NAG FL Interface
g08aef (test_​friedman)

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1 Purpose

g08aef performs the Friedman two-way analysis of variance by ranks on k related samples of size n.

2 Specification

Fortran Interface
Subroutine g08aef ( x, ldx, k, n, w1, w2, fr, p, ifail)
Integer, Intent (In) :: ldx, k, n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(ldx,n)
Real (Kind=nag_wp), Intent (Out) :: w1(k), w2(k), fr, p
C Header Interface
#include <nag.h>
void  g08aef_ (const double x[], const Integer *ldx, const Integer *k, const Integer *n, double w1[], double w2[], double *fr, double *p, Integer *ifail)
The routine may be called by the names g08aef or nagf_nonpar_test_friedman.

3 Description

The Friedman test investigates the score differences between k matched samples of size n, the scores in the ith sample being denoted by
(Thus the sample scores may be regarded as a two-way table with k rows and n columns.) The hypothesis under test, H0, often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis H1 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, rij 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 ti=j=1nrij, for i=1,2,,k.
  3. (c)The Friedman test statistic F is computed, where
    F=12nk(k+1) i=1k{ti-12n(k+1)}2.  
g08aef returns the value of F, and also an approximation, p, to the significance of this value. (F approximately follows a χk-12 distribution, so large values of F imply rejection of H0). H0 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., Siegel (1956)); for k=2 the Sign test (see g08aaf) or Wilcoxon test (see g08agf) is in any case more appropriate.

4 References

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

5 Arguments

1: x(ldx,n) Real (Kind=nag_wp) array Input
On entry: x(i,j) must be set to the value, xij, of observation j in sample i, for i=1,2,,k and j=1,2,,n.
2: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g08aef is called.
Constraint: ldxk.
3: k Integer Input
On entry: k, the number of samples.
Constraint: k2.
4: n Integer Input
On entry: n, the size of each sample.
Constraint: n1.
5: w1(k) Real (Kind=nag_wp) array Workspace
6: w2(k) Real (Kind=nag_wp) array Workspace
7: fr Real (Kind=nag_wp) Output
On exit: the value of the Friedman test statistic, F.
8: p Real (Kind=nag_wp) Output
On exit: the approximate significance, p, of the Friedman test statistic.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n1.
On entry, ldx=value and k=value.
Constraint: ldxk.
On entry, k=value.
Constraint: k2.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

For estimates of the accuracy of the significance p, see g01ecf. The χ2 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.
g08aef is not threaded in any implementation.

9 Further Comments

The time taken by g08aef is approximately proportional to the product nk.
If k=2, the Sign test (see g08aaf) or Wilcoxon test (see g08agf) is more appropriate.

10 Example

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

10.1 Program Text

Program Text (g08aefe.f90)

10.2 Program Data

Program Data (g08aefe.d)

10.3 Program Results

Program Results (g08aefe.r)