Reliabilities are often reported as though they were invariable characteristics of tests. Of course, they are not. They depend not only on the construction of the test, but also on the distribution of the examinee sample tested. Conventionally, only person separation reliability is reported, but item separation statistics are also useful indicators. They tell how well this sample of examinees have spread out the items along the measure of the test, and so defined a meaningful variable.
The Table below aids interpreting and predicting reliabilities. Its underlying components are:
Observed SD = the observed standard deviation of reported measures, for examinees or for items.
Root Mean-Square Error (RMSE) = "average" measurement error of reported measures.
True SD = standard deviation of reported measures corrected for measurement error inflation.
Observed SD and RMSE are calculated directly from the reported
measures and their standard errors. Then,
(True SD)^2 = (Observed SD)^2 - (RMSE)^2
Separation Ratio:
G = (True SD)/(RMSE) is a ratio scale index comparing the "true"
spread of the measures with their measurement error. It indicates the
measure of spread of this sample of examinees (or test items) in units
of the test error in their measures.
Separation Reliability:
G^2/(1+G^2) = (True SD)^2/(Observed SD)^2 = KR-20 or Alpha. This is a
correlation coefficient, the ratio of true measure variance to
observed measure variance. When G=1, True SD = RMSE, and reliability
is 0.5. A reliability less than 0.5 implies that the differences
between measures are mainly due to measurement error.
Discernible Strata: (4G+1)/3
The functional range of measures is around 4 True SD. Inflate this by
1 RMSE to allow for the error in the observed measures. Set a
significant difference between two measures at 3 RMSE. Then there are
(4 True SD + RMSE)/(3 RMSE) = (4G+1)/3
significantly different levels of measures in the functional range.
See discussion at RMT 16:3 p. 888
-----------------------------------------------------------------
Separation KR-20, Alpha: % Variance: Distinct Strata:
Ratio: G G^2/(1+G^2) Not Due Error/Due Error (4G+1)/3
-----------------------------------------------------------------
0 .00 0/100 1
1 .50 50/50 1
1.5 .70 70/30 2
2 .80 80/20 3
3 .90 90/10 4
4 .94 94/6 5
5 .96 96/4 7
6 .97 97/3 8
7 .98 98/2 9
-----------------------------------------------------------------
William P. Fisher, Jr.
Wright, B. D., & Masters, G. N. (1982, pp. 92, 105-106). Rating scale analysis: Rasch measurement. Chicago, Illinois: MESA Press.
Reliability, Separation, Strata Statistics, W Fisher Jr … Rasch Measurement Transactions, 1992, 6:3 p. 238
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