frank.flm: F rank functional GLM

Description

Multiple testing in permutation inference for the general linear model (GLM)

Usage

frank.flm(
  nsim,
  formula.full,
  formula.reduced,
  curve_sets,
  factors = NULL,
  savefuns = TRUE,
  lm.args = NULL,
  GET.args = NULL,
  mc.cores = 1,
  mc.args = NULL,
  cl = NULL,
  method = c("best", "mlm", "lm", "ne")
)

Value

A global_envelope object, which can be printed and plotted directly.

Arguments

nsim: The number of random permutations.
formula.full: The formula specifying the general linear model, see formula in lm.
formula.reduced: The formula of the reduced model with nuisance factors only. This model should be nested within the full model.
curve_sets: A named list of sets of curves giving the dependent variable (Y), and possibly additionally factors whose values vary across the argument values of the functions. The dimensions of the elements should match with each other. Note that factors that are fixed across the functions can be given in the argument factors. Also fdata objects allowed.
factors: A data frame of factors. An alternative way to specify factors when they are constant for all argument values of the functions. The number of rows of the data frame should be equal to the number of curves. Each column should specify the values of a factor.
savefuns: Logical or "return". If TRUE, then the functions from permutations are saved to the attribute simfuns. If "return", then the function returns the permutations in a curve_set, instead of the result of the envelope test on those; this can be used by partial_forder.
lm.args: A named list of additional arguments to be passed to lm. See details.
GET.args: A named list of additional arguments to be passed to global_envelope_test, e.g. typeone specifies the type of multiple testing control, FWER or FDR. See global_envelope_test for the defaults and available options.
mc.cores: The number of cores to use, i.e. at most how many child processes will be run simultaneously. Must be at least one, and parallelization requires at least two cores. On a Windows computer mc.cores must be 1 (no parallelization). For details, see mclapply, for which the argument is passed. Parallelization can be used in generating simulations and in calculating the second stage tests.
mc.args: A named list of additional arguments to be passed to mclapply. Only relevant if mc.cores is more than 1.
cl: Allows parallelization through the use of parLapply (works also in Windows), see the argument cl there, and examples.
method: For advanced use.

Details

The function frank.flm performs a nonparametric test of significance of a covariate in the functional GLM. Similarly as in the graphical functional GLM (graph.flm), the Freedman-Lane algorithm (Freedman and Lane, 1983) is applied to permute the functions (to obtain the simulations under the null hypothesis of "no effects"); consequently, the test achieves the desired significance level only approximately. If the reduced model contains only a constant, then the algorithm corresponds to simple permutation of raw data. In contrast to the graphical functional GLM, the F rank functional GLM is based on the F-statistics that are calculated at each argument value of the functions. The global envelope test is applied to the observed and simulated F-statistics. The test is able to find if the factor of interest is significant and also which argument values of the functional domain are responsible for the potential rejection.

The specification of the full and reduced formulas is important. The reduced model should be nested within the full model. The full model should include in addition to the reduced model the interesting factors whose effects are under investigation.

There are different versions of the implementation depending on the application.

If there are no extra arguments given by the user in lm.args, then a fast implementation by solving the normal equations is used to directly compute the F-statistics.
If all the covariates are constant across the functions, but there are some extra arguments, then a linear model is fitted separately by least-squares estimation to the data at each argument value of the functions fitting a multiple linear model by lm. The possible extra arguments passed in lm.args to lm must be of the form that lm accepts for fitting a multiple linear model. In the basic case, no extra arguments are needed.
If some of the covariates vary across the space and there are user specified extra arguments given in lm.args, then the implementation fits a linear model at each argument value of the functions using lm, which can be rather slow. The arguments lm.args are passed to lm for fitting each linear model.

By default the fastest applicable method is used. This can be changed by setting method argument. The cases above correspond to "ne", "mlm" and "lm". Changing the default can be useful for checking the validity of the implementation.

References

Mrkvička, T., Myllymäki, M., Kuronen, M. and Narisetty, N. N. (2022) New methods for multiple testing in permutation inference for the general linear model. Statistics in Medicine 41(2), 276-297. doi: 10.1002/sim.9236

Freedman, D., & Lane, D. (1983) A nonstochastic interpretation of reported significance levels. Journal of Business & Economic Statistics 1(4), 292-298. doi:10.2307/1391660

Examples

Run this code

data("GDPtax")
factors.df <- data.frame(Group = GDPtax$Group, Tax = GDPtax$Profittax)
nsim <- 999
nsim <- 19
res.tax_within_group <- frank.flm(nsim = nsim,
  formula.full = Y~Group+Tax+Group:Tax,
  formula.reduced = Y~Group+Tax,
  curve_sets = list(Y=GDPtax$GDP),
  factors = factors.df)
plot(res.tax_within_group)

# Image set examples
data("abide_9002_23")
iset <- abide_9002_23$curve_set
# \dontshow{
# Cut the data to reduce time
iset$r <- iset$r[1:29,]
iset$funcs <- iset$funcs[1:29, ]
# }
res.F <- frank.flm(nsim = 19, # Increase nsim for serious analysis!
  formula.full = Y ~ Group + Age + Sex,
  formula.reduced = Y ~ Age + Sex,
  curve_sets = list(Y = iset),
  factors = abide_9002_23[['factors']],
  GET.args = list(type = "area"))
plot(res.F)

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