Threshold Wise Testing procedure for testing functional analysis of variance

The function implements the Threshold 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 threshold-wise error rate.

Usage

TWTaov(formula, dx = NULL, B = 1000L, method = c("residuals", "responses"))

twt_aov(
  formula,
  dx = NULL,
  n_perm = 1000L,
  method = c("residuals", "responses")
)

Arguments

formula: An object of class stats::formula (or one that can be coerced to that class) specifying the model to be fitted in a symbolic fashion. The output variable (left-hand side) of the formula can be either a matrix of dimension \(n \times J\) containing the pointwise evaluations of \(n\) functions on the same grid of \(J\) points, or an object of class fda::fd.
dx: A numeric value specifying the discretization step of the grid used to evaluate functional data when it is provided as objects of class fda::fd. Defaults to NULL, in which case a default value of 0.01 is used which corresponds to a grid of size 100L. Unused if functional data is provided in the form of matrices.
B: An integer value specifying the number of iterations of the MC algorithm to evaluate the p-value of the permutation tests. Defaults to 1000L.
method: A string specifying the method used to calculate the p-value of permutation tests. Choices are either "residuals" which performs permutation of residuals under the reduced model according to the Freedman and Lane scheme or "responses", which performs permutation of the responses, according to the Manly scheme. Defaults to "residuals".
n_perm: An integer value specifying the number of permutations for the permutation tests. Defaults to 1000L.

Value

An object of class fanova containing 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.
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).
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.

Optionally, the list may contain the following components:

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 \((j,j+1,...,j+(p-i))\); this component is present only if correction is set to "IWT".
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 \((j,j+1,...,j+(p-i))\); this component is present only if correction is set to "IWT".
heatmap_matrix_F: Heatmap matrix of p-values of functional F-test (used only for plots); this component is present only if correction is set to "IWT".
heatmap_matrix_factors: Heatmap matrix of p-values of functional F-tests on each factor of the analysis of variance (used only for plots); this component is present only if correction is set to "IWT".
Global_pval_F: Global p-value of the overall test F; this component is present only if correction is set to "Global".
Global_pval_factors: Global p-value of test F involving each factor separately; this component is present only if correction is set to "Global".

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

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

# Performing the TWT
TWT_result <- TWTaov(temperature ~ groups, B = 100L)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Threshold-wise tests ────────────────────────────────────────────────────────
#> 
#> ── Threshold-Wise Testing completed ────────────────────────────────────────────

# Summary of the TWT results
summary(TWT_result)
#> $call
#> twt_aov(formula = formula, dx = dx, n_perm = B, method = method)
#> 
#> $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 TWT results
layout(1)
plot(TWT_result)


# All graphics on the same device
layout(matrix(1:4, nrow = 2, byrow = FALSE))
plot(
  TWT_result,
  main = 'NASA data',
  plot_adjpval = TRUE,
  xlab = 'Day',
  xrange = c(1, 365)
)