Description of MRPP

Abstract

Multiresponse permutation procedures (MRPP) are a class of permutation methods of one or more dimensions for distinguishing possible differences among two or more groups. To motivate MRPP, initially consider samples of independent and identically distributed univariate random variables of sizes n₁, ...,n_g, namely,

$$({y_{11}}, \cdots ,{y_{{n_1}1}}), \cdots ,({y_{1g}}, \cdots ,{y_{{n_g}g}}),$$

from g populations with cumulative distribution functions F¹(x), ...,F_g(x), respectively. For simplicity, suppose that population i is normal with mean μ_i and variance σ² (i = 1, ...,g). This is the standard one-way classification model with g groups. In the classical test of a null hypothesis of no group differences, one tests H⁰: μ¹ = ... =μ^g versus H¹: μⁱ≠μ^j for some i≠j using the F statistic given by

$$F = \frac{{M{S_{between}}}}{{M{S_{within}}}}$$

where

$$M{S_{between}} = M{S_{treatment}} = \frac{1}{{g - 1}}S{S_{between,}}$$

$$S{S_{between}} = \sum\limits_{i = 1}^g {{n_i}} {({\bar y_i} - \bar y)^2},$$