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. Cristante, E. Robusto Mathematical Social Sciences 38 1999 259 –274
4. The interaction of the two binomial conditions ‘independence by equal probability’
In terms of the manifestation of the binomial properties in the persons’ responses, in Table 3 different cases are considered in order to describe the interaction between such
properties. Some illustrative numerical examples will be made for n 52. Case 1 – Independence-Equal probability. Both the binomial conditions are satisfied.
As it appears in cell 1 of Table 3, l 5 l , that is to say there is no implication among
g l
responses of persons who belong to a subgroup and there is also similarity among responses. For n 52 persons and a 232 table where both persons have probability 0.50
to endorse an item, the joint probabilities of the entries 0, 0, 0, 1 or 1, 0 and 1, 1 are 0.25, 0.50 and 0.25, when the binomial conditions are satisfied.
Case 2 – Independence-Unequal probability. Only one of the binomial conditions is satisfied. The heterogeneity of the response probabilities generate a response distribution
which is more peaked than the binomial and, as it appears in cell 5 of Table 3, l . l .
g l
If the case is n 52 and for instance person 1 and person 2 have probabilities 0.10 and 0.90 to endorse an item respectively, the joint probabilities in the 232 table are 0.09,
0.82 and 0.09. The probability distribution is more peaked than the binomial. When there is independence among responses, the cases in cells 2 and 6 of Table
3 are not admissible; the cells are structurally empty. Case 3 – Dependence-Equal probability. Only one of the binomial conditions is
satisfied. Independence is violated. As it appears in cells 3 and 4 of Table 3, the l parameter is affected by the sign of the covariation. For n 52, when both person 1 and
person 2 have probability 0.50 to endorse an item, if the joint probabilities are 0.20, 0.60 and 0.20, the covariation is negative, l . l and the probability distribution is more
g l
peaked than the binomial. However if the covariation is positive and the joint probability values are, for instance, 0.30, 0.40 and 0.30, l
, l and the probability distribution is
g l
flatter than the binomial. Case 4 – Dependence-Unequal probability. No one of the binomial conditions is
satisfied. As it appears in Table 3, cells 7 and 8, also in this case the l parameter is affected by the sign of the covariation. For n 52, if, for instance, the probabilities of
Table 3 Interaction of independence by probability
Independence No violation
Violation Negative
Positive Negative
Positive covariation
covariation covariation
covariation Probability
Equal 1
2 3
4 l 5 l
– l . l
l ,
l
g l
g l
g l
Unequal 5
6 7
8 l . l
– l . l
l ,
l
g l
g l
g l
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
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