GET (version 0.1-8)

graph.flm: Graphical functional GLM

Description

Non-parametric graphical tests of significance in functional general linear model (GLM)

Usage

graph.flm(
  nsim,
  formula.full,
  formula.reduced,
  curve_sets,
  factors = NULL,
  contrasts = FALSE,
  savefuns = FALSE,
  lm.args = NULL,
  GET.args = NULL,
  mc.cores = 1L,
  mc.args = NULL,
  cl = NULL,
  fast = TRUE
)

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 all the factors. The dimensions of the elements should match with each other, i.e. the factor values should be given for each argument value and each function. If factors are given in the argument factors, then can also be just the curve set representing Y. Also fdata objects allowed.

factors

A data frame of factors. An alternative way to specify factors when they are constant for all argument values. 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.

contrasts

Logical. FALSE and TRUE specify the two test functions as described in description part of this help file.

savefuns

Logical. If TRUE, then the functions from permutations are saved to the attribute simfuns.

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.

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.

fast

Logical. See details.

Value

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

Details

The function graph.flm performs the graphical functional GLM of Mrkvi<U+010D>ka et al. (2019). This is a nonparametric graphical test of significance of a covariate in functional GLM. The test is able to find not only if the factor of interest is significant, but also which functional domain is responsible for the potential rejection. In the case of functional multi-way main effect ANOVA or functional main effect ANCOVA models, the test is able to find which groups differ (and where they differ). In the case of functional factorial ANOVA or functional factorial ANCOVA models, the test is able to find which combination of levels (which interactions) differ (and where they differ). The described tests are global envelope tests applied in the context of GLMs. 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 approximately achieves the desired significance level.

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. The implementation to find the coefficients of the interesting factors is based on dummy.coef and the restrictions there apply.

There are different versions of the implementation depending on the application. Given that the argument fast is TRUE, then

• If all the covariates are constant across the functions, i.e. they can be provided in the argument factors, 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 setting fast = FALSE, it is possible to use the slow version for any case. Usually this is not desired.

References

Mrkvi<U+010D>ka, T., Roskovec, T. and Rost, M. (2019) A nonparametric graphical tests of significance in functional GLM. Methodology and Computing in Applied Probability. doi: 10.1007/s11009-019-09756-y

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

# NOT RUN {
data(rimov)
res <- graph.flm(nsim=19, # Increase the number of simulations for serious analysis!
                 formula.full = Y~Year,
                 formula.reduced = Y~1,
                 curve_sets = list(Y=rimov), factors = data.frame(Year = 1979:2014))
plot(res)

# Test if there is a change in the slope in 1994,
# i.e. the full model is T = a + b*year + c*year:group,
res <- graph.flm(nsim = 19, # Increase the number of simulations for serious analysis!
                 formula.full = Y ~ Year + Year:Group,
                 formula.reduced = Y ~ Year,
                 curve_sets = list(Y=rimov),
                 factors = data.frame(Year = 1979:2014,
                                     Group = factor(c(rep(1,times=24), rep(2,times=12)),
                                                    levels=1:2)),
                 contrasts = FALSE)
plot(res)

# }
# NOT RUN {
nsim <- 999
# }
# NOT RUN {
data(GDPtax)
factors.df <- data.frame(Group = GDPtax$Group, Tax = GDPtax$Profittax)
res.tax_within_group <- graph.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 data examples

data(abide_9002_23)
iset <- abide_9002_23$curve_set
# }
# NOT RUN {
# Figure of an image in the group 1 and group 2
plot(iset, idx=c(1, 27))

# Testing the discrete factor 'group' with contrasts
# (Use contrasts = FALSE for 'means')
res <- graph.flm(nsim = 19, # Increase nsim for serious analysis!
                 formula.full = Y ~ Group + Sex + Age,
                 formula.reduced = Y ~ Sex + Age,
                 curve_sets = list(Y = iset),
                 factors = abide_9002_23[['factors']],
                 contrasts = TRUE,
                 GET.args = list(type = "area"))
plot(res)

# Testing the continuous factor 'age'
res.a <- graph.flm(nsim = 19, # Increase nsim for serious analysis!
                 formula.full = Y ~ Group + Sex + Age,
                 formula.reduced = Y ~ Group + Sex,
                 curve_sets = list(Y = iset),
                 factors = abide_9002_23[['factors']],
                 GET.args = list(type = "area"))
plot(res.a)
# }