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Inflated Discrete Beta Regression Models for Likert and Discrete Rating Scale Outcomes

Discrete ordinal responses such as Likert scales are regularly proposed in questionnaires and used as dependent variable in modeling. The response distribution for such scales is always discrete, with bounded support and often skewed. In addition, one particular level of the scale is frequently inflated as it cumulates respondents who invariably choose that particular level (typically the middle or one extreme of the scale) without hesitation with those who chose that alternative but might have selected a neighboring one. The inflated discrete beta regression (IDBR) model addresses those four critical characteristics that have never been taken into account simultaneously by existing models. The mean and the dispersion of rates are jointly regressed on covariates using an underlying beta distribution. The probability that choosers of the inflated level invariably make that choice is also regressed on covariates. Simulation studies used to evaluate the statistical properties of the IDBR model suggest that it produces more precise predictions than competing models. The ability to jointly model the location and dispersion of (the distribution of) an ordinal response, as well as to characterize the profile of subject selecting an"inflated"alternative are the most relevant features of the IDBR model. It is illustrated with the analysis of the political positioning on a"left-right"scale of the Belgian respondents in the 2012 European Social Survey.


Introduction
Whatever the subject of interest, discrete scales are used everywhere in questionnaires. It could be Likert scales from Strongly disagree to Strongly agree, rating scales from 0 to 10 or the "left-right" political scale such as for question B19 of the European Social Survey (ESS, 2012). In applications, such data are generally analyzed using classical linear regression. Yet, this approach has several statistical limits: rates on such scales are always bounded, often skewed and frequently inflated at one of the bounds of the scale. Last but not least, the discreteness of the scale is not recognized in estimation and predictions when using that model.
Among the alternatives to linear regression, discrete response models naturally address the discreteness of the scale and the existence of boundaries.
Nevertheless, skewed and inflated responses are less effectively handled by ordered logit or probit regressions. The multinomial versions of those models perform better there but are heavily parametrized and complicated to interpret. Another alternative to linear regression are the various versions of beta regression proposed in the literature for modeling rates and proportions (e.g., Brehm & Gates, 1993;Paolino, 2001;Ferrari & Cribari-Neto, 2004;Branscum, Johnson, & Thurmond, 2007;Simas, Barreto-Souza, & Rocha, 2010). The beta distribution is bounded and can be skewed but makes the assumption of continuity. As far as we know, none of these methods has been adapted to discrete scales and their quality on such scales has never been studied. Inflated beta regressions also exist for continuous outcomes (e.g., Ospina & Ferrari, 2010;Wieczorek & Hawala, 2011;Ospina & Ferrari, 2012). Nevertheless, the inflation is considered outside the support of the beta distribution which is not suitable to deal with discrete scales.
The model proposed here addresses the four expounded limits by adapting the beta regression proposed by Simas et al. (2010). The discrete scale is assumed to be the observed counterpart of an underlying beta distribution where the mean and the dispersion are jointly regressed on covariates. The likelihood function is adapted to suit the discreteness of the scale. Skewness and boundedness are intrinsic features of the beta distribution. Finally, when one of the points of the support is inflated, a switch to an inflated discrete beta regression is made by jointly modeling an additional mass of probability on the inflated point that can be anywhere on the scale. The next section contains the theoretical description of the discrete beta regression model, while section 3 describes the required amendments to deal with inflation. The fourth section is related to the estimation procedure and contains simulation studies of its properties. The fifth section compares the quality of predictions of the inflated discrete beta regression model with those of competing models. Finally, the sixth section illustrates our model on one question of the European Social Survey for which the inflated point is in the middle of an odd-level scale.

Discrete Beta Regression
Let y ⋆ i be the choice of individual i on a K-level discrete scale Y ⋆ . This scale is considered as a random variable with unknown distribution but known support, Y ⋆ = {a, a + h ⋆ , ..., b − h ⋆ , b}: thus, we assume that each point on this scale are equally spaced and possibly labeled ("Strongly agree", etc.). In order to match the discrete scale with the support of the beta distribution, its support is rescaled into the unit interval using Observed discrete data and proxy interval considered under the beta density as described in equation (1).
The discrete beta regression model is defined in two steps. Firstly, the rescaled discrete scale is linked to the continuous beta distribution using where U ∼ Be(p, q) with p > 0, q > 0 and u ∈ (0, 1). Figure 1 illustrates the idea behind this link: the black area under the curve of the beta density is used as a proxy for the probability mass in the bar.
Secondly, the parameters of the beta distribution are regressed on covariates. It is based on an alternative parametrization of the distribution enabling to model location and dispersion separately. The location parameter µ is the mean of the beta distribution given by The dispersion parameter φ is proportional to the variance of the distribution, Both quantities take values in (0, 1). Link functions g 1 and g 2 are used to relate µ and φ to covariates, They are assumed to be strictly monotonic and twice differentiable. Logit links will be used in applications. The utility functions f 1 and f 2 will be assumed linear in this paper but this is not compulsory. We assume that the matrices ∂f 1 (X, β)/∂β and ∂f 2 (Z, θ)/∂θ have full rank. The covariates matrices X and Z could be (partially) the same but this is not compulsory.
The contribution of the i th observation to the likelihood in the discrete beta regression is given by where y i is the observed point on the reduced scale, p( , .] is the beta function. When logit links and linear utility functions are used, this general class of discrete beta regression can be written with In contrast to the continuous beta regression model proposed by Simas et al. (2010), the discreteness of the scale is embodied in the model formulation. A parametrization of the beta distribution in term of dispersion is also preferred as consulting experience suggests that this concept is generally easier to explain to non-statisticians. Nevertheless, since the Simas' precision parameter is defined by ϕ = p + q, one has φ = 1 ϕ+1 . Thus, if one uses a logarithm link in Simas' model and a logit link in (3) with linear utility functions on both sides, then the θ's are simply the opposite of Simas' precision parameters. Consequently, users can easily switch from dispersion to precision by changing the signs of the parameters.

Inflated Discrete Beta Regression
In discrete ordinal responses, an unusual inflation in the distribution of the responses is regularly observed at one of the point of the scale's support, say the k th one where y = kh with k ∈ {1, 2, ..., K} and h = 1/K. It might be one of the bounds of the scales or the central level of odd-level scales.
In such situation, one can suspect that choosers of that point are composed of two types of respondents: those who invariably choose that particular level without hesitation and those who choose that alternative but might have chosen the neighboring points on the scale as well. In the following, the probability related to the former is noted π and is regressed on covariates whereas the behavior of the latter is described by the discrete beta regression (DBR) model defined in section 2.
Both steps of modeling described before have to be adapted. Thus, one where U denotes the latent beta of the DBR component and I(.) is an indicator function. The probability π can also be related to covariates using g 0 (π) = f 0 (W, γ) , g 0 : (0, 1) → R , where g 0 and f 0 denote the link and utility functions respectively.
The contribution of the i th observation to the likelihood of the inflated discrete beta regression (IDBR) is where elements of the DBR remain defined as in equation (4). This general class of inflated discrete beta regression can be written as g −1 0 (f 0 (w i , γ)) = 1 1+exp(−w ′ i γ) when logit link and linear utility function are used. In order to determine if an inflation point must be included in the model, users are advised to visualize the empirical distribution of the response within classes of potential covariates and to wonder if the choice of the inflated point on the scale might correspond to a specific behavior or pattern of respondents for whom the proposition is clearly not an option or is a systematic choice. A post-modeling comparison of the DBR and IDBR models can also be conducted using the Bayes factor or other criteria.
Zero-inflated, one-inflated and zero-and-one-inflated beta regression model already exist for continuous outcomes (Ospina & Ferrari, 2010;Wieczorek & Hawala, 2011;Ospina & Ferrari, 2012). Contrary to the IDBR, those models do not take into account the discreteness of the scale and are based on a disjoint support: the inflation is observed outside the support of the beta distribution. The IDBR is more comparable to the zero-inflated Poisson regression proposed by Lambert (1992) where 0 is contained in the support of the Poisson distribution. Furthermore, the inflation can be set anywhere (where found appropriate) on the scale in the IDBR model. This will be illustrated in the case study in section 6.

Estimation
A Bayesian estimation of the model can be obtained by sampling its joint posterior using a Metropolis algorithm with adaptive scale parameters (Atchade & Rosenthal, 2005). For this purpose, the authors developed a Fortran code called from R. Since there is no theoretical constraint on the regression parameters, non informative uniform priors were considered. With logit links for g 0 , g 1 and g 2 , the values of regression parameters for standardized covariates are just constrained to take values in [−10; 10] since changes outside this interval do not affect the first four decimal places of π, µ and φ, see (2), (3) and (5) respectively. After a burn in period of 1000 iterations, convergence is monitored using the Gelman diagnostic tool (Gelman & Rubin, 1992) on 3 parallel Markov chains Monte Carlo (MCMC) of length 1000 each.
The initial values for the regression parameters in the first chain were selected using marginal mean, variance and proportion estimates: , whereȳ = 1 n n i=1 y i and n is the sample size. The idea is to set those intercepts to values that approximate the marginal distribution of the response.
The other parameters are set to 0.
The initial values for the second chain approximately correspond to an uniform distribution for a response on (0, 1) (and, thus, with mean 1/2 and variance 1/12) showing no inflation: For the third chain, the initial values of the parameters were estimated using the continuous beta regression of Simas et al. (2010) for location and dispersion and the frequentist estimation of a binary logit model for the inflation.
The median of the 3000 draws following the burn in is used as a point estimate for each parameter and to evaluate the bias, the standard deviation and the root mean squared error in simulations. The credibility intervals considered are the highest posterior density (HPD) intervals calculated using the function HPDinterval in the R package coda (Plummer, Best, Cowles, & Vines, 2006).
The quality of our estimation scheme was evaluated in details under various simulation settings: inflated and non inflated scales, varying sample sizes, high correlations between the covariates and omission of significant explanatory variables. Within those simulations, the acceptance rates, the effective sample sizes and the convergence diagnostics were considered as well as the bias and the precision of parameter estimates. The covering rates of HPD intervals and their length were monitored too. Tables 1 and 2 provide a selected summary of those simulation results; detailed results are available on request.
The results presented in table 1 are based on 500 simulated datasets with a 6-level discrete scale outcome and seven covariates: variables V1 to V4 were generated from independent normal distributions with mean equals to 3 and variance equals to 1; dummies D1 to D3 were generated with probabilities of success equal to 0.5. The marginal distribution of the rates across the 500 simulations is illustrated on the top left part of figure 2 with purely  As expected, tables 1 and 2 show that the bias, the standard deviation (Emp. SD), the root mean squared error (RMSE) and the length of HPD intervals decrease with sample size. The coverage of the credibility intervals are close to their nominal level even with moderate sample sizes.

Predictions
To sample the predictive distribution of the response of subject i, one can rely on the MCMC sample for the regression parameters and the following algorithm. For iteration l of the L = 3000 draws after burn in and for observation or scenario i: 1. Draw p i,l ∼ Uni [0, 1] and setŷ i,l = kh if p i,l ≤ [1 + exp (−w ′ iγl )] −1 and proceed with step 2 otherwise; and round it to the upper bound of the proxy interval under the beta density:ŷ i,l = h ⌈u i,l /h⌉.
The empirical distribution of Y i can be approximated using the empirical distribution of the Monte Carlo sample {ŷ i,l } L l=1 . A point prediction can be obtained from it by reporting its mode (say). Subset of values can also be constructed to predict Y i . Forcing it to be an interval does not lead to good results in term of covering rate and interval length. Instead, we recommend to produce potentially disconnected regions that turn to have nice covering rates and smaller length. This method allows the (1 − α) HPD to be disjoint if the inflated point has an estimated predictive probability greater than α.
In such cases, the k th point of the scale should be part of the (1 − α) HPD predictive region and added to the (1 − α −π i ) HPD predictive interval  were made using parameter estimates from dataset s. Figure 2 illustrates the marginal distribution of the outcome in the four simulation settings.  Table 3 reports the percentage of cases correctly predicted in the different simulation settings. Discrete response models (IDBR, clm and multinom) perform slightly better than the continuous ones (lm and betareg). Linear regression model (lm) is always the worse choice even when the outcome has been generated by it and simply rounded. The IDBR proves to be more accurate on datasets generated with inflation related to specific covariate pattern. When it is not, the IDBR is equivalent to its discrete competitor in term of proportion of correct predictions.  Table 4 reports coverages and the length of the 95% credible intervals.
The coverages are generally close to their nominal values or tend to exceed them. The lengths have been computed by subtracting the upper and lower bound. When disjoint intervals (resulting from a separated inflation response) were produced using IDBR, the minimum distance between two consecutive points on the scale h has been added to the main interval length to account for the addition of the inflation point. In order to make the interpretation comparable with different numbers of levels, all lengths have been rescaled to a maximum of 1 in the table. IDBR produces much narrower intervals than competing models even when the prediction region is disjoint: it indicates a better precision in predictions. When an inflation is introduced in the generating process, we get a large percentage of disjoint intervals (34.6 to 37.1% in these simulations). This is not the case with datasets generated by the linear regression model even when there is an inflation due to rounding.
Then, the presence of numerous disjoint intervals might be the symptom of subjects always selecting the inflated point among the proposed choices for particular covariate patterns and the relevance of the inflation submodel. Rates in figure 3 bring the inflation clearly out. About 35% of the respondents choose the central level of the scale, the only one which is neither on the left nor on the right. The hypothesis of the inflated discrete beta regression (IDBR) model is that those respondents might be divided in two groups:

Discussion
In this paper, we address four critical characteristics of Likert and rating scale outcomes that have never been taken into account simultaneously by existing models. Discreteness, boundedness and potential skewness are handled by the discrete beta regression (DBR) model described in section 2.The ability to identify likely invariant choosers of a particular level in the scale is one of the relevant features of the inflated discrete beta regression (IDBR) model described in section 3.The excellent statistical properties of the latter tool suggest that it is a valuable alternative to the existing (and less comprehensive) methods to model selection behaviors and to make predictions.
The DBR and IDBR models produce valuable outputs in both interpre-tation and prediction processes. Identifying the covariates influencing the location of the outcome is a classical model output, but understanding what determines unanimity or indecisiveness in subgroups of respondents might be determinant when for example marketers launch a campaign or doctors evaluate patient's pain. Distinguishing between invariant and less resolute choosers of a particular alternative in a Likert scale could for example save money to announcers by not sending ads to likely invariable non buyers.
Compared with competing models, the IDBR model produces discrete predictions with high proportion of cases correctly predicted and the narrowest credible intervals.
Extension of IDBR to multiple inflations (e.g.: both bounds are inflated) is straightforward but has not been tested yet. Adaptation of the IDBR to hierarchical experiment such as conjoint analysis or psychological measurements would have numerous applications. It will be our next step of research.