Local testing procedures for functional-on-scalar linear models

The function implements local testing procedures for testing mean differences between multiple functional populations. 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)
partition closed testing (controlling the FWER on a partition)
functional Benjamini Hochberg (controlling the FDR)

Usage

functional_lm_test(
  formula,
  correction,
  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 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.
correction: A string specifying the method used to calculate the adjusted p-value function. Choices are either "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 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".
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. Choices are either "Integral", in which case the statistic is defined as the integral of the F-test statistic over the domain, or "Max", in which case the statistic is defined as the maximum of the F-test statistic over the domain. Defaults to "Integral".

Value

An object of class flm 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_part: 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_part: 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_part: 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_part 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".
Global_pval_F: Global p-value of the overall test F; this component is present only if correction is set to "Global".
Global_pval_part: 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.

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

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

# Performing the TWT
TWT_result <- TWTlm(temperature ~ groups, B = 100L)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Threshold-wise tests ────────────────────────────────────────────────────────
#> 
#> ── Threshold-Wise Testing completed ────────────────────────────────────────────
# Summary of the TWT results
summary(TWT_result)
#> $call
#> twt_lm(formula = formula, dx = dx, n_perm = B, method = method)
#> 
#> $ttest
#>             Minimum p-value    
#> (Intercept)               0 ***
#> 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 ***
#>