- Background
- Formula
- See also
- References
- 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 
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
where 
and 
are the measurement errors associated with the estimates and .
The estimated correlation between two sets of estimates is
which, assuming the errors are uncorrelated with each other and with the true attribute values, gives
where 
is the separation index of the set of estimates of , which is analogous to Cronbach's alpha; that is, in terms of classical test theory, is analogous to a reliability coefficient. Specifically, the separation index is given as follows:
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 
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 
with independent errors, then 
estimates the true correlation between and .
See also
- Regression dilution
- Errors-in-variables model
References
- Jensen, A.R. (1998). The g Factor: The Science of Mental Ability Praeger, Connecticut, US. {{ISBN|0-275-96103-6}}
- Spearman, C. (1904) "The Proof and Measurement of Association between Two Things". The American Journal of Psychology, 15 (1), 72–101 {{JSTOR|1412159}}
- Specific
External links
- Disattenuating correlations
- Disattenuation of correlation and regression coefficients: Jason W. Osborne
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