Interval-wise testing procedure for testing functional-on-scalar linear models

The function is used to fit and test functional linear models. It can be used to carry out regression, and analysis of variance. It implements the interval-wise testing procedure (IWT) for testing the significance of the effects of scalar covariates on a functional population.

Usage

IWTlm(
  formula,
  dx = NULL,
  B = 1000L,
  method = c("residuals", "responses"),
  recycle = TRUE
)

iwt_lm(
  formula,
  dx = NULL,
  n_perm = 1000L,
  method = c("residuals", "responses"),
  recycle = TRUE
)

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.
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.
n_perm: An integer value specifying the number of permutations for the permutation tests. Defaults to 1000L.

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: 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_part: A numeric matrix with one row per model term containing the unadjusted p-value functions of the per-term t-tests.
adjusted_pval_part: A numeric matrix with one row per model term containing the adjusted p-value functions of the per-term t-tests.
data_eval: A numeric matrix containing the functional response 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_part: An array of dimensions \((L+1) \times p \times p\) of p-values of the per-term interval-wise t-tests. Element \((l,i,j)\) contains the p-value of the joint test on term \(l\) 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_part: A numeric vector of global p-values of the per-term t-tests. Present only if correction is "Global".

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

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

# Performing the IWT
IWT_result <- IWTlm(temperature ~ groups, B = 2L)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Interval-wise tests ─────────────────────────────────────────────────────────
#> 
#> ── Interval-Wise Testing completed ─────────────────────────────────────────────
# Summary of the IWT results
summary(IWT_result)
#> $call
#> functional_lm_test(formula = formula, correction = "IWT", dx = dx, 
#>     B = n_perm, method = method, recycle = recycle)
#> 
#> $ttest
#>             Minimum p-value    
#> (Intercept)               0 ***
#> groups                    0 ***
#> 
#> $R2
#>               Range of functional R-squared
#> Min R-squared                  0.0003189364
#> Max R-squared                  0.2476892354
#> 
#> $ftest
#>   Minimum p-value    
#> 1               0 ***
#> 

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