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# LCONCORDANCE procedure

Calculates Lin’s concordance correlation coefficient (R.W. Payne & M.S. Dhanoa).

### Options

`PRINT` = string token Controls printed output (`concordance`); default `conc` Defines the sets of measurements when they are all supplied in a single `DATA` variate Saves Lin’s the concordance coefficient Saves the lower confidence limit for the coefficient Saves the upper confidence limit for the coefficient Saves the correlation coefficient Saves the bias correction factor Saves the Z transformation of the coefficient Saves the standard deviation of the Z transformation Defines the size of the confidence interval; default 0.95 i.e. 95% Defines the set of measurements to be used as the control if there are more than two variates or groups; default 1

### Parameter

`DATA` = variates List of variates specifying the sets of measurements to be compared, or a single variate containing all the measurements (the `GROUPS` option must then be set to indicate the set to which each unit belongs)

### Description

Lin’s concordance correlation coefficient measures how well a new set of observations reproduce an original set. So, for example, it can be used to assess the effectiveness of new instruments or measurement methods.

The coefficient is calculated by multiplying two components. The first is the ordinary Pearson correlation coefficient, which essentially assesses the linear relationship between the two sets of measurements. However, for the second set to reproduce the first, the slope of the line relating the two sets should be one, and the line should go through the origin. These other aspects are assessed by the second component, which is known as Cb.

The measurements are supplied using the `DATA` parameter. You can set this to a list of variates, one for each measurement. Alternatively, you can put them all into a single variate, and set the `GROUPS` option to a factor to identify which measurement is stored in each unit of the variate. (`LCONCORDANCE` then assumes that the individuals that were measured are recorded in the same order within each set of measurements.) If there are more than two sets of measurements, `LCONCORDANCE` takes one of these as the control (i.e. the standard) set, and compares the others with this. By default the control is first variate if `DATA` has been set to a list of variates, or the set corresponding to the reference level of the `GROUPS` factor (see the `FACTOR` directive) if there was a single variate. However, you can define a different control by setting the `REFERENCELEVEL` option, to a scalar to indicate the number of the variate within the list of `DATA` variates of the level of the GROUPS factor. Alternatively, if the `GROUPS` factor has labels, you can set `REFERENCELEVEL` to a text.

Lin (1989, 2000) has shown that, if the coefficient is given an inverse hypobolic tangent transformation (i.e. a Z-transformation), the result has an approximate Normal distribution. `LCONCORDANCE` uses this to produce a confidence interval for the coefficient. The size of the interval is specified by the `CIPROBABILITY` option; the default is 0.95 (i.e. 95%).

By default, the concordance coefficient, the lower and upper confidence limits, the correlation coefficient and Cb are printed. However, you can set option `PRINT=*` to suppress this. The `CONCORDANCE`, `LOWER`, `UPPER`, `CORRELATION`, `CB`, `ZTRANSFORMATION` and `ZSD` parameters allow the coefficient and all the associated information to be saved.

Options: `PRINT`, `GROUPS`, `CONCORDANCE`, `LOWER`, `UPPER`, `CORRELATION`, `CB`, `ZTRANSFORMATION`, `ZSD`, `CIPROBABILITY`, `REFERENCELEVEL`.

Parameter: `DATA`.

### Method

The coefficient ρc is derived by Lin (1989) by considering how well the relationship between the measurements is represented by a line through the origin at an angle of 45 degrees (as would be generated if the two measurements generated identical results):

ρc = 1 – dc2 / du2

where dc2 is the expected squared perpendicular deviation from the line, and du2 is the expected squared perpendicular deviation from the line when the measurements are uncorrelated.

This can be written as

ρc = ρ × Cb

The term ρ is the standard Pearson product-moment correlation coefficient, while Cb is a bias correction factor which is calculated by

Cb = 2 / (v + 1/v + u2)

v = s1 / s2

u = (m1m2) / √(s1 × s2)

where mi and si (i = 1,2) are the mean and standard deviation of the ith set of measurements.

The Z-transformation is

Z = 0.5 × (log(1 + ρc) / log(1 – ρc))

The standard deviation of the Z-transformed coefficient is calculated as defined by Lin (2000).

### Action with `RESTRICT`

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

Lin, L.I. (1989). A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45, 255-268.

Lin, L.I. (2000). A note on the concordance correlation coefficient. Biometrics, 56, 324-325.

Procedures: `BLANDALTMAN`, `SLCONCORDANCE`, `CMHTEST`, `FCORRELATION`, `KCONCORDANCE`, `KTAU`, `SPEARMAN`.

Commands for: Basic and nonparametric statistics.

### Example

```CAPTION 'LCONCORDANCE example',\
!t('Data from Muller & Buttner (1994, Statistics in Medicine, 13,',\
'2465-2476); also see Nickerson (1997, Biometrics, 53, 1503-1507).');\
STYLE=meta,plain
VARIATE [VALUES=4.8,5.6,6.0,6.4,6.5,6.6,6.8,7.0,7.0,7.2,7.4,7.6,\
7.7,7.7,8.2,8.2,8.3,8.5,9.3,10.2,10.4,10.6,11.4] Trial1
&       [VALUES=5.8,5.1,7.7,7.8,7.6,8.1,8.0,8.1,6.6,8.1,9.5,9.6,\
8.5,9.5,9.1,10.,9.1,10.8,11.5,11.5,11.2,11.5,12.0] Trial2
LCONCORDANCE Trial1,Trial2
```