NAG Library Routine Document

G08AEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G08AEF performs the Friedman two-way analysis of variance by ranks on

k

related samples of size

n

2 Specification

SUBROUTINE G08AEF (

X, LDX, K, N, W1, W2, FR, P, IFAIL)

INTEGER	LDX, K, N, IFAIL
REAL (KIND=nag_wp)	X(LDX,N), W1(K), W2(K), FR, P

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 Parameters

1: X(LDX,N) – REAL (KIND=nag_wp) arrayInput: 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 – INTEGERInput: 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 – INTEGERInput: On entry: $k$ , the number of samples.
Constraint: $K \geq 2$ .
4: N – INTEGERInput: On entry: $n$ , the size of each sample.
Constraint: $N \geq 1$ .
5: W1(K) – REAL (KIND=nag_wp) arrayWorkspace
6: W2(K) – REAL (KIND=nag_wp) arrayWorkspace
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 – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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 < 1

$IFAIL = 2$

On entry,

LDX < K

$IFAIL = 3$

On entry,

K \leq 1

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 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.

9 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.

NAG Library Routine DocumentG08AEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08AEF