F . Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
267
endorsing an item are 0.60 and 0.40 and the joint probabilities are 0.20, 0.40, 0.20 and 0.20 respectively for the entries 0, 0, 0, 1, 1, 0 and 1, 1 in a 232 table, the
covariation is negative, l . l and the probability distribution is more peaked than the
g l
binomial. However if the probabilities are 0.30, 0.30, 0.10 and 0.30, the covariation is positive, l
, l and the probability distribution is flatter than the binomial.
g l
On the base of the previous considerations and examples, it is to notice that when l
g
, l cells 4 and 8 in Table 3 there is no doubt about violation of independence
l
among subjects’ responses; whereas when l . l cells 3, 5 and 7 in Table 3
g l
three different interpretations are possible. In order to decide about dependence or independence among responses when l . l , inspection of the sign of the covariation
g l
and of the equal probability property are needed. If equal probability and negative covariation coexist, there is dependence among responses, but if unequal probability and
negative covariation interact in the data cells 5 and 7 in Table 3 it might be either a case of dependence or a case of independence. In this situation a hypothesis of
independence could be made on the base of the level of inequality of the probabilities of the persons’ responses. The more heterogeneous the probability among responses is, the
more likely it is that the responses are independent. If n 52 and if for instance person 1 has probability 0.10 and person 2 has probability 0.90 to endorse an item, there is a
strong tendency to independence which could be easily illustrated by means of the joint probabilities in a 232 table.
5. Estimation procedures of b and l parameters
g g
For estimating both the b and l parameters, the procedure suggested by Andrich et
g g
al. 1982 to estimate the parameters of Eq. 3 is adopted here for the parameters of Eq. 7. Such a procedure is an extension of the method for dichotomous data described by
Andrich 1996. The procedure, already mentioned in a previous paragraph, when adapted to the particular purposes of model 7, consists in conditioning out the item
parameter d so that the two subgroup parameters b
and l are estimated simul-
i g
g
taneously, but independently, of the item parameter. The method used involves considering subgroups in pairs.
For any two subgroups g and q g ± q of the matrix of Fig. 2 and for a particular item i, the Eq. 8 is given
P X , X d , b , l , n , b , l , n
u
hs d
j
gi qi
i g
g g
q q
q
exp x n 2 x l 2 x d 1 x n 2 x l 2 x b 1 x 1 x d
f s d
s d
s d g
g g
g g
g i q
q q
q q
q g
q i
]]]]]]]]]]]]]]]]] 5
8 g g
gi qi
where X , X represents the ordered pair of responses X
of subgroup g and X of
s d
gi qi
gi qi
subgroup q to item i; g and g
are normalizing factors.
gi qi
For all items with the same total score s received from subgroup g and subgroup q, it follows that
268 F
. Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
P X , X b , l , n , b , l , n , s 5 x 1 x u
hs d
j
g q
g g
g q
q q
g q
exp x n 2 x l 2 x b 1 x n 2 x l 2 x b
f s d
s d
g
g g
g g
g g
q q
q q
q q
]]]]]]]]]]]]]]]]] 5
9
U
O
exp c n 2 c l 2 cb 1 s 2 c n 2 s 1 c l 2 s 2 c b
s d
s d
f s d
s d
g
g g
g q
q q
c 5L
where L 5max 0, s 2 n and U 5 min s, n .
s d
s d
q g
The item parameter d drops out of expression 7. Considering all possible scores s
i
[ 0,m 1 m and all possible subgroup pairs, it follows that the likelihood function L
s d
g q
is given by Eq. 10
h
exp h 2 1
O
t l 2 r b s
d
s d
F G
g g
g g
g 51
]]]]]]]]] L 5
10
n 1n
s g
q
U h
h k
gq
P P P O
e
S D
gqcs q .g
g 51 s 50
x 5L k
k
where t 5 o
x n 2 x ; r 5 o
x ; k is the number of items; h is the number of
s d
g i 51
gi g
gi g
i 51 gi
s
subgroups; k denotes the number of items which received a total score of s from
gq
subgroups g and q; e 5 exp c n 2 c l 2 cb 1 s 2 c n 2 s 1 c l 2 s 2 c b .
s d
s d
f s d
s d
g
gqcs g
g g
q q
q
The next step consists in the maximization of the log likelihood function by setting the first derivatives of this function with respect to l
and b equal to zero. The
g g
equations obtained, both for l and b , are solved using a modified Newton-Raphson
g g
algorithm. The derivation of the two equations for the parameters of Eq. 4 is an application of the method presented in Andrich et al. 1982. The item parameter d is
i
then estimated unconditionally taking as known the estimated parameters b and l . The
g g
RUMM v. 2.7 computer program by Andrich et al. 1996 is suggested for the estimation of the parameters.
6. Test of fit procedures
