frank.fanova: Rank envelope F-test

Description

A one-way functional ANOVA based on the rank envelope applied to F values

Usage

frank.fanova(
  nsim,
  curve_set,
  groups,
  variances = "equal",
  test.equality = c("mean", "var", "cov"),
  cov.lag = 1,
  savefuns = FALSE,
  ...
)

Arguments

nsim: The number of random permutations.
curve_set: The original data (an array of functions) provided as a curve_set object or a fdata object of fda.usc. The curve set should include the argument values for the functions in the component r, and the observed functions in the component obs.
groups: The original groups (a factor vector representing the assignment to groups).
variances: Either "equal" or "unequal". If "equal", then the traditional F-values are used. If "unequal", then the corrected F-values are used. The current implementation uses lm to get the corrected F-values.
test.equality: A character with possible values mean (default), var and cov. If mean, the functional ANOVA is performed to compare the means in the groups. If var, then the equality of variances of the curves in the groups is tested by performing the graphical functional ANOVA test on the functions $$Z_{ij}(r) = T_{ij}(r) - \bar{T}_j(r).$$ If cov, then the equality of lag cov.lag covariance is tested by performing the fANOVA with $$W_{ij}(r) = \sqrt{|V_{ij}(r)|\cdot sign(V_{ij}(r))},$$ where $$V_{ij}(r) = (T_{ij}(r) - \bar{T}_j(r))((T_{ij}(r+s) - \bar{T}_j(r+s))).$$ See Mrkvicka et al. (2020) for more details.
cov.lag: The lag of the covariance for testing the equality of covariances, see test.equality.
savefuns: Logical. If TRUE, then the functions from permutations are saved to the attribute simfuns.
...: Additional parameters to be passed to global_envelope_test. For example, the type of multiple testing control, FWER or FDR must be set by typeone. And, if typeone = "fwer", the type of the global envelope can be chosen by specifying the argument type. See global_envelope_test for the defaults and available options. (The test here uses alternative="two.sided" and nstep=1 (when relevant), but all the other specifications are to be given in ....)

Details

The test assumes that there are $J$ groups which contain $n_1,\dots,n_J$ functions $T_{ij}, i=\dots,J, j=1,\dots,n_j$. The functions should be given in the argument x, and the groups in the argument groups. The test assumes that there exists non random functions $\mu(r)$ and $\mu_i(r)$ such that $$T_{ij}(r) =\mu(r) + \mu_i(r) + e_{ij}(r), i=1, \dots, J, j=1, \dots , n_j$$ where $e_{ij}(r)$ are independent and normally distributed. The test vector is $$\mathbf{T} = (F(r_1), F(r_2), \dots , F(r_K)),$$ where $F(r_i)$ stands for the F-statistic. The simulations are performed by permuting the test functions. Further details can be found in Mrkvička et al. (2020).

The argument variances="equal" means that equal variances across groups are assumed. The correction for unequal variances can be done by using the corrected F-statistic (option variances="unequal").

Unfortunately this test is not able to detect which groups are different from each other.

References

Mrkvička, T., Myllymäki, M., Jilek, M. and Hahn, U. (2020) A one-way ANOVA test for functional data with graphical interpretation. Kybernetika 56 (3), 432-458. doi: 10.14736/kyb-2020-3-0432

Examples

Run this code

data("rimov")
groups <- factor(c(rep(1, times=12), rep(2, times=12), rep(3, times=12)))
res <- frank.fanova(nsim = 2499, curve_set = rimov, groups = groups)
res <- frank.fanova(nsim = 4, curve_set = rimov, groups = groups, alpha = 0.2)
plot(res)

data("imageset3")
res2 <- frank.fanova(nsim = 19, # Increase nsim for serious analysis!
                     curve_set = imageset3$image_set,
                     groups = imageset3$Group)
plot(res2)
plot(res2, fixedscales=FALSE)