Disattenuation

In measurement and statistics, disattenuation of a correlation between two sets of parameters or measures is the estimation of the correlation in a manner that accounts for measurement error contained within the estimates of those parameters.

Background
Correlations between parameters are diluted or weakened by measurement error. Disattenuation provides for a more accurate estimate of the correlation between the parameters by accounting for this effect.

Let $$\beta_n$$ and $$\theta_n$$ be the measures of two attributes of person n and let $$\hat{\beta_n}$$ be a maximum likelihood estimate of $$\beta_n$$ derived from application of a measurement model, such as the Rasch model. Also, let:



\hat{\beta_n} = \beta_n + \epsilon_n $$

where $$\epsilon_n$$ is the measurement error associated with the estimate $$\hat{\beta_n}$$, as per Fisher information theory. Specifically, the variance of the maximum likelihood estimate of a person parameter is given by the negative of the reciprocal of the second derivative of the log-likelihood function with respect to the parameter. The person parameter represents the theoretical measure of the person for the relevant attribute or trait.

Derivation of the formula
The correlation between two sets of estimates is



\mbox{corr}(\hat{\beta_n},\hat{\theta_n})= \frac{\mbox{cov}(\hat{\beta_n},\hat{\theta_n})}{\sqrt{\mbox{var}[\hat{\beta}]\mbox{var}[\hat{\theta}}]} $$



=\frac{\mbox{cov}(\beta_n+\epsilon_1, \theta_n+\epsilon_2)}{\sqrt{\mbox{var}[\beta_n+\epsilon_1]\mbox{var}[\theta_n+\epsilon_2]}}, $$

which, assuming the errors are uncorrelated with each other and with the estimates, gives



\mbox{corr}(\hat{\beta_n},\hat{\theta_n})= \frac{\mbox{cov}(\beta_n,\theta_n)}{\sqrt{(\mbox{var}[\beta_n]+\mbox{var}[\epsilon_1])(\mbox{var}[\theta_n]+\mbox{var}[\epsilon_2])}} $$



=\frac{\mbox{cov}(\beta_n,\theta_n)}{\sqrt{(\mbox{var}[\beta_n]\mbox{var}[\theta_n])}}.\frac{\sqrt{\mbox{var}[\beta]\mbox{var}[\theta]}}{\sqrt{(\mbox{var}[\beta_n]+\mbox{var}[\epsilon_1])(\mbox{var}[\theta_n]+\mbox{var}[\epsilon_2])}} $$



=\rho \sqrt{R_\beta R_\theta}, $$

where $$R_\beta$$ is the separation index of the set of estimates of $$\beta_n$$, $$n = 1,...,N$$, which is analogous to Cronbach's alpha; this is, in terms of Classical test theory, $$R_\beta$$ is analogous to a reliability coefficient. Specifically, the separation index is given as follows:



R_\beta=\frac{\mbox{var}[\beta]}{\mbox{var}[\beta]+\mbox{var}[\epsilon_1]}=\frac{\mbox{var}[\hat{\beta}]-\mbox{var}[\epsilon_1]}{\mbox{var}[\hat{\beta}]}, $$

where the mean squared standard error of person estimate gives an estimate of the variance of the errors, $$\epsilon_n$$, across persons. 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 two sets of parameters or measures is therefore



\rho = \frac{\mbox{corr}(\hat{\beta_n},\hat{\theta_n})}{\sqrt{R_\beta R_\theta}}. $$

That is, the disattenuated correlation is obtained by dividing the correlation between the estimates by the square root of the product of the separation indices of the two sets of estimates. Expressed in terms of Classical test theory, the correlation is divided by the square root of the product of the reliability coefficients of two tests.