78 P. Cerchiello, M. Iannario, and D. Piccolo
In fact, better solutions are obtained when we introducecovariatesfor relating
both feeling and uncertainty to the subject’s characteristic. Generally, covariates im-
prove the model fitting, discriminate among different sub-populations and are able
to make more accurate predictions. Moreover, this circumstanceshould enhance the
interpretation of parameters’ estimates and the discussion of possible scenarios.
Following a general paradigm [14, 18], we relateπandξparameters to the
subjects’ covariates through a logistic function. The chosen mapping is the simplest
one among the many transformations of real variables into the unit interval anda
posterioriit provides evidence of ease of interpretation for the problems we will be
discussing.
When we introduce covariates into aMU Brandom variable, we define these
structures asCUB(p,q)models characterised by a general parameter vectorθ =
(π, ξ)′via the logistic mappings:
(π|yi)=1
1 +e−yiβ; (ξ|wi)=1
1 +e−wiγ; i= 1 , 2 ,...,n. (2)Here, we denote byyi andwithe subject’s covariates for explainingπiandξi,
respectively. Notice that (2) allows the consideration of models without covariates
(p=q= 0 ); moreover, the significant set of covariates may or may not present
some overlapping [11, 13, 19].
Finally, inferential issues forCUBmodels are tackled by maximum likelihood
(ML) methods, exploiting the E-M algorithm [16, 17]. The related asymptotic infer-
ence may be applied using the approximate variance and covariance matrix of the ML
estimators [14]. This approach has been successfully applied in several fields, espe-
cially in relation to evaluations of goods and services [6] and other fields of analysis
such as social analysis [10, 11], medicine [7],sensometric studies [19] and linguistics
[1].
The models we have introduced are able to fit and explain the behaviour of a
univariate rating variable while we realise that the expression of a complete ranking
list ofmobjects/items/services bynsubjects should require a multivariate setting.
Thus, the analysis that will be pursued in this paper should be interpreted as a marginal
if we studied the rank distributions of a single item without reference to the ranks
expressed towards the remaining ones.
Then in the following section, we analyse both the different items and injuries;
afterwards we propose a complex map that summarises the essential information
without distortion or inaccuracy.
3 Assessing risk perception: some empirical evidence
3.1 Data analysis
A cross-sectional study was performed in a printing press factory in Northern Italy
that manufactures catalogues, books and reproductions of artworks. The staff of the
factory consists of 700 employees (300 office workers and 400 blue-collar workers).