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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_ranks_and_scores (g01dh)

## Purpose

nag_stat_ranks_and_scores (g01dh) computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.

## Syntax

[r, ifail] = g01dh(scores, ties, x, 'n', n)
[r, ifail] = nag_stat_ranks_and_scores(scores, ties, x, 'n', n)

## Description

nag_stat_ranks_and_scores (g01dh) computes one of the following scores for a sample of observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
1. Rank Scores
The ranks are assigned to the data in ascending order, that is the $i$th observation has score ${s}_{i}=k$ if it is the $k$th smallest observation in the sample.
2. Normal Scores
The Normal scores are the expected values of the Normal order statistics from a sample of size $n$. If ${x}_{i}$ is the $k$th smallest observation in the sample, then the score for that observation, ${s}_{i}$, is $E\left({Z}_{k}\right)$ where ${Z}_{k}$ is the $k$th order statistic in a sample of size $n$ from a standard Normal distribution and $E$ is the expectation operator.
3. Blom, Tukey and van der Waerden Scores
These scores are approximations to the Normal scores. The scores are obtained by evaluating the inverse cumulative Normal distribution function, ${\Phi }^{-1}\left(·\right)$, at the values of the ranks scaled into the interval $\left(0,1\right)$ using different scaling transformations.
The Blom scores use the scaling transformation $\frac{{r}_{i}-\frac{3}{8}}{n+\frac{1}{4}}$ for the rank ${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. Thus the Blom score corresponding to the observation ${x}_{i}$ is
 $si = Φ-1 ri - 38 n+14 .$
The Tukey scores use the scaling transformation $\frac{{r}_{i}-\frac{1}{3}}{n+\frac{1}{3}}$; the Tukey score corresponding to the observation ${x}_{i}$ is
 $si = Φ-1 ri - 13 n+13 .$
The van der Waerden scores use the scaling transformation $\frac{{r}_{i}}{n+1}$; the van der Waerden score corresponding to the observation ${x}_{i}$ is
 $si = Φ-1 ri n+1 .$
The van der Waerden scores may be used to carry out the van der Waerden test for testing for differences between several population distributions, see Conover (1980).
4. Savage Scores
The Savage scores are the expected values of the exponential order statistics from a sample of size $n$. They may be used in a test discussed by Savage (1956) and Lehmann (1975). If ${x}_{i}$ is the $k$th smallest observation in the sample, then the score for that observation is
 $si = EYk = 1n + 1n-1 + ⋯ + 1n-k+1 ,$
where ${Y}_{k}$ is the $k$th order statistic in a sample of size $n$ from a standard exponential distribution and $E$ is the expectation operator.
Ties may be handled in one of five ways. Let ${x}_{t\left(\mathit{i}\right)}$, for $\mathit{i}=1,2,\dots ,m$, denote $m$ tied observations, that is ${x}_{t\left(1\right)}={x}_{t\left(2\right)}=\cdots ={x}_{t\left(m\right)}$ with $t\left(1\right). If the rank of ${x}_{t\left(1\right)}$ is $k$, then if ties are ignored the rank of ${x}_{t\left(j\right)}$ will be $k+j-1$. Let the scores ignoring ties be ${s}_{t\left(1\right)}^{*},{s}_{t\left(2\right)}^{*},\dots ,{s}_{t\left(m\right)}^{*}$. Then the scores, ${s}_{t\left(\mathit{i}\right)}$, for $\mathit{i}=1,2,\dots ,m$, may be calculated as follows:
• – if averages are used, then ${s}_{t\left(i\right)}=\sum _{j=1}^{m}{s}_{t\left(j\right)}^{*}/m$;
• – if the lowest score is used, then ${s}_{t\left(i\right)}={s}_{t\left(1\right)}^{*}$;
• – if the highest score is used, then ${s}_{t\left(i\right)}={s}_{t\left(m\right)}^{*}$;
• – if ties are to be broken randomly, then ${s}_{t\left(i\right)}={s}_{t\left(I\right)}^{*}$ where $I\in \left\{\text{random permutation of ​}1,2,\dots ,m\right\}$;
• – if ties are to be ignored, then ${s}_{t\left(i\right)}={s}_{t\left(i\right)}^{*}$.

## References

Blom G (1958) Statistical Estimates and Transformed Beta-variables Wiley
Conover W J (1980) Practical Nonparametric Statistics Wiley
Lehmann E L (1975) Nonparametrics: Statistical Methods Based on Ranks Holden–Day
Savage I R (1956) Contributions to the theory of rank order statistics – the two-sample case Ann. Math. Statist. 27 590–615
Tukey J W (1962) The future of data analysis Ann. Math. Statist. 33 1–67

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{scores}$ – string (length ≥ 1)
Indicates which of the following scores are required.
${\mathbf{scores}}=\text{'R'}$
The ranks.
${\mathbf{scores}}=\text{'N'}$
The Normal scores, that is the expected value of the Normal order statistics.
${\mathbf{scores}}=\text{'B'}$
The Blom version of the Normal scores.
${\mathbf{scores}}=\text{'T'}$
The Tukey version of the Normal scores.
${\mathbf{scores}}=\text{'V'}$
The van der Waerden version of the Normal scores.
${\mathbf{scores}}=\text{'S'}$
The Savage scores, that is the expected value of the exponential order statistics.
Constraint: ${\mathbf{scores}}=\text{'R'}$, $\text{'N'}$, $\text{'B'}$, $\text{'T'}$, $\text{'V'}$ or $\text{'S'}$.
2:     $\mathrm{ties}$ – string (length ≥ 1)
Indicates which of the following methods is to be used to assign scores to tied observations.
${\mathbf{ties}}=\text{'A'}$
The average of the scores for tied observations is used.
${\mathbf{ties}}=\text{'L'}$
The lowest score in the group of ties is used.
${\mathbf{ties}}=\text{'H'}$
The highest score in the group of ties is used.
${\mathbf{ties}}=\text{'N'}$
The nonrepeatable random number generator is used to randomly untie any group of tied observations.
${\mathbf{ties}}=\text{'R'}$
The repeatable random number generator is used to randomly untie any group of tied observations.
${\mathbf{ties}}=\text{'I'}$
Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.
Constraint: ${\mathbf{ties}}=\text{'A'}$, $\text{'L'}$, $\text{'H'}$, $\text{'N'}$, $\text{'R'}$ or $\text{'I'}$.
3:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The sample of observations, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{r}\left({\mathbf{n}}\right)$ – double array
Contains the scores, ${s}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, as specified by scores.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{scores}}\ne \text{'R'}$, $\text{'N'}$, $\text{'B'}$, $\text{'T'}$, $\text{'V'}$ or $\text{'S'}$, or ${\mathbf{ties}}\ne \text{'A'}$, $\text{'L'}$, $\text{'H'}$, $\text{'N'}$, $\text{'R'}$ or $\text{'I'}$, or ${\mathbf{n}}<1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

For ${\mathbf{scores}}=\text{'R'}$, the results should be accurate to machine precision.
For ${\mathbf{scores}}=\text{'S'}$, the results should be accurate to a small multiple of machine precision.
For ${\mathbf{scores}}=\text{'N'}$, the results should have a relative accuracy of at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(100×\epsilon ,{10}^{-8}\right)$ where $\epsilon$ is the machine precision.
For ${\mathbf{scores}}=\text{'B'}$, $\text{'T'}$ or $\text{'V'}$, the results should have a relative accuracy of at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(10×\epsilon ,{10}^{-12}\right)$.

If more accurate Normal scores are required nag_stat_normal_scores_exact (g01da) should be used with appropriate settings for the input argument etol.

## Example

This example computes and prints the Savage scores for a sample of five observations. The average of the scores of any tied observations is used.
```function g01dh_example

fprintf('g01dh example results\n\n');

scores = 'Savage';
ties   = 'Average';
x      = [2;     0;     2;     2;     0];

[r, ifail] = g01dh( ...
scores, ties, x);

% Display results
fprintf('Scores: %s\n', scores);
fprintf('Ties  : %s\n', ties);
fprintf('Ranks : \n\n');
disp(r);

```
```g01dh example results

Scores: Savage
Ties  : Average
Ranks :

1.4500
0.3250
1.4500
1.4500
0.3250

```