Attenuation

Attenuation is a statistical concept that refers to underestimating the correlation between two different measures because of measurement error. Because no test or other measurement of any construct has perfect reliability, the validity of the scores between predictor and criterion will decrease. Hence, when correlating scores from two survey instruments, the obtained correlation may be substantively lower if the score reliabilities from both instruments are suspect. Therefore, Charles Spearman proposed the following “correction for attenuation” formula, estimating the correlation between two measures if the scores on both had perfect reliability:

In this formula, rxyc is the correlation between the predictor (x) and the criterion (y) corrected for attenuation; rxy is the correlation between the predictor and criterion scores; rxx is the reliability of the predictor scores; and ryy, represents the reliability of the criterion scores.

Suppose the correlation between scores on self-esteem and anger scales is .30. If the reliability (e.g. Cronbach's alpha) of the scores from the self-esteem inventory is .80 and the reliability of the scores from the anger inventory is .90, then the correction for attenuation would be equal to the following:

Because the reliabilities of the scores from the self-esteem and anger scales are high, there is little correction. However, suppose the score reliabilities for the anger and self-esteem inventories are extremely low (e.g. .40). The correction for attenuation would escalate to .75. If the square root of the product of the reliabilities were less than .30, then the correction for attenuation would be greater than 1.0!

However, rather than correcting for score unreliability in both measures, there are times in which one would correct for score unreliability for either the predictor or criterion variables. For example, suppose the correlation between scores from a job interview (x) and from a personnel test (y) is equal to .25, and assume that the reliability of the personnel test is .70. If one corrected only for the score unreliability of the criterion, then the following equation would be used:

In this case, the correction for attenuation would equal .30. One could also use a similar equation for [Page 37]correcting the predictor variable. For example, suppose the correlation between scores from a personnel test (x) and the number of interviews completed in a week (y) is equal to .20 and the score reliability of the personnel test is .60. The correction for attenuation would equal .26, using the following equation for correcting only for the score reliability of the predictor variable:

Paul Muchinsky summarized the recommendations for applying the correction for attenuation. First, the corrected correlations should neither be tested for statistical significance nor should they be compared with uncorrected validity coefficients. Second, the correction for attenuation does not increase predictive validity of test scores. Donald Zimmerman and Richard Williams indicated that the correction for attenuation is useful given high score reliabilities and large sample sizes. Although the correction for attenuation has been used in a variety of situations (e.g. meta-analysis), various statisticians have suggested caution in interpreting its results.