naginterfaces.library.stat.ranks_​and_​scores

naginterfaces.library.stat.ranks_and_scores(scores, ties, x)

ranks_and_scores computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.

For full information please refer to the NAG Library document for g01dh

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01dhf.html

Parameters

scoresstr, length 1

Indicates which of the following scores are required.

$scores=\text{'R'}$

The ranks.

$scores=\text{'N'}$

The Normal scores, that is the expected value of the Normal order statistics.

$scores=\text{'B'}$

The Blom version of the Normal scores.

$scores=\text{'T'}$

The Tukey version of the Normal scores.

$scores=\text{'V'}$

The van der Waerden version of the Normal scores.

$scores=\text{'S'}$

The Savage scores, that is the expected value of the exponential order statistics.

tiesstr, length 1

Indicates which of the following methods is to be used to assign scores to tied observations.

$ties=\text{'A'}$

The average of the scores for tied observations is used.

$ties=\text{'L'}$

The lowest score in the group of ties is used.

$ties=\text{'H'}$

The highest score in the group of ties is used.

$ties=\text{'N'}$

The nonrepeatable random number generator is used to randomly untie any group of tied observations.

$ties=\text{'R'}$

The repeatable random number generator is used to randomly untie any group of tied observations.

$ties=\text{'I'}$

Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.

xfloat, array-like, shape $\left(n\right)$

The sample of observations, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Returns

rfloat, ndarray, shape $\left(n\right)$

Contains the scores, $s_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, as specified by $scores$.

Raises

NagValueError

(errno $1$)

On entry, $scores=⟨value⟩$.

Constraint: $scores=\text{'R'}$, $\text{'N'}$, $\text{'B'}$, $\text{'T'}$, $\text{'V'}$ or $\text{'S'}$.

(errno $1$)

On entry, $ties=⟨value⟩$.

Constraint: $ties=\text{'A'}$, $\text{'L'}$, $\text{'H'}$, $\text{'R'}$ or $\text{'I'}$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

ranks_and_scores 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, ${\Phi }^{-1}(\cdot )$, at the values of the ranks scaled into the interval $(0,1)$ using different scaling transformations.

   The Blom scores use the scaling transformation $\frac{r_i-\frac{3}{8}}{n+\frac{1}{4}}$ for the rank $r_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. Thus the Blom score corresponding to the observation $x_i$ is

   $$s_i={\Phi }^{-1}\left(\frac{r_i-\frac{3}{8}}{n+\frac{1}{4}}\right)\text{.}$$

   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

   $$s_i={\Phi }^{-1}\left(\frac{r_i-\frac{1}{3}}{n+\frac{1}{3}}\right)\text{.}$$

   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

   $$s_i={\Phi }^{-1}\left(\frac{r_i}{n+1}\right)\text{.}$$

   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

   $$s_i=E\left(Y_k\right)=\frac{1}{n}+\frac{1}{n-1}+\cdots +\frac{1}{n-k+1}\text{,}$$

   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 $x_{t\left(\text{i}\right)}$, for $\text{i}=1,2,\dots ,m$, denote $m$ tied observations, that is $x_{t\left(1\right)}=x_{t\left(2\right)}=\cdots =x_{t\left(m\right)}$ with $t\left(1\right)<t\left(2\right)<\cdots <t\left(m\right)$. 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 $s_{t\left(1\right)}^{\ast },s_{t\left(2\right)}^{\ast },\dots ,s_{t\left(m\right)}^{\ast }$. Then the scores, $s_{t\left(\text{i}\right)}$, for $\text{i}=1,2,\dots ,m$, may be calculated as follows:

• if averages are used, then $s_{t\left(i\right)}={\sum }_{j=1}^m{s}_{t\left(j\right)}^{\ast }/m$;

• if the lowest score is used, then $s_{t\left(i\right)}=s_{t\left(1\right)}^{\ast }$;

• if the highest score is used, then $s_{t\left(i\right)}=s_{t\left(m\right)}^{\ast }$;

• if ties are to be broken randomly, then $s_{t\left(i\right)}=s_{t\left(I\right)}^{\ast }$ where $I\in \{\text{random permutation of}1,2,\dots ,m\}$;

• if ties are to be ignored, then $s_{t\left(i\right)}=s_{t\left(i\right)}^{\ast }$.

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