Mixture models for ordinal responses to account for uncertainty of choice

Abstract

In CUB models the uncertainty of choice is explicitly modelled as a Combination of discrete Uniform and shifted Binomial random variables. The basic concept to model the response as a mixture of a deliberate choice of a response category and an uncertainty component that is represented by a uniform distribution on the response categories is extended to a much wider class of models. The deliberate choice can in particular be determined by classical ordinal response models as the cumulative and adjacent categories model. Then one obtains the traditional and flexible models as special cases when the uncertainty component is irrelevant. It is shown that the effect of explanatory variables is underestimated if the uncertainty component is neglected in a cumulative type mixture model. Visualization tools for the effects of variables are proposed and the modelling strategies are evaluated by use of real data sets. It is demonstrated that the extended class of models frequently yields better fit than classical ordinal response models without an uncertainty component.

Appendix

1.1 Identifiability

We assume that the number of categories is greater than 2 ($k>2$) and that there is an effect of a continuous covariate x, that is $\gamma \ne 0$. Let the CUP model with the cumulative logit model in the preference part be represented by two parameterizations, that is, for all x and r one has

$$\begin{aligned} \pi F(\gamma _{0r}+x\gamma ) + (1-\pi )r/k = \tilde{\pi }F(\tilde{\gamma }_{0r}+x\tilde{\gamma }) + (1-\tilde{\pi })r/k. \end{aligned}$$

There are values $\Delta _{0r}, \Delta$ such that $\tilde{\gamma }_{0r}=\gamma _{0r}+\Delta _{0r}$, $\tilde{\gamma }=\gamma +\Delta$. With $\eta _r(x)=\gamma _{0r}+x\gamma$ one obtains for all x and r

$$\begin{aligned} \pi F(\eta _r(x)) - \tilde{\pi }F(\eta _r(x)+\Delta _{0r}+ x\Delta ) = (\pi -\tilde{\pi })r/k. \end{aligned}$$

Let us consider now the specific values $x_z= - \gamma _{0r}/\gamma + z/\gamma$ yielding for all values z and r

$$\begin{aligned} \pi F(z) - \tilde{\pi }F(z+\Delta _{0r}+ x_z\Delta ) = (\pi -\tilde{\pi })r/k. \end{aligned}$$

By building the difference between these equations for values z and $z-1$ one obtains for all values z

$$\begin{aligned} \pi (F(z) - F(z-1)) = \tilde{\pi }(F(z+\Delta _{0r}+ x_z\Delta ) - F(z-1+\Delta _{0r}+ x_z\Delta )). \end{aligned}$$

(6)

The equation has to hold in particular for values $z=1,2,\ldots$. Since the logistic distribution function $F(\eta )= \exp (\eta )/(1+\exp (\eta ))$ is strictly monotonic and the derivative $F'(\eta )= \exp (\eta )/(1+\exp (\eta )^2)$ is different for all values $\eta$ it follows that $\Delta _{0r}=\Delta =0$ and $\pi =\tilde{\pi }$.

If the support of the covariate is finite one can consider different z- values. If $x \in [l,u]$ ($\gamma$ positive) one considers the transformed values $z_i= \gamma l+\gamma _{0r}+\gamma (u-l)i/M$, for $i=1,\dots ,M$, where M is any natural number. Then for all transformed values $x_{z_i}= - \gamma _{0r}/\gamma + z_i/\gamma$ one has $x_{z_i} \in [l,u]$. Thus, Eq. (6) has to hold for M different values $z_i$. Since M can be any natural number the same argument as before yields $\Delta _{0r}=\Delta =0$ and $\pi =\tilde{\pi }$.

1.2 Estimation

The general CUP model is determined by the probability

$$\begin{aligned} P(r_{i}|\varvec{x}_i)= \pi _i P_M(r_{i}|\varvec{x}_i) + (1- \pi _i)P_U(r_{i}), \end{aligned}$$

where the first mixture component follows an ordinal model and the second represents the discrete uniform distribution.

For given data $(r_{i},\varvec{x}_{i})$, $i=1,\dots ,n$, and collecting all parameters of the ordinal model used in the first mixture component in the parameter $\varvec{\theta }$, the log-likelihood to be maximized is

$$\begin{aligned} l(\varvec{\theta })=\sum _{i=1}^n \log (\pi _i P_M(r_i|\varvec{x}_i) + (1- \pi _i)P_U(r_i)). \end{aligned}$$

The usual way to obtain estimates is to consider it as a problem with incomplete data and solve the maximization problem by using the EM algorithm. Therefore, let $z_{i}$ denote the unknown mixture components with $z_{i}=1$ indicating that observation i is from the first mixture component, $z_{i}=0$ indicates that it is from the second mixture component. Then the complete density for $(r_i,z_i)$ is

$$\begin{aligned} P(r_i, z_i \vert \varvec{x}_{i}, \varvec{\theta })= P(r_i \vert z_{i}, \varvec{x}_{i}, \varvec{\theta })P( z_{i} )=P_M(r_i|\varvec{x}_i)^{z_i}P_U(r_i)^{z_i-1}\pi _i^{z_i}(1-\pi _i)^{z_i-1} \end{aligned}$$

yielding the complete log-likelihood

$$\begin{aligned} l_c(\varvec{\theta })&= \sum _{i=1}^{n} \log (P(r_i, z_i \vert \varvec{x}_{i}, \varvec{\theta }))\\&= \sum _{i=1}^{n} z_i (\log (P_M(r_i|\varvec{x}_i))+ \log (\pi _i))+ (1-z_i)(\log (P_U(r_i))+ \log (1-\pi _i)). \end{aligned}$$

The EM algorithm treats $z_{i}$ as missing data and maximizes the log-likelihood iteratively by using an expectation and a maximization step. During the E-step the conditional expectation of the complete log-likelihood given the observed data $\varvec{r}$ and the current estimate $\varvec{\theta }^{(s)}$,

$$\begin{aligned} M(\varvec{\theta }|\varvec{\theta }^{(s)})={\text {E}}\left( l_c(\varvec{\theta })|\varvec{r}, \varvec{\theta }^{(s)}\right) \end{aligned}$$

has to be computed. Because $l_c(\varvec{\theta })$ is linear in the unobservable data $z_{i}$, it is only necessary to estimate the current conditional expectation of $z_{i}$. From Bayes’s theorem follows

$$\begin{aligned} E(z_{i}|\varvec{y},\varvec{\theta })&= P\left( z_{i}=1| r_i,\varvec{x}_{i},\varvec{\theta }\right) \\&=P\left( r_i|z_{i}=1,\varvec{x}_{i},\varvec{\theta }\right) P(z_{i}=1|\varvec{x}_{i},\varvec{\theta })/P(r_i|\varvec{x}_{i},\varvec{\theta }) \\&=\pi _i P_M(r_i|\varvec{x}_{i},\varvec{\theta })/P( r_i|\varvec{x}_{i},\varvec{\theta })= \hat{z}_{i}. \end{aligned}$$

This is the posterior probability that the observation $r_i$ belongs to the first component of the mixture. For the s-th iteration one obtains

$$\begin{aligned} M(\varvec{\theta }|\varvec{\theta }^{(s)})&= \sum _{i=1}^{n} \hat{z}_{i}^{(s)}\left( \log (\pi _{i})+ \log (P_M(r_i \vert \varvec{x}_{i},\varvec{\theta })\right) \\&\quad +(1-\hat{z}_{i}^{(s)})\left( \log (1-\pi _{i})+ \log (P_U(r_i)\right) \\&= \underbrace{\sum _{i=1}^{n} \hat{z}_{i}^{(s)}\log (\pi _{i})+ (1-\hat{z}_{i}^{(s)})(\log (1-\pi _{i}))}_{M_{1}} \\&\quad + \underbrace{\sum _{i=1}^{n} \hat{z}_{i}^{(s)}\log (P_M(r_i \vert \varvec{x}_{i},\varvec{\theta }))+(1-\hat{z}_{i}^{(s)})\log (P_U(r_i))}_{M_{2}}. \end{aligned}$$

Thus, for given $\varvec{\theta }^{(s)}$ one computes in the E-step the weights $\hat{z}_{i}^{(s)}$ and in the M-step maximizes $M(\varvec{\theta }|\varvec{\theta }^{(s)})$ (or rather $M_{1}$ and $M_{2}$). If the mixture probabilities do not depend on covariates, that is, $\pi _i=\pi$, one obtains

$$\begin{aligned} \pi ^{(s+1)}= \frac{1}{n}\sum _{i=1}^{n}\hat{z}_{i}^{(s)}\quad \text {and} \quad \varvec{\theta }^{(s+1)}= {\text {argmax}}_{\varvec{\theta }}\sum _{i=1}^{n}\hat{z}_{i}^{(s)}\log (P_M(r_i|\varvec{x}_{i},\varvec{\theta })). \end{aligned}$$

The E- and M-steps are repeated alternatingly until the difference $L(\varvec{\theta }^{(s+1)})-L(\varvec{\theta }^{(s)})$ is small enough to assume convergence. Computation of $\varvec{\theta }^{(s+1)}$ can be based on familiar maximization tools, because one maximizes a weighted log-likelihood of an ordinal model with known weights. In the case where only intercepts are component-specific, the derivatives are very similar to the score function used in a Gauss-Hermite quadrature and a similar EM algorithm applies with an additional calculation of the mixing distribution (see Aitkin 1999).

Dempster et al. (1977) showed that under weak conditions the EM algorithm finds a local maximum of the likelihood function $L(\varvec{\theta })$. Hence it is sensible to use different start values $\varvec{\theta }^{(0)}$ to find the solution of the maximization problem.

If covariates determine the probability that observation i belongs to the first mixture component in the form of a logit model, $\pi _{i}({\varvec{\beta }})=1/(1+\exp ({-\varvec{z}_{i}^{T}{\varvec{\beta }}}))$, $M_{1}$ is the weighted log-likelihood of a binary logit model. Then $M_{1}$ and $M_{2}$ are maximized separately to obtain the next iteration. The simple update $\pi ^{(s+1)}= \sum _{i=1}^{n}\hat{z}_{i}^{(s)}/n$ is replaced by

$$\begin{aligned} {\varvec{\beta }}^{(s+1)}= {\text {argmax}}_{{\varvec{\beta }}}\sum _{i=1}^{n} \hat{z}_{i}^{(s)}\log (\pi _{i}({\varvec{\beta }}))+ (1-\hat{z}_{i}^{(s)})(\log (1-\pi _{i}({\varvec{\beta }}))). \end{aligned}$$

As default value for the stopping of the iterations we used the difference in two consecutive likelihoods; if it was below $10^{-6}$ the algorithm was stopped.