LCONCORDANCE procedure

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

Options

Parameter

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 C_b.

   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 ρ_c 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):

ρ_c = 1 - d_c² / d_u²

where d_c² is the expected squared perpendicular deviation from the line, and d_u² is the expected squared perpendicular deviation from the line when the measurements are uncorrelated.

   This can be written as

ρ_c = ρ × C_b

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 + u²)

v = s₁ / s₂

u = (m₁ - m₂) / √(s₁ × s₂)

where m_i and s_i (i = 1,2) are the mean and standard deviation of the i^th set of measurements.

   The Z-transformation is

Z = 0.5 × (log(1 + ρ_c) / log(1 - ρ_c))

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.

References

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.