(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 $F R$ is computed, where

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

g08aec returns the value of

F R

, and also an approximation,

p

, to the significance of this value. (

F R

approximately follows a

χ_{k - 1}^{2}

distribution, so large values of

F R

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

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

Constraint: $n \geq 1$ .
3: $x [k \times tdx]$ – const double Input: On entry: $x [(i - 1) \times tdx + j - 1]$ must be set to the value, $x_{i j}$ , of observation $j$ in sample $i$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, n$ .
4: $tdx$ – Integer Input: On entry: the stride separating matrix column elements in the array x.

Constraint: $tdx \geq n$ .
5: $fr$ – double * Output: On exit: the value of the Friedman test statistic, $F R$ .
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

NE_2_INT_ARG_LT: On entry, $tdx = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $tdx \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LE: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .
NE_INT_ARG_LT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: 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

g08aec is not threaded in any implementation.

9 Further Comments

The time taken by g08aec is approximately proportional to the product

n k

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.

g08ae: FL CL CPP AD

NAG CL Interfaceg08aec (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 CL Interface
g08aec (test_friedman)