Skip to contents

Interval Wise Testing procedure for testing functional analysis of variance

IWTaov.Rd

The function implements the Interval 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 interval-wise error rate.

Usage

IWTaov(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. 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.

recycle

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

Value

IWTaov returns 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.

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

  • 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).

  • 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 \((j,j+1,...,j+(p-i))\).

  • 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.

  • 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).

References

Pini, A., & Vantini, S. (2017). Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2), 407-424.

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

Examples

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

# Performing the IWT
IWT.result <- IWTaov(temperature ~ groups, B = 10L)
#> Error in eval(predvars, data, env): object 'groups' not found

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

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

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