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
5.5 Estimation
The ConQuest software uses maximum likelihood methods to provide estimates of g, S, and x.
Combining the conditional item response model (6) and the population model (12) we obtain
the unconditional or marginal response model,
and it follows that the likelihood is,
( ) = ( ) ( )
ò
fx x; x,g , S fx x;
x | q fqq; g, S dq,
(13)
q
N
Õ fx n
n=
1
L = ( x ; , , ) xgS
where N is the total number of sampled students.
, (14)
Differentiating with respect to each of the parameters and defining the marginal posterior as
h ( q ; W , x, g, S | x ) =
q
n n n
( ) ( )
( )
fxxn; x| qn fqqn;
Wn,
g, S
f x ; W , xgS , ,
n n
provides the following system of likelihood equations:
and
T I M S S D A T A B A S E U S E R G U I D E 5 - 5
x
(15)
N é
ù
A¢ å êxn
– ò Ez( z| qn) hq( qn; Yn, x,g , S | xn)
dqnú=
0 ,
(16)
n=
1 ëê
q
ûú
n
gˆ = q
,
æ ö
ç ÷
è ø
æ
N
N
T
T ö
å nWn çåWnWn
÷
è ø
n=
1 n=
1
-1
(17)
N
ˆ 1
T
S = ò ( qn -gWn)
( qn - gWn) hq( qn; Yn, x, g, S | xn)
dqn,
(18)
N
å
n=
1 q n
( ) = ( ) å ¢ ( + )
where Ez z| qn Y qn, x zexp z bqn Ax
zÎW
[ ]
and qn ò qnhqqn; Yn, x,g, S| xn
dqn
.
= ( )
qn
The system of equations defined by (16), (17), and (18) is solved using an EM algorithm
(Dempster, Laird, and Rubin, 1977) following the approach of Bock and Aitken (1981).
(19)
(20)