• 1 Background
• 2 Formula
• 3 See also
• 4 References
• 5 External links

Background

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.

Formula

Let ${\displaystyle \beta }$ and ${\displaystyle \theta }$ 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 ${\displaystyle {\hat {\beta }}}$ and ${\displaystyle {\hat {\theta }}}$ be estimates of ${\displaystyle \beta }$ and ${\displaystyle \theta }$ derived either directly by observation-with-error or from application of a measurement model, such as the Rasch model. Also, let

Let ${\displaystyle \beta }$ and ${\displaystyle \mathrm{\theta <!--\; \theta \; -->}}Let$${\displaystyle \beta }$ and ${\displaystyle \theta }$ 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 ${\displaystyle {\hat {\beta }}}$ and ${\displaystyle {\hat {\theta }}}$ be estimates of ${\displaystyle \beta }$ and ${\displaystyle \theta }$ derived either directly by observation-with-error or from application of a measurement model, such as the Rasch model. Also, let

${\displaystyle \epsilon _{\beta }}$ and ${\displaystyle \epsilon _{\theta }}$ are the measurement errors associated with the estimates ${\displaystyle {\hat {\beta }}}$ and ${\displaystyle {\hat {\theta }}}$.

The estimated correlation between two sets of estimates is

${\displaystyle \operatorname {corr} ({\hat {\beta }},{\hat {\theta }})={\frac {\operatorname {cov} (\beta ,\theta )}{\sqrt {(\operatorname {var} [\beta ]+\operatorname {var} [\epsilon _{\beta }])(\operatorname {var} [\theta ]+\operatorname {var} [\epsilon _{\theta }])}}}}$