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codeGeneral.R
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codeGeneral.R
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### Function: expit
# This function calculates expit
expit = function(x){
exp(x)/(1+exp(x))
}
### Function: calculateEstimate
# This function calculates the estimator based on the observed data
# Input: data, a data frame containing the observed data;
# estimationMethod, a function which indicates the method of estimation;
# ..., further arguments for the estimator function.
# Output: estimate, an estimated value for the treatment effect.
calculateEstimate = function(data,
estimationMethod,
...){
ellipsis_args = as.list(substitute(list(...)))[-1L]
all_args = c(list(data=data), ellipsis_args)
# Estimate the treatment effect using the function 'estimationMethod'
estimate = do.call(what = estimationMethod,
args = all_args[intersect(x=names(all_args),
y= formalArgs(estimationMethod))])[[1]]
estimate
}
### Function: calculateVariance
# This function calculates the variance of the estimator based on observed data
# Input: data, a data frame containing the observed data;
# estimationMethod, a function which indicates the method of estimation;
# bootstraps, the number of bootstraps;
# ..., further arguments for the estimator function.
# Output: variance, estimated variance of the estimated the treatment effect.
calculateVariance = function(data,
estimationMethod,
bootstraps = 2500,
...){
ellipsis_args = as.list(substitute(list(...)))[-1L]
# Use bootstrap to estimate variance
set.seed(123, "L'Ecuyer")
med.boot = mclapply(1:bootstraps, function(i) {
dataNew = data[sample(nrow(data), nrow(data), replace = TRUE), ]
all_args = c(list(data=dataNew, estimationMethod=estimationMethod), ellipsis_args)
estimate = do.call(what = calculateEstimate,
args = all_args)
estimate
}, mc.cores = 16)
# Calculate variance based on the different bootstrap estimates
variance = var(unlist(med.boot))
variance
}
### Function: calculateCovariance
# This function calculates the covariance of the estimator
# and the previous estimators based on observed data
# Input: data, a data frame containing the observed data;
# estimationMethod, a function which indicates the method of estimation;
# bootstraps, the number of bootstraps;
# ..., further arguments for the estimator function.
# Output: covariance, a vector of the estimated covariances between the
# current estimate and previous estimates, and the estimated variance
# of the current estimate.
calculateCovariance = function(data,
estimationMethod,
bootstraps = 2500,
#analysisTimes,
previousDatasets=list(),
parametersPreviousEstimators = NULL,
...
){
ellipsis_args = as.list(substitute(list(...)))[-1L]
analysisNumber = length(previousDatasets)+1
# Use bootstrap to estimate variance and covariances
set.seed(123, "L'Ecuyer")
med.boot = mclapply(1:bootstraps, function(i) {
#dataNew = data[sample(nrow(data), nrow(data), replace = TRUE), ]
selection = sample(data$id, nrow(data), replace = TRUE)
selectionFreq = as.data.frame(table(selection))
sub_sample_df = data[data$id%in%selection,]
match_freq = selectionFreq[match(sub_sample_df$id, selectionFreq$selection),]
sub_sample_df$Freq = match_freq$Freq
selected_rows = rep(1:nrow(sub_sample_df), sub_sample_df$Freq)
dataNew = sub_sample_df[selected_rows,]
estimate=c()
all_args = c(list(data = dataNew, estimationMethod = estimationMethod), ellipsis_args)
estimate[analysisNumber] = do.call(what = calculateEstimate,
args = all_args)[[1]]
for(j in 1:(analysisNumber-1)){
# Make a dataset with data available at analysis j and
# indicate which participants had the primary endpoint observed
# at analysis j (i.e., are part of cohort 1)
dataNew1 = previousDatasets[[j]]
sub_sample_df = dataNew1[dataNew1$id%in%selection,]
match_freq = selectionFreq[match(sub_sample_df$id, selectionFreq$selection),]
sub_sample_df$Freq = match_freq$Freq
selected_rows = rep(1:nrow(sub_sample_df), sub_sample_df$Freq)
dataNew1 = sub_sample_df[selected_rows,]
if(is.list(parametersPreviousEstimators)==TRUE){
ellipsis_args[names(parametersPreviousEstimators[[j]])]=parametersPreviousEstimators[[j]]
all_args = c(list(data = dataNew1, estimationMethod = estimationMethod), ellipsis_args)
estimate[j] = do.call(what = calculateEstimate,
args = all_args)[[1]]
}else{
all_args = c(list(data = dataNew1, estimationMethod = estimationMethod), ellipsis_args)
estimate[j] = do.call(what = calculateEstimate,
args = all_args)[[1]]
}
}
estimate
}, mc.cores = 16)
# Calculate covariances and variance based on the different bootstrap estimates
dataCov = data.frame(matrix(unlist(med.boot), nrow=length(med.boot), byrow=TRUE))
covariance = cov(dataCov)[analysisNumber,]
covariance
}
### Function: updateEstimate
# This function orthogonalizes the original estimates.
# Input: covMatrix, covariance matrix of original estimates up to current analysis;
# originalEstimates, original estimates up to current analysis.
# Output: a list with (1) estimateUpdated, the updated/orthogonalized
# estimate at current analysis;
# (2) varianceUpdated, the variance of the updated/orthogonalized
# estimate at current analysis.
updateEstimate = function(covMatrixOriginal,
estimatesOriginal){
# k is the analysis for which we want to update the estimate
k = length(estimatesOriginal)
# covMatrixProjection is the kxk covariance matrix of
# (theta_k-theta_1, ..., theta_k-theta_{k-1}, theta_k)
covMatrixProjection = matrix(1, k, k)*covMatrixOriginal[k,k] -
as.vector(rep(1, k))%*%t(as.vector(c(covMatrixOriginal[1:k-1,k], 0))) -
t(as.vector(rep(1, k))%*%t(as.vector(c(covMatrixOriginal[1:k-1,k], 0)))) +
rbind(cbind(covMatrixOriginal[1:k-1,1:k-1], 0), 0)
# A is the Cholesky decomposition of covMatrixProjection
A = chol(covMatrixProjection)
# Atilde is a kx(k-1) matrix with the first k-1 columns of A
Atilde = A[,-k]
# Ak is a kx1 vector equal to the kth column of A
Ak = A[,k]
# W equals the vector # (theta_k-theta_1, ..., theta_k-theta_{k-1}, theta_k)
W = c(estimatesOriginal[k] - estimatesOriginal[1:k-1], estimatesOriginal[k])
# scale are the orthogonalizing 'coefficients'
scale = solve(A)%*%(diag(k)-Atilde%*%solve(t(Atilde)%*%Atilde)
%*%t(Atilde))%*%Ak
# updatedEstimate is the updated/orthogonalized estimate at analysis k
estimateUpdated = t(scale)%*%W
# updatedVariance is the variance of the
# updated/orthogonalized estimate at analysis k
varianceUpdated = t(scale)%*%t(A)%*%A%*%scale
list(cbind(estimateUpdated, varianceUpdated))
}
### Function: interimDecision
# This function makes a decision based on a test statistic
# Input: testStatistic, a numeric value for the calculated test statistic;
# alpha, the (total) siginificance level alpha;
# alternative, a character string specifying the alternative hypothesis;
# boundaries, type of alpha spending function to use (see ldBounds in ldbounds package);
# plannedAnalyses, total number of planned analyses (interim and final);
# previousInformationTimes, information times of estimates up to the previous analysis;
# currentInformationTime, information time of current analysis.
# Output: decision, a logical value whether the null hypothesis is rejected or not.
interimDecision = function(testStatistic,
alpha = 0.05,
alternative = "two.sided",
boundaries = 1,
plannedAnalyses,
previousInformationTimes,
currentInformationTime
){
# Number of current analysis
analysisNumber=length(previousInformationTimes)+1
# Make decision based on test statistics and bounds
if(alternative=="two.sided"){
if(analysisNumber==plannedAnalyses){
bound=ldBounds(t= c(previousInformationTimes, 1),
iuse = c(boundaries, boundaries),
asf = NULL,
alpha = c(alpha/2, alpha/2))$upper.bounds[analysisNumber]
}else{
bound=ldBounds(t= c(previousInformationTimes, currentInformationTime),
iuse = c(boundaries, boundaries),
asf = NULL,
alpha = c(alpha/2, alpha/2))$upper.bounds[analysisNumber]
}
decision = as.logical(abs(testStatistic)>bound)
}else{
if(analysisNumber==plannedAnalyses){
bound=bounds(t= c(previousInformationTimes, 1),
iuse = boundaries,
asf = NULL,
alpha = alpha)$upper.bounds[analysisNumber]
}else{
bound=bounds(t= c(previousInformationTimes, currentInformationTime),
iuse = boundaries,
asf = NULL,
alpha = alpha)$upper.bounds[analysisNumber]
}
if(alternative=="less"){
decision = as.logical(testStatistic<(-bound))
}else{
decision = as.logical(testStatistic>bound)
}
}
decision
}
### Function: interimInformation
# This function calculates the information based on observed data for chosen estimator
# Input: data, a data frame containing the observed data;
# totalInformation, a value indicating the total/maximum information;
# analysisNumber, number of analysis;
# estimationMethod, method of estimation for the treatment effect;
# update, a character string indicating whether information needs to be calculated
# based on updated estimate ("yes") or based on original estimate ("no");
# previousEstimatesOriginal, vector of original estimates up to the previous analysis;
# previousCovMatrixOriginal, covariance matrix of original estimates up to the previous analysis;
# ..., further arguments for the estimator function or calculation of variance (ie, number of bootstraps).
# Output: a list with (1) information, information available in current data;
# (2) informationTime, information time (information available relative to total information)
interimInformation = function(data,
totalInformation,
analysisNumber,
#analysisTimes,
previousDatasets=list(),
estimationMethod,
update,
previousEstimatesOriginal=c(),
previousCovMatrixOriginal=c(),
parametersPreviousEstimators = NULL,
...){
ellipsis_args = as.list(substitute(list(...)))[-1L]
# Estimate (original) treatment effect and corresponding variance
all_args = c(list(data=data, estimationMethod=estimationMethod), ellipsis_args)
estimate = do.call(what = calculateEstimate,
args = all_args)
if(update == "no" | analysisNumber==1){
variance = do.call(what = calculateVariance,
args = all_args)
# Calculate (original) information and information time
information = 1/variance
informationTime = information/totalInformation
}else{
all_args_cov = c(all_args,
list(previousDatasets=previousDatasets))
all_args_cov = c(all_args_cov,
list(parametersPreviousEstimators = parametersPreviousEstimators))
covariance = do.call(what = calculateCovariance,
args = all_args_cov)
# Update covariance matrix of original estimates
covMatrix = cbind(rbind(previousCovMatrixOriginal, covariance[-analysisNumber]), covariance)
# Update/orthogonalize the original estimate at the (interim) analysis
# based on the original covariance matrix and
# the original estimates
updated = updateEstimate(
covMatrixOriginal=covMatrix,
estimatesOriginal=c(previousEstimatesOriginal, estimate))
information = 1/updated[[1]][1,2]
informationTime = information/totalInformation
}
list(information, informationTime)
}
### Function: calculateCorrectionTerm
calculateCorrectionTerm = function(data,
y0_formula,
y1_formula){
number = length(names(select(data, contains(".r_"))))
n1=length(which(data[,paste(".r_",number,sep='')]==1&data[,"treatment"]==1))
n0=length(which(data[,paste(".r_",number,sep='')]==1&data[,"treatment"]==0))
p0=dim(model.matrix(y0_formula, data))[2]
p1=dim(model.matrix(y1_formula, data))[2]
correctionTerm = (1/(n1-p1)+1/(n0-p0))/(1/(n1-1)+1/(n0-1))
correctionTerm
}
### Function: interimAnalysis
# This function performs an interim analysis based on observed data
# Input: data, a data frame containing the observed data;
# totalInformation, a value indicating the total/maximum information;
# estimationMethod, method of estimation for the treatment effect;
# previousEstimatesOriginal, vector of original estimates up to the previous analysis;
# previousCovMatrixOriginal, covariance matrix of original estimates up to the previous analysis;
# previousInformationTimesOriginal, information times of original estimates up to the previous analysis;
# previousInformationTimesUpdated, information times of updated estimates up to the previous analysis;
# null.value, a number indicating the value of theta under the null hypothesis,
# alpha, the (total) siginificance level alpha;
# alternative, a character string specifying the alternative hypothesis,
# must be one of "two.sided" (default), "greater" or "less";
# boundaries, type of alpha spending function to use (see bounds in ldbounds package);
# plannedAnalyses, total number of analyses (interim and final);
# ..., further arguments for the estimator function or calculation of covariance (ie, number of bootstraps).
# Output: a list with (1) estimateOriginal, original estimate at current analysis;
# (2) standardErrorOriginal, estimated standard error of original estimate at current analysis;
# (3) testStatisticOriginal, test statistics corresponding with original estimate at current analysis;
# (4) decisionOriginal, decision corresponding with original estimate at current analysis;
# (5) informationTimeOriginal, information time corresponding with original estimate at current analysis;
# (6) estimateUpdated, updated/orthogonalized estimate at current analysis;
# (7) standardErrorUpdated, estimated standard error of updated/orthogonalized
# estimate at current analysis;
# (8) testStatisticUpdated, test statistics corresponding with updated estimate at current analysis;
# (9) decisionUpdated, decision corresponding with updated estimate at current analysis;
# (10) informationTimeUpdated, information time corresponding with updated estimate at current analysis;
# (11) covMatrixOriginal, covariance matrix of original estimates up to current analysis.
interimAnalysis = function(data,
totalInformation,
estimationMethod,
previousEstimatesOriginal=c(),
previousCovMatrixOriginal=c(),
previousInformationTimesOriginal=c(),
previousInformationTimesUpdated=c(),
#analysisTimes,
previousDatasets=list(),
null.value = 0,
alpha = 0.025,
alternative = "two.sided",
boundaries = 1,
plannedAnalyses,
parametersPreviousEstimators = NULL,
correction="no",
...){
# Number of current analysis
analysisNumber = length(previousInformationTimesOriginal)+1
ellipsis_args = as.list(substitute(list(...)))[-1L]
# Estimate (original) treatment effect and corresponding variance
# based on the estimator in Appendix C
all_args = c(list(data=data, estimationMethod=estimationMethod), ellipsis_args)
estimateOriginal = do.call(what = calculateEstimate,
args = all_args)
# Update covariance matrix of original estimates
if(analysisNumber==1){
variance = do.call(what = calculateVariance,
args = all_args)
covMatrixOriginal = variance
standardErrorOriginal = sqrt(variance)
}else{
all_args_cov = c(all_args,
list(previousDatasets=previousDatasets))
all_args_cov = c(all_args_cov,
list(parametersPreviousEstimators = parametersPreviousEstimators))
covariance = do.call(what = calculateCovariance,
args = all_args_cov)
covMatrixOriginal = cbind(rbind(previousCovMatrixOriginal, covariance[-analysisNumber]), covariance)
standardErrorOriginal = sqrt(covariance[analysisNumber])
}
if(correction=="yes"){
correctionTerm = do.call(what = calculateCorrectionTerm,
args = all_args[intersect(x=names(all_args),
y= formalArgs(calculateCorrectionTerm))])
}else{
correctionTerm = 1
}
# Calculate (original) SE, information, information time and test statistic
standardErrorOriginal = standardErrorOriginal*sqrt(correctionTerm)
informationOriginal=1/(standardErrorOriginal)^2
informationTimeOriginal = informationOriginal/totalInformation
testStatisticOriginal = estimateOriginal/standardErrorOriginal
if(analysisNumber==1){
# Calculate updated variance and
# calculate (updated) SE, information, information time and test statistic
estimateUpdated = estimateOriginal
standardErrorUpdated = standardErrorOriginal
informationUpdated = informationOriginal
informationTimeUpdated = informationTimeOriginal
testStatisticUpdated = testStatisticOriginal
}else{
# Update/orthogonalize the original estimate at the (interim) analysis
# based on the original covariance matrix and
# the original estimates
updated = updateEstimate(
covMatrixOriginal=covMatrixOriginal,
estimatesOriginal=c(previousEstimatesOriginal, estimateOriginal))
estimateUpdated = updated[[1]][1,1]
standardErrorUpdated = sqrt(updated[[1]][1,2])*sqrt(correctionTerm)
informationUpdated=1/(standardErrorUpdated)^2
informationTimeUpdated = informationUpdated/totalInformation
testStatisticUpdated = estimateUpdated/standardErrorUpdated
}
decisionOriginal = interimDecision(testStatistic=testStatisticOriginal,
alpha = alpha,
alternative = alternative,
boundaries = boundaries,
plannedAnalyses=plannedAnalyses,
previousInformationTimes=previousInformationTimesOriginal,
currentInformationTime=informationTimeOriginal
)
decisionUpdated = interimDecision(testStatistic=testStatisticUpdated,
alpha = alpha,
alternative = alternative,
boundaries = boundaries,
plannedAnalyses=plannedAnalyses,
previousInformationTimes=previousInformationTimesUpdated,
currentInformationTime=informationTimeUpdated)
list(estimateOriginal=estimateOriginal,
standardErrorOriginal=standardErrorOriginal,
testStatisticOriginal=testStatisticOriginal,
decisionOriginal=decisionOriginal,
informationTimeOriginal=informationTimeOriginal,
estimateUpdated=estimateUpdated,
standardErrorUpdated=standardErrorUpdated,
testStatisticUpdated=testStatisticUpdated,
decisionUpdated=decisionUpdated,
informationTimeUpdated=informationTimeUpdated,
covMatrixOriginal=covMatrixOriginal)
}
### Function: standardization
# This function implements the estimator in Appendix C of the paper
# Input: data, a data frame containing the observed data;
# y0_formula, the model to be fitted for the outcome under control;
# y1_formula, the model to be fitted for the outcome under treatment;
# family, a description of the error distribution and link function to be used in the model.
# Output: a list with
# (1) estimate, original estimate based on estimator in Appendix C;
# (2) y1_pred, predictions of the outcome under treatment for all recruited participants;
# (3) y0_pred, predictions of the outcome under control for all recruited participants;
standardization = function(data,
y0_formula,
y1_formula,
family){
# Make a dataset with current cohort 1 data;
# i.e., the cohort of patients used to fit working models
number = length(names(select(data, contains(".r_"))))
dataCoh1 = data[which(data[,paste(".r_",number,sep='')]==1),]
# Fit working models under control and treatment
y0_mod = glm(y0_formula, family = family, data = dataCoh1[dataCoh1$treatment==0,])
y1_mod = glm(y1_formula, family = family, data = dataCoh1[dataCoh1$treatment==1,])
# Make predictions under control and treatment for all patients in the dataset
y0_pred = predict(y0_mod, newdata = data, type = "response")
y1_pred = predict(y1_mod, newdata = data, type = "response")
# Estimate treatment effect
estimate = mean(y1_pred) - mean(y0_pred)
list(estimate, y1_pred, y0_pred)
}
### Function: data_at_time_t
# This function determines the dataset at time 'analysis_time'
# Input: data, a data frame containing all observed data;
# id_column, column name of identifier;
# analysis_time, time of the analysis;
# enrollment_time, column name of the enrollment times;
# treatment_column, column name of treatment variable;
# covariate_columns, column names of covariates;
# outcome_columns, column names of outcomes;
# outcome_times, column names of outcome times.
# Output: data, a data frame with the observed data at time 'analysis_time'
data_at_time_t <-
function(
data,
id_column,
analysis_time,
enrollment_time,
treatment_column,
covariate_columns,
outcome_columns,
outcome_times
) {
# Select only relevant columns
data <-
data[ which(data[, enrollment_time] <= analysis_time),
c(id_column, covariate_columns, enrollment_time,
treatment_column, outcome_times, outcome_columns)
]
r_matrix <- 1*(!is.na(data[, outcome_columns]))
r_matrix[which(data[, outcome_times] > analysis_time, arr.ind = TRUE)] <- NA
for(i in 1:length(outcome_times)){
data[which(data[, outcome_times[i]] > analysis_time),
outcome_columns[i]] <- NA
}
data <-
data.frame(
data,
setNames(
object = data.frame(r_matrix),
nm = paste0(".r_", 1:length(outcome_columns))
)
) %>%
as_tibble()
return(data)
}