#' Power simulations for cluster-randomized trials: Stepped Wedge Design, Continuous Outcome.
#'
#' This set of functions utilize iterative simulations to determine
#' approximate power for stepped wedge cluster-randomized controlled trials. Users
#' can modify a variety of parameters to suit the simulations to their
#' desired experimental situation.
#'
#' Runs power simulations for stepped wedge cluster-randomized controlled trials
#' with continuous outcome. The stepped wedge trial design is a type of cross-over
#' design in which clusters change treatments in waves. Initially all the
#' clusters recieve the same standard treatment, and at the end of the trial all
#' of the clusters will be recieving the treatment of interest. More than one
#' cluster can change treatments in a wave, but the order in which clusters
#' change treatments is randomly determined. The outcome of interest is assessed
#' in each cluster during each wave.
#'
#' Users must specify the desired number of simulations, number of subjects per
#' cluster, number of clusters per arm, expected means for each arm,
#' within-cluster variance, between-cluster variance, significance level,
#' analytic method, progress updates, and simulated data set output may also be
#' specified.
#'
#'
#' @param nsim Number of datasets to simulate; accepts integer (required).
#' @param nsubjects Number of subjects per cluster; accepts either a scalar (equal cluster sizes)
#' or a vector of length \code{nclusters} (user-defined size for each cluster) (required).
#' @param nclusters Number of clusters; accepts non-negative integer scalar (required).
#' @param mu0 Expected baseline mean; accepts numeric, default 0. Required..
#' @param mu1 Expected post-treatment mean; accepts numeric. Required.
#' @param steps Number of crossover steps; a baseline step (all clusters in non-treatment group) is assumed.
#' Accepts positive scalar (indicating the total number of steps; clusters per step is obtained by
#' \code{nclusters / steps}) or a vector of non-negative integers corresponding either to the number
#' of clusters to be crossed over at each time point (e.g c(2,4,4,2); nclusters = 10) or the cumulative
#' number of clusters crossed over by a given time point (e.g. c(2,4,8,10); nclusters = 10) (required).
#' @param sigma_sq Within-cluster variance; accepts non-negative numeric scalar (indicating equal within-cluster variances for both
#' treatment groups) or a vector of length 2 specifying within-cluster variances for the non-treatment and treatment groups,
#' respectively (required).
#' @param sigma_b_sq Between-cluster variance; accepts non-negative numeric scalar (indicating equal
#' between-cluster variances for both treatment groups) or a vector of length 2 specifying treatment-specific
#' between-cluster variances (required).
#' @param alpha Significance level. Default = 0.05.
#' @param method Analytical method, either Generalized Linear Mixed Effects Model (GLMM) or
#' Generalized Estimating Equation (GEE). Accepts c('glmm', 'gee') (required); default = 'glmm'.
#' @param quiet When set to FALSE, displays simulation progress and estimated completion time; default is FALSE.
#' @param allSimData Option to output list of all simulated datasets; default = FALSE.
#' @param poorFitOverride Option to override \code{stop()} if more than 25\%
#' of fits fail to converge; default = FALSE.
#' @param lowPowerOverride Option to override \code{stop()} if the power
#' is less than 0.5 after the first 50 simulations and every ten simulations
#' thereafter. On function execution stop, the actual power is printed in the
#' stop message. Default = FALSE. When TRUE, this check is ignored and the
#' calculated power is returned regardless of value.
#' @param timelimitOverride Logical. When FALSE, stops execution if the estimated completion time
#' is more than 2 minutes. Defaults to TRUE.
#' @param seed Option to set.seed. Default is NULL.
#'
#'
#' @return A list with the following components
#' \itemize{
#' \item Character string indicating total number of simulations and simulation type
#' \item Number of simulations
#' \item Data frame with columns "Power" (Estimated statistical power),
#' "lower.95.ci" (Lower 95% confidence interval bound),
#' "upper.95.ci" (Upper 95% confidence interval bound)
#' \item Analytic method used for power estimation
#' \item Significance level
#' \item Vector containing user-defined cluster sizes
#' \item Vector containing user-defined number of clusters
#' \item Data frame reporting ICC, within & between cluster variances for
#' Treatment/Non-Treatment groups at each time point
#' \item Vector containing expected means for each arm based on user inputs
#' \item Data frame containing mean response values for each treatment group at each time point
#' \item Matrix showing cluster crossover at each time point
#' \item Data frame with columns:
#' "Estimate" (Estimate of treatment effect for a given simulation),
#' "Std.err" (Standard error for treatment effect estimate),
#' "Test.statistic" (z-value (for GLMM) or Wald statistic (for GEE)),
#' "p.value",
#' "sig.val" (Is p-value less than alpha?)
#' \item If \code{allSimData = TRUE}, a list of data frames, each containing:
#' "y" (Simulated response value),
#' "trt" (Indicator for treatment group),
#' "time.point" (Indicator for step; "t1" = time point 0)
#' "clust" (Indicator for cluster),
#' "period" (Indicator for at which step a cluster crosses over)
#' }
#'
#' If \code{nofit = T}, a data frame of the simulated data sets, containing:
#'
#' \itemize{
#' \item "arm" (Indicator for treatment arm)
#' \item "cluster" (Indicator for cluster)
#' \item "y1" ... "yn" (Simulated response value for each of the \code{nsim} data sets).
#' }
#'
#' @examples
#'
#' # Estimate power for a trial with 3 steps and 9 clusters in arm 1 (often the
#' # standard-of-care or 'control' arm) at the initiation of the study. Those
#' # clusters have 14 subjects each, with sigma_b = 1 and sigma_b_sq = 1. We
#' # have estimated arm outcome means of 1 and 2.1 in the first and second arms,
#' # respectively, and 100 simulated data sets analyzed by the GLMM method. Using seed = 123,
#' # the resulting power should be 0.82.
#'
#' \dontrun{
#' normal.sw.rct = cps.sw.normal(nsim = 100, nsubjects = 14, nclusters = 9,
#' mu0 = 1, mu1 = 2.1, steps = 3, sigma_sq = 1,
#' sigma_b_sq = 1, alpha = 0.05, method = 'glmm',
#' seed = 123)
#' }
#'
#' @author Alexander R. Bogdan
#' @author Alexandria C. Sakrejda (\email{acbro0@@umass.edu})
#' @author Ken Kleinman (\email{ken.kleinman@@gmail.com})
#'
#' @export
cps.sw.normal = function(nsim = NULL,
nsubjects = NULL,
nclusters = NULL,
mu0 = 0,
mu1 = NULL,
steps = NULL,
sigma_sq = NULL,
sigma_b_sq = NULL,
alpha = 0.05,
method = 'glmm',
quiet = FALSE,
allSimData = FALSE,
poorFitOverride = FALSE,
lowPowerOverride = FALSE,
timelimitOverride = TRUE,
seed = NULL) {
if (!is.null(seed)) {
set.seed(seed = seed)
}
# Create vectors to collect iteration-specific values
est.vector = vector("numeric", length = nsim)
se.vector = vector("numeric", length = nsim)
stat.vector = vector("numeric", length = nsim)
pval.vector = vector("numeric", length = nsim)
converge = vector("logical", length = nsim)
simulated.datasets = list()
# Set start.time for progress iterator & initialize progress bar
start.time = Sys.time()
prog.bar = progress::progress_bar$new(
format = "(:spin) [:bar] :percent eta :eta",
total = nsim,
clear = FALSE,
width = 100
)
prog.bar$tick(0)
# Create wholenumber function
is.wholenumber = function(x, tol = .Machine$double.eps ^ 0.5)
abs(x - round(x)) < tol
# Validate NSIM, NSUBJECTS, NCLUSTERS, mu0, mu1
sim.data.arg.list = list(nsim, nclusters, nsubjects, mu0, mu1)
sim.data.args = unlist(lapply(sim.data.arg.list, is.null))
if (sum(sim.data.args) > 0) {
stop(
"nsim, nclusters, nsubjects, mu0, and mu1 must all be specified. Please review your input values."
)
}
min1.warning = " must be an integer greater than or equal to 1"
if (!is.wholenumber(nsim) || nsim < 1) {
stop(paste0("NSIM", min1.warning))
}
if (!is.wholenumber(nclusters) || nclusters < 1) {
stop(paste0("NCLUSTERS", min1.warning))
}
if (!is.wholenumber(nsubjects) || nsubjects < 1) {
stop(paste0("NSUBJECTS", min1.warning))
}
if (length(nclusters) != 1) {
stop("NCLUSTERS must be a scalar (Total number of clusters)")
}
# Set sample sizes for each cluster (if not already specified)
if (length(nsubjects) == 1) {
nsubjects[1:nclusters] = nsubjects
}
if (!length(nsubjects) %in% c(1, nclusters)) {
stop(
"NSUBJECTS must either be a scalar (indicating equal cluster sizes) or a vector of length NCLUSTERS (specifying cluster sizes for each cluster)."
)
}
# Validate STEPS
if (!is.wholenumber(steps) || steps < 0) {
stop("All values supplied to STEPS must be non-negative integers")
}
if (length(steps) == 1) {
if (!is.wholenumber(nclusters / steps)) {
stop(
"nclusters/steps must be a whole number. See documentation for steps parameter in '?cps.sw.normal'"
)
}
}
if (length(steps) > 1) {
if (sum(steps) != nclusters && max(steps) != nclusters) {
stop(
"Total number of clusters specified by STEPS must either sum to NCLUSTERS or increase monotonically such that max(STEPS) == NCLUSTERS"
)
}
}
if (length(steps) == 1) {
crossover.ind = 1:steps
step.increment = nclusters / steps
step.index = seq(from = step.increment, to = nclusters, by = step.increment)
### Need to figure out how to allocate clusters when nclusters %% steps != 0
}
# Create indexing vector for when SUM(STEPS) == NCLUSTERS & MAX(STEPS) == NCLUSTERS
if (sum(steps) == nclusters) {
step.index = sapply(1:length(steps), function(x)
sum(steps[0:x]))
crossover.ind = 1:length(steps)
}
if (max(steps) == nclusters) {
crossover.ind = 1:length(steps)
step.index = steps
}
# Create vector to store group means at each time point
values.vector = cbind(c(rep(0, length(step.index) * 2)))
# Validate ALPHA
min0.warning = " must be a numeric value greater than 0"
if (!is.numeric(alpha) || alpha < 0 || alpha > 1) {
stop("ALPHA must be a numeric value between 0 - 1")
}
# Validate sigma_sq, SIGMA_B0, SIGMA_B1
sigma_b_sq.warning = " must be a scalar (equal between-cluster variance for both treatment and non-treatment groups)
or a vector of length 2, specifying unique between-cluster variances for the treatment and non-treatment groups."
if (!is.numeric(sigma_sq) || any(sigma_sq < 0)) {
stop("All values supplied to sigma_sq must be numeric values > 0")
}
if (!length(sigma_sq) %in% c(1, 2)) {
stop(
"sigma_sq must be a scalar (equal within-cluster variance for both treatment and non-treatment groups)
or a vector of length 2, specifying unique within-cluster variances for the treatment and non-treament groups."
)
}
if (!is.numeric(sigma_b_sq) || any(sigma_b_sq < 0)) {
stop("All values supplied to sigma_b_sq must be numeric values >= 0")
}
if (!length(sigma_b_sq) %in% c(1, 2)) {
stop("sigma_b_sq", sigma_b_sq.warning)
}
# Set sigma_sq & sigma_b_sq (if not already set)
if (length(sigma_sq) == 1) {
sigma_sq[2] = sigma_sq
}
# Note: If user-defined, sigma_b_sq[2] is additive
if (length(sigma_b_sq) == 2) {
sigma_b_sq[2] = sigma_b_sq[1] + sigma_b_sq[2]
}
if (length(sigma_b_sq) == 1) {
sigma_b_sq[2] = sigma_b_sq
}
# Validate METHOD, QUIET, allSimData
if (!is.element(method, c('glmm', 'gee'))) {
stop(
"METHOD must be either 'glmm' (Generalized Linear Mixed Model)
or 'gee'(Generalized Estimating Equation)"
)
}
if (!is.logical(quiet)) {
stop(
"QUIET must be either TRUE (No progress information shown) or FALSE (Progress information shown)"
)
}
if (!is.logical(allSimData)) {
stop(
"allSimData must be either TRUE (Output all simulated data sets) or FALSE (No simulated data output"
)
}
# Create indicators for CLUSTER, STEP (period) & CROSSOVER (trt)
clust = unlist(lapply(1:nclusters, function(x)
rep(x, length.out = nsubjects[x])))
period = NULL
k = 1 # iterator
for (i in 1:nclusters) {
if ((i - 1) %in% step.index) {
k = k + 1
}
period = append(period, rep(crossover.ind[k], length.out = nsubjects[i]))
}
d = data.frame(period = period, clust = clust)
# Create crossover indicators and melt output columns into single column
for (j in 1:(length(crossover.ind) + 1)) {
d[[paste0("t", j)]] = ifelse(j > d[, 'period'], 1, 0)
}
sim.dat = tidyr::gather(d, key = 'time.point', value = 'trt', c(colnames(d)[-c(1, 2)]))
#sim.dat['time.point'] = as.numeric(gsub("t", "", sim.dat[, 'time.point']))
sim.dat['y'] = 0
sim.dat.holder <- list()
## Create simulation & analysis loop
for (i in 1:nsim) {
# Create vectors of cluster effects
ntrt.cluster.effects = stats::rnorm(nclusters, mean = 0, sd = sqrt(sigma_b_sq[1]))
trt.cluster.effects = stats::rnorm(nclusters, mean = 0, sd = sqrt(sigma_b_sq[2]))
# Add subject specific effects & cluster effects
for (j in 1:nclusters) {
# Assign non-treatment subject & cluster effects
sim.dat['y'] = ifelse(
sim.dat[, 'clust'] == j & sim.dat[, 'trt'] == 0,
stats::rnorm(sum(sim.dat[, 'clust'] == j &
sim.dat[, 'trt'] == 0), mu0, sqrt(sigma_sq[1])) +
ntrt.cluster.effects[j],
sim.dat[, 'y']
)
# Assign treatment subject & cluster effects
sim.dat['y'] = ifelse(
sim.dat[, 'clust'] == j & sim.dat[, 'trt'] == 1,
stats::rnorm(
sum(sim.dat[, 'clust'] == j &
sim.dat[, 'trt'] == 1),
mu1,
sqrt(sigma_sq[2])
) +
trt.cluster.effects[j],
sim.dat[, 'y']
)
}
# Add subject-specific error terms
sim.dat['y'] = sim.dat['y'] + stats::rnorm(nrow(sim.dat), 0, 1)
# Calculate mean values for each group at each time point, for a given simulation
iter.values = cbind(stats::aggregate(y ~ trt + time.point, data = sim.dat, mean)[, 3])
values.vector = values.vector + iter.values
sim.dat$period <- as.factor(sim.dat$period)
sim.dat$clust <- as.factor(sim.dat$clust)
sim.dat$time.point <- as.factor(sim.dat$time.point)
sim.dat$trt <- as.factor(sim.dat$trt)
sim.dat.holder[[i]] <- sim.dat
# Store simulated data sets if allSimData == TRUE
if (allSimData == TRUE) {
simulated.datasets[[i]] = list(sim.dat)
}
}
###################################################################
### DEV NOTE: Hussey & Hughes (2007) does not specify degrees of freedom for significance testing
### GLMM DF = NCLUSTERS[Total # of clusters] - length(STEP.INDEX)[# of steps] - 2[# of treatment groups]
###################################################################
models <- list()
# Set warnings to OFF
# Note: Warnings will still be stored in 'warning.list'
options(warn = -1)
for (i in 1:nsim) {
# Fit GLMM (lmer)
if (method == 'glmm') {
my.mod = lme4::lmer(y ~ trt + time.point + (1 |
clust), data = sim.dat.holder[[i]])
models[[i]] = my.mod
glmm.values <- summary(my.mod)$coefficients
p.val = 2 * stats::pt(-abs(glmm.values['trt1', 't value']), df = nclusters - length(step.index) - 2)
est.vector[i] = glmm.values['trt1', 'Estimate']
se.vector[i] = glmm.values['trt1', 'Std. Error']
stat.vector[i] = glmm.values['trt1', 't value']
pval.vector[i] = p.val
converge[i] = is.null(my.mod@optinfo$conv$lme4$messages)
}
# Set warnings to ON
options(warn = 0)
# Fit GEE (geeglm)
if (method == 'gee') {
sim.dat = dplyr::arrange(sim.dat, clust)
my.mod = geepack::geeglm(
y ~ trt + time.point,
data = sim.dat.holder[[i]],
id = clust,
corstr = "exchangeable"
)
gee.values = summary(my.mod)$coefficients
est.vector[i] = gee.values['trt', 'Estimate']
se.vector[i] = gee.values['trt', 'Std.err']
stat.vector[i] = gee.values['trt', 'Wald']
pval.vector[i] = gee.values['trt', 'Pr(>|W|)']
converge[i] = ifelse(summary(my.mod)$error == 0, TRUE, FALSE)
}
# option to stop the function early if fits are singular
if (poorFitOverride == FALSE && converge[i] == FALSE) {
if (sum(converge == FALSE, na.rm = TRUE) > (nsim * .25)) {
stop(
"more than 25% of simulations are singular fit: check model specifications"
)
}
}
# stop the loop if power is <0.5
if (lowPowerOverride == FALSE && i > 50 && (i %% 10 == 0)) {
sig.val.temp <-
ifelse(pval.vector < alpha, 1, 0)
pval.power.temp <- sum(sig.val.temp, na.rm = TRUE) / i
if (pval.power.temp < 0.5) {
stop(
paste(
"Calculated power is < ",
pval.power.temp,
". Set lowPowerOverride == TRUE to run the simulations anyway.",
sep = ""
)
)
}
}
# Update progress information
if (i == 1) {
avg.iter.time = as.numeric(difftime(Sys.time(), start.time, units = 'secs'))
time.est = avg.iter.time * (nsim - 1) / 60
hr.est = time.est %/% 60
min.est = round(time.est %% 60, 2)
if (min.est > 2 && timelimitOverride == FALSE){
stop(paste0("Estimated completion time: ",
hr.est,
'Hr:',
min.est,
'Min'
))
}
if (quiet == FALSE) {
message(
paste0(
'Begin simulations :: Start Time: ',
Sys.time(),
' :: Estimated completion time: ',
hr.est,
'Hr:',
min.est,
'Min'
)
)
}
}
# Iterate progress bar
prog.bar$update(i / nsim)
Sys.sleep(1 / 100)
if (i == nsim) {
total.est = as.numeric(difftime(Sys.time(), start.time, units = 'secs'))
hr.est = total.est %/% 3600
min.est = total.est %/% 60
sec.est = round(total.est %% 60, 2)
message(
paste0(
"Simulations Complete! Time Completed: ",
Sys.time(),
"\nTotal Runtime: ",
hr.est,
'Hr:',
min.est,
'Min:',
sec.est,
'Sec'
)
)
}
}
## Output objects
# Create object containing summary statement
summary.message = paste0(
"Monte Carlo Power Estimation based on ",
nsim,
" Simulations: Stepped Wedge Design, Continuous Outcome"
)
# Create method object
long.method = switch(method, glmm = 'Generalized Linear Mixed Model',
gee = 'Generalized Estimating Equation')
# Store simulation output in data frame
cps.model.est = data.frame(
Estimate = as.vector(unlist(est.vector)),
Std.err = as.vector(unlist(se.vector)),
Test.statistic = as.vector(unlist(stat.vector)),
p.value = as.vector(unlist(pval.vector)),
converge = as.vector(converge)
)
cps.model.est[, 'sig.val'] = ifelse(cps.model.est[, 'p.value'] < alpha, 1, 0)
# Calculate and store power estimate & confidence intervals
cps.model.temp <- dplyr::filter(cps.model.est, converge == TRUE)
power.parms <- confintCalc(alpha = alpha,
nsim = nsim,
p.val = cps.model.temp[, 'p.value'])
rownames(power.parms) <- "post-treatment"
# Create object containing treatment & time-specific differences
values.vector = values.vector / nsim
group.means = data.frame(
Time.point = c(0, rep(1:(
length(step.index) - 1
), each = 2), length(step.index)),
Treatment = c(0, rep(c(0, 1), length.out = (
length(step.index) - 1
) * 2), 1),
Mean = round(values.vector, 3)
)
# Create object containing cluster sizes
cluster.sizes = nsubjects
# Create object containing number of clusters
n.clusters = t(data.frame(
"Non.Treatment" = c("n.clust" = nclusters[1]),
"Treatment" = c("n.clust" = nclusters[2])
))
# Create object containing variance parameters for each group at each time point
var.parms = t(data.frame(
'Non.Treatment' = c('sigma_sq' = sigma_sq[1], 'sigma_b_sq' = sigma_b_sq[1]),
'Treatment' = c('sigma_sq' = sigma_sq[2], 'sigma_b_sq' = sigma_b_sq[2])
))
# Create crossover matrix output object
crossover.mat = apply(as.matrix(c(0, step.index)), 1,
function(x)
c(rep(1, length.out = x), rep(0, length.out = nclusters - x)))
# Create list containing all output (class 'crtpwr') and return
complete.output = structure(
list(
"overview" = summary.message,
"nsim" = nsim,
"power" = power.parms,
"method" = long.method,
"alpha" = alpha,
"cluster.sizes" = cluster.sizes,
"n.clusters" = n.clusters,
"variance.parms" = var.parms,
"inputs" = c(mu0, mu1),
"means" = group.means,
"model.estimates" = cps.model.est,
"sim.data" = simulated.datasets,
"crossover.matrix" = crossover.mat,
"convergence" = converge
),
class = 'crtpwr'
)
return(complete.output)
}
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