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Local testing procedures for functional-on-scalar linear models

Source: R/functional-lm-test.R
functional_lm_test.Rd

Implements local testing procedures for testing the significance of the effects of scalar covariates on a functional response in a functional-on-scalar linear model. 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_lm_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 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

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.

See also

iwt_lm(), twt_lm() and global_lm() for calling a specific correction directly. plot.flm() for plotting the results and summary.flm() for summarizing the results.

Examples

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

# Performing the TWT
TWT_result <- functional_lm_test(
  temperature ~ groups,
  correction = "TWT",
  B = 10L
)
#> 
#> ── Point-wise tests ────────────────────────────────────────────────────────────
#> 
#> ── Threshold-wise tests ────────────────────────────────────────────────────────
#> 
#> ── Threshold-Wise Testing completed ────────────────────────────────────────────
# Summary of the TWT results
summary(TWT_result)
#> $call
#> functional_lm_test(formula = temperature ~ groups, correction = "TWT", 
#>     B = 10L)
#> 
#> $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 ***
#>