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Global testing procedure for testing functional-on-scalar linear models

Globallm.Rd

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 global testing procedure for testing the significance of the effects of scalar covariates on a functional population.

Usage

Globallm(
  formula,
  B = 1000L,
  dx = NULL,
  recycle = TRUE,
  method = "residuals",
  stat = "Integral"
)

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. Defaults to 1000L.

dx

Used only if an fd object is provided. In this case, dx is the size of the discretization step of the gridused to evaluate functional data. If set to NULL, a grid of size 100L is used. Defaults to NULL.

recycle

Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Defaults to TRUE.

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.

stat

Type of test statistic used for the global test. Possible values are: 'Integral' (default) for the integral over the domain of the F- and t-test statistics; 'Max' for max over the domain of the F- and t-test statistics.

Value

An object of class IWTlm. The function summary is used to obtain and print a summary of the results. This object is a list containing 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).

  • Global_pval_F: Global p-value of the overall test F.

  • Global_pval_part: Global p-value of t-test involving each covariate separately.

  • 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-suared.

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.

See also

See summary.IWTlm for summaries and plot.IWTlm for plotting the results. See ITPlmbspline for a functional linear model test based on an a-priori selected B-spline basis expansion. See also IWTaov to fit and test a functional analysis of variance applying the IWT, and IWT1, IWT2 for one-population and two-population tests.

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 IWT
Global.result <- Globallm(temperature ~ groups, B = 1000)
#> Error in eval(predvars, data, env): object 'groups' not found
# Summary of the IWT results
summary(Global.result)
#> Error: object 'Global.result' not found

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

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