User Guide for the TIMSS International Database.pdf - TIMSS and ...

S C A L I N G C H A P T E R 5
of the coding was used in the scaling, so this facility in the model and scaling software was not
used in TIMSS.
Letting q be the latent variable the item response probability model is written as:
T
exp bij
aij
Pr ( Xij = ; A,b,
| )
( q + x)
1 x q = K
,
i
(2)
T
exp b q + a x
and a response vector probability model as
f
å
k = 1
( ik ij )
[ ]
T
( x q)= ( q ) x ( bq + A )
; x| Y , x exp x ,
(3)
ì
T
with Y(
q, x)= íå
exp z bq + Ax
îzÎW
[ ( ) ]
where W is the set of all possible response vectors.
T I M S S D A T A B A S E U S E R G U I D E 5 - 3
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ý
þ
-
1
, (4)
5.3 The Multidimensional Random Coefficients Multinomial
Logit Model
The multidimensional form of the model is a straightforward extension of the model that
assumes that a set of D traits underlie the individuals’ responses. The D latent traits define a
D-dimensional latent space, and the individuals’ positions in the D-dimensional latent space
are represented by the vector q=( q1, q2, K, qD
) . The scoring function of response category
k in item i now corresponds to a D by 1 column vector rather than a scalar as in the
unidimensional model. A response in category k in dimension d of item i is scored bikd. The
T
scores across D dimensions can be collected into a column vector bik = ( bik1, bik2, K , bikD)
,
T
again be collected into the scoring sub-matrix for item i, Bi = ( bi1, bi2, K , biD)
, and then
T T T collected into a scoring matrix B= ( B1 B2 BI)
T
, , K , for the whole test. If the item parameter
vector, x, and the design matrix, A, are defined as they were in the unidimensional model the
probability of a response in category k of item i is modeled as
( ij
) =
Pr X = 1;
A, B,
x| q
And for a response vector we have:
exp(
bijq+ a¢
ijx)
i
exp(
b q+ a¢
x)
K
å
k = 1
ik ik
[ ]
( ) = ( ) ¢ ( + )
f x; x| q Y q, x exp x Bq Ax
,
ì
T
with Y(
qx , ) = íå
exp z Bq+ Ax
îzÎW
[ ( ) ]
The difference between the unidimensional model and the multidimensional model is that the
ability parameter is a scalar, q, in the former, and a D by one-column vector, q, in the latter.
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ý
þ
-1
.
(5)
(6)
(7)