R/IWTaov.R
In alessiapini/fdatest: Interval Wise Testing for Functional Data
Defines functions
IWTaov
Documented in
IWTaov
#' @title Interval Wise Testing procedure for testing functional analysis of variance
#'
#' @description The function implements the Interval Wise Testing procedure for testing mean differences between several
#' functional populations in a one-way or multi-way functional analysis of variance framework.
#' Functional data are tested locally and unadjusted and adjusted p-value
#' functions are provided. The unadjusted p-value function controls the point-wise error rate.
#' The adjusted p-value function controls the
#' interval-wise error rate.
#'
#' @param formula An object of class "\code{\link{formula}}" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
#' The output variable of the formula can be either a matrix of dimension \code{c(n,J)} collecting the pointwise evaluations of \code{n} functional data on the same grid of \code{J} points, or a \code{fd} object from the package \code{fda}.
#'
#' @param B The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is \code{B=1000}.
#'
#' @param method Permutation method used to calculate the p-value of permutation tests. Choose "\code{residuals}" for the permutations of residuals under the reduced model, according to the Freedman and Lane scheme, and "\code{responses}" for the permutation of the responses, according to the Manly scheme.
#'
#' @param dx Used only if a \code{fd} object is provided. In this case, \code{dx} is the size of the discretization step of the grid used to evaluate functional data.
#' If set to \code{NULL}, a grid of size 100 is used. Default is \code{NULL}.
#'
#' @param recycle Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Default is \code{TRUE}.
#'
#'
#'
#' @return \code{IWTaov} returns an object of \code{\link{class}} "\code{IWTaov}". The function \code{summary} is used to obtain and print a summary of the results.
#' An object of class "\code{IWTaov}" is a list containing at least the following components:
#' \item{call}{The matched call.}
#' \item{design_matrix}{The design matrix of the functional-on-scalar linear model.}
#' \item{unadjusted_pval_F}{Evaluation on a grid of the unadjusted p-value function of the functional F-test.}
#' \item{pval_matrix_F}{Matrix of dimensions \code{c(p,p)} of the p-values of the intervalwise F-tests. The element \code{(i,j)} of matrix \code{pval.matrix} contains the p-value of the test of interval indexed by \code{(j,j+1,...,j+(p-i))}.}
#' \item{adjusted_pval_F}{Evaluation on a grid of the adjusted p-value function of the functional F-test.}
#' \item{unadjusted_pval_factors}{Evaluation on a grid of the unadjusted p-value function of the functional F-tests on each factor of the analysis of variance (rows).}
#' \item{pval.matrix.factors}{Array of dimensions \code{c(L+1,p,p)} of the p-values of the multivariate F-tests on factors. The element \code{(l,i,j)} of array \code{pval.matrix} contains the p-value of the joint NPC test on factor \code{l} of the components \code{(j,j+1,...,j+(p-i))}.}
#' \item{adjusted.pval.factors}{adjusted p-values of the functional F-tests on each factor of the analysis of variance (rows) and each basis coefficient (columns).}
#' \item{data.eval}{Evaluation on a fine uniform grid of the functional data obtained through the basis expansion.}
#' \item{coeff.regr.eval}{Evaluation on a fine uniform grid of the functional regression coefficients.}
#' \item{fitted.eval}{Evaluation on a fine uniform grid of the fitted values of the functional regression.}
#' \item{residuals.eval}{Evaluation on a fine uniform grid of the residuals of the functional regression.}
#' \item{R2.eval}{Evaluation on a fine uniform grid of the functional R-squared of the regression.}
#' \item{heatmap.matrix.F}{Heatmap matrix of p-values of functional F-test (used only for plots).}
#' \item{heatmap.matrix.factors}{Heatmap matrix of p-values of functional F-tests on each factor of the analysis of variance (used only for plots).}
#'
#' @seealso See \code{\link{summary.IWTaov}} for summaries and \code{\link{plot.IWTaov}} for plotting the results.
#' See \code{\link{ITPaov}} for a functional analysis of variance test based on B-spline basis expansion.
#' See also \code{\link{IWTlm}} to fit and test a functional-on-scalar linear model applying the IWT, and \code{\link{IWT1}}, \code{\link{IWT2}} for one-population and two-population tests.
#'
#'
#' @examples
#' # Importing the NASA temperatures data set
#' data(NASAtemp)
#' temperature <- rbind(NASAtemp$milan,NASAtemp$paris)
#' groups <- c(rep(0,22),rep(1,22))
#'
#' # Performing the IWT
#' IWT.result <- IWTaov(temperature ~ groups,B=1000)
#'
#' # Summary of the ITP results
#' summary(IWT.result)
#'
#' # Plot of the IWT results
#' layout(1)
#' plot(IWT.result)
#'
#' # All graphics on the same device
#' layout(matrix(1:4,nrow=2,byrow=FALSE))
#' plot(IWT.result,main='NASA data', plot.adjpval = TRUE,xlab='Day',xrange=c(1,365))
#'
#' @references
#' Pini, A., & Vantini, S. (2017). Interval-wise testing for functional data. \emph{Journal of Nonparametric Statistics}, 29(2), 407-424
#'
#' Pini, A., Vantini, S., Colosimo, B. M., & Grasso, M. (2018). Domain‐selective functional analysis of variance for supervised statistical profile monitoring of signal data. \emph{Journal of the Royal Statistical Society: Series C (Applied Statistics)} 67(1), 55-81.
#'
#' Abramowicz, K., Hager, C. K., Pini, A., Schelin, L., Sjostedt de Luna, S., & Vantini, S. (2018).
#' Nonparametric inference for functional‐on‐scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament. \emph{Scandinavian Journal of Statistics} 45(4), 1036-1061.
#'
#' D. Freedman and D. Lane (1983). A Nonstochastic Interpretation of Reported Significance Levels. \emph{Journal of Business & Economic Statistics} 1.4, 292-298.
#'
#' B. F. J. Manly (2006). Randomization, \emph{Bootstrap and Monte Carlo Methods in Biology}. Vol. 70. CRC Press.
#'
#' @export
IWTaov <- function(formula,B=1000,method='residuals',dx=NULL,recycle=TRUE){
pval.correct <- function(pval.matrix){
matrice_pval_2_2x <- cbind(pval.matrix,pval.matrix)
p <- dim(pval.matrix)[2]
matrice_pval_2_2x <- matrice_pval_2_2x[,(2*p):1]
corrected.pval <- numeric(p)
corrected.pval.matrix <- matrix(nrow=p,ncol=p)
corrected.pval.matrix[p,] <- pval.matrix[p,p:1]
for(var in 1:p){
pval_var <- matrice_pval_2_2x[p,var]
inizio <- var
fine <- var #inizio fisso, fine aumenta salendo nelle righe
for(riga in (p-1):1){
fine <- fine + 1
pval_cono <- matrice_pval_2_2x[riga,inizio:fine]
pval_var <- max(pval_var,pval_cono,na.rm=TRUE)
corrected.pval.matrix[riga,var] <- pval_var
}
corrected.pval[var] <- pval_var
}
corrected.pval <- corrected.pval[p:1]
corrected.pval.matrix <- corrected.pval.matrix[,p:1]
return(corrected.pval.matrix)
}
stat_lm_glob <- function(anova){
result <- summary.lm(anova)$f[1]
return(result)
}
stat_aov_part <- function(anova){
result <- summary(anova)[[1]][,4]
result <- result[-length(result)]
return(result)
}
extract.residuals = function(anova){
return(anova$residuals)
}
extract.fitted = function(anova){
return(anova$fitted)
}
#extract.pval <- function(anova){
# result <- summary(anova)[[1]][,5]
# result <- result[-length(result)]
# return(result)
#}
env <- environment(formula)
variables = all.vars(formula)
y.name = variables[1]
covariates.names <- colnames(attr(terms(formula),"factors"))
#data.all = model.frame(formula)
cl <- match.call()
data <- get(y.name,envir = env)
if(is.fd(data)){ # data is a functional data object
rangeval <- data$basis$rangeval
if(is.null(dx)){
dx <- (rangeval[2]-rangeval[1])*0.01
}
abscissa <- seq(rangeval[1],rangeval[2],by=dx)
coeff <- t(eval.fd(fdobj=data,evalarg=abscissa))
}else if(is.matrix(data)){
coeff <- data
}else{
stop("First argument of the formula must be either a functional data object or a matrix.")
}
#design.matrix = model.matrix(formula)
#mf = model.frame(formula)
#data = model.response(mf)
dummynames.all <- colnames(attr(terms(formula),"factors"))
formula.const <- deparse(formula[[3]],width.cutoff = 500L) #extracting the part after ~ on formula. this will not work if the formula is longer than 500 char
formula.discrete <- as.formula(paste('coeff ~',formula.const),env=environment())
design.matrix = model.matrix(formula.discrete)
mf = model.frame(formula.discrete)
#var.names = variables[-1]
#nvar = length(var.names)
n <- dim(coeff)[1]
J <- dim(coeff)[2]
p <- dim(coeff)[2]
npt <- J
print('Point-wise tests')
#univariate permutations
coeffnames <- paste('coeff[,',as.character(1:p),']',sep='')
formula.coeff <- paste(coeffnames,'~',formula.const)
formula.coeff <- sapply(formula.coeff,as.formula,env=environment())
aovcoeff1 <- aov(formula.coeff[[1]],data=mf)
var.names <- rownames(summary(aovcoeff1)[[1]])
df.vars <- summary(aovcoeff1)[[1]][,1]
df.residuals <- df.vars[length(df.vars)]
var.names <- var.names[-length(var.names)]
nvar = length(var.names)
for(ii in 1:nvar){
var.names[ii] <- gsub(' ' , '',var.names[ii])
}
index.vars <- cbind(c(2,(cumsum(df.vars)+2)[-length(df.vars)]),cumsum(df.vars)+1)
regr0 = lm.fit(design.matrix,coeff)
#pval_parametric <- sapply(aov0,extract.pval)
MS0 <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MS0[var,] <- colSums(rbind(regr0$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T0_part <- MS0[1:nvar,] / matrix(MS0[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE)
Sigma <- chol2inv(regr0$qr$qr)
resvar <- colSums(regr0$residuals^2)/regr0$df.residual
if(nvar >1){
T0_glob <- colSums((regr0$fitted - matrix(colMeans(regr0$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
}else if(nvar==1){ #only one factor -> the permutation of the residuals is equivalent to the one of responses
method <- 'responses'
T0_glob <- colSums((regr0$fitted - matrix(colMeans(regr0$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
}else if(nvar==0){
method = 'responses' # model with only intercept -> the permutation of the residuals is equivalent to the one of responses
T0_glob = numeric(p)
}
#calculate residuals
if(method=='residuals'){
#n residuals for each coefficient of basis expansion (1:p)
#and for each partial test + global test (nvar+1)
#saved in array of dim (nvar+1,n,p)
design.matrix.names2 = design.matrix
var.names2 = var.names
if(length(grep('factor',formula.const))>0){
index.factor = grep('factor',var.names)
replace.names = paste('group',(1:length(index.factor)),sep='')
var.names2[index.factor] = replace.names
colnames(design.matrix.names2) = var.names2
}
residui = array(dim=c(nvar,n,p))
fitted_part = array(dim=c(nvar,n,p)) # fitted values of the reduced model (different for each test)
formula.coeff_part = vector('list',nvar)
regr0_part = vector('list',nvar)
dummy.interaz <- grep(':',dummynames.all)
#coeff.perm_part = array(dim=c(nvar+1,n,p))
for(ii in 1:(nvar)){ #no test on intercept
var.ii = var.names2[ii]
variables.reduced = var.names2[-which(var.names2==var.ii)] #removing the current variable to test
if(length(grep(':',var.ii))>0){ # testing interaction
#print('interaz')
var12 <- strsplit(var.ii,':')
var1 <- var12[[1]][1]
var2 <- var12[[1]][2]
dummy.test1 <- grep(var1,dummynames.all)
dummy.test2 <- grep(var2,dummynames.all)
dummy.test <- intersect(dummy.test1,dummy.test2)
dummynames.reduced <- dummynames.all[-dummy.test]
}else{
#print('nointeraz')
dummy.test <- grep(var.ii,dummynames.all)
dummy.test <- setdiff(dummy.test,dummy.interaz)
dummynames.reduced <- dummynames.all[-dummy.test]
}
if(nvar>1){
formula.temp = paste(dummynames.reduced,collapse=' + ')
}else{
formula.temp = '1' #removing the only variable -> reduced model only has intercept term
}
formula.coeff.temp <- paste(coeffnames,'~',formula.temp)
formula.coeff_part[[ii]] <- sapply(formula.coeff.temp,as.formula,env=environment())
regr0_part[[ii]] = lapply(formula.coeff_part[[ii]],lm)
residui[ii,,] = simplify2array(lapply(regr0_part[[ii]],extract.residuals))
fitted_part[ii,,] = simplify2array(lapply(regr0_part[[ii]],extract.fitted))
}
}
T_glob <- matrix(ncol=p,nrow=B)
T_part = array(dim=c(B,nvar,p))
for (perm in 1:B){
# the F test is the same for both methods
if(nvar >0){
permutazioni <- sample(n)
coeff_perm <- coeff[permutazioni,]
}else{ # testing intercept -> permute signs
signs <- rbinom(n,1,0.5)*2 - 1
coeff_perm <- coeff*signs
}
regr_perm = lm.fit(design.matrix,coeff_perm)
Sigma <- chol2inv(regr_perm$qr$qr)
resvar <- colSums(regr_perm$residuals^2)/regr0$df.residual
if(nvar > 0)
T_glob[perm,] <- colSums((regr_perm$fitted - matrix(colMeans(regr_perm$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
# partial tests: differ depending on the method
if(method=='responses'){
MSperm <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MSperm[var,] <- colSums(rbind(regr_perm$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T_part[perm,,] <- MSperm[1:nvar,] / matrix(MSperm[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE)
}else if(method=='residuals'){
residui_perm = residui[,permutazioni,]
aov_perm_part = vector('list',nvar)
for(ii in 1:(nvar)){
coeff_perm = fitted_part[ii,,] + residui_perm[ii,,]
regr_perm = lm.fit(design.matrix,coeff_perm)
MSperm <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MSperm[var,] <- colSums(rbind(regr_perm$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T_part[perm,ii,] <- (MSperm[1:nvar,] / matrix(MSperm[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE))[ii,]
}
}
}
pval_glob <- numeric(p)
pval_part = matrix(nrow=nvar,ncol=p)
for(i in 1:p){
pval_glob[i] <- sum(T_glob[,i]>=T0_glob[i])/B
pval_part[,i] = colSums(T_part[,,i]>=matrix(T0_part[,i],nrow=B,ncol=nvar,byrow=TRUE))/B
}
#combination
print('Interval-wise tests')
#asymmetric combination matrix:
matrice_pval_asymm_glob <- matrix(nrow=p,ncol=p)
matrice_pval_asymm_glob[p,] <- pval_glob[1:p]
T0_2x_glob <- c(T0_glob,T0_glob)
T_2x_glob <- cbind(T_glob,T_glob)
matrice_pval_asymm_part <- array(dim=c(nvar,p,p))
T0_2x_part <- cbind(T0_part,T0_part)
T_2x_part = array(dim=c(B,nvar,p*2))
for(ii in 1:(nvar)){
matrice_pval_asymm_part[ii,p,] <- pval_part[ii,1:p]
T_2x_part[,ii,] <- cbind(T_part[,ii,],T_part[,ii,])
}
maxrow <- 1
if(recycle==TRUE){
for(i in (p-1):maxrow){
for(j in 1:p){
inf <- j
sup <- (p-i)+j
T0_temp <- sum(T0_2x_glob[inf:sup])
T_temp <- rowSums(T_2x_glob[,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_glob[i,j] <- pval_temp
for(ii in 1:(nvar )){
T0_temp <- sum(T0_2x_part[ii,inf:sup])
T_temp <- rowSums(T_2x_part[,ii,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_part[ii,i,j] <- pval_temp
}
}
print(paste('creating the p-value matrix: end of row ',as.character(p-i+1),' out of ',as.character(p),sep=''))
}
}else{
for(i in (p-1):maxrow){ # rows
for(j in 1:i){ # columns
inf <- j
sup <- (p-i)+j
T0_temp <- sum(T0_2x_glob[inf:sup])
T_temp <- rowSums(T_2x_glob[,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_glob[i,j] <- pval_temp
for(ii in 1:(nvar )){
T0_temp <- sum(T0_2x_part[ii,inf:sup])
T_temp <- rowSums(T_2x_part[,ii,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_part[ii,i,j] <- pval_temp
}
}
print(paste('creating the p-value matrix: end of row ',as.character(p-i+1),' out of ',as.character(p),sep=''))
}
}
corrected.pval.matrix_glob <- pval.correct(matrice_pval_asymm_glob)
corrected.pval_glob <- corrected.pval.matrix_glob[1,]
corrected.pval_part = matrix(nrow=nvar,ncol=p)
corrected.pval.matrix_part = array(dim=c(nvar,p,p))
for(ii in 1:(nvar)){
corrected.pval.matrix_part[ii,,] = pval.correct(matrice_pval_asymm_part[ii,,])
corrected.pval_part[ii,] <- corrected.pval.matrix_part[ii,1,]
}
coeff.regr = regr0$coeff
coeff.t <- coeff.regr
fitted.regr = regr0$fitted.values
fitted.t <- fitted.regr
rownames(corrected.pval_part) = var.names
rownames(coeff.t) = colnames(design.matrix)
rownames(coeff.regr) = colnames(design.matrix)
rownames(pval_part) = var.names
residuals.t = coeff - fitted.t
ybar.t = colMeans(coeff)
R2.t = colSums((fitted.t - matrix(data=ybar.t,nrow=n,ncol=npt,byrow=TRUE))^2)/colSums((coeff - matrix(data=ybar.t,nrow=n,ncol=npt,byrow=TRUE))^2)
print('Interval-Wise Testing completed')
IWTresult <- list(call=cl,
design_matrix=design.matrix,
unadjusted_pval_F=pval_glob,
pval_matrix_F=matrice_pval_asymm_glob,
adjusted_pval_F=corrected.pval_glob,
unadjusted_pval_factors=pval_part,
pval_matrix_factors=matrice_pval_asymm_part,
adjusted_pval_factors=corrected.pval_part,
data.eval=coeff,
coeff.regr.eval=coeff.t,
fitted.eval=fitted.t,
residuals.eval=residuals.t,
R2.eval=R2.t
#pval_parametric=pval_parametric
)
class(IWTresult) = 'IWTaov'
return(IWTresult)
}
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Defines functions IWTaov
Documented in IWTaov
#' @title Interval Wise Testing procedure for testing functional analysis of variance
#'
#' @description The function implements the Interval Wise Testing procedure for testing mean differences between several
#' functional populations in a one-way or multi-way functional analysis of variance framework.
#' Functional data are tested locally and unadjusted and adjusted p-value
#' functions are provided. The unadjusted p-value function controls the point-wise error rate.
#' The adjusted p-value function controls the
#' interval-wise error rate.
#'
#' @param formula An object of class "\code{\link{formula}}" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
#' The output variable of the formula can be either a matrix of dimension \code{c(n,J)} collecting the pointwise evaluations of \code{n} functional data on the same grid of \code{J} points, or a \code{fd} object from the package \code{fda}.
#'
#' @param B The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is \code{B=1000}.
#'
#' @param method Permutation method used to calculate the p-value of permutation tests. Choose "\code{residuals}" for the permutations of residuals under the reduced model, according to the Freedman and Lane scheme, and "\code{responses}" for the permutation of the responses, according to the Manly scheme.
#'
#' @param dx Used only if a \code{fd} object is provided. In this case, \code{dx} is the size of the discretization step of the grid used to evaluate functional data.
#' If set to \code{NULL}, a grid of size 100 is used. Default is \code{NULL}.
#'
#' @param recycle Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Default is \code{TRUE}.
#'
#'
#'
#' @return \code{IWTaov} returns an object of \code{\link{class}} "\code{IWTaov}". The function \code{summary} is used to obtain and print a summary of the results.
#' An object of class "\code{IWTaov}" is a list containing at least the following components:
#' \item{call}{The matched call.}
#' \item{design_matrix}{The design matrix of the functional-on-scalar linear model.}
#' \item{unadjusted_pval_F}{Evaluation on a grid of the unadjusted p-value function of the functional F-test.}
#' \item{pval_matrix_F}{Matrix of dimensions \code{c(p,p)} of the p-values of the intervalwise F-tests. The element \code{(i,j)} of matrix \code{pval.matrix} contains the p-value of the test of interval indexed by \code{(j,j+1,...,j+(p-i))}.}
#' \item{adjusted_pval_F}{Evaluation on a grid of the adjusted p-value function of the functional F-test.}
#' \item{unadjusted_pval_factors}{Evaluation on a grid of the unadjusted p-value function of the functional F-tests on each factor of the analysis of variance (rows).}
#' \item{pval.matrix.factors}{Array of dimensions \code{c(L+1,p,p)} of the p-values of the multivariate F-tests on factors. The element \code{(l,i,j)} of array \code{pval.matrix} contains the p-value of the joint NPC test on factor \code{l} of the components \code{(j,j+1,...,j+(p-i))}.}
#' \item{adjusted.pval.factors}{adjusted p-values of the functional F-tests on each factor of the analysis of variance (rows) and each basis coefficient (columns).}
#' \item{data.eval}{Evaluation on a fine uniform grid of the functional data obtained through the basis expansion.}
#' \item{coeff.regr.eval}{Evaluation on a fine uniform grid of the functional regression coefficients.}
#' \item{fitted.eval}{Evaluation on a fine uniform grid of the fitted values of the functional regression.}
#' \item{residuals.eval}{Evaluation on a fine uniform grid of the residuals of the functional regression.}
#' \item{R2.eval}{Evaluation on a fine uniform grid of the functional R-squared of the regression.}
#' \item{heatmap.matrix.F}{Heatmap matrix of p-values of functional F-test (used only for plots).}
#' \item{heatmap.matrix.factors}{Heatmap matrix of p-values of functional F-tests on each factor of the analysis of variance (used only for plots).}
#'
#' @seealso See \code{\link{summary.IWTaov}} for summaries and \code{\link{plot.IWTaov}} for plotting the results.
#' See \code{\link{ITPaov}} for a functional analysis of variance test based on B-spline basis expansion.
#' See also \code{\link{IWTlm}} to fit and test a functional-on-scalar linear model applying the IWT, and \code{\link{IWT1}}, \code{\link{IWT2}} for one-population and two-population tests.
#'
#'
#' @examples
#' # Importing the NASA temperatures data set
#' data(NASAtemp)
#' temperature <- rbind(NASAtemp$milan,NASAtemp$paris)
#' groups <- c(rep(0,22),rep(1,22))
#'
#' # Performing the IWT
#' IWT.result <- IWTaov(temperature ~ groups,B=1000)
#'
#' # Summary of the ITP results
#' summary(IWT.result)
#'
#' # Plot of the IWT results
#' layout(1)
#' plot(IWT.result)
#'
#' # All graphics on the same device
#' layout(matrix(1:4,nrow=2,byrow=FALSE))
#' plot(IWT.result,main='NASA data', plot.adjpval = TRUE,xlab='Day',xrange=c(1,365))
#'
#' @references
#' Pini, A., & Vantini, S. (2017). Interval-wise testing for functional data. \emph{Journal of Nonparametric Statistics}, 29(2), 407-424
#'
#' Pini, A., Vantini, S., Colosimo, B. M., & Grasso, M. (2018). Domain‐selective functional analysis of variance for supervised statistical profile monitoring of signal data. \emph{Journal of the Royal Statistical Society: Series C (Applied Statistics)} 67(1), 55-81.
#'
#' Abramowicz, K., Hager, C. K., Pini, A., Schelin, L., Sjostedt de Luna, S., & Vantini, S. (2018).
#' Nonparametric inference for functional‐on‐scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament. \emph{Scandinavian Journal of Statistics} 45(4), 1036-1061.
#'
#' D. Freedman and D. Lane (1983). A Nonstochastic Interpretation of Reported Significance Levels. \emph{Journal of Business & Economic Statistics} 1.4, 292-298.
#'
#' B. F. J. Manly (2006). Randomization, \emph{Bootstrap and Monte Carlo Methods in Biology}. Vol. 70. CRC Press.
#'
#' @export
IWTaov <- function(formula,B=1000,method='residuals',dx=NULL,recycle=TRUE){
pval.correct <- function(pval.matrix){
matrice_pval_2_2x <- cbind(pval.matrix,pval.matrix)
p <- dim(pval.matrix)[2]
matrice_pval_2_2x <- matrice_pval_2_2x[,(2*p):1]
corrected.pval <- numeric(p)
corrected.pval.matrix <- matrix(nrow=p,ncol=p)
corrected.pval.matrix[p,] <- pval.matrix[p,p:1]
for(var in 1:p){
pval_var <- matrice_pval_2_2x[p,var]
inizio <- var
fine <- var #inizio fisso, fine aumenta salendo nelle righe
for(riga in (p-1):1){
fine <- fine + 1
pval_cono <- matrice_pval_2_2x[riga,inizio:fine]
pval_var <- max(pval_var,pval_cono,na.rm=TRUE)
corrected.pval.matrix[riga,var] <- pval_var
}
corrected.pval[var] <- pval_var
}
corrected.pval <- corrected.pval[p:1]
corrected.pval.matrix <- corrected.pval.matrix[,p:1]
return(corrected.pval.matrix)
}
stat_lm_glob <- function(anova){
result <- summary.lm(anova)$f[1]
return(result)
}
stat_aov_part <- function(anova){
result <- summary(anova)[[1]][,4]
result <- result[-length(result)]
return(result)
}
extract.residuals = function(anova){
return(anova$residuals)
}
extract.fitted = function(anova){
return(anova$fitted)
}
#extract.pval <- function(anova){
# result <- summary(anova)[[1]][,5]
# result <- result[-length(result)]
# return(result)
#}
env <- environment(formula)
variables = all.vars(formula)
y.name = variables[1]
covariates.names <- colnames(attr(terms(formula),"factors"))
#data.all = model.frame(formula)
cl <- match.call()
data <- get(y.name,envir = env)
if(is.fd(data)){ # data is a functional data object
rangeval <- data$basis$rangeval
if(is.null(dx)){
dx <- (rangeval[2]-rangeval[1])*0.01
}
abscissa <- seq(rangeval[1],rangeval[2],by=dx)
coeff <- t(eval.fd(fdobj=data,evalarg=abscissa))
}else if(is.matrix(data)){
coeff <- data
}else{
stop("First argument of the formula must be either a functional data object or a matrix.")
}
#design.matrix = model.matrix(formula)
#mf = model.frame(formula)
#data = model.response(mf)
dummynames.all <- colnames(attr(terms(formula),"factors"))
formula.const <- deparse(formula[[3]],width.cutoff = 500L) #extracting the part after ~ on formula. this will not work if the formula is longer than 500 char
formula.discrete <- as.formula(paste('coeff ~',formula.const),env=environment())
design.matrix = model.matrix(formula.discrete)
mf = model.frame(formula.discrete)
#var.names = variables[-1]
#nvar = length(var.names)
n <- dim(coeff)[1]
J <- dim(coeff)[2]
p <- dim(coeff)[2]
npt <- J
print('Point-wise tests')
#univariate permutations
coeffnames <- paste('coeff[,',as.character(1:p),']',sep='')
formula.coeff <- paste(coeffnames,'~',formula.const)
formula.coeff <- sapply(formula.coeff,as.formula,env=environment())
aovcoeff1 <- aov(formula.coeff[[1]],data=mf)
var.names <- rownames(summary(aovcoeff1)[[1]])
df.vars <- summary(aovcoeff1)[[1]][,1]
df.residuals <- df.vars[length(df.vars)]
var.names <- var.names[-length(var.names)]
nvar = length(var.names)
for(ii in 1:nvar){
var.names[ii] <- gsub(' ' , '',var.names[ii])
}
index.vars <- cbind(c(2,(cumsum(df.vars)+2)[-length(df.vars)]),cumsum(df.vars)+1)
regr0 = lm.fit(design.matrix,coeff)
#pval_parametric <- sapply(aov0,extract.pval)
MS0 <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MS0[var,] <- colSums(rbind(regr0$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T0_part <- MS0[1:nvar,] / matrix(MS0[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE)
Sigma <- chol2inv(regr0$qr$qr)
resvar <- colSums(regr0$residuals^2)/regr0$df.residual
if(nvar >1){
T0_glob <- colSums((regr0$fitted - matrix(colMeans(regr0$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
}else if(nvar==1){ #only one factor -> the permutation of the residuals is equivalent to the one of responses
method <- 'responses'
T0_glob <- colSums((regr0$fitted - matrix(colMeans(regr0$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
}else if(nvar==0){
method = 'responses' # model with only intercept -> the permutation of the residuals is equivalent to the one of responses
T0_glob = numeric(p)
}
#calculate residuals
if(method=='residuals'){
#n residuals for each coefficient of basis expansion (1:p)
#and for each partial test + global test (nvar+1)
#saved in array of dim (nvar+1,n,p)
design.matrix.names2 = design.matrix
var.names2 = var.names
if(length(grep('factor',formula.const))>0){
index.factor = grep('factor',var.names)
replace.names = paste('group',(1:length(index.factor)),sep='')
var.names2[index.factor] = replace.names
colnames(design.matrix.names2) = var.names2
}
residui = array(dim=c(nvar,n,p))
fitted_part = array(dim=c(nvar,n,p)) # fitted values of the reduced model (different for each test)
formula.coeff_part = vector('list',nvar)
regr0_part = vector('list',nvar)
dummy.interaz <- grep(':',dummynames.all)
#coeff.perm_part = array(dim=c(nvar+1,n,p))
for(ii in 1:(nvar)){ #no test on intercept
var.ii = var.names2[ii]
variables.reduced = var.names2[-which(var.names2==var.ii)] #removing the current variable to test
if(length(grep(':',var.ii))>0){ # testing interaction
#print('interaz')
var12 <- strsplit(var.ii,':')
var1 <- var12[[1]][1]
var2 <- var12[[1]][2]
dummy.test1 <- grep(var1,dummynames.all)
dummy.test2 <- grep(var2,dummynames.all)
dummy.test <- intersect(dummy.test1,dummy.test2)
dummynames.reduced <- dummynames.all[-dummy.test]
}else{
#print('nointeraz')
dummy.test <- grep(var.ii,dummynames.all)
dummy.test <- setdiff(dummy.test,dummy.interaz)
dummynames.reduced <- dummynames.all[-dummy.test]
}
if(nvar>1){
formula.temp = paste(dummynames.reduced,collapse=' + ')
}else{
formula.temp = '1' #removing the only variable -> reduced model only has intercept term
}
formula.coeff.temp <- paste(coeffnames,'~',formula.temp)
formula.coeff_part[[ii]] <- sapply(formula.coeff.temp,as.formula,env=environment())
regr0_part[[ii]] = lapply(formula.coeff_part[[ii]],lm)
residui[ii,,] = simplify2array(lapply(regr0_part[[ii]],extract.residuals))
fitted_part[ii,,] = simplify2array(lapply(regr0_part[[ii]],extract.fitted))
}
}
T_glob <- matrix(ncol=p,nrow=B)
T_part = array(dim=c(B,nvar,p))
for (perm in 1:B){
# the F test is the same for both methods
if(nvar >0){
permutazioni <- sample(n)
coeff_perm <- coeff[permutazioni,]
}else{ # testing intercept -> permute signs
signs <- rbinom(n,1,0.5)*2 - 1
coeff_perm <- coeff*signs
}
regr_perm = lm.fit(design.matrix,coeff_perm)
Sigma <- chol2inv(regr_perm$qr$qr)
resvar <- colSums(regr_perm$residuals^2)/regr0$df.residual
if(nvar > 0)
T_glob[perm,] <- colSums((regr_perm$fitted - matrix(colMeans(regr_perm$fitted),nrow=n,ncol=p,byrow=TRUE))^2)/ ((nvar)*resvar)
# partial tests: differ depending on the method
if(method=='responses'){
MSperm <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MSperm[var,] <- colSums(rbind(regr_perm$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T_part[perm,,] <- MSperm[1:nvar,] / matrix(MSperm[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE)
}else if(method=='residuals'){
residui_perm = residui[,permutazioni,]
aov_perm_part = vector('list',nvar)
for(ii in 1:(nvar)){
coeff_perm = fitted_part[ii,,] + residui_perm[ii,,]
regr_perm = lm.fit(design.matrix,coeff_perm)
MSperm <- matrix(nrow=nvar+1,ncol=p)
for(var in 1:(nvar+1)){
MSperm[var,] <- colSums(rbind(regr_perm$effects[index.vars[var,1]:index.vars[var,2],]^2))/df.vars[var]
}
# test statistic:
T_part[perm,ii,] <- (MSperm[1:nvar,] / matrix(MSperm[nvar+1,],nrow=nvar,ncol=p,byrow=TRUE))[ii,]
}
}
}
pval_glob <- numeric(p)
pval_part = matrix(nrow=nvar,ncol=p)
for(i in 1:p){
pval_glob[i] <- sum(T_glob[,i]>=T0_glob[i])/B
pval_part[,i] = colSums(T_part[,,i]>=matrix(T0_part[,i],nrow=B,ncol=nvar,byrow=TRUE))/B
}
#combination
print('Interval-wise tests')
#asymmetric combination matrix:
matrice_pval_asymm_glob <- matrix(nrow=p,ncol=p)
matrice_pval_asymm_glob[p,] <- pval_glob[1:p]
T0_2x_glob <- c(T0_glob,T0_glob)
T_2x_glob <- cbind(T_glob,T_glob)
matrice_pval_asymm_part <- array(dim=c(nvar,p,p))
T0_2x_part <- cbind(T0_part,T0_part)
T_2x_part = array(dim=c(B,nvar,p*2))
for(ii in 1:(nvar)){
matrice_pval_asymm_part[ii,p,] <- pval_part[ii,1:p]
T_2x_part[,ii,] <- cbind(T_part[,ii,],T_part[,ii,])
}
maxrow <- 1
if(recycle==TRUE){
for(i in (p-1):maxrow){
for(j in 1:p){
inf <- j
sup <- (p-i)+j
T0_temp <- sum(T0_2x_glob[inf:sup])
T_temp <- rowSums(T_2x_glob[,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_glob[i,j] <- pval_temp
for(ii in 1:(nvar )){
T0_temp <- sum(T0_2x_part[ii,inf:sup])
T_temp <- rowSums(T_2x_part[,ii,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_part[ii,i,j] <- pval_temp
}
}
print(paste('creating the p-value matrix: end of row ',as.character(p-i+1),' out of ',as.character(p),sep=''))
}
}else{
for(i in (p-1):maxrow){ # rows
for(j in 1:i){ # columns
inf <- j
sup <- (p-i)+j
T0_temp <- sum(T0_2x_glob[inf:sup])
T_temp <- rowSums(T_2x_glob[,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_glob[i,j] <- pval_temp
for(ii in 1:(nvar )){
T0_temp <- sum(T0_2x_part[ii,inf:sup])
T_temp <- rowSums(T_2x_part[,ii,inf:sup])
pval_temp <- sum(T_temp>=T0_temp)/B
matrice_pval_asymm_part[ii,i,j] <- pval_temp
}
}
print(paste('creating the p-value matrix: end of row ',as.character(p-i+1),' out of ',as.character(p),sep=''))
}
}
corrected.pval.matrix_glob <- pval.correct(matrice_pval_asymm_glob)
corrected.pval_glob <- corrected.pval.matrix_glob[1,]
corrected.pval_part = matrix(nrow=nvar,ncol=p)
corrected.pval.matrix_part = array(dim=c(nvar,p,p))
for(ii in 1:(nvar)){
corrected.pval.matrix_part[ii,,] = pval.correct(matrice_pval_asymm_part[ii,,])
corrected.pval_part[ii,] <- corrected.pval.matrix_part[ii,1,]
}
coeff.regr = regr0$coeff
coeff.t <- coeff.regr
fitted.regr = regr0$fitted.values
fitted.t <- fitted.regr
rownames(corrected.pval_part) = var.names
rownames(coeff.t) = colnames(design.matrix)
rownames(coeff.regr) = colnames(design.matrix)
rownames(pval_part) = var.names
residuals.t = coeff - fitted.t
ybar.t = colMeans(coeff)
R2.t = colSums((fitted.t - matrix(data=ybar.t,nrow=n,ncol=npt,byrow=TRUE))^2)/colSums((coeff - matrix(data=ybar.t,nrow=n,ncol=npt,byrow=TRUE))^2)
print('Interval-Wise Testing completed')
IWTresult <- list(call=cl,
design_matrix=design.matrix,
unadjusted_pval_F=pval_glob,
pval_matrix_F=matrice_pval_asymm_glob,
adjusted_pval_F=corrected.pval_glob,
unadjusted_pval_factors=pval_part,
pval_matrix_factors=matrice_pval_asymm_part,
adjusted_pval_factors=corrected.pval_part,
data.eval=coeff,
coeff.regr.eval=coeff.t,
fitted.eval=fitted.t,
residuals.eval=residuals.t,
R2.eval=R2.t
#pval_parametric=pval_parametric
)
class(IWTresult) = 'IWTaov'
return(IWTresult)
}
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