# IWT2: Two population Interval Wise Testing procedure In alessiapini/fdatest: Interval Wise Testing for Functional Data

## Description

The function implements the Interval Wise Testing procedure for testing mean differences between two functional populations. 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.

## Usage

 1 2 IWT2(data1, data2, mu = 0, B = 1000, paired = FALSE, dx = NULL, recycle = TRUE, alternative = "two.sided")

## Arguments

 data1 First population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a fd object from the package fda. If pointwise evaluations are provided, data2 is a matrix of dimensions c(n1,J), with J evaluations on columns and n1 units on rows. data2 Second population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a fd object from the package fda. If pointwise evaluations are provided, data2 is a matrix of dimensions c(n1,J), with J evaluations on columns and n2 units on rows. mu Functional mean difference under the null hypothesis. Three possibilities are available for mu: a constant (in this case, a constant function is used); a J-dimensional vector containing the evaluations on the same grid which data are evaluated; a fd object from the package fda containing one function. The default is mu=0. B The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is B=1000. paired Flag indicating whether a paired test has to be performed. Default is FALSE. dx Used only if a fd object is provided. In this case, dx is the size of the discretization step of the grid used to evaluate functional data. If set to NULL, a grid of size 100 is used. Default is NULL. recycle Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Default is TRUE. alternative A character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".

## Value

IWT2 returns an object of class "IWT2". An object of class "IWT2" is a list containing at least the following components:

 test String vector indicating the type of test performed. In this case equal to "2pop". mu Evaluation on a grid of the functional mean difference under the null hypothesis (as entered by the user). unadjusted_pval Evaluation on a grid of the unadjusted p-value function. pval_matrix Matrix of dimensions c(p,p) of the p-values of the intervalwise tests. The element (i,j) of matrix pval.matrix contains the p-value of the test contains the p-value of the test of interval indexed by (j,j+1,...,j+(p-i)). adjusted_pval Evaluation on a grid of the adjusted p-value function. data.eval Evaluation on a grid of the functional data. ord_labels Vector of labels indicating the group membership of data.eval

## 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. (2017). Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2), 407-424