NAG Library Function Document

nag_friedman_test (g08aec)

+− 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

1 Purpose

nag_friedman_test (g08aec) performs the Friedman two-way analysis of variance by ranks on

k

related samples of size

n

2 Specification

#include <nag.h>

#include <nagg08.h>

void	nag_friedman_test (Integer k, Integer n, const double x[], Integer tdx, double fr, double p, NagError *fail)

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 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} .

nag_friedman_test (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

or 4, tables should be consulted (e.g., n of Siegel (1956)); for

k = 2

the Sign test (see nag_sign_test (g08aac)) or Wilcoxon test (see nag_wilcoxon_test (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 – IntegerInput: On entry: $k$ , the number of samples.
Constraint: $k \geq 2$ .
2: n – IntegerInput: On entry: the size of each sample, $n$ .
Constraint: $n \geq 1$ .
3: x[ $k \times tdx$ ] – const doubleInput: 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 – IntegerInput: 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 3.6 in the Essential Introduction).

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 nag_prob_chi_sq (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

Not applicable.

9 Further Comments

The time taken by nag_friedman_test (g08aec) is approximately proportional to the product

n k

k = 2

, the Sign test (see nag_sign_test (g08aac)) or Wilcoxon test (see nag_wilcoxon_test (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.

NAG Library Function Documentnag_friedman_test (g08aec)

+− 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 Library Function Document

nag_friedman_test (g08aec)