Threshold Wise Testing procedure for testing functional analysis of variance

The function implements the Threshold 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 threshold-wise error rate.

Usage

TWTaov(formula, B = 1000L, method = "residuals", dx = NULL)

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. The output variable of the formula can be either a matrix of dimension c(n,J) collecting the pointwise evaluations of n functional data on the same grid of J points, or a fd object from the package fda.
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.
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.

Value

An object of class IWTaov. The function summary is used to obtain and print a summary of the results. An object of class "IWTaov" 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.
unadjusted_pval_F: Evaluation on a grid of the unadjusted p-value function of the functional F-test.
adjusted_pval_F: Evaluation on a grid of the adjusted p-value function of the functional F-test.
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).
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.

References

Abramowicz, K., Pini, A., Schelin, L., Stamm, A., & Vantini, S. (2022). “Domain selection and familywise error rate for functional data: A unified framework. Biometrics 79(2), 1119-1132.

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.

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 TWT
TWT.result <- TWTaov(temperature ~ groups, B = 100L)
#> Error in eval(predvars, data, env): object 'groups' not found

# Summary of the ITP results
summary(TWT.result)
#> Error: object 'TWT.result' not found

# Plot of the TWT results
layout(1)
plot(TWT.result)
#> Error: object 'TWT.result' not found

# All graphics on the same device
layout(matrix(1:4, nrow = 2, byrow = FALSE))
plot(
  TWT.result, 
  main = 'NASA data', 
  plot_adjpval = TRUE, 
  xlab = 'Day', 
  xrange = c(1, 365)
)
#> Error: object 'TWT.result' not found