Interval Testing Procedure for testing unctional analysis of variance with B-spline basis

The function implements the Interval Testing Procedure for testing for significant differences between several functional population evaluated on a uniform grid, in a functional analysis of variance setting. Data are represented by means of the B-spline basis and the significance of each basis coefficient is tested with an interval-wise control of the Family Wise Error Rate. The default parameters of the basis expansion lead to the piece-wise interpolating function.

Usage

ITPaovbspline(
  formula,
  order = 2,
  nknots = dim(stats::model.response(stats::model.frame(formula)))[2],
  B = 1000,
  method = "residuals"
)

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.
order: Order of the B-spline basis expansion. The default is order=2.
nknots: Number of knots of the B-spline basis expansion. The default is nknots=dim(data1)[2].
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.

Value

ITPaovbspline returns an object of class "ITPaov". The function summary is used to obtain and print a summary of the results. An object of class "ITPaov" 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.
basis: String vector indicating the basis used for the first phase of the algorithm. In this case equal to "B-spline".
coeff: Matrix of dimensions c(n,p) of the p coefficients of the B-spline basis expansion. Rows are associated to units and columns to the basis index.
coeff_regr: Matrix of dimensions c(L+1,p) of the p coefficients of the B-spline basis expansion of the intercept (first row) and the L effects of the covariates specified in formula. Columns are associated to the basis index.
pval_F: Unadjusted p-values of the functional F-test for each basis coefficient.
pval_matrix_F: Matrix of dimensions c(p,p) of the p-values of the multivariate F-tests. The element (i,j) of matrix pval_matrix contains the p-value of the joint NPC test of the components (j,j+1,...,j+(p-i)).
adjusted_pval_F: Adjusted p-values of the functional F-test for each basis coefficient.
pval_factors: Unadjusted p-values of the functional F-tests on each factor of the analysis of variance, separately (rows) and each basis coefficient (columns).
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

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.

Examples

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

# Performing the ITP
ITP_result <- ITPaovbspline(temperature ~ groups,B=5,nknots=20,order=3)
#> Warning: `ITPaovbspline()` was deprecated in fdatest 2.2.0.
#> ℹ Please use `IWTaov()` instead.
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#> ── Creating the p-value matrix: end of row 218 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 219 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 220 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 221 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 222 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 223 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 224 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 225 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 226 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 227 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 228 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 229 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 230 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 231 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 232 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 233 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 234 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 235 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 236 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 237 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 238 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 239 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 240 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 241 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 242 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 243 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 244 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 245 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 246 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 247 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 248 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 249 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 250 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 251 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 252 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 253 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 264 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 270 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 281 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 312 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 313 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 314 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 322 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 323 out of 365 ──────────────────────
#> 
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#> 
#> ── Creating the p-value matrix: end of row 325 out of 365 ──────────────────────
#> 
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#> 
#> ── Creating the p-value matrix: end of row 327 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 328 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 329 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 330 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 331 out of 365 ──────────────────────
#> 
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#> 
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 335 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 336 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 337 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 338 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 339 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 340 out of 365 ──────────────────────
#> 
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#> 
#> ── Creating the p-value matrix: end of row 342 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 343 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 344 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 345 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 346 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 347 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 348 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 349 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 350 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 351 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 352 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 353 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 354 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 355 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 356 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 357 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 358 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 359 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 360 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 361 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 362 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 363 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 364 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 365 out of 365 ──────────────────────
#> 
#> ── Interval-Wise Testing completed ─────────────────────────────────────────────

# Summary of the ITP results
summary(ITP_result)
#> $call
#> iwt_aov(formula = formula, dx = dx, n_perm = B, method = method, 
#>     recycle = recycle)
#> 
#> $factors
#>        Minimum p-value    
#> groups               0 ***
#> 
#> $R2
#>               Range of functional R-squared
#> Min R-squared                  3.390203e-05
#> Max R-squared                  5.399620e-01
#> 
#> $ftest
#>   Minimum p-value    
#> 1               0 ***
#> 

# Plot of the ITP results
plot(
  ITP_result,
  main = "NASA data",
  plot_adjpval = TRUE,
  xlab = 'Day',
  xrange = c(1, 365)
)