Correction for attenuation is a statistical procedure developed by Charles Spearman in 1904 that is used to "rid a correlation coefficient from the weakening effect of measurement error" (Jensen, 1998), a phenomenon known as regression dilution. In measurement and statistics, the correction is also called disattenuation. The correction assures that the correlation across data units (for example, people) between two sets of variables is estimated in a manner that accounts for error contained within the measurement of those variables.[1]
Estimates of correlations between variables are diluted (weakened) by measurement error. Disattenuation provides for a more accurate estimate of the correlation by accounting for this effect.
Let and
be the true values of two attributes of some person or statistical unit. These values are variables by virtue of the assumption that they differ for different statistical units in the population. Let
and
be estimates of
and
derived either directly by observation-with-error or from application of a measurement model, such as the Rasch model. Also, let
Let and
The estimated correlation between two sets of estimates is
where the mean squared standard error of person estimate gives an estimate of the variance of the errors, . The standard errors are normally produced as a by-product of the estimation process (see Rasch model estimation).
The disattenuated estimate of the correlation between the two sets of parameter estimates is therefore
That is, the disattenuated correlation estimate is obtained by dividing the correlation between the estimates by the geometric mean of the separation indices of the two sets of estimates. Expressed in terms of classical test theory, the correlation is divided by the geometric mean of the reliability coefficients of two tests.
Given two random variables and
. The standard errors are normally produced as a by-product of the estimation process (see Rasch model estimation).
The disattenuated estimate of the correlation between the two sets of parameter estimates is therefore
That is, the disattenuated correlation estimate is obtained by dividing the correlation between the estimates by the geometric mean of the separation indices of the two sets of estimates. Expressed in terms of classical test theory, the correlation is divided by the geometric mean of the reliability coefficients of two tests.
Given two random variables and geometric mean of the separation indices of the two sets of estimates. Expressed in terms of classical test theory, the correlation is divided by the geometric mean of the reliability coefficients of two tests.
Given two random variables and
measured as
Given two random variables
and
measured as
and
with measured correlation
and a known reliability for each variable,
and
, the estimated correlation between
and
corrected for attenuation is
How well the variables are measured affects the correlation of X and Y. The correction for attenuation tells one what the estimated correlation is expected to be if one could measure X′ and Y′ with perfect reliability.
Thus if and
are taken to be imperfect measurements of underlying variables
and
are taken to be imperfect measurements of underlying variables
and
with independent errors, then
estimates the true correlation between
and
.