Calculates the sample size for Lin’s concordance correlation coefficient (R.W. Payne).

### Options

`PRINT` = string token |
What to print (`replication` , `power` ); default `repl` , `powe` |
---|---|

`PROBABILITY` = scalar |
Significance level at which the non-reproducibility is to be tested; default 0.05 |

`POWER` = scalar |
The required power (i.e. probability of detection) of the test; default 0.9 |

`REPLICATION` = variate |
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

`CORRELATION` = scalars |
Correlation for two samples with the smallest amount of non-reproducibility required to be detected |
---|---|

`CONCORDANCE` = scalars |
Value of Lin’s concordance for two samples with the smallest amount of non-reproducibility required to be detected |

`MEANSHIFT` = scalars |
Value of the shift in means (divided by the harmonic mean of the standard deviations) for two samples with the smallest amount of non-reproducibility required to be detected |

`SDRATIO` = scalars |
Value of the ratio of the standard deviations for two samples with the smallest amount of non-reproducibility required to be detected |

`NREPLICATES` = scalars |
Saves the required number of replicates |

`VREPLICATION` = variates |
Numbers of replicates for which powers have been calculated |

`VPOWER` = variates |
Power (i.e. probability of detection) for the various numbers of replicates |

### Description

Lin’s concordance correlation coefficient can be used to assess how well a new method of measurement reproduces the results provided by a standard method. To do this, you measure the same set of units using the two methods, and calculate the concordance between the resulting two sets of measurements. The methods are regarded as equivalent if the coefficient is greater than some threshold. SLCONCORDANCE helps you to decide how many units need to be measured to make a reliable assessment.

The concordance coefficient is defined by the equation

ρ* _{c}* = ρ ×

*C*

_{b}(see Lin 1989, 2000 or procedure `LCONCORDANCE`

). The term ρ is the standard Pearson product-moment correlation coefficient, while *C _{b}* is a bias correction factor which is calculated by

*C _{b}* = 2 / (

*v*+ 1/

*v*+

*u*

^{2})

*v* = *s*_{1} / *s*_{2}

*u* = (*m*_{1} – *m*_{2}) / √(*s*_{1} × *s*_{2})

where *m _{i}* and

*s*(

_{i}*i*= 1,2) are the mean and standard deviation of the

*i*

^{th}set of measurements. The quantity

*u*represents the shift in the mean between the two sets of measurements divided by the harmonic mean of their standard deviations, while

*v*is the ratio of the two standard deviations.

If the coefficient is given a Z-transformation, the result has an approximate Normal distribution, with a standard deviation that depends on ρ* _{c}*, ρ and

*u*(see Lin 1989, 2000). So, to calculate the sample size,

`SLCONCORDANCE`

needs to know the values of these quantities for two sets of measurements displaying the smallest amount of non-reproducibility that is required to be detected. The correlation coefficient (ρ) is specified by the `CORRELATION`

parameter, the concordance coefficient by the `CONCORDANCE`

parameter, and *u*by the

`MEANSHIFT`

parameter. Alternatively, you can omit either `CONCORDANCE`

or `MEANSHIFT`

provided you specify the ratio of the standard deviations, *v*, using the

`SDRATIO`

parameter. (`SLCONCORDANCE`

can then calculate the omitted quantity using the equations in the previous paragraph.)The significance level for the test is specified by the `PROBABILITY`

option (default 0.05 i.e. 5%). This is for a one-sided test, on the basis that you would not reject a new method for being too similar to the standard method. (Note, this also corresponds to a test for non-inferiority; see the Methods section of the documentation for procedure `STTEST`

.) The required probability for detecting non-reproducibility (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 to measure using each method (i.e. the sample size); |
---|---|

`power` |
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`

, `RATIOREPLICATION`

, `REPLICATION`

.

Parameters: `CORRELATION`

, `CONCORDANCE`

, `MEANSHIFT`

, `SDRATIO`

, `NREPLICATES`

, `VREPLICATION`

, `VPOWER`

.

### Method

The calculation uses the fact that the transformation

*Z* = 0.5 × (log(1 + ρ* _{c}*) / log(1 – ρ

*))*

_{c}has an approximate Normal distribution, with a standard deviation defined by Lin (2000). Note, the results produced by `SLCONCORDANCE`

do not match those of Lin (1992) firstly because of the correction to the equation for the standard deviation noted by Lin (2000), and secondly because the equation for *n* on page 601 of Lin (1992) should read

*n* = ( (`EDNORMAL`

(1-β) + `EDNORMAL`

(1-α)) * *S* / (*Z* – *Z _{c}*

_{,a}) )

^{2}+ 2.

### References

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

Lin, L.I. (1992). Assay validation using the concordance correlation coefficient. *Biometrics*, 48, 599-604.

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

### See also

Procedure: `LCONCORDANCE`

.

Commands for: Design of experiments.

### Example

CAPTION 'SLCONCORDANCE example',!t(\ 'Sample size required to detect non-reproducibility represented by',\ 'a correlation of 0.95, a concordance of 0.9 and a shift in means',\ '(scaled by harmonic average s.d.) of 0.1.'); STYLE=meta,plain SLCONCORDANCE CORRELATION=0.95; CONCORDANCE=0.9; MEANSHIFT=0.1