design matrix of "SWD"
(designMat_SWD <- samplingDataCRT::designMatrix(nC = I, nT = K, 
                                                nSw = Sw, design = "SWD"))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    1    1    1    1    1    1
[2,]    0    0    1    1    1    1    1
[3,]    0    0    0    1    1    1    1
[4,]    0    0    0    0    1    1    1
[5,]    0    0    0    0    0    1    1
[6,]    0    0    0    0    0    0    1

3.3 Determine different fidelity patterns

Fidelity refers to the degree to which an intervention was implemented as it was prescribed or intended. We aim to include different patterns of how fidelity might increase over time to estimate the respective effects on power of the study. To describe hypothetical fidelity patterns of increasing fidelity (slow, linear, fast) different mathematical functions (i.e. logistic, linear and exponential curves) are implemented. By considering different values for the slope parameter we can cover a range of fidelity patterns. The slope parameter ranges form \((0,\infty)\), where a slope parameter near to \(0\) indicates a increase far away from a linear (fast increase upper left corner, slow increase right bottom corner curve) and a great slope parameter near to linear (see Figure 3.1 ).

library(ggplot2)
library(gridExtra)

#study design
#############
#number of measurement
K<-7  
#points of time afterintervention
T.points<-K-1

#parameter Fidelity specification
###############################
Fid.T1<-0.2
Fid.End<-1

####set several slopes of increasing fidelity
#######################################
slope.seq<-round(exp(1)^(seq(-2,2,1)),2)
nr.sl<-length(slope.seq)

####fidelity patterns determined by several slopes within slow and fast increase
##############################################################################
res.plot.Patterns<-NULL
for(sl in slope.seq){
 res<-fidelitysim::find.Fidelity.log(time.points=T.points, Fid.End, Fid.T1, 
                                     par.slope=sl)
 res<-data.frame(res, FUN="log", slope=sl)
 res.plot.Patterns<-rbind(res.plot.Patterns, res)
 res<-fidelitysim::find.Fidelity.exp(time.points=T.points, Fid.End, Fid.T1, 
                                     par.slope=sl)
 res<-data.frame(res, FUN="exp", slope=sl)
 res.plot.Patterns<-rbind(res.plot.Patterns, res)
      
}

#fidelity pattern for linear increase
#####################################
res.lin<-fidelitysim::find.Fidelity.linear(time.points=T.points, Fid.End, Fid.T1)
res.plot.Patterns<-rbind(res.plot.Patterns, data.frame(res.lin, FUN="linear", 
                                                       slope=1))

Figure 3.1: Patterns of a fidelity increase (fast, linear or slow) over 6 times points

For our calculation within the simulation we use the determined fractional values of intervention effects to define the degree of deviation from 100% implementation within the design matrix. Three functions in {fidelitysim} provide the implementation of the different patterns of fidelity.

3.3.1 Slow increase of fidelity using exponential function

Here we show an example how to determine the design matrix for a cluster randomized parallel design study (with same design as above) with existing implementation error, where fidelity starts with 40% after implementation and reach after a slow increase 80% at the end of the study. A great slope parameter is chosen which reflect a slow increase more closed to the linear increase.

#study design
#############
I <- 6 # number of clusters
J <- 10 # number of individuals per cluster (used later in simulation-step)
K <- 7 # number of time points
Sw<- round(I/2) # number of cluster within the control group

#parameter Fidelity specification
###############################
Fid.T1<-0.4
Fid.End<-0.8

#parameter tunes the slope for the log and exp functions
slope.seq<-5
  
#exponential function to determine slow increase
(res.exp<-fidelitysim::find.Fidelity.exp(time.points=K, Fid.End, Fid.T1, 
                                         par.slope=slope.seq))
     time Fidelity.Prozent
[1,]    1               40
[2,]    2               44
[3,]    3               49
[4,]    4               55
[5,]    5               62
[6,]    6               70
[7,]    7               80
#determine correspondingdesign matrix
(A1.exp <-fidelitysim::implemMatrix.parallel(nC=I, nT=K, nSw=Sw, 
                                pattern=res.exp[,"Fidelity.Prozent"]/100))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]  0.0 0.00 0.00 0.00 0.00  0.0  0.0
[2,]  0.0 0.00 0.00 0.00 0.00  0.0  0.0
[3,]  0.0 0.00 0.00 0.00 0.00  0.0  0.0
[4,]  0.4 0.44 0.49 0.55 0.62  0.7  0.8
[5,]  0.4 0.44 0.49 0.55 0.62  0.7  0.8
[6,]  0.4 0.44 0.49 0.55 0.62  0.7  0.8

3.3.2 Linear increase of fidelity using linear function

Here we show an example how to determine the design matrix for a cluster randomized parallel design study (with same design as above) with existing implementation error, where fidelity starts with 40% after implementation and reach after a linear increase 80% at the end of the study.

#parameter Fidelity specification
###############################
Fid.T1<-0.4
Fid.End<-0.8


#slope for linear function
m<-(Fid.T1-Fid.End)/(1-(K-1))

#linear increase
(res.lin<-fidelitysim::find.Fidelity.linear(time.points=K, Fid.End, Fid.T1))
     time Fidelity.Prozent
[1,]    1               40
[2,]    2               47
[3,]    3               53
[4,]    4               60
[5,]    5               67
[6,]    6               73
[7,]    7               80
# design matrix of a learning impelementation pattern, linear
(A1.lin <-fidelitysim::implemMatrix.parallel(nC=I, nT=K, nSw=round(I/2), 
                                pattern=res.lin[,"Fidelity.Prozent"]/100))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]  0.0 0.00 0.00  0.0 0.00 0.00  0.0
[2,]  0.0 0.00 0.00  0.0 0.00 0.00  0.0
[3,]  0.0 0.00 0.00  0.0 0.00 0.00  0.0
[4,]  0.4 0.47 0.53  0.6 0.67 0.73  0.8
[5,]  0.4 0.47 0.53  0.6 0.67 0.73  0.8
[6,]  0.4 0.47 0.53  0.6 0.67 0.73  0.8

3.3.3 Fast increase of fidelity using logarithmic function

Here we show an example how to determine the design matrix for a cluster randomized stepped wedge design study (with same design as above) with existing implementation error, where fidelity starts with 20% after implementation and reach after a fast increase 100% at the end of the study. A small slope parameter is chosen for determining the fidelity curve, which reflect a very fast increase.

Sw <- 1 # number of cluster switches per time point can be manually set

#parameter Fidelity specification
###############################
Fid.T1<-0.2
Fid.End<-1

#parameter tunes the slope for the log and exp functions
slope.seq<-0.2

#logistic function to determine fast increase
(res.log<-fidelitysim::find.Fidelity.log(time.points=K-1, Fid.End, Fid.T1, 
                                         par.slope=slope.seq))
     time Fidelity.Prozent
[1,]    1               20
[2,]    2               95
[3,]    3               97
[4,]    4               98
[5,]    5               99
[6,]    6              100
(A1.log <-samplingDataCRT::implemMatrix.SWD(nC=I, nT=K, nSw=Sw, 
                                pattern=res.log[,"Fidelity.Prozent"]/100))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0  0.2 0.95 0.97 0.98 0.99 1.00
[2,]    0  0.0 0.20 0.95 0.97 0.98 0.99
[3,]    0  0.0 0.00 0.20 0.95 0.97 0.98
[4,]    0  0.0 0.00 0.00 0.20 0.95 0.97
[5,]    0  0.0 0.00 0.00 0.00 0.20 0.95
[6,]    0  0.0 0.00 0.00 0.00 0.00 0.20