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NNS ANOVA: Nonparametric Analysis of Variance

Source: R/ANOVA.R
NNS.ANOVA.Rd

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.

Usage

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.

Value

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.

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2017) "Continuous CDFs and ANOVA with NNS" doi:10.2139/ssrn.3007373

Author

Fred Viole, OVVO Financial Systems

Examples

 if (FALSE) { # \dontrun{
### 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)
} # }