Varying uncertainty in CUB models

Abstract

This paper presents a generalization of a mixture model used for the analysis of ratings and preferences by introducing a varying uncertainty component. According to the standard mixture model, called CUB model, the response probabilities are defined as a convex combination of shifted Binomial and discrete Uniform random variables. Our proposal introduces uncertainty distributions with different shapes, which could capture response style and indecision of respondents with greater effectiveness. Since we consider several alternative specifications that are nonnested, we suggest the implementation of a Vuong test for choosing among them. In this regard, some simulation experiments and real case studies confirm the usefulness of the approach.

Appendix

EM algorithm

The EM algorithm may be effectively implemented for a finite mixture (McLachlan and Peel 2000) by the following steps, where $p_r^V$ has to be specified on a priori ground. Hereafter, for a given m, we will denote $\varvec{\theta }=(\pi ,\xi )'$ and set a small tolerance $\epsilon$ (${=}10^{-6}$, for instance).

• 0. $k=0$; $\varvec{\theta }^{(0)}=(\pi ^{(0)},\xi ^{(0)})'$;   $\ell ^{(0)}=\ell (\varvec{\theta }^{(0)})$.

• 1. $bb^{(k)}=b_r(\xi ^{(k)});\,\tau ^{(k)}=\left[ 1+p_r^V\, \frac{1-\pi ^{\left( k \right) }}{\pi ^{\left( k\right) }\,b_r\left( r;\xi ^{\left( k\right) }\right) } \right] ^{-1},\quad r=1,2,\ldots ,m$.;

• 2. $\overline{R}_n(\varvec{\theta }^{(k)})= \frac{\sum \nolimits _{r=1}^{m}\,r\,n_r\,\tau (r; \varvec{\theta }^{(k)})}{\sum \nolimits _{r=1}^{m} n_r\,\tau (r; \varvec{\theta }^{(k)})}$.

• 3. $\pi ^{(k+1)}=(1/n)\sum _{r=1}^{m}\,n_r\,\tau (r; \varvec{\theta }^{(k)})$;   $\xi ^{(k+1)}=\frac{m-\overline{R}_n(\varvec{\theta }^{(k)})}{m-1}$.

• 4.    $\varvec{\theta }^{(k+1)}=(\pi ^{(k+1)},\xi ^{(k+1)})';\,\,\quad \ell ^{(k+1)}=\ell (\varvec{\theta }^{(k+1)})$.

• 5.
  $$\begin{aligned} \left\{ \begin{array}{ll} \text {if }\mid \ell ^{(k+1)}-\ell ^{(k)}\mid \ge \epsilon ,\,k \rightarrow k+1; \hbox {go to} 1;\\ \text {if }\mid \ell ^{(k+1)}-\ell ^{(k)}\mid < \epsilon , \hat{\varvec{\theta }}=\varvec{\theta }^{(k+1)}; \hbox {stop}.\\ \end{array} \right. \end{aligned}$$