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

i

th sample being denoted by

x_{i 1}, x_{i 2}, \dots, x_{i n} .

(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

(a)The scores in each column are ranked, $r_{i j}$ denoting the rank within column $j$ of the observation in row $i$ . Average ranks are assigned to tied scores.
(b)The ranks are summed over each row to give rank sums $t_{i} = \sum_{j = 1}^{n} r_{i j}$ , for $i = 1, 2, \dots, k$ .
(c)The Friedman test statistic $F$ is computed, where

$F = \frac{12}{n k (k + 1)} \sum_{i = 1}^{k} {t_{i} - \frac{1}{2} n (k + 1)}^{2} .$

g08aef returns the value of

F

, and also an approximation,

p

, to the significance of this value. (

F

approximately follows a

χ_{k - 1}^{2}

distribution, so large values of

F

imply rejection of

H_{0}

H_{0}

is rejected by a test of chosen size

α

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, $x_{i j}$ , of observation $j$ in sample $i$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, n$ .
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: $ldx \geq k$ .
3: $k$ – Integer Input: On entry: $k$ , the number of samples.

Constraint: $k \geq 2$ .
4: $n$ – Integer Input: On entry: $n$ , the size of each sample.

Constraint: $n \geq 1$ .
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

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $ldx = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $ldx \geq k$ .

$ifail = 3$: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: 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

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.

g08aef is not threaded in any implementation.

9 Further Comments

The time taken by g08aef is approximately proportional to the product

n k

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.

g08ae: FL CL CPP AD PY MB

NAG FL Interfaceg08aef (test_​friedman)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g08aef (test_friedman)