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 !New Castle UK - September 4, 2013

A new class of models for rating data

Marica Manisera & Paola Zuccolotto University of Brescia, Italy Newcastle UK - September 4, 2013

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)

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

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

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)

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)

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

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

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”

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

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

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

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

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

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”

Personal job situation (different l for each Country)

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

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 !!!!!

www.syrtoproject.eu !!

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