# ITPaovbspline: Interval Testing Procedure for testing unctional analysis of... In alessiapini/fdatest: Interval Wise Testing for Functional Data

## Description

The function implements the Interval Testing Procedure for testing for significant differences between several functional population evaluated on a uniform grid, in a functional analysis of variance setting. Data are represented by means of the B-spline basis and the significance of each basis coefficient is tested with an interval-wise control of the Family Wise Error Rate. The default parameters of the basis expansion lead to the piece-wise interpolating function.

## Usage

 ```1 2 3``` ```ITPaovbspline(formula, order = 2, nknots = dim(model.response(model.frame(formula))), B = 1000, method = "residuals") ```

## Arguments

 `formula` An object of class "`formula`" (or one that can be coerced to that class): a symbolic description of the model to be fitted. `order` Order of the B-spline basis expansion. The default is `order=2`. `nknots` Number of knots of the B-spline basis expansion. The default is `nknots=dim(data1)`. `B` The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is `B=1000`. `method` Permutation method used to calculate the p-value of permutation tests. Choose "`residuals`" for the permutations of residuals under the reduced model, according to the Freedman and Lane scheme, and "`responses`" for the permutation of the responses, according to the Manly scheme.

## Value

`ITPaovbspline` returns an object of `class` "`ITPaov`". The function `summary` is used to obtain and print a summary of the results. An object of class "`ITPaov`" is a list containing at least the following components:

 `call` The matched call. `design.matrix` The design matrix of the functional-on-scalar linear model. `basis` String vector indicating the basis used for the first phase of the algorithm. In this case equal to `"B-spline"`. `coeff` Matrix of dimensions `c(n,p)` of the `p` coefficients of the B-spline basis expansion. Rows are associated to units and columns to the basis index. `coeff.regr` Matrix of dimensions `c(L+1,p)` of the `p` coefficients of the B-spline basis expansion of the intercept (first row) and the `L` effects of the covariates specified in `formula`. Columns are associated to the basis index. `pval.F` Unadjusted p-values of the functional F-test for each basis coefficient. `pval.matrix.F` Matrix of dimensions `c(p,p)` of the p-values of the multivariate F-tests. The element `(i,j)` of matrix `pval.matrix` contains the p-value of the joint NPC test of the components `(j,j+1,...,j+(p-i))`. `adjusted.pval.F` Adjusted p-values of the functional F-test for each basis coefficient. `pval.factors` Unadjusted p-values of the functional F-tests on each factor of the analysis of variance, separately (rows) and each basis coefficient (columns). `pval.matrix.factors` Array of dimensions `c(L+1,p,p)` of the p-values of the multivariate F-tests on factors. The element `(l,i,j)` of array `pval.matrix` contains the p-value of the joint NPC test on factor `l` of the components `(j,j+1,...,j+(p-i))`. `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). `data.eval` Evaluation on a fine uniform grid of the functional data obtained through the basis expansion. `coeff.regr.eval` Evaluation on a fine uniform grid of the functional regression coefficients. `fitted.eval` Evaluation on a fine uniform grid of the fitted values of the functional regression. `residuals.eval` Evaluation on a fine uniform grid of the residuals of the functional regression. `R2.eval` Evaluation on a fine uniform grid of the functional R-squared of the regression. `heatmap.matrix.F` Heatmap matrix of p-values of functional F-test (used only for plots). `heatmap.matrix.factors` Heatmap matrix of p-values of functional F-tests on each factor of the analysis of variance (used only for plots).

## 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.

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. 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. Scandinavian Journal of Statistics 45(4), 1036-1061.

D. Freedman and D. Lane (1983). A Nonstochastic Interpretation of Reported Significance Levels. Journal of Business & Economic Statistics 1.4, 292-298.

B. F. J. Manly (2006). Randomization, Bootstrap and Monte Carlo Methods in Biology. Vol. 70. CRC Press.

## See Also

See `summary.ITPaov` for summaries and `plot.ITPaov` for plotting the results. See `IWTaov` for a functional analysis of variance test that is not based on an a-priori selected basis expansion. See also `ITPlmbspline` to fit and test a functional-on-scalar linear model applying the ITP, and `ITP1bspline`, `ITP2bspline`, `ITP2fourier`, `ITP2pafourier` for one-population and two-population tests.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```# Importing the NASA temperatures data set data(NASAtemp) temperature <- rbind(NASAtemp\$milan,NASAtemp\$paris) groups <- c(rep(0,22),rep(1,22)) # Performing the ITP ITP.result <- ITPaovbspline(temperature ~ groups,B=1000,nknots=20,order=3) # Summary of the ITP results summary(ITP.result) # Plot of the ITP results layout(1) plot(ITP.result) # All graphics on the same device layout(matrix(1:4,nrow=2,byrow=FALSE)) plot(ITP.result,main='NASA data', plot.adjpval = TRUE,xlab='Day',xrange=c(1,365)) ```

alessiapini/fdatest documentation built on Oct. 30, 2020, 8:15 a.m.