Quality & Quantity 33: 45–58, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
On the Problem of Identiﬁcation in Multiplicative
Intensity-Rate Models with Multiple Interactions
Department of Statistics, Uppsala University PO Box 513, 751 20 Uppsala, Sweden
Abstract. In this paper we examine a multiplicative intensity model in which a covariate interacts
with two other covariates in the same model. We demonstrate, analytically, that in such situations a
log-linear parameterization based on two pairs of baseline levels cannot be transformed, uniquely, to
the, otherwise equivalent, multiplicative parameterization. We show that the problem lies in an over-
sight of the conditional independence between the two covariates interacting with a common third co-
variate. As a solution, therefore, we propose an approach that takes due account of such dependence.
Our proposed approach uses a common baseline level for the three covariates involved in interaction
while estimating the corresponding relative intensities. The issues addressed are illustrated with a
demographic data set involving the estimation of rates of transition to parenthood.
Key words: intensity rate, baseline level, multiplicative intensity model, log-linear intensity model,
multiple interactions, identiﬁcation, complete independence, conditional independence.
In an attempt to investigate sex-differentials in the intensity of ﬁrst-birth to cohabit-
ing and married Swedish adults, Bernhardt and Bjerén (1990) used a multiplicative
intensity model of the type discussed in Breslow and Day (1975) and reviewed
in Hoem (1987). The model controls for ﬁve sociodemographic variables – Sex,
Education, Residence, Age,andDuration.
In their ﬁnal analysis, the authors found
a ﬁve-factor two-interaction model that ﬁts the data ‘best’. This ﬁnal model is such
that Education and Duration act independently while Sex interacts with both Resi-
dence and Age. The relative intensities resulting from such a model and displayed
in their Table 4 (page 13) are as shown in Table I.
We shall postpone details on how the values in the table are obtained to later
sections. For the moment it sufﬁces with the interpretation. According to Table
I(a), the low-educated (men or women) are about twice (1.91 times) as likely to
have ﬁrst-birth as those with middle-level education when the other covariates are
controlled for. Those with high-level education, on the other hand, have about the
same intensity (1.03 times) as those with middle-level education.
Similarly, Table I(b) shows that men residing in Värmland (except Torsby) are
more than 5 times as likely to have ﬁrst-birth as men residing in Torsby, while
men residing in other parts of Sweden (outside Värmland) have intensity that is