betaregscale: Beta Regression for Interval-Censored Scale-Derived Outcomes

Maximum-likelihood beta regression for scores recorded on a bounded scale $\{0, 1, \ldots, K\}$ (pain rating scales, Likert-type items, ratings), where $K =$ ncuts is the maximum score; a scale that starts at 1 is shifted to start at 0. A score is read as a coarsened observation of a latent $Y \in (0, 1)$ with a beta distribution: each score maps to a cell $[l_i, u_i]$ of $(0, 1)$ and contributes the beta probability of that cell to the likelihood. This is the interval beta regression model of Lopes (2023). The package fits fixed- and variable-dispersion models (brs) and mixed models with random intercepts and slopes (brsmm), and provides simulation, score probabilities, bootstrap, cross-validation and marginal effects.

Model

For observation $i$, $$Y_i \sim \mathrm{Beta}(a_i, b_i), \qquad g_1(\mu_i) = x_i^\top \beta, \qquad g_2(\phi_i) = z_i^\top \gamma,$$ where $(a_i, b_i)$ follow from $(\mu_i, \phi_i)$ under one of three parameterisations (brs_repar). The default, repar = 2, uses the mean $\mu$ and the dispersion $\phi = 1/(1 + a + b)$, written $\sigma$ in Lopes (2023). The score $s_i$ gives the cell $[l_i, u_i]$ and the censoring type $\delta_i$ (brs_check), and the log-likelihood is $$\ell(\beta, \gamma) = \sum_{\delta_i = 0} \log f(y_i) + \sum_{\delta_i = 1} \log F(u_i) + \sum_{\delta_i = 2} \log\{1 - F(l_i)\} + \sum_{\delta_i = 3} \log\{F(u_i) - F(l_i)\},$$ with $f$ and $F$ the beta density and distribution function. Scores 0 and $K$ are left- and right-censored ($\delta = 1, 2$), other scores interval-censored ($\delta = 3$); values already in $(0, 1)$ are exact ($\delta = 0$).

Main functions

brs_check, brs_prep: Scores (or analyst-supplied intervals) to cells and censoring types.
brs: Model fit; brs_fit_fixed and brs_fit_var are the fixed- and variable-dispersion workers.
brsmm: Mixed model with Gaussian random intercepts and slopes (Laplace, adaptive Gauss-Hermite or quasi-Monte Carlo integration).
brs_sim: Simulate scores from the model.
brs_predict_scoreprob: Predicted probability of each score.
brs_bootstrap, brs_cv, brs_marginaleffects, brs_table: Parametric bootstrap, cross-validation, average marginal effects and model comparison tables.

Fits of class "brs" and "brsmm" have methods for print, summary, coef, vcov, confint, logLik, AIC, BIC, nobs, anova, fitted, residuals, predict, plot and autoplot. As in betareg, coef() and vcov() take model = c("full", "mean", "precision"). Every fit checks its gradient, Hessian and likelihood clamps and warns in one line when something is wrong ('Fit diagnostics' in brs).

Relation to other approaches

The model is the interval beta regression (model M2) of Lopes (2023). That dissertation compares it with beta regression on the rescaled scores (betareg, M1) and with the quasi-beta regression of Bonat et al. (2019), fitted with mcglm (Bonat and Jørgensen, 2016), which specifies only the mean and the variance (M3). In that study's simulations M3 had Wald coverage closest to 95%, M1 and M2 fell below 90% in several scenarios with more than 500 observations and dispersion above 0.2, and M3 tended to underestimate the covariate effects; on its knee-surgery data M1 and M2 gave similar time effects. These are results of that study and of the implementation used then, not properties guaranteed by the package (a Monte Carlo study of the current code is in vignette("brs-advanced-workflows")). The interval model is useful when the coarsening of the scale is part of the question: it gives score probabilities, treats the borders of the scale as censoring and supports likelihood-ratio tests.

References

Lopes, J. E. (2023). Modelos de regressao beta para dados de escala. Master's dissertation, Universidade Federal do Parana, Curitiba. URI: https://hdl.handle.net/1884/86624.

Ferrari, S. L. P., and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501

Bonat, W. H., Petterle, R. R., Hinde, J., and Demétrio, C. G. B. (2019). Flexible quasi-beta regression models for continuous bounded data. Statistical Modelling, 19(6), 617–633.

Bonat, W. H., and Jørgensen, B. (2016). Multivariate covariance generalized linear models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 65(5), 649–675.

Author

Maintainer: José Evandeilton Lopes evandeilton@gmail.com (ORCID)

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