Local testing procedures for the functional analysis of variance

Implements local testing procedures for comparing the mean functions of multiple functional populations in a one-way or multi-way functional analysis of variance (FANOVA) 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 can be computed according to the following methods:

global testing (controlling the FWER weakly)
interval-wise testing (controlling the interval-wise error rate)
threshold-wise testing (controlling the FWER asymptotically)

Usage

functional_anova_test(
  formula,
  correction = c("IWT", "TWT", "Global"),
  dx = NULL,
  B = 1000L,
  method = c("residuals", "responses"),
  recycle = TRUE,
  stat = c("Integral", "Max")
)

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 response (left-hand side) 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.
correction: A string specifying the method used to calculate the adjusted p-value function. Choices are "Global" for global testing, "IWT" for interval-wise testing, or "TWT" for threshold-wise testing.
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 permutations used to evaluate the p-values of the permutation tests. Defaults to 1000L. Passed as n_perm in iwt_aov(), twt_aov() and global_aov().
method: A string specifying the permutation scheme. "residuals" permutes residuals under the reduced model (Freedman-Lane scheme); "responses" permutes the responses (Manly scheme). Defaults to "residuals".
recycle: A boolean value specifying whether the recycled version of the interval-wise testing procedure should be used. See Pini and Vantini (2017) for details. Defaults to TRUE.
stat: A string specifying the test statistic used for the global test. "Integral" uses the integral of the F-statistic over the domain; "Max" uses the maximum. Defaults to "Integral".

Value

An object of class faov containing the following components:

call: The matched call.
design_matrix: The design matrix of the functional ANOVA model.
unadjusted_pval_F: A numeric vector of length \(J\) containing the unadjusted p-value function of the global F-test evaluated on the grid.
adjusted_pval_F: A numeric vector of length \(J\) containing the adjusted p-value function of the global F-test evaluated on the grid.
unadjusted_pval_factors: A numeric matrix with one row per factor containing the unadjusted p-value functions of the per-factor F-tests.
adjusted_pval_factors: A numeric matrix with one row per factor containing the adjusted p-value functions of the per-factor F-tests.
data_eval: A numeric matrix containing the functional data evaluated on the grid.
coeff_regr_eval: A numeric matrix containing the functional regression coefficients evaluated on the grid.
fitted_eval: A numeric matrix containing the fitted values of the functional regression evaluated on the grid.
residuals_eval: A numeric matrix containing the residuals of the functional regression evaluated on the grid.
R2_eval: A numeric vector containing the functional R-squared evaluated on the grid.

Optionally, the list may contain the following components:

pval_matrix_F: A matrix of dimensions \(p \times p\) of p-values of the interval-wise F-tests. Element \((i,j)\) contains the p-value of the test on the interval \((j, j+1, \ldots, j+(p-i))\). Present only if correction is "IWT".
pval_matrix_factors: An array of dimensions \(L \times p \times p\) of p-values of the per-factor interval-wise F-tests. Element \((l,i,j)\) contains the p-value of the joint test on factor \(l\) and interval \((j, j+1, \ldots, j+(p-i))\). Present only if correction is "IWT".
global_pval_F: Global p-value of the overall F-test. Present only if correction is "Global".
global_pval_factors: A numeric vector of global p-values of the per-factor F-tests. Present only if correction is "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.

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.

Examples

temperature <- rbind(NASAtemp$milan[, 1:100], NASAtemp$paris[, 1:100])
groups <- c(rep(0, 22), rep(1, 22))

# Performing the TWT for multiple populations
TWT_result <- functional_anova_test(
  temperature ~ groups,
  correction = "TWT",
  B = 10L
)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Threshold-wise tests ────────────────────────────────────────────────────────
#> 
#> ── Threshold-Wise Testing completed ────────────────────────────────────────────

# Plotting the results of the TWT
plot(
  TWT_result,
  xrange = c(0, 12),
  main = "TWT results for testing mean differences"
)


# Selecting the significant components at 5% level
which(TWT_result$adjusted_pval_F < 0.05)
#>  [1] 29 30 40 46 47 48 49 50 55 57 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
#> [26] 76 77 80 81 84 85 86 87 88 89 90 91 92 93 94 95 96 97

# Performing the IWT for multiple populations
IWT_result <- functional_anova_test(
  temperature ~ groups,
  correction = "IWT",
  B = 10L
)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Interval-wise tests ─────────────────────────────────────────────────────────
#> 
#> ── Interval-Wise Testing completed ─────────────────────────────────────────────

# Plotting the results of the IWT
plot(
  IWT_result,
  xrange = c(0, 12),
  main = "IWT results for testing mean differences"
)


# Selecting the significant components at 5% level
which(IWT_result$adjusted_pval_F < 0.05)
#> [1] 69 70 71 72 88 89 90