Threshold-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 Threshold-wise testing procedure (TWT) for testing the significance of the effects of scalar covariates on a functional population.

Usage

TWTlm(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. 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 fd (see 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.
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: 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.

Value

An object of class TWTlm. 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: Call of the function.
design_matrix: Design matrix of the linear model.
unadjusted_pval_F: Unadjusted p-value function of the F test.
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).
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-squared.

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)
# Defining the covariates
temperature <- rbind(NASAtemp$milan, NASAtemp$paris)
groups <- c(rep(0, 22), rep(1, 22))

# Performing the TWT
TWT.result <- TWTlm(temperature ~ groups, B = 100L)
#> Error in eval(predvars, data, env): object 'groups' not found
# Summary of the TWT results
summary(TWT.result)
#> Error: object 'TWT.result' not found

# Plot of the TWT results
layout(1)
plot(
  TWT.result, 
  main = 'NASA data', 
  plot_adjpval = TRUE, 
  xlab = 'Day', 
  xrange = c(1, 365)
)
#> Error: object 'TWT.result' not found

# All graphics on the same device
layout(matrix(1:6, nrow = 3, byrow = FALSE))
plot(
  TWT.result, 
  main = 'NASA data', 
  plot_adjpval = TRUE, 
  xlab = 'Day', 
  xrange = c(1, 365)
)
#> Error: object 'TWT.result' not found