# NNS ANOVA: Nonparametric Analysis of Variance Performs a distribution-free ANOVA using partial-moment statistics to assess differences between control and treatment groups. Depending on the setting of `means.only`, the procedure tests either differences in central tendency (means or medians) or differences across the full empirical distributions. ```r NNS.ANOVA( control, treatment, means.only = FALSE, medians = FALSE, confidence.interval = 0.95, tails = "Both", pairwise = FALSE, plot = TRUE, robust = FALSE ) ``` ## Arguments - `control`: Numeric vector of control group observations - `treatment`: Numeric vector of treatment group observations - `means.only`: Logical; `FALSE` (default) uses full distribution analysis. Set `TRUE` for mean-only comparison - `medians`: Logical; `FALSE` (default) uses means. Set `TRUE` for median-based analysis - `confidence.interval`: Numeric [0,1]; confidence level for effect size bounds (e.g., 0.95) - `tails`: Character; specifies CI tail(s): "both", "left", or "right" - `pairwise`: logical; `FALSE` (default) Returns pairwise certainty tests when set to `pairwise = TRUE`. - `plot`: Logical; `TRUE` (default) generates distribution plot - `robust`: logical; `FALSE` (default) Generates 100 independent random permutations to test results, and returns / plots 95 percent confidence intervals along with robust central tendency of all results for pairwise analysis only. ## Returns Returns a list containing: * `Control_Statistic`: Mean/median of control group * `Treatment_Statistic`: Mean/median of treatment group * `Grand_Statistic`: Grand mean/median * `Control_CDF`: CDF value at grand statistic (control) * `Treatment_CDF`: CDF value at grand statistic (treatment) * `Certainty`: Probability that the groups are the **same** (means-only or full distribution depending on `means.only`). * `Effect_Size_LB`: Lower bound of treatment effect (if CI requested) * `Effect_Size_UB`: Upper bound of treatment effect (if CI requested) * `Confidence_Level`: Confidence level used (if CI requested) ## Details The key output is the `Certainty` metric, a calibrated probability in $[0, 1]$ representing the likelihood that the groups being compared are the *same* with respect to the chosen comparison mode: * If `means.only = TRUE`: `Certainty` is the probability that the group **means** (or medians, if `medians = TRUE`) are the same. * If `means.only = FALSE`: `Certainty` is the probability that the two **entire distributions** are the same. This makes `Certainty` the conceptual inverse of a classical p-value. A *low* Certainty (e.g., < 0.10) indicates strong evidence of difference, while a *high* Certainty (e.g., > 0.90) indicates strong evidence of similarity. ## Examples ```r ## Not run: ### Binary analysis and effect size set.seed(123) x <- rnorm(100) ; y <- rnorm(100) NNS.ANOVA(control = x, treatment = y) ### Two variable analysis with no control variable A <- cbind(x, y) NNS.ANOVA(A) ### Medians test NNS.ANOVA(A, means.only = TRUE, medians = TRUE) ### Multiple variable analysis with no control variable set.seed(123) x <- rnorm(100) ; y <- rnorm(100) ; z <- rnorm(100) A <- cbind(x, y, z) NNS.ANOVA(A) ### Different length vectors used in a list x <- rnorm(30) ; y <- rnorm(40) ; z <- rnorm(50) A <- list(x, y, z) NNS.ANOVA(A) ## End(Not run) ``` ## References Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995) Viole, F. (2017) "Continuous CDFs and ANOVA with NNS" tools:::Rd_expr_doi("10.2139/ssrn.3007373") ## Author(s) Fred Viole, OVVO Financial Systems

NNS.ANOVA function