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# KTAU procedure

Calculates Kendall’s rank correlation coefficient τ (R.W. Payne & D.B. Baird).

### Options

`PRINT` = string tokens Output required (`correlations`, `probabilities`); default `corr`, `prob` Defines the sample membership if only one variate is specified by `DATA` Scalar to save the rank correlation coefficient if there are two samples, or symmetric matrix to save the coefficients between all pairs of samples if there are several Scalar to save the probability for the correlation coefficient if there are two samples, or symmetric matrix to save the probabilities for all pairs of samples if there are several Scalar to save a transformation of tau that approximately follows a Normal distribution with mean zero and variance if there are two samples, or symmetric matrix to save the transformation for all pairs of samples if there are several

### Parameter

`DATA` = variates List of variates containing the data for each sample, or a single variate containing the data from all the samples (the `GROUPS` option must then be set to indicate the sample to which each unit belongs)

### Description

`KTAU` calculates Kendall’s rank correlation coefficient (known as τ i.e. tau) between pairs of samples. The samples can be stored in different variates and supplied in a list with the `DATA` parameter. Alternatively, they can all be placed in a single variate, and the `GROUPS` option set to a factor to indicate the sample to which each unit belongs.

The `PRINT` option controls the printed output, with settings:

    `correlations` to print the correlations between the samples; and to print the corresponding probabilities (calculated under the assumption, or null hypothesis, that there is no association between the samples).

By default these are both printed.

The `CORRELATIONS` option allows the correlations to be saved, in a scalar if there are only two samples or in a symmetric matrix if there are three or more. Similarly, the probabilities can be saved using the `PROBABILITIES` option. Also, you can use the `NORMAL` option to save a transformation of τ that approximately follows a Normal distribution with mean zero and variance; this provides reasonably accurate probabilities when the number of units N is no smaller than 8 (see Kendall 1948).

Options: `PRINT`, `GROUPS`, `CORRELATIONS`, `PROBABILITIES`, `NORMAL`.

Parameter: `DATA`.

### Method

Kendall’s rank correlation coefficient τ is a measure of association between the rankings of two variables measured on N individuals. It is calculated as

τ = S / √(NC1 × NC2)

S is defined as the sum of

`SIGN`(xixj) × `SIGN`(yiyj)

over all pair of distinct units i and j. NC1 and NC2 are the number of valid comparisons (removing ties and missing values) that can be made amongst the first and second set of samples, respectively. (See Siegel 1956, pages 213-223.)

The transformation of τ into a Normal random variable is given by

τ / √( (2 × (2 × N + 5)) / (9 × N × (N – 1)) )

The probabilities are calculated using procedure `PRKTAU`.

### Action with `RESTRICT`

If any of the variates in `DATA` is restricted, the statistic is calculated only for the set of units not excluded by the restriction.

Kendall, M.G. (1948). Rank Correlation Methods. Griffin, London.

Siegel, S. (1956). Nonparametric Statistics for the Behavioural Sciences. McGraw-Hill, New York.

Procedures: `PRKTAU`, `CMHTEST`, `FCORRELATION`, `KCONCORDANCE`, `LCONCORDANCE`, `SPEARMAN`.

Commands for Basic and nonparametric statistics.

### Example

```CAPTION  'KTAU example',!t(\
'Data from Siegel, (1956, Nonparametric Statistics, pages 205 & 208).',\
'2 variables are measured on N individuals.'); STYLE=meta,plain
VARIATE  [VALUE= 82, 98, 87, 40,116,113,111, 83, 85,126,106,117] Authority
&        [VALUE= 42, 46, 39, 37, 65, 88, 86, 56, 62, 92, 54, 81] Status
&        [VALUE=  0,  0,  1,  1,  3,  4,  5,  6,  7,  8,  8, 12] Yielding
PRINT    Authority,Status,Yielding; DECIMALS=0
KTAU     [CORRELATIONS=Correlation; PROBABILITIES=Probability]\
Authority,Status
PRINT    Correlation,Probability
KTAU     [CORRELATIONS=Correlation; PROBABILITIES=Probability]\
Status,Yielding
PRINT    Correlation,Probability
```