Compares population Frechet means and variances using the statistic of Dubey and Muller (2019), equation (14). The asymptotic chi-squared calibration targets equality of those summaries, not arbitrary equality of distributions. The permutation version requires exchangeable observations under the stronger null of identical group distributions, and targets alternatives detected by the same mean-and-variance statistic.
Usage
riem.fanova(..., maxiter = 50, eps = 1e-05, geometry = NULL)
riem.fanovaP(..., maxiter = 50, eps = 1e-05, nperm = 99, geometry = NULL)Arguments
- ...
At least two compatible
riemdataobjects, each containing at least three observations. Observations must be independent within and across groups for the documented calibration; paired, repeated or clustered observations require a separate restricted resampling procedure.- maxiter
Positive integer iteration budget for each Frechet mean.
- eps
Finite positive tolerance for the shared mean solver.
- geometry
A geometry name or saved specification with a compatible mean and distance implementation. By default all groups must resolve to the same geometry.
- nperm
Positive integer number of random label permutations.
Value
An htest object with statistic, p-value, null/alternative,
resolved geometry, calibration, group sizes and mean diagnostics.
The permutation version additionally retains permuted statistics,
nperm, and mc_se.
Details
The asymptotic result requires unique population/sample means, positive variances of the squared distances to each group mean, suitable moment/metric-entropy conditions, and group proportions bounded away from zero. The cited paper gives sufficient bounded-space conditions; accepting an input manifold does not establish them for every population distribution. The function rejects failed mean fits and a degenerate empirical denominator.
For group proportions \(\lambda_j=n_j/n\), the two denominators are \(\sum_j\lambda_j/\widehat\sigma_j^2\) and \(\sum_j\lambda_j^2\widehat\sigma_j^2\). Distances are jointly rescaled before forming the statistic; this leaves it unchanged and avoids unit-scale overflow. Numerical rescaling does not transform the underlying geometry.
riem.fanovaP refits all group means for every sampled label assignment.
It uses \((1+\#\{T_b\geq T_{obs}\})/(nperm+1)\), including ties. The pooled
fit is unchanged by label permutation and is reused. Permutations follow R's
current random-number state. The returned Monte Carlo standard error is a
plug-in diagnostic for simulation variability, not inferential uncertainty
in the scientific effect.
References
Dubey P, Müller H (2019). “Fréchet analysis of variance for random objects.” Biometrika, 106(4), 803–821. ISSN 0006-3444, 1464-3510.
Examples
X <- wrap.euclidean(matrix(c(-1, 0, 2, 3), ncol = 1))
Y <- wrap.euclidean(matrix(c(0, 1, 4, 8), ncol = 1))
riem.fanova(X, Y)
#>
#> Frechet Analysis of Variance on euclidean
#>
#> data: Supplied groups
#> Tn = 3.6818, df = 1, p-value = 0.05501
#> alternative hypothesis: true equal_frechet_means_and_variances is 0
#>
set.seed(17)
riem.fanovaP(X, Y, nperm = 19)
#>
#> Frechet Analysis of Variance on euclidean (permutation calibration)
#>
#> data: Supplied groups
#> Tn = 3.6818, df = 1, p-value = 0.35
#> alternative hypothesis: true exchangeable_group_distributions is 0
#>