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## ACR Upper percentiles critical value for test of single multivariate normal outlier.
## From the method given by Wilks (1963) and approaching to a F distribution
## function by the Yang and Lee (1987) formulation, we provide an m-file to
## get the critical value of the maximun squared Mahalanobis distance to detect
## outliers from a normal multivariate sample.
##
## Syntax: function x = ACR(p,n,alpha)
## $$ The function's name is giving as a gratefull to Dr. Alvin C. Rencher for his
## unvaluable contribution to multivariate statistics with his text 'Methods of
## Multivariate Analysis'.$$
##
## Inputs:
##' @param p number of independent variables.
##' @param n sample size.
##' @param alpha significance level (default = 0.05).
##
## Output:
##' @return ACR value of the maximum squared Mahalanobis distance.
##
## We can generate all the critical values of the maximun squared Mahalanobis
## distance presented on the Table XXXII of by Barnett and Lewis (1978) and
## Table A.6 of Rencher (2002). Also with any given significance level (alpha).
##
## Example: For p = 3; n = 25; alpha=0.01;
##
## Calling on Matlab the function:
## ACR(p,n,alpha)
##
## Answer is:
##
## 13.1753
##
## Created by A. Trujillo-Ortiz, R. Hernandez-Walls, A. Castro-Perez and K. Barba-Rojo
## Facultad de Ciencias Marinas
## Universidad Autonoma de Baja California
## Apdo. Postal 453
## Ensenada, Baja California
## Mexico.
## atrujo@uabc.mx
##
## Copyright. August 20, 2006.
##
## To cite this file, this would be an appropriate format:
## Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez and K. Barba-Rojo. (2006).
## ACR:Upper percentiles critical value for test of single multivariate normal outlier.
## A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/
## fileexchange/loadFile.do?objectId=12161
##
## References:
## Barnett, V. and Lewis, T. (1978), Outliers on Statistical Data.
## New-York:John Wiley & Sons.
## Rencher, A. C. (2002), Methods of Multivariate Analysis. 2nd. ed.
## New-Jersey:John Wiley & Sons. Chapter 13 (pp. 408-450).
## Wilks, S. S. (1963), Multivariate Statistical Outliers. Sankhya,
## Series A, 25: 407-426.
## Yang, S. S. and Lee, Y. (1987), Identification of a Multivariate
## Outlier. Presented at the Annual Meeting of the American
## Statistical Association, San Francisco, August 1987.
##
ACR <- function(p,n,alpha){
if (missing(alpha)) alpha <- 0.05
if (alpha<=0 | alpha>=1) stop("Significance level must be between 0 and 1")
if (missing(p)|missing(n)) stop("Requires args p and n")
a <- alpha
## F distribution critical value with p and n-p-1 degrees of freedom using the Bonferroni correction
Fc <- qf(1-a/n,p,n-p-1)
ACR <- (p*(n-1)^2*Fc)/(n*(n-p-1)+(n*p*Fc))
return(ACR)
} ##ACR
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