When two sets of measures, {x} and {y}, are correlated, measurement error lowers the correlation coefficient below the level it would have reached had the measures been precise. The reliability (Cronbach Alpha, KR-20, Rasch, etc.) of a set of measures is the proportion of observed variance not due to measurement error, rxx for set {x} and ryy for set {y}. Measurement error can be removed from a correlation coefficient, rxy, to estimate the correlation coefficient disattenuated of measurement error, Rxy, by the formula (Spearman 1904, 1910):
Rxy = rxy / sqrt (rxx ryy)
For two sets of person scores or measures, use the person "test" reliabilities.
For two sets of item p-values or measures, use the item (not the "test") reliabilities.
If you have the standard error of each score or measure, then the reliability of the set of scores or measures is:
rxx = [(observed variance of the measures) - sum(SE² of each measure)/(count of measures)] / [(observed variance of the measures)]
Disattenuated values greater than 1.00 indicate that measurement error is not randomly distributed. Report the disattenuated correlation as 1.0.
Muchinsky (1996) summarizes features of the disattenuated correlation coefficient:
1.Disattenuation does not change the quality of the measures or their predictive power.
2.Disattenuated correlations are not directly comparable with uncorrected correlations.
3.Disattenuated correlations are not suited to statistical hypothesis testing.
4.Disattenuation is not a substitute for precise measurement.
5.But, disattenuation tells us whether the correlation between two sets of measures is low because of measurement error or because the two sets are really uncorrelated.
Randall E. Schumacker
Muchinsky P.M. (1996) The correction for attenuation. Educational & Psychological Measurement 56:1, 63-75.
Spearman C. (1904) The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101.
Spearman C. (1910) Correlation calculated from faulty data. British Journal of Psychology, 3, 271-295
Zimmerman, D. W., & Williams, R. H. (1997). Properties of the Spearman correction for attenuation for normal and realistic non-normal distributions. Applied Psychological Measurement, 21, 253-270.
The reliabilities, rxx and ryy can be computed from tables of measures with standard errors:
rxx = ( S.D.(measures for set(x))**2 - RMSE(set(x))**2 ) / S.D.(measures for set(x))**2
ryy = ( S.D.(measures for set(y))**2 - RMSE(set(y))**2 ) / S.D.(measures for set(y))**2
Table of Disattenuated of Correlation Coefficients | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reliability (Test 1) multiplied by Reliability (Test 2) |
Reported Test 1 x Test 2 Correlation Coefficient | ||||||||||||||||||
.05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .45 | .50 | .55 | .60 | .65 | .70 | .75 | .80 | .85 | .90 | .95 | |
.05 | .22 | .45 | .67 | .89 | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
.10 | .16 | .32 | .47 | .63 | .79 | .95 | - | - | - | - | - | - | - | - | - | - | - | - | - |
.15 | .13 | .26 | .39 | .52 | .65 | .77 | .90 | - | - | - | - | - | - | - | - | - | - | - | - |
.20 | .11 | .22 | .34 | .45 | .56 | .67 | .78 | .89 | - | - | - | - | - | - | - | - | - | - | - |
.25 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | - | - | - | - | - | - | - | - | - | - |
.30 | .09 | .18 | .27 | .37 | .46 | .55 | .64 | .73 | .82 | .91 | - | - | - | - | - | - | - | - | - |
.35 | .08 | .17 | .25 | .34 | .42 | .51 | .59 | .68 | .76 | .85 | .93 | - | - | - | - | - | - | - | - |
.40 | .08 | .16 | .24 | .32 | .40 | .47 | .55 | .63 | .71 | .79 | .87 | .95 | - | - | - | - | - | - | - |
.45 | .07 | .15 | .22 | .30 | .37 | .45 | .52 | .60 | .67 | .75 | .82 | .89 | .97 | - | - | - | - | - | - |
.50 | .07 | .14 | .21 | .28 | .35 | .42 | .49 | .57 | .64 | .71 | .78 | .85 | .92 | .99 | - | - | - | - | - |
.55 | .07 | .13 | .20 | .27 | .34 | .40 | .47 | .54 | .61 | .67 | .74 | .81 | .88 | .94 | - | - | - | - | - |
.60 | .06 | .13 | .19 | .26 | .32 | .39 | .45 | .52 | .58 | .65 | .71 | .77 | .84 | .90 | .97 | - | - | - | - |
.65 | .06 | .12 | .19 | .25 | .31 | .37 | .43 | .50 | .56 | .62 | .68 | .74 | .81 | .87 | .93 | .99 | - | - | - |
.70 | .06 | .12 | .18 | .24 | .30 | .36 | .42 | .48 | .54 | .60 | .66 | .72 | .78 | .84 | .90 | .96 | - | - | - |
.75 | .06 | .12 | .17 | .23 | .29 | .35 | .40 | .46 | .52 | .58 | .64 | .69 | .75 | .81 | .87 | .92 | .98 | - | - |
.80 | .06 | .11 | .17 | .22 | .28 | .34 | .39 | .45 | .50 | .56 | .61 | .67 | .73 | .78 | .84 | .89 | .95 | - | - |
.85 | .05 | .11 | .16 | .22 | .27 | .33 | .38 | .43 | .49 | .54 | .60 | .65 | .71 | .76 | .81 | .87 | .92 | .98 | - |
.90 | .05 | .11 | .16 | .21 | .26 | .32 | .37 | .42 | .47 | .53 | .58 | .63 | .69 | .74 | .79 | .84 | .90 | .95 | - |
.95 | .05 | .10 | .15 | .21 | .26 | .31 | .36 | .41 | .46 | .51 | .56 | .62 | .67 | .72 | .77 | .82 | .87 | .92 | .97 |
"The correlation coefficient corrected for attenuation between two tests x and y is the correlation between their true scores [or true measures]. If, on the basis of a sample of examinees, the corrected coefficient is near unity, the experimenter concludes that the two tests are measuring the same trait." (p. 117) in Joreskog, K.G.(1971) Statistical analysis of sets of congeneric tests, Psychometrica 36, 109-133
Disattenuating correlation coefficients. Schumacker RE, Muchinsky PM. … Rasch Measurement Transactions, 1996, 10:1 p.479
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