A beta regression analysis of COVID-19 mortality in Brazil.

Brazil was one of the countries most impacted by the COVID-19 pandemic, with a cumulative total of nearly 700,000 deaths by early 2023. The country's federative units were unevenly affected by the pandemic and adopted mitigation measures of different scopes and intensity. There was intense conflict between the federal government and state governments over the relevance and extent of such measures. We build a simple regression model with good predictive power on state COVID-19 mortality rates in Brazil. Our results reveal that the federative units' urbanization rate and per capita income are important for determining their mean mortality rate and that the number of physicians per 100,000 inhabitants is important for modeling the mortality rate precision. Based on the fitted model, we obtain approximations for the levels of administrative efficiency of local governments in dealing with the pandemic.

Beta distribution misspecification tests with application to Covid-19 mortality rates in the United States.

The beta distribution is routinely used to model variables that assume values in the standard unit interval, (0, 1). Several alternative laws have, nonetheless, been proposed in the literature, such as the Kumaraswamy and simplex distributions. A natural and empirically motivated question is: does the beta law provide an adequate representation for a given dataset? We test the null hypothesis that the beta model is correctly specified against the alternative hypothesis that it does not provide an adequate data fit. Our tests are based on the information matrix equality, which only holds when the model is correctly specified. They are thus sensitive to model misspecification. Simulation evidence shows that the tests perform well, especially when coupled with bootstrap resampling. We model state and county Covid-19 mortality rates in the United States. The misspecification tests indicate that the beta law successfully represents Covid-19 death rates when they are computed using either data from prior to the start of the vaccination campaign or data collected when such a campaign was under way. In the latter case, the beta law is only accepted when the negative impact of vaccination reach on death rates is moderate. The beta model is rejected under data heterogeneity, i.e., when mortality rates are computed using information gathered during both time periods.

Bartlett-corrected tests for varying precision beta regressions with application to environmental biometrics.

Beta regressions are commonly used with responses that assume values in the standard unit interval, such as rates, proportions and concentration indices. Hypothesis testing inferences on the model parameters are typically performed using the likelihood ratio test. It delivers accurate inferences when the sample size is large, but can otherwise lead to unreliable conclusions. It is thus important to develop alternative tests with superior finite sample behavior. We derive the Bartlett correction to the likelihood ratio test under the more general formulation of the beta regression model, i.e. under varying precision. The model contains two submodels, one for the mean response and a separate one for the precision parameter. Our interest lies in performing testing inferences on the parameters that index both submodels. We use three Bartlett-corrected likelihood ratio test statistics that are expected to yield superior performance when the sample size is small. We present Monte Carlo simulation evidence on the finite sample behavior of the Bartlett-corrected tests relative to the standard likelihood ratio test and to two improved tests that are based on an alternative approach. The numerical evidence shows that one of the Bartlett-corrected typically delivers accurate inferences even when the sample is quite small. An empirical application related to behavioral biometrics is presented and discussed.

Modified likelihood ratio tests for unit gamma regressions.

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model's parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.

Inflated Kumaraswamy distributions.

The Kumaraswamy distribution is useful for modeling variables whose support is the standard unit interval, i.e., (0, 1). It is not uncommon, however, for the data to contain zeros and/or ones. When that happens, the interest shifts to modeling variables that assume values in [0, 1), (0, 1] or [0, 1]. Our goal in this paper is to introduce inflated Kumaraswamy distributions that can be used to that end. We consider inflation at one of the extremes of the standard unit interval and also the more challenging case in which inflation takes place at both interval endpoints. We introduce inflated Kumaraswamy distributions, discuss their main properties, show how to estimate their parameters (point and interval estimation) and explain how testing inferences can be performed. We also present Monte Carlo evidence on the finite sample performances of point estimation, confidence intervals and hypothesis tests. An empirical application is presented and discussed.

On nonlinear beta regression residuals.

We proposed a new residual to be used in linear and nonlinear beta regressions. Unlike the residuals that had already been proposed, the derivation of the new residual takes into account not only information relative to the estimation of the mean submodel but also takes into account information obtained from the precision submodel. This is an advantage of the residual we introduced. Additionally, the new residual is computationally less intensive than the weighted residual. Recall that the computation of the latter involves an n×n matrix, where n is the sample size. Obviously, that can be a problem when the sample size is very large. In contrast, our residual does not suffer from that. It can be easily computed even in large samples. Finally, our residual proved to be able to identify atypical observations as well as the weighted residual. We also propose new thresholds for residual plots and a scheme for the choice of starting values to be used in maximum likelihood point estimation in the class of nonlinear beta regression models. We report Monte Carlo simulation results on the behavior of different residuals. We also present and discuss two empirical applications; one uses the proportion of killed grasshoppers in an assay on the grasshopper Melanopus sanguinipes with the insecticide carbofuran and the synergist piperonyl butoxide, which enhances the toxicity of the insecticide, and the other uses simulated data. The results favor the new methodology we introduce.