#' Interval-wise testing procedure for testing functional-on-scalar linear
#' models
#'
#' The function is used to fit and test functional linear models. It can be used
#' to carry out regression, and analysis of variance. It implements the
#' interval-wise testing procedure (IWT) for testing the significance of the
#' effects of scalar covariates on a functional population.
#'
#' @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.
#' Example: y ~ A + B where: y is a matrix of dimension n * p containing the
#' point-wise evaluations of the n functional data on p points or an object of
#' class \code{fd} (see \code{fda} package) containing the functional data set
#' A, B are n-dimensional vectors containing the values of two covariates.
#' Covariates may be either scalar or factors.
#' @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 step size for the point-wise evaluations of functional data. dx is
#' only used ia an object of class 'fd' is provided as response in the
#' formula.
#' @param recycle flag specifying whether the recycled version of IWT has to be
#' used.
#'
#' @return \code{IWTlm} returns an object of \code{\link{class}} "\code{IWTlm}".
#' The function \code{summary} is used to obtain and print a summary of the
#' results. An object of class "\code{ITPlm}" is a list containing at least
#' the following components:
#'
#' - `call`: Call of the function.
#' - `design_matrix`: Design matrix of the linear model.
#' - `unadjusted_pval_F`: Unadjusted p-value function of the F test.
#' - `pval_matrix_F`: Matrix of dimensions `c(p,p)` of the p-values of the
#' interval-wise F-tests. The element \eqn{(i,j)} of matrix `pval_matrix_F`
#' contains the p-value of the test on interval \eqn{(j,j+1,...,j+(p-i))}.
#' - `adjusted_pval_F`: Adjusted p-value function of the F test.
#' - `unadjusted_pval_part`: Unadjusted p-value functions of the functional
#' t-tests on each covariate, separately (rows) on each domain point
#' (columns).
#' - `pval_matrix_part`: Array of dimensions `c(L+1,p,p)` of the p-values of
#' the interval-wise t-tests on covariates. The element \eqn{(l,i,j)} of array
#' `pval_matrix_part` contains the p-value of the test of covariate `l` on
#' interval \eqn{(j,j+1,...,j+(p-i))}.
#' - `adjusted_pval_part`: Adjusted p-values of the functional t-tests on each
#' covariate (rows) on each domain point (columns).
#' - `data.eval`: Evaluation of functional data.
#' - `coeff.regr.eval`: Evaluation of the regression coefficients.
#' - `fitted.eval`: Evaluation of the fitted values.
#' - `residuals.eval`: Evaluation of the residuals.
#' - `R2.eval`: Evaluation of the functional R-suared.
#'
#' @seealso See \code{\link{summary.IWTlm}} for summaries and
#' \code{\link{plot.IWTlm}} for plotting the results. See
#' \code{\link{ITPlmbspline}} for a functional linear model test based on an
#' a-priori selected B-spline basis expansion. See also \code{\link{IWTaov}}
#' to fit and test a functional analysis of variance applying the IWT, and
#' \code{\link{IWT1}}, \code{\link{IWT2}} for one-population and
#' two-population tests.
#'
#' @references
#' A. Pini and S. Vantini (2017). The Interval Testing Procedure: Inference for
#' Functional Data Controlling the Family Wise Error Rate on Intervals.
#' Biometrics 73(3): 835–845.
#'
#' 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
#' @examples
#' # Defining the covariates
#' temperature <- rbind(NASAtemp$milan, NASAtemp$paris)
#' groups <- c(rep(0, 22), rep(1, 22))
#'
#' # Performing the IWT
#' IWT.result <- IWTlm(temperature ~ groups, B = 2L)
#' # Summary of the IWT results
#' summary(IWT.result)
#'
#' # Plot of the IWT results
#' graphics::layout(1)
#' plot(
#' IWT.result,
#' main = 'NASA data',
#' plot_adjpval = TRUE,
#' xlab = 'Day',
#' xrange = c(1, 365)
#' )
#'
#' # All graphics on the same device
#' graphics::layout(matrix(1:6, nrow = 3, byrow = FALSE))
#' plot(
#' IWT.result,
#' main = 'NASA data',
#' plot_adjpval = TRUE,
#' xlab = 'Day',
#' xrange = c(1, 365)
#' )
IWTlm <- function(formula,
B = 1000L,
method = "residuals",
dx = NULL,
recycle = TRUE) {
cl <- match.call()
coeff <- formula2coeff(formula, dx = dx)
design_matrix <- formula2design_matrix(formula, coeff)
nvar <- dim(design_matrix)[2] - 1
var_names <- colnames(design_matrix)
p <- dim(coeff)[2]
n <- dim(coeff)[1]
# Univariate permutations
regr0 <- stats::lm.fit(design_matrix, coeff)
# Test statistics
Sigma <- chol2inv(regr0$qr$qr)
resvar <- colSums(regr0$residuals^2) / regr0$df.residual
se <- sqrt(
matrix(
diag(Sigma),
nrow = nvar + 1,
ncol = p,
byrow = FALSE
) * matrix(
resvar,
nrow = nvar + 1,
ncol = p,
byrow = TRUE
)
)
T0_part <- abs(regr0$coeff / se)^2
if (nvar > 0) {
T0_glob <- colSums((regr0$fitted - matrix(
colMeans(regr0$fitted),
nrow = n,
ncol = p,
byrow = TRUE
))^2) / (nvar * resvar)
} else {
method <- 'responses'
T0_glob <- numeric(p)
T0_part <- t(as.matrix(T0_part))
}
# Compute 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)
# Extracting the part after ~ on formula.
# This will not work if the formula
# is longer than 500 char
formula_const <- deparse(formula[[3]], width.cutoff = 500L)
design_matrix_names2 <- design_matrix
var_names2 <- var_names
coeffnames <- paste('coeff[,', as.character(1:p), ']', sep = '')
formula_temp <- coeff ~ design_matrix
mf_temp <- cbind(
stats::model.frame(formula_temp)[-((p + 1):(p + nvar + 1))],
as.data.frame(design_matrix[, -1])
)
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 + 1, n, p))
fitted_part <- array(dim = c(nvar + 1, n, p))
formula_coeff_part <- vector('list', nvar + 1)
regr0_part <- vector('list', nvar + 1)
# The first one is the intercept. Treated as special case after loop
for (ii in 2:(nvar + 1)) {
var_ii <- var_names2[ii]
variables_reduced <- var_names2[-c(1, which(var_names2 == var_ii))]
if (nvar > 1) {
formula_temp <- paste(variables_reduced, collapse = ' + ')
} else {
# Removing the unique variable -> reduced model only has intercept ter
formula_temp <- '1'
}
formula_temp2 <- coeff ~ design_matrix_names2
mf_temp2 <- cbind(
stats::model.frame(formula_temp2)[-((p + 1):(p + nvar + 1))],
as.data.frame(design_matrix_names2[, -1])
)
formula_coeff_temp <- paste(coeffnames, '~', formula_temp)
formula_coeff_part[[ii]] <- sapply(formula_coeff_temp, stats::as.formula)
regr0_part[[ii]] <- lapply(formula_coeff_part[[ii]], stats::lm, data = mf_temp2)
residui[ii, , ] <- simplify2array(lapply(regr0_part[[ii]], extract_residuals))
fitted_part[ii, , ] <- simplify2array(lapply(regr0_part[[ii]], extract_fitted))
}
ii <- 1 # intercept
formula_temp <- paste(formula_const, ' -1', sep = '')
formula_coeff_temp <- paste(coeffnames, '~', formula_temp)
formula_coeff_part[[ii]] <- sapply(formula_coeff_temp, stats::as.formula)
regr0_part[[ii]] <- lapply(formula_coeff_part[[ii]], stats::lm, data = mf_temp)
residui[ii, , ] <- simplify2array(lapply(regr0_part[[ii]], extract_residuals))
fitted_part[ii, , ] <- simplify2array(lapply(regr0_part[[ii]], extract_fitted))
}
cli::cli_h1("Point-wise tests")
# CMC algorithm
T_glob <- matrix(ncol = p, nrow = B)
T_part <- array(dim = c(B, nvar + 1, 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 { # Test on intercept permuting signs
signs <- stats::rbinom(n, 1, 0.5) * 2 - 1
coeff_perm <- coeff * signs
}
regr_perm <- stats::lm.fit(design_matrix, coeff_perm)
Sigma <- chol2inv(regr_perm$qr$qr)
resvar <- colSums(regr_perm$residuals^2) / regr_perm$df.residual
if (nvar > 0) {
T_glob[perm, ] <- colSums((
regr_perm$fitted - matrix(
colMeans(regr_perm$fitted),
nrow = n,
ncol = p,
byrow = TRUE
)
)^2) / (nvar * resvar)
}
# Partial tests: differ depending on the method
if (method == 'responses') {
se <- sqrt(
matrix(
diag(Sigma),
nrow = nvar + 1,
ncol = p,
byrow = FALSE
) * matrix(
resvar,
nrow = nvar + 1,
ncol = p,
byrow = TRUE
)
)
T_part[perm, , ] <- abs(regr0$coeff / se)^2
} else if (method == 'residuals'){
residui_perm <- residui[, permutazioni, ]
regr_perm_part <- vector('list', nvar + 1)
for (ii in 1:(nvar + 1)) {
coeff_perm <- fitted_part[ii, , ] + residui_perm[ii, , ]
regr_perm <- stats::lm.fit(design_matrix, coeff_perm)
Sigma <- chol2inv(regr_perm$qr$qr)
resvar <- colSums(regr_perm$residuals^2) / regr_perm$df.residual
se <- sqrt(
matrix(
diag(Sigma),
nrow = nvar + 1 ,
ncol = p,
byrow = FALSE
) * matrix(
resvar,
nrow = nvar + 1,
ncol = p,
byrow = TRUE
)
)
T_part[perm, ii, ] <- abs(regr_perm$coeff / se)[ii, ]^2
}
}
}
pval_glob <- numeric(p)
pval_part <- matrix(nrow = nvar + 1, 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 + 1,
byrow = TRUE
)) / B
}
cli::cli_h1("Interval-wise tests")
matrice_pval_asymm_glob <- matrix(nrow = p, ncol = p)
matrice_pval_asymm_glob[p, ] <- pval_glob[1:p]
pval_2x_glob <- c(pval_glob, pval_glob)
T0_2x_glob <- c(T0_glob, T0_glob)
T_2x_glob <- cbind(T_glob, T_glob)
matrice_pval_asymm_part <- array(dim = c(nvar + 1, p, p))
pval_2x_part <- cbind(pval_part, pval_part)
T0_2x_part <- cbind(T0_part, T0_part)
T_2x_part = array(dim = c(B, nvar + 1, p * 2))
for (ii in 1:(nvar + 1)) {
matrice_pval_asymm_part[ii, p, ] <- pval_part[ii, 1:p]
T_2x_part[, ii, ] <- cbind(T_part[, ii, ], T_part[, ii, ])
}
if (recycle) {
for (i in (p - 1):1) {
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 + 1)) {
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
}
}
cli::cli_h1("Creating the p-value matrix: end of row {p - i + 1} out of {p}")
}
} else {
for (i in (p - 1):1) {
for (j in 1:i) {
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 + 1)) {
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
}
}
cli::cli_h1("Creating the p-value matrix: end of row {p - i + 1} out of {p}")
}
}
corrected.pval.matrix_glob <- pval_correct(matrice_pval_asymm_glob)
corrected.pval_glob <- corrected.pval.matrix_glob[1, ]
corrected.pval_part <- matrix(nrow = nvar + 1, ncol = p)
corrected.pval.matrix_part <- array(dim = c(nvar + 1, p, p))
for (ii in 1:(nvar + 1)) {
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
fitted.t <- fitted.regr
rownames(corrected.pval_part) <- var_names
rownames(coeff.t) <- var_names
rownames(coeff.regr) <- var_names
rownames(pval_part) <- var_names
data.eval <- coeff
residuals.t <- data.eval - fitted.t
ybar.t <- colMeans(data.eval)
npt <- p
R2.t <- colSums((fitted.t - matrix(
data = ybar.t,
nrow = n,
ncol = npt,
byrow = TRUE
))^2) / colSums((data.eval - matrix(
data = ybar.t,
nrow = n,
ncol = npt,
byrow = TRUE
))^2)
cli::cli_h1("Interval-Wise Testing completed")
out <- 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_part = pval_part,
pval_matrix_part = matrice_pval_asymm_part,
adjusted_pval_part = corrected.pval_part,
data.eval = coeff,
coeff.regr.eval = coeff.t,
fitted.eval = fitted.t,
residuals.eval = residuals.t,
R2.eval = R2.t
)
class(out) <- "IWTlm"
out
}
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