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Interval-wise testing procedure for testing functional-on-scalar linear models

IWTlm.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 interval-wise testing procedure (IWT) for testing the significance of the effects of scalar covariates on a functional population.

Usage

IWTlm(formula, B = 1000L, method = "residuals", dx = NULL, recycle = TRUE)

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.

recycle

flag specifying whether the recycled version of IWT has to be used.

Value

IWTlm returns an object of class "IWTlm". 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.

  • pval_matrix_F: Matrix of dimensions c(p,p) of the p-values of the interval-wise F-tests. The element \((i,j)\) of matrix pval_matrix_F contains the p-value of the test on interval \((j,j+1,...,j+(p-i))\).

  • 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).

  • pval_matrix_part: Array of dimensions c(L+1,p,p) of the p-values of the interval-wise t-tests on covariates. The element \((l,i,j)\) of array pval_matrix_part contains the p-value of the test of covariate l on interval \((j,j+1,...,j+(p-i))\).

  • 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-suared.

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.

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.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

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

# Performing the IWT
IWT.result <- IWTlm(temperature ~ groups, B = 2L)
#> Error in eval(predvars, data, env): object 'groups' not found
# Summary of the IWT results
summary(IWT.result)
#> Error: object 'IWT.result' not found

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

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