# NAG Library Routine Document

## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine g08aef ( x, ldx, k, n, w1, w2, fr, p,
 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
#include <nagmk26.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)

## 3Description

The Friedman test investigates the score differences between $k$ matched samples of size $n$, the scores in the $i$th sample being denoted by
 $xi1,xi2,…,xin.$
(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, ${r}_{ij}$ 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 ${t}_{\mathit{i}}=\sum _{j=1}^{n}{r}_{\mathit{i}j}$, for $\mathit{i}=1,2,\dots ,k$.
(c) The Friedman test statistic $F$ is computed, where
 $F=12nkk+1 ∑i=1kti-12nk+12.$
g08aef returns the value of $F$, and also an approximation, $p$, to the significance of this value. ($F$ approximately follows a ${\chi }_{k-1}^{2}$ distribution, so large values of $F$ imply rejection of ${H}_{0}$). ${H}_{0}$ is rejected by a test of chosen size $\alpha$ if $p<\alpha$. The approximation $p$ is acceptable unless $k=4$ and $n<5$, or $k=3$ and $n<10$, or $k=2$ and $n<20$; for $k=3\text{​ or ​}4$, 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.
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

## 5Arguments

1:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value, ${x}_{\mathit{i}\mathit{j}}$, of observation $\mathit{j}$ in sample $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$ and $\mathit{j}=1,2,\dots ,n$.
2:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g08aef is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{k}}$.
3:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of samples.
Constraint: ${\mathbf{k}}\ge 2$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the size of each sample.
Constraint: ${\mathbf{n}}\ge 1$.
5:     $\mathbf{w1}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
6:     $\mathbf{w2}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{fr}$ – Real (Kind=nag_wp)Output
On exit: the value of the Friedman test statistic, $F$.
8:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the approximate significance, $p$, of the Friedman test statistic.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{k}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 2$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For estimates of the accuracy of the significance $p$, see g01ecf. The ${\chi }^{2}$ approximation is acceptable unless $k=4$ and $n<5$, or $k=3$ and $n<10$, or $k=2$ and $n<20$.

## 8Parallelism and Performance

g08aef is not threaded in any implementation.

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.

## 10Example

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.

### 10.1Program Text

Program Text (g08aefe.f90)

### 10.2Program Data

Program Data (g08aefe.d)

### 10.3Program Results

Program Results (g08aefe.r)