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
The test of fit for the subgroup location parameter b is based on the item-subgroup
g
interaction residuals. The procedure employed involves the prediction of the scores 0, 1, . . . , n for each subgroup in each item, given the model of Eq. 7 and the parameter
estimates. Tests of fit for specific subgroups are obtained by summing and transforming standardized residuals across items.
With the estimated parameters b , l and d , the probability of any outcome is
g g
i
obtained by Eq. 7. Then the expectation value and the variance of the score of subgroup g on item i are given by the usual expressions
n
E X 5 x 5
O
xP ,
s d
gi xgi
x 50 n
where P 5 P X 5 x is the probability obtained with Eq. 7 and V X 5 x 5
o
s
s d
s d
xgi gi
gi x 50
2 n
2
x P 2
o xP
. The standardized residual between the observed and predicted
d
s d
xgi x 50
xgi
scores is given by
F . Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
269
x 2 E X
f s dg
gi
]]]] z 5
. ]]
gi
V X
s d
gi
œ
This residual has a mean of 0 and a variance of 1. The sum of the statistic z
across either persons or items is equal to zero. Therefore,
gi
considering in this context the test of fit for the parameter b , the sum of the squares of
g
the residuals across k items is used to derive the fit statistic:
2 2
z 5
O
z
g gi
k
with expected value
2 2
2
E z 5 E
O
z 5
O
E z
s d s d
g gi
gi
S D
k k
and variance
2 2
2
V z 5 V
O
z 5
O
V z
s d s d
g gi
gi
S D
k k
.
2
A standardized residual is calculated for z by Eq. 11
g 2
2
z 2 E z
f s dg
g g
]]]] S 5
11 ]]
1 2
V z
s d
g
œ
2 2
Since z 0, the minimum value S is 2 E z , while its maximum is in principle 1`;
s d
g 1
g
this makes the distribution asymmetrical. A transformation that makes the distribution more symmetrical is the ratio of Eq. 12
2 2
2
E z ln z 2 ln E z
s df s dg
g g
g
]]]]]] S 5
12 ]]
2 2
V z
s d
g
œ
the S ratio is symmetrical around 0 and has a mean of 0 and a variance of 1. The shape
2
of the distribution approximates the standard normal distribution.
7. An illustrative application: stress and resources of handicapped children’s parents
An application of the RDSM with dyadic data is presented here. The data correspond to the responses of 44 parent couples of pathologic children to a questionnaire devised to
measure parents’ stress and resources in coping with the family and the child’s problems. The scale called ‘Questionnaire on resources and stress’ QRS constructed by
Friedrich et al. 1983 is a dichotomous item scale. Here it is applied in the Italian version Saviolo Negrin and Cristante, 1996, which is the result of many studies
intended to validate the questionnaire for the Italian population. It consists of three
270 F
. Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
subscales, named ‘Family problems’ QRS , ‘Pessimism’ QRS and ‘Child’s
1 2
problems’ QRS , where each subscale contains 24 items. In the validation process with
3
the Italian subjects, Rasch analyses were carried out for each subscale; the fit to the model was satisfactory for all items. For illustrative purposes, only the results
concerning the QRS subscale are shown and discussed in this context.
1
The QRS items were applied to 44 couples, that is 88 parents of children affected by
1
asthma or thalassemia. The responses of the 88 subjects, analyzed in previous studies, showed local independence on the base of Rasch analyses. The number of subjects is
rather small, nonetheless it should be noted that this case is concerned with measuring and studying the effects of small subgroups couples of people in the context where the
items, as mentioned above, have already been checked and validated for the required properties.
The subjects were from different Italian regions both in the north and south of the country and belonged to the middle class social level; their age ranged from 31 to 64;
the age of their pathologic children ranged from 6 to 25. The following main questions were posed: a what positions do the 44 parent couples occupy on the QRS
1
dimension?; b are the responses of the partners of each couple statistically independent or dependent?; c how similar are the responses of the parents in each couple? In order
to estimate the positions of the couples on the dimension and to assess the dependence independence of the responses, Eq. 7 was applied. Whereas the equi-probability of the
responses was tested by means of a Chi-square; due to the small frequencies a continuity correction was used. In Table 4 the parent couple identification number and the location
parameter estimate b on the QRS dimension are shown. The l dependence parameter
g 1
g
and the Chi-square statistic are also presented. The fit to the model of the couples’ responses was elaborated with Eq. 12 and the Chi-square statistic was obtained as
explained in Footnote 1. The location estimate b describes the level of stress and the
g
family resources as perceived by the couple. A low estimated value describes inadequate resources and a consequent high level of stress, on the contrary a high estimated value
corresponds to adequate resources and, as a consequence, a low level of stress. In Table 4, it can be noted, for instance, that parent couple number 44 perceives
satisfactory resources in the family and manifests a low level of stress, since b 5
44
1.599. On the contrary, couple number 31 perceives inadequate resources and expresses a high level of stress since b
5 2 2.146.
31
The fit to the model is satisfactory except for couple 15 S 51.75, obtained by means
2
of Eq. 12, p 50.04, denoting an anomalous response pattern to the 24 items which are indicators of the QRS dimension.
1
The interpretation of the parameter l in terms of independence dependence between
g
parents’ responses must take into consideration also the equality inequality of the responses within each couple. An analysis of the difference between parents’ responses
was carried out by means of the Chi-square statistic and the statistical significance of the difference was checked p ,0.05. In Table 5 the interaction between independence
dependence and equality inequality is shown; in the four cells the 43 couples couple 15 have been excluded in these results are distributed considering the limiting value
l 5 0.69 for dependence and Chi-square53.84, d.f.51 and p50.05 as the critical value
l
above which inequality between responses is accepted.
F . Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
271
Table 4 Stress and resources perceived by pathologic children’s parents.
Parental couple Location parameter
Dependence Equi-probability of
number estimate b
parameter estimate responding
g
l Chi-square
g
1 0.835
0.69 2.13
2 1.178
0.70 2.13
3 0.025
20.75 0.18
4 0.386
20.19 2.13
5 0.868
0.70 2.13
6 0.216
20.67 1.11
7 0.923
0.27 0.83
8 1.062
20.71 1.25
9 0.814
0.69 2.13
10 0.289
0.72 5.42
11 20.186
0.29 0.67
12 0.794
21.41 0.40
13 0.645
20.25 1.43
14 20.416
0.70 0.06
15 0.036
0.09 1.42
16 1.118
0.70 1.11
17 0.141
20.11 0.91
18 20.204
20.25 0.13
19 20.648
21.46 0.06
20 0.374
20.92 0.63
21 0.269
20.52 0.22
22 20.075
21.96 0.18
23 0.000
21.06 0.10
24 0.455
0.13 0.22
25 0.027
20.79 0.18
26 21.238
0.69 0.04
27 0.284
0.70 0.08
28 20.202
20.49 0.36
29 0.682
20.30 0.29
30 20.482
20.87 0.12
31 22.146
20.21 0.05
32 20.927
20.76 0.21
33 0.405
0.46 1.70
34 21.235
0.17 0.07
35 21.446
0.23 0.34
36 20.648
21.46 0.06
37 20.345
0.48 0.12
38 20.139
22.10 0.18
39 21.219
0.09 0.96
40 0.011
22.02 0.18
41 20.740
0.49 0.48
42 20.953
20.79 0.04
43 0.686
20.79 0.83
44 1.599
20.90 0.67
, p ,0.05, d.f.51.
272 F
. Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274 Table 5
a
Equality inequality by independence dependence of responses of 43 parent couples Parents’ responses
Independent Dependent
Parents’ responses Equal
1, 2, 5, 9, 14, 16, 3, 4, 6, 7, 8, 11,
26, 27 12, 13, 17, 18, 19,
20, 21, 22, 23, 24, 25, 28, 29 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44
Unequal 10
a
The numbers in the cells correspond to the identification couple numbers of Table 4.
Data in Table 5 point out the following main results: a the parents, who on the base of Rasch analyses all showed a good fit to the model, when analyzed in couples, show
that independence – one of the fundamental properties of the Rasch models – is violated in 79 of couples. That is, when analyzed individually the 86 subjects respond
independently, whereas when considered in couples they are no longer independent in most cases. Only 9 out of the 43 couples give independent answers to the items. b As
expected, the majority of couples, 34 out of 43, give dependent and similar equal responses to the items characteristic of the QRS dimension. In other words, the two
1
parents not only express similar opinions in relation to their family resources and stress, but the father’s opinion implies the mother’s and vice versa. Such an implication might
be interpreted as a sort of reciprocal conditioning of the two parents. c Both the binomial properties, independence and equal probability, are not violated in the
responses of eight couples; that is, these parents give similar responses, but there is no implication and no reciprocal conditioning. d Only the members of couple 10 give
different responses and moreover do not influence each other when answering. For this couple the probability of endorsing an item is 0.09 for person 1 and 0.42 for person 2. A
hypothesis of independence can be made.
8. Summary and final remarks
