Attenuation, Correction For

Attenuation is a term used to describe the reduction in the magnitude of the correlation between the scores of two measurement instruments that is caused by their unreliabilities. Charles Spearman first recognized the value of correcting for attenuation by noting that we are interested in determining the true relationship between the constructs we study, not the relationship between flawed empirical measures of these constructs. His solution was to estimate the correlation between two variables using perfectly reliable empirical measures. He developed the following formula:

where rxx and ryy equal the reliability coefficients of the two instruments, and rxy equals the obtained correlation between the scores of the two instruments. It is assumed that X and Y are imperfect measures of underlying constructs X′ and Y′, containing independent, random errors, and rx′y′ equals the true correlation between X′ and Y′. If rxx equals .61, ryy equals .55, and rxy equals .43, then [Page 51]

The use of the formula allows the researcher to answer the question, What would the correlation (i.e., rx′y′) be if both of the empirical measures were error free? The example illustrates the considerable reduction in the size of rxy caused by the unreliabilities of the scores for the two instruments.

Although correcting for the attenuation of both empirically measured constructs is useful for investigating theoretical problems, it is more difficult to justify for practical applications. For instance, when predicting who will succeed in higher-education programs or who will benefit from special education services, we are limited by the fallible instruments at hand; after all, the application of the correction for attenuation does not make the empirical scores more reliable than they really are! Although it may not be appropriate to correct for attenuation of a predictor variable, it may well be justifiable to adjust for the inaccuracy of criterion measures. For instance, why should inaccurate graduate grade point averages be allowed to make Graduate Record Examinations scores appear less valid than they really are? For this single correction problem, the formula is as follows:

Confirmatory factor analysis (CFA) provides a second way to correct for attenuation. In CFA, the measurement error of each latent variable is explicitly modeled. In research comparing the results of correcting for attenuation via the two approaches, Fan found highly comparable results for the same data. That is, both approaches provided nearly identical point estimates and confidence intervals for the relationship between the true scores of his variables. Nevertheless, it should be mentioned that the CFA approach might be less applicable, given the constraints of modeling item-level data (e.g., extreme item skewness and kurtosis, different item distributions, and item unreliability).