A new class of models for rating data - Marica Manisera, Paola Zuccolotto, September 4, 2013. 2013 International Conference of the Royal Statistical Society

1. A new class of models
for rating data
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and Policy Interventions
Marica Manisera & Paola Zuccolotto
University of Brescia
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New Castle UK - September 4, 2013

2. A new class of models
for rating data
Marica Manisera & Paola Zuccolotto
University of Brescia, Italy
Newcastle UK - September 4, 2013

3. Introduction
Rating data are often encountered, e.g. when individuals’
perceptions are measured by their observed responses to
survey questions with ordinal response scales
In this contribution, rating data are modelled using the
new class of the Nonlinear CUB models (Manisera &
Zuccolotto, 2013) introduced to generalize the standard CUB
(Piccolo, 2003; D’Elia & Piccolo, 2005)

4. CUB models
The response of each subject is interpreted as the
combination of
• a feeling attitude towards the item being evaluated
• an intrinsic uncertainty related to the circumstances
surrounding the discrete choice
The feeling and uncertainty components are considered
in the CUB models by a mixture of a Discrete Uniform U
and a Shifted Binomial V random variables

5. CUB models
The observed rating r (r = 1,…,m) is the realization of a
discrete random variable R with probability distribution
given by
with
thus the parametric space is the (left open) unit square
feeling uncertainty

6. CUB models
Parameters are related to the latent components of the
responses:
1 − ξ  feeling with the item
1 − π  uncertainty of the choice
There is a one-to-one correspondence between a CUB
random variable and θ=(π, ξ)'
 each CUB model can be represented as a
point in the unit square (with coordinates 1−π, 1−ξ)

7. CUB models
The CUB models have been extended in several directions
(Iannario & Piccolo, 2011)
Inferential issues (specification, estimation, validation)
have been obtained also for the extended CUB models
A program in R is freely available (Iannario & Piccolo, 2009)

8. Nonlinear CUB models
The probability distribution of the discrete random
variable R is given by
It depends on a new parameter
and l is a step function mapping
into
feeling uncertainty

9. Nonlinear CUB models
This formulation is derived as a special case of a general
framework for the decision process, which is assumed to
be composed of two approaches:
- the feeling approach, proceeding through T
consecutive steps (feeling path): the last rating is
obtained by summarizing T consecutive elementary judgments
and transforming them into a Likert-scaled rating
- the uncertainty approach, leading to formulate a
completely random rating
The final rating is derived by the feeling or the
uncertainty approach, with given probabilities

10. Nonlinear CUB models - Example
A person is asked to rate his/her job satisfaction on a
Likert scale from 1 to m=5. His/her final decision could
derive from an unconscious decision process
Feeling approach (probability π ): he/she asks
him/herself for T=m−1=4 times “Am I satisfied? Yes or
No?”. Only a positive response allows to move from one
rating to the next one. At each step, a provisional rating is
given by 1 plus the no. of “Yes” up to that step. At the end,
the last rating of the feeling path is given by 1 plus the
Total no. of “Yes”

11. Nonlinear CUB models
Uncertainty approach (probability 1 − π ): for many
reasons, the respondent gives a completely random
rating
This example corresponds to the decision process
modelled by the CUB models (T=m−1)
In the Nonlinear CUB, a number of basic judgments
T>m − 1 is required since the person could need to
accumulate more than one “Yes” to move from one rating
to the next one

12. Nonlinear CUB models
The l function is essential in the decision process
When T=m − 1 , CUB and Nonlinear CUB coincide
The number gs of “Yes”
needed to move from
rating s to s+1 univocally
determines l
g1,…, gm can be
considered parameters
of the model

13. Transition probabilities
The transition probability φt(s) is the probability of
moving to rating s+1 at step t+1 of the feeling path, given
that the rating at step t is s, with s=1,..,m
They describe the state of mind of the respondents about
the response scale. In general, we consider the average
transition probability φ (s)(averaging over t)
In Manisera and Zuccolotto (2013) , we defined the
decision process linear or nonlinear according to whether
φt(s) is constant or non-constant for different t and s

14. Nonlinear CUB models
Also, we showed that, while CUB meets a sufficient
condition for linearity, Nonlinear CUB do not. This is the
reason for their name (we also gave a graphical
explanation)
When comparing different Nonlinear CUB, ξ cannot be
fairly compared, since the l function can be different
 we introduce a new parameter
µ= expected number of one-rating-point increments
during the feeling path

15. Nonlinear CUB models: Estimation
Following CUB, for Nonlinear CUB ML estimates can also
be obtained
We are still working on this issue, since it is a difficult task
and presents identifiability problems
Nevertheless, it is possible to obtain naïve estimates
resorting to the algorithm used to estimate CUB:
• fix maxs(gs)
• maximize logLik with respect to ξ and π constrained to all the
configurations g1, …, gm
• choose the best model according to a proper goodness-of-fit criterion
Results of simulations are encouraging

16. Case study
Data come from Standard Eurobarometer 78, a sample
survey covering the national population of citizens of the
27 European Union Member States
(the average number of interviewees over the 27 Countries is 986)
Focus on one question:
How would you judge the current situation in your
personal job situation?
with possible responses:
1=“very bad” 2=“rather bad“
3=“rather good“ 4=“very good”

18. Personal job situation (different l for each Country)

19. Conclusions and future research
Results show how the Nonlinear CUB models allow us to
model rating data resulting from cognitive mechanisms
with non-constant transition probabilities
The inferential issues of the Nonlinear CUB models are
now our main challenge
Also, the Nonlinear CUB models could be extended:
• to include subjects’ covariates to relate feeling to
respondents’ features
• in a multivariate framework

20. Some references
D'Elia A., Piccolo D. (2005) A mixture model for preference data
analysis. Comput Stat Data An, 49, 917-934.
Iannario M., Piccolo D. (2009) A program in R for CUB models
inference, Version 2.0, available at
http://www.dipstat.unina.it/CUBmodels1/
Iannario M., Piccolo D. (2011) CUB Models: Statistical Methods and
Empirical Evidence, in: Kenett, R. S. and Salini, S. (eds.), Modern
Analysis of Customer Surveys, Wiley, NY, 231–254.
Manisera M., Zuccolotto P. (2013) Modelling rating data with
Nonlinear CUB models, Submitted.
Piccolo D. (2003) On the moments of a mixture of uniform and
shifted binomial random variables, Quaderni di Statistica, 5, 85–
104.

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This project is funded by the European Union
under the 7th Framework Programme (FP7-SSH/2007-2013)
Grant Agreement n° 320270
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www.syrtoproject.eu
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This documents reflects only the author's view. The European Union is not liable for any use that may be made of the information
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