# Test equality of probability vectors Perform optimal transport (OT) barycenter based tests for equality of probability vectors in a one-way layout. ```r ot_barycenter_test( samples, costm, null.mu = NULL, w = NULL, num.sim = 1000, solver = ot_test_lp_solver(), is.metric = is_metric_cost_mat(costm, tol.ti = Inf), verbose = FALSE ) ``` ## Arguments - `samples`: matrix (row-wise) or nested list containing $K$ count vectors. A count vector is a vector of length $N$ that contains the number of times a sample was observed at the respective points. - `costm`: semi-metric cost matrix $c \in \mathbb{R}^{N \times N}$. - `null.mu`: probability measures $\mu$ underlying the null distribution. Must be of the same structure as `samples`. - `w`: weight vector $w \in \mathbb{R}^K_+$. - `num.sim`: number of samples to draw from the limiting null distribution. - `solver`: the LP solver to use, see `ot_test_lp_solver`. - `is.metric`: value indicating whether $c$ is a metric cost matrix, see `is_metric_cost_mat`. - `verbose`: logical value indicating whether additional information should be printed. ## Returns An object of class `"ot_barycenter_test"` containing: ||| |:--|:--| |`mu`|empirical version of $\mu$ that is based on `samples`| |`n`|the sample sizes| |`p.value`|the $p$ -value| |`statistic`|the value of the test statistic| |`null.samples`|samples drawn from the null distribution| ## Details Denote with $\mu^1, \ldots, \mu^K$ the probability measures that underlie the samples contained in `samples`. To test for the one-way null hypothesis $H_0 : \mu^1 = \ldots = \mu^K$, this test employs the OT barycenter statistic which is defined as c("$\n$", "$ T^B(\\mu) := \\sqrt{\\rho_n} B_c^w(\\mu^1, \\ldots, \\mu^K)\\,,\n$") where $\rho_n$ is a scaling factor and $B_c^w$ is the OT barycenter functional, see `ot_barycenter`. The test is based on the asymptotic distribution of $T^B$ under under the null, for more details see the reference. These simulations can be done in parallel via `future::plan` and the progress can be shown with `progressr::with_progress`. Especially for large $N$ and $K$, simulating a sufficient number of samples from the limiting null distribution might take a while. Consider using `FDOTT` instead. ## Examples ```r # enable txt progressbar progressr::handlers("txtprogressbar") # enable parallel computation if (requireNamespace("future")) { future::plan(future::multisession) } K <- 3 N <- 2 costm <- cost_matrix_lp(1:N) # use higher number to better approximate null distribution and get more accurate p-value num.sim <- 10 n <- c(300, 360, 200) # underlying probability vectors mu <- matrix(1 / N, K, N, TRUE) # to run this, a LP solver must be available for ROI (ROI.plugin.glpk by default) if (requireNamespace("ROI.plugin.glpk")) { solver <- ot_test_lp_solver("glpk") set.seed(123) samples <- tab_sample(n, mu) progressr::with_progress({ res <- ot_barycenter_test(samples, costm, num.sim = num.sim, solver = solver) }) print(res) } # measures are not equal anymore mu[2, ] <- 1:N / sum(1:N) if (requireNamespace("ROI.plugin.glpk")) { solver <- ot_test_lp_solver("glpk") set.seed(123) samples <- tab_sample(n, mu) progressr::with_progress({ res2 <- ot_barycenter_test(samples, costm, num.sim = num.sim, solver = solver) }) print(res2) } ``` ## References TODO ## See Also `FDOTT` with `H0 = "="`.

ot_barycenter_test function