NAG CL Interface
g08aec (test_​friedman)

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1 Purpose

g08aec performs the Friedman two-way analysis of variance by ranks on k related samples of size n .

2 Specification

#include <nag.h>
void  g08aec (Integer k, Integer n, const double x[], Integer tdx, double *fr, double *p, NagError *fail)
The function may be called by the names: g08aec, nag_nonpar_test_friedman or nag_friedman_test.

3 Description

The Friedman test investigates the score differences between k matched samples of size n , the scores in the i th sample being denoted by:
x i1 , x i2 , , x in .  
(Thus the sample scores may be regarded as a two-way table with k rows and n columns.) The hypothesis under test, H 0 , often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis H 1 that they come from different populations.
The test is based on the observed distribution of score rankings between the matched observations in different samples.
The test proceeds as follows:
  1. (a)The scores in each column are ranked, r ij denoting the rank within column j of the observation in row i . Average ranks are assigned to tied scores.
  2. (b)The ranks are summed over each row to give rank sums t i = j=1 n r ij , for i=1,2,,k.
  3. (c)The Friedman test statistic FR is computed, where
    FR = 12 nk (k+1) i=1 k { t i - 1 2 n(k+1)} 2 .  
g08aec returns the value of FR , and also an approximation, p , to the significance of this value. ( FR approximately follows a χ k-1 2 distribution, so large values of FR imply rejection of H 0 ). H 0 is rejected by a test of chosen size α if p<α . The approximation p is acceptable unless k=4 and n<5 , or k=3 and n<10 , or k=2 and n<20 ; for k=3 or 4, tables should be consulted (e.g., n of Siegel (1956)); for k=2 the Sign test (see g08aac) or Wilcoxon test (see g08agc) is in any case more appropriate.

4 References

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

5 Arguments

1: k Integer Input
On entry: k, the number of samples.
Constraint: k2.
2: n Integer Input
On entry: the size of each sample, n .
Constraint: n1 .
3: x[k×tdx] const double Input
On entry: x[(i-1)×tdx+j-1] must be set to the value, x ij , of observation j in sample i , for i=1,2,,k and j=1,2,,n.
4: tdx Integer Input
On entry: the stride separating matrix column elements in the array x.
Constraint: tdxn .
5: fr double * Output
On exit: the value of the Friedman test statistic, FR .
6: p double * Output
On exit: the approximate significance, p , of the Friedman test statistic.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

On entry, tdx=value while n=value . These arguments must satisfy tdxn .
Dynamic memory allocation failed.
On entry, k=value.
Constraint: k2.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

For estimates of the accuracy of the significance p , see g01ecc. The χ 2 approximation is acceptable unless k=4 and n<5 , or k=3 and n<10 , or k=2 and n<20 .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g08aec is not threaded in any implementation.

9 Further Comments

The time taken by g08aec is approximately proportional to the product nk .
If k=2 , the Sign test (see g08aac) or Wilcoxon test (see g08agc) is more appropriate.

10 Example

This example is taken from page 169 of Siegel (1956). The data relate to training scores of three matched samples of 18 rats, trained under three different patterns of reinforcement.

10.1 Program Text

Program Text (g08aece.c)

10.2 Program Data

Program Data (g08aece.d)

10.3 Program Results

Program Results (g08aece.r)