key: cord-0479188-o5grsrxy
authors: Bourguignon, Marcelo; Gallardo, Diego I.; Saulo, Helton
title: A parametric quantile beta regression for modeling case fatality rates of COVID-19
date: 2021-10-09
journal: nan
DOI: nan
sha: 707b0a0461f2f059896301f15eded863ee9597d2
doc_id: 479188
cord_uid: o5grsrxy

Motivated by the case fatality rate (CFR) of COVID-19, in this paper, we develop a fully parametric quantile regression model based on the generalized three-parameter beta (GB3) distribution. Beta regression models are primarily used to model rates and proportions. However, these models are usually specified in terms of a conditional mean. Therefore, they may be inadequate if the observed response variable follows an asymmetrical distribution, such as CFR data. In addition, beta regression models do not consider the effect of the covariates across the spectrum of the dependent variable, which is possible through the conditional quantile approach. In order to introduce the proposed GB3 regression model, we first reparameterize the GB3 distribution by inserting a quantile parameter and then we develop the new proposed quantile model. We also propose a simple interpretation of the predictor-response relationship in terms of percentage increases/decreases of the quantile. A Monte Carlo study is carried out for evaluating the performance of the maximum likelihood estimates and the choice of the link functions. Finally, a real COVID-19 dataset from Chile is analyzed and discussed to illustrate the proposed approach.

The new coronavirus respiratory syndrome disease pandemic has affected several countries around the world; see Vitenu-Sackey and Barfi (2021) . In particular, the COVID-19 pandemic has hit Chile hard in the past few months. Some authors have studied different aspects of COVID-19 in Chile. For instance, Tariq et al. (2021) estimated the reproduction number throughout the epidemic and studied the effectiveness of lockdowns. Guerrero-Nancuante and Manríquez (2020) studied the course of in Chile based on the generalized Susceptible-Exposed-Infectious-Removed model, and Barría-Sandoval et al. (2021) discussed different time series methodologies to predict the number of confirmed cases and deaths. In practice, these works can be useful in an epidemic, especially because they provide the basis for making decisions by policy-makers, e.g. directing health care resources to certain areas or identifying how long social distancing policies may need to be in effect.

This work focuses on a measure of COVID-19 mortality risk known as case fatality rate (CFR) . Such a measure is particularly important for health policies, and it is computed as the ratio between confirmed deaths and confirmed cases. It is one of the indicators that serve to monitor the severity of the pandemic; see https://ourworldindata.org/. In terms of stochastic representation, the CFR can be written as

where V ∈ R + and W ∈ R + are two random variables representing the number of confirmed COVID-19-related deaths and COVID-19 cases with no death result, respectively. The sum W + V represents the number of confirmed COVID-19 cases, and the ratio (1) has support in the unit interval (0, 1). Malik (1967) and Ahuja (1969) both showed that if V and W are independent random variables following standard (scale parameter equal to 1) gamma distributions with shape parameters α > 0 and β > 0, i.e. V ∼ GA(α, 1) and

W ∼ GA(β, 1), then

namely, Y is beta distributed with shape parameters α, β > 0 and d = stands for equality in distribution. It should be noted that stochastic representations are important since they may justify some models arising naturally in real situations, as seen above.

The modeling of CFR of COVID-19 can be approached by regression models that assume a unit interval response, such as the widely used beta regression model. This model was proposed by Ferrari and Cribari-Neto (2004) , where the authors consider an alternative parameterization of the beta distribution in terms of the mean and a precision parameter.

In the beta regression model, the mean is related to a set of covariates through a linear predictor. Nevertheless, it is well known that the widely popular mean regression model could be inadequate if the observed response variable follows an asymmetrical distribution (see Galarza et al. (2017) and Sánchez et al. (2021) ), which is quite common for rates and

proportions. In such a situation, quantile regression models (Koenker, 2005) can be more suitable. Quantile regression is a methodology used for understanding the conditional distribution of a response variable given the values of some covariates at different levels (quantiles), thus providing users with a more complete picture. Despite this, as far as we know, a specific parametric quantile regression model to describe data observed in the unitary interval at different quantiles that satisfies the stochastic representation in (1) has never been considered in the literature. This paper works in this direction.

Although the beta regression is the standard model for quantifying the influence of covariates on the mean of the response variable in the unit interval, it is not useful for quantifying such an influence on the quantiles of the dependent variable. This occurs because the cumulative distribution function (CDF) of the beta distribution does not have an invertible closed form, which hinders its utilization for parametric quantile regression purposes.

A natural approach to model CFR in observational studies is to apply a logarithmic (1). Such criticism can be extended to the CDF-quantile family of two-parameter distributions (Smithson and Shou, 2017) .

On the other hand, a probability distribution related to the beta model that has received little attention in the literature was introduced by Libby and Novick (1982) , named the Libby-Novick beta distribution or generalized beta with three parameters (GB3) distribution. This distribution is a generalization of the standard beta distribution and can be an interesting and useful model for modeling double bounded data satisfying (1), which is the main feature of CFR data. The GB3 distribution is a three-parameter generalization of the usual two-parameter beta distribution, and it is perhaps the simplest generalization of the two-parameter beta distribution that allows for significant additional flexibility; see Ristić et al. (2015) . The GB3 distribution offers more flexibility for modeling real data than the beta distribution because the additional shape parameter can control the skewness and kurtosis simultaneously, varying tail weights and providing more entropy; see Cordeiro et al. (2014) . Furthermore, as said earlier, the GB3 distribution presents the stochastic representation in (1). Despite all these characteristics, little attention in the statistical literature has been devoted to the GB3 distribution.

In this work, motivated by the features of COVID-19 CFR data, we propose and study a new parametric quantile regression model based on the GB3 distribution and verify that the proposed quantile regression model is suitable for modeling double bounded data satisfying the stochastic representation in (1). To the best of our knowledge, a specific parametric quantile regression model to describe data of the type V /(W + V ) at different levels has never been considered in the literature. The quantile approach allows for capturing the influence of the covariates on the spectrum of the dependent variable, in addition to coping with outliers; see Koenker (2005) , Hao and Naiman (2007) , and Waldmann (2018) .

Furthermore, the proposed quantile regression model allows for the quantiles of the data to be described on their original scale, unlike the existing models that employ a logarithmic transformation of the CFR. Similarly to the works of Noufaily and Jones (2013) and Saulo et al. (2021) , who introduced quantile parameters in the generalized gamma and logsymmetric distributions, respectively, we introduce a reparameterization of the GB3 model by inserting a quantile parameter before developing the GB3 quantile regression model.

Our proposed approach relates the quantiles of (1), q(V /(W + V )|X), to a set of explanatory variables X = (X 1 , . . . , X k ), providing a full parametric elegant quantile regression model; see Berger et al. (2019) for a similar approach related to the ratio of two positively correlated biomarkers. We also propose a simple interpretation of the predictor-response relationship in terms of percentage increases/decreases of the quantile. We analyze COVID-19 data from Chile and find that the CFR is explained by population density, positivity for tests, and the percentage of the population fully vaccinated.

The rest of the paper unfolds as follows. In Section 2, we describe the classical GB3 distribution and propose a reparameterization of this distribution in terms of a quantile parameter. In Section 3, we introduce the GB3 quantile regression model and discuss the estimation of the model parameters by the maximum likelihood (ML) method. We also discuss residual analysis and covariate selection in this section. In Section 4, we carry out a Monte Carlo simulation study to evaluate the recovery of the parameters and assess the choice of the link functions. In Section 5, we apply the GB3 quantile regression model to a real COVID-19 CFR dataset. Finally, in Section 6, we provide some concluding remarks.

In this section, we shall describe the GB3 distribution and introduce the quantile-based reparameterization of this distribution, which will be useful subsequently for developing the GB3 quantile regression model.

A random variable Y follows the generalized beta distribution with three parameters, α > 0, β > 0, and λ > 0, denoted by Y ∼ GB3(λ, α, β), if its CDF is given by

where

The probability density function (PDF) associated with Equation (2) is

We can see that Equation (3) reduces to the beta distribution when λ = 1.

This property is similar to the one enjoyed by the beta distribution. According to Cordeiro et al. (2014) , taking α = β, the shape parameter λ gives tail weights of the PDF to the extreme right, and skewness and kurtosis increase when λ approaches zero for 0 < λ < 1. On the other hand, the PDF (3) becomes more symmetric when λ approaches one, and for λ > 1, the parameter λ gives tail weights of the PDF to the extreme left when λ → ∞. Furthermore, the skewness and kurtosis increase when λ tends to infinity.

Remark 2.1. (Relation to the generalized beta distribution of the first kind model) The generalized beta distribution of the first kind (McDonald and Xu, 1995) , GB1(µ, σ, ν, τ ) , is defined by the following PDF:

where α = µ(1 − σ 2 )/σ 2 > 0 and β = (1 − µ)(1 − σ 2 )/σ 2 > 0; see Stasinopoulos and Rigby (2007) . Comparing with (3), we can see that the GB3(α, β, λ) model is a special type of the GB1(µ, σ, ν, τ ) model as follows:

and additional details can be found in their paper.

In a similar way to the classical beta distribution, if X 1 ∼ GA(α, θ 1 ) and X 2 ∼ GA(β, θ 2 ) are independent gamma distributions, then the random variable (Libby and Novick, 1982 )

is distributed according to a GB3 distribution, where λ = θ 1 /θ 2 .

Note that, as I x (α, β) corresponds to the CDF of the usual beta distribution, the τ -th quantile of the GB3 distribution can be represented as

where z α,β (τ ) denotes the τ -th quantile of the beta distribution with parameters α and β.

To compute the quantile z α,β (τ ) as required, it is not necessary to implement such complex formulae as (2). Instead, one can use the qbeta(·) function in R (R Core Team, 2021).

To introduce the proposed quantile regression model, we shall reparametrize (3) in terms of the τ -th quantile µ = q(τ ; λ, α, β) such that λ can be written (from Eq. (6)) as follows

, 0 < µ < 1.

This parametrization has not been proposed in the statistical literature. Under this new parametrization, the PDF and CDF of the GB3 distribution can be written, respectively, as

Hereafter, we shall use the notation Y ∼ GB3(µ, α, β), where µ ∈ (0, 1) is the quantile parameter, α > 0 and β > 0 are the shape parameters, and τ ∈ (0, 1) is assumed to be known. Figure 1 displays different shapes of the GB3 PDF for different combinations of parameters. It is noteworthy that the GB3 is very flexible since its density assumes different forms. 8

In this section, we shall introduce the GB3 quantile regression model and discuss parameter estimation based on the ML method. We also consider residuals analysis and covariate selection for the proposed model.

Let Y 1 , . . . , Y n be independent random variables such that each Y i , for i = 1, . . . , n, has PDF defined in (7),

, for a fixed (known) probability τ ∈ (0, 1) associated with the quantile of interest. Suppose µ(τ ), α(τ ), and β(τ ) satisfy the following functional relations:

where

are the vectors of the unknown regression coefficients, which are assumed to be function-

. . , z il ) ⊤ , and w i = (w i1 , . . . , w im ) ⊤ are the observations of the k, l, and m known regressors, for i = 1, . . . , n. Furthermore, we assume that the covariate matrices X = (x 1 , . . . , x n ) ⊤ , Z = (z 1 , . . . , z n ) ⊤ , and W = (w 1 , . . . , w n ) ⊤ have ranks k, l, and m, respectively. The link functions g 1 : (0, 1) → R, g 2 : R + → R, and g 3 : R + → R in (8) must be strictly monotone, positive, and at least twice differentiable, with g −1 1 (·), g −1 2 (·), and g −1 3 (·) being the inverse functions of g 1 (·), g 2 (·), and g 3 (·), respectively. For g 1 (·), the most common choices are the logit, probit, loglog, and complementary loglog (cloglog) link functions, whereas for g 2 (·) and g 3 (·) the most common choice is the log link.

The interpretation of the regression coefficients related to the quantile µ i can be done in terms of percentage change. Consider θ j (τ ) as the j-th regression coefficient where j denotes the exclusion of the j-th element. Then, x i(j) and θ (j) (τ ) denote the vector of covariates excluding x ij and the vector of regression coefficients excluding θ j (τ ), respectively. Note that when x ij increases by one, holding x i(j) fixed, we obtain

Therefore, for any j increasing x ij by one unit, the quantile µ i can be expressed as a percentage change as follows:

That is, (10) provides the percentage increase (or decrease if the value of β j is negative) in the quantile µ i when x ij is increased by one unit. For x ij dichotomous, (10) provides the percentage increase (or decrease if β j is negative) in the quantile µ i when x ij changes from category 0 to 1; see Weisberg (2014) . Note that when x i = log(z i ), z i can be replaced by c z i , and (9) can be rewritten as

For example, if z i is increased by 5%, then c = 1.05 and the percentage change (10) is computed taking into account (11) with log(1.05) ≈ 0.5. However, this interpretation not only depends on θ j (τ ) but also on the remaining coefficients.

To provide an alternative and more useful interpretation, we can use a Taylor series expansion of first order to approximate log[g −1 (x)] to zero. For the cases where g 1 (·) corresponds to the logit, probit, loglog, and cloglog links, we have that log[g −1 (x)] ≈ a 0 + a 1 x, where a 0 = − log(2) and a 1 = 1/2 for logit; a 0 = − log(2) and a 1 = 2/π for probit; a 0 = −e and a 1 = 1 for loglog; and a 0 = log(1−exp(−e)) and a 1 = [1+exp(−e)] −1 −1 for cloglog. Therefore,

and

which only depends on θ j (τ ). For this reason, 100 × (exp [−a 1 θ j (τ )] − 1) % can be interpreted as the approximate percentage increase (or decrease) in the response variable when the j-th covariate related to the quantile is increased by one unit and the remaining covariates are fixed.

Additionally, by the stochastic representation in Equation (5), we have that, conditional on z ⊤ i and w ⊤ i , the latent variables X 1 and X 2 are GA(α i (τ ), θ 1 ) and GA(β i (τ ), θ 2 ) distributed, respectively. It is well known that the means of such distributions are α i (τ )/θ 1 and β i (τ )/θ 2 , respectively. Therefore, an increment of one unit in the j-th covariate of z ⊤ i produces the following relative change in the mean of X 1 :

In other words, 100 ×[exp(ν j (τ )) − 1] % represents the relative increment (or decrement) in the mean of X 1 when the j-th covariate increases one unit and the rest of the covariates are maintained as fixed. Similarly, it is possible to show that 100×[exp(η j (τ )) − 1] % represents the increment/decrement in the mean of X 2 when the j-th covariate of w ⊤ i increases one unit and the rest of the covariates are maintained. In the CFR of COVID-19, this allows for the interpretation of the relative changes directly in terms of the number of deaths and cases of COVID-19 for the population density, average of positivity for the tests in a determined past period, and percentage of the population fully vaccinated in Chile, for instance. This interpretation is valid for any τ ∈ (0, 1). Note that it is possible that a covariate could be related to the total deaths and/or total cases of COVID-19, but not related to the quantile of the CFR of COVID-19, and vice versa. Henceforth, in order to avoid overloading the notation, we omit τ in θ, ν and η and in µ i , α i and β i .

Let Y 1 , . . . , Y n be a set of n independent random variables such that Y i ∼ GB3(µ i , α i , β i ), and y 1 , . . . , y n be the corresponding observations. The corresponding likelihood function

where µ i , α i , and β i are given in (8) and

.

By taking the logarithm of (13), we obtain the log-likelihood function for Θ,

where

The score vectorl(Θ) is obtained by differentiating the log-likelihood function in (14) with respect to each component of Θ. By equating the score vector to zero, we obtain the ML estimate of Θ. However, as the ML estimators of Θ do not have closed form expressions, the parameters must be estimated using standard numerical optimization algorithms. We use the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method;

see Mittelhammer et al. (2000, p. 199) . Numerical maximization of the log-likelihood function is accomplished by using the R software (available at http://www.r-project.org).

The computational program is available from authors upon request.

Under mild regularity conditions (Cox and Hinkley, 1979 ) and when n is large, the asymptotic distribution of the ML estimators Θ = ( θ ⊤ , ν ⊤ , η ⊤ ) ⊤ is approximately multivariate normal (of dimension k + l + m) with mean vector Θ = (θ ⊤ , ν ⊤ , η ⊤ ) ⊤ and variance-

is the expected Fisher information matrix. Unfortunately, there is no closed form expression for K(Θ). Nevertheless, a consistent estimator of the expected Fisher information matrix is given by

which is the estimated observed Fisher information matrix. Therefore, for large n, we can approximate K(Θ) by J( Θ).

To assess the goodness of fit and departures from the assumptions of the model, we

shall use the randomized quantile (Dunn and Smyth, 1996) (RQ) residual given by

where Φ −1 is the inverse function of the standard normal CDF and F (·) is the CDF fitted to the data. This residual follows approximately a standard normal distribution when the model is correctly specified. Then, we can use quantile-quantile (QQ) plots to assess the fit.

A critical point of the model is the selection of the covariates included in each component of the model: µ i , α i , and β i . If we dispose of r covariates for each individual and considering that the intercept is included in all components, we can arrange such covariates in 2 3r ways,

i.e., such quantity of models. A way to select a combination of covariates is to fit the 2 3r models and select the one with minimum Akaike information criteria (AIC; Akaike, 1974) or Bayesian information criteria (BIC; Schwarz, 1978) , for instance. However, such method demands a high computational cost and some covariates could not be significant for some significance levels. To avoid these problems, an alternative is to fit the model with all covariates in the three components and successively eliminate the covariate (which can be in any of the 3 components of the model) with the lower significance (i.e., the greater pvalue based on the asymptotic distribution of the ML estimator). Under this scheme, we compute the estimates for 3r models at maximum.

In this section, we present two simulation studies to assess different aspects of the GB3 regression model. The first study assesses the recovery parameters of the model under different combinations of sample size and τ . The second study evaluates the performance of the model selection using information criteria to select a determined link to the quantile parameter in the GB3 regression model. For all cases, we considered the design matrices 

In this study, we focused on the properties of the ML estimators of the GB3 regression model. The data generation and parameter estimation were based on the logit link. For each combination of τ and n, we drew 1,000 samples of size n and computed the ML estimates. We present the estimated bias (bias), the mean of the estimated standard errors (SE), the root of the estimated mean squared error (RMSE), and the 95% coverage probabilities (CP) based on the normal distribution. Table 1 summarizes the results. Note that, in general, within each linear predictor, the intercepts presented the greater bias.

However, such bias is reduced as the sample size increases. We also note that SE and RMSE decrease when n increases, suggesting that the standard errors are well estimated and the estimators are consistent. Finally, the CP's seem poor for small sample sizes, especially for the parameters related to the quantile. However, this is natural considering that we are estimating eight parameters based on 100 observations. We also remark that such CP's improve considerably for n = 200 and n = 305. This result suggests that one should be careful with the asymptotic normal distribution of the ML estimator for small and moderate sample sizes. 

In this section, we show that, given the "true link" used for the quantile, the performance of the ML estimators of the GB3 regression model is good. However, the correct selection of the link can be a critical point in a real data application. We considered only the estimated log-likelihood (LL) function for model selection and the logit, probit, loglog, and cloglog link functions. The data generation was performed based on the GB3 model and a specified 

In this section, we present and discuss a real data application related to the CFR of COVID-19 in different communes of Chile to illustrate the performance of the GB3 quantile regression model. In order to estimate the parameters of the model, we adopted the ML method (as discussed in Subsection 3.3) and all computations were performed using the function optim(·) in R.

Chile is administratively divided into 348 communes, which belong to 16 regions. We considered the cases and deaths of COVID-19 reported by the Chilean Ministry of Health for each commune between May 24 and July 23, 2021, but only for communes with at least 50 cases and 1 death, totaling 335,727 cases and 8,692 deaths in 305 communes. The communes were divided into three zones: North, Center, and South (see Figure 2 ). This division is typically used in Chile and is supported by climatological characteristics from the different zones. We also considered the following information for each commune:

• cfr: CFR (confirmed deaths/confirmed cases) between May 24 and July 23, 2021. Mean=0.026, Median=0.025, standard deviation=0.012, minimum=0.004, and max-imum=0.099; see Figure 3 . Figure 4 : Positivity rate, population density, and vaccination for communes in the North, Center, and South zones of Chile. 

We considered modeling the τ -th quantile of the CFR, say cfr(τ ), such that cfr i (τ ) ∼ GB3(µ i (τ ), α i (τ ), β i (τ )), for i = 1, . . . , n, with g 1 (µ i (τ )) = θ 0 (τ ) + θ 1 (τ ) log(dens i ) + θ 2 (τ )posit i + θ 3 (τ )vaccine i , g 2 (α i (τ )) = ν 0 (τ ) + ν 1 (τ ) log(dens i ) + ν 2 (τ )posit i + ν 3 (τ )vaccine i ,

We considered the quantiles {0.1, 0.25, 0.50, 0.75, 0.90} and used the four link functions previously discussed (logit, probit, loglog, and cloglog) . For each combination of τ and link function, we selected the covariates based on the method discussed in Section 3.5.

Considering that the terms related to the intercept, log(dens), posit, and vaccine are variables 0 to 3, results of AIC and BIC criteria and selected covariates for each component are presented in Table 4 . Note that the values of AIC and BIC are similar for the four considered links. Thus, we used the logit link (because it is the traditional link used) to continue the analysis. In addition, based on the results, the final model was g 1 (µ i (τ )) = θ 0 (τ ) + θ 1 (τ ) log(dens i ) + θ 3 (τ )vaccine i , g 2 (α i (τ )) = ν 0 (τ ) + ν 1 (τ ) log(dens i ) + ν 2 (τ )posit i , g 3 (β i (τ )) = η 0 (τ ).

In other words, the quantile was modeled by log(dens) and vaccine, α was modeled by log(dens) and posit, whereas β only considered the intercept. Figure 6 plots the estimated parameters of the proposed model across such quantiles. We observe that the estimates of θ 0 (τ ) increase across τ , implying that the τ -th quantile of CFR increases as τ increases for the case when log(dens) and vaccine are fixed at 0. On the other hand, the estimates of θ 1 (τ ) decrease across τ , i.e., the τ -th quantile of CFR decreases as ldens increases. This is expected because denser cities have greater and better access to health centers. Similarly, we can conclude that the τ -th quantile of CFR decreases as vaccine increases. Note that from the discussion provided in Equation (12), as 1 − exp(−θ 1 (τ = 0.5)/2) = −0.023 and 1 − exp(θ 3 (τ = 0.5)/2) = −0.009, we also concluded that the median of the average CFR:

• decreases by approximately 2.3% when log(dens) is increased by one unit and vaccination is fixed;

• decreases by approximately 0.9% when the vaccination is increased by 1% in the population and log(dens) is fixed.

Additionally, considering the estimates for τ = 0.5, we have that exp(ν 1 ) = exp(0.3302) − 1 = 0.391. Therefore, the deaths related to COVID-19 in Chile in the two last months increase by 39.1% if the log(dens) is increased by one unit. On the other hand, as the β parameter is constant, variations in density, vaccination, and positivity do not produce significant variations in the total COVID-19 cases in Chile. Figure 7 shows the QQ plots of the RQ residuals for the GB3 quantile regression model in Table 5 . The proposed model provided a satisfactory fit to the data as most empirical quantiles agree with the theoretical ones. Finally, Figure 8 shows the predicted median in the two last months for the CFR in the three zones if the vaccination process is increased by 30% in each commune (capped at 100%). Comparing Figures 3 and 8 , we observe a large decrease in the predicted median for the average CFR in the two last months, showing that the vaccination is an effective way to reduce COVID-19 deaths. In this paper, we proposed a new parametric quantile regression model for dealing with data in the interval (0, 1). The proposed model is based on a reparameterization of the GB3 distribution, which has the distribution quantile as one of its parameters. In particular, we proposed a regression model to describe data of the type V /(W + V ) at different quantiles. The estimates of the model parameters have a simple interpretation in terms of percentage increases/decreases of the quantile q(V /(W + V )|X). We allowed a regression structure for the µ, α, and β parameters and used maximum likelihood inference to estimate the model parameters, which can be easily computed using the R software. Therefore, our model required less computational cost. We carried out a Monte Carlo simulation study to evaluate the recovery of the parameters and assess the choice of the link functions. The numerical results showed a good performance of the maximum likelihood estimates. We also applied the proposed model to a real COVID-19 dataset from Chile. In particular, we modeled the case fatality rate (CFR) of COVID-19 and used the population density, positivity rate, and percentage of the population fully vaccinated as covariates.

The main conclusions were that the median of the CFR in the two last months decreased by approximately 2.3% when the logarithm of the population density was increased by one unit (a denser region has greater and better access to health centers) and decreased by approximately 0.9% when the vaccination process was increased by 1%. The results

were quite favorable to the proposed GB3 quantile regression model and emphasized the importance of using a quantile approach. As part of future research, we envisage to propose zero-or-one inflated quantile regression models based on the GB3 model.

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