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Ch. 5: Ability Scale
I. Introduction
A. In CTT, X (number-right-score) is an unbiased estimate of a personās true score
T,
ļ
E(X) T.
B. In IRT, a personās ability, , is monotonically related to the personās true score.
Monotonic relationship is a non-linear and strictly increasing relationship.
C. General procedure in estimating the true ability of a person.
1. Obtain item responses of 1s and 0s (binary item).
2. When item parameters are known, use one of the estimation methods
(MLE, BME, EAP, WLE).
3. When item parameters are unknown, use joint MLE or MMLE for item
parameters and then, use one of MLE, BME, EAP, and WLE.
4. The estimated or transformed
ļ©
can be reported.
II. Nature of ability scale
A. In CTT,
1. E(X) T
2. X/N = proportion-correct in criterion-referenced test.
3. Norm-reference test: stanines, percentiles,
T-score, ACT, SAT, GRE.
4. X is test- and person-dependent.
B. In IRT
1. (ability, trait, or proficiency level) is test-independent.
2. may be an ordinal scale, but has a limited ratio scale interpretation.
III. Transformation of
A. Linear transformation of , b, and a.
Let * = , b* =
b
, and a* = a
/
, then 3-pl is,
P(
*
) = c +
)]()[(/
1
1
bDa
e
c
= c +
][/
1
1
bDa
e
c
= c +
][./
1
1
bDa
e
c
= c +
)(
1
1
bDa
e
c
= P( ).
P( ) is invariant under the linear transformation of , b, and a. ļ
Indeterminancy. (Woodcock-Johnsonās scale p. 80).
B. Non-linear transformation of and b: partial ratio-scale interpretation only for
the Rasch model.
1. Let * =
D
e
and b* =
Db
e
where D = 1, a = 1.
2. P( *) =
)(
)(
1bD
bD
e
e
=
DbD
DbD
e
e
1
=
DbD
DbD
ee
ee
/1
/
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