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# SCORRELATION procedure

Calculates the sample size to detect specified correlations (R.W. Payne).

### Options

`PRINT` = string token What to print (`replication`, `power`); default `repl`, `powe` Significance level at which the correlation or difference between correlations is to be tested; default 0.05 The required power (i.e. probability of detection) of the test; default 0.9 Whether to a one- or two-sided test is to be made (`onesided`, `twosided`); default `ones` Ratio of replication sample2:sample1 (i.e. the size of sample for group 2 should be `RATIOREPLICATION` times the size of sample for group 1); default 1 Replication values for which to calculate and print or save the power; default * takes 11 replication values centred around the required number of replicates

### Parameters

`COR1` = scalars Anticipated correlation in group 1 Anticipated correlation in group 2 Saves the required number of replicates Numbers of replicates for which powers have been calculated Power (i.e. probability of detection) for the various numbers of replicates

### Description

`SCORRELATION` may be useful when you wish to assess the correlation between two variables within a single group of subjects, or when you wish to compare the correlations between two groups of subjects. The correlation in this case is the product moment correlation coefficient, as calculated by the `CORRELATION` function (and so the variables are assumed to have Normal distributions).

If there is a single group of subjects the correlation is specified (in a scalar) by the `COR1` parameter, and the assumption is that we wish to assess whether this is non-zero. With two groups the correlations are specified by the `COR1` and `COR2` parameters (again in scalars). Generally equal sample sizes are assumed for the two groups. However, you can set the `RATIOREPLICATION` option to a scalar, `R` say, to indicate that the size of the second sample should be `R` times the size of the first sample. The `NREPLICATES` parameter allows you to save the required size of the first sample.

The significance level for the test is specified by the `PROBABILITY` option (default 0.05 i.e. 5%). By default this is for a one-sided test, but you can set option `TMETHOD=twosided` for a two-sided test. The required probabilty for detection of the correlation or difference in correlations (that is, the power of the test) is specified by the `POWER` option (default 0.9).

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

    `replication` to print the required number of replicates in each sample (i.e. the size of each sample); to print a table giving the power (i.e. probability of detection) provided by a range of numbers of replicates.

By default both are printed.

The replications and corresponding powers can also be saved, in variates, using the `VREPLICATION` and `VPOWER` parameters. The `REPLICATION` option can specify the replication values for which to calculate and print or save the power; if this is not set, the default is to take 11 replication values centred around the required number of replicates.

Options: `PRINT`, `PROBABILITY`, `POWER`, `TMETHOD`, `RATIOREPLICATION`, `REPLICATION`.

Parameters: `COR1`, `COR2`, `NREPLICATES`, `VREPLICATION`, `VPOWER`.

### Method

With a single group, suppose that the sample correlation is r and the number of subjects is n. `SCORRELATION` uses the fact that, under the null hypothesis of a zero correlation, the variable

t = r × √((n – 2) / (1 – r2))

has a t distribution on n-2 degrees of freedom.

With two groups, `SCORRELATION` uses Fisher’s Z transformation:

z = 0.5 × log((1 + r)/( 1 – r))

Provided the sample sizes are reasonably large, z can be assumed to have a Normal distribution with variance 1/(n-3).

Directive: `CORRELATE`.

Procedure: `FCORRELATION`.

Commands for: Design of experiments.

### Example

```CAPTION      'SCORRELATION example',\
'1) one sample, anticipated correlation 0.5.'; STYLE=meta,plain
SCORRELATION [PRINT=replication,power; TMETHOD=twosided] 0.5
CAPTION      '2) two samples, anticipated correlations 0.2 and 0.6.'
SCORRELATION [PRINT=replication,power; TMETHOD=twosided] 0.2; COR2=0.6
```
Updated on June 18, 2019