ITPaovbspline: Interval Testing Procedure for testing Functional analysis of...

In fdatest: Interval Testing Procedure for Functional Data

Interval Testing Procedure for testing Functional analysis of variance with B-spline basis

Description

ITPaovbspline is used to fit and test functional analysis of variance. The function implements the Interval Testing Procedure for testing for significant differences between several functional population evaluated on a uniform grid. 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

ITPaovbspline(formula, order = 2, 
              nknots = dim(model.response(model.frame(formula)))[2], 
              B = 10000, 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 dim(model.response(model.frame(formula)))[2].
B	The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is B=10000.
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 "ITPlm" 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	Uncorrected 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)).
corrected.pval.F	Corrected p-values of the functional F-test for each basis coefficient.
pval.factors	Uncorrected 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)).
corrected.pval.factors	Corrected 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).

Author(s)

Alessia Pini, Simone Vantini

References

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.

A. Pini and S. Vantini (2013). The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. MOX-report 13/2013, Politecnico di Milano.

K. Abramowicz, S. De Luna, C. Häger, A. Pini, L. Schelin, and S. Vantini (2015). Distribution-Free Interval-Wise Inference for Functional-on-Scalar Linear Models. MOX-report 3/2015, Politecnico di Milano.

See Also

See summary.ITPaov for summaries and plot.ITPaov for plotting the results.

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

# 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))

fdatest documentation built on May 4, 2022, 9:06 a.m.