A new class of regression model for a bounded response with application in the study of the incidence rate of colorectal cancer

Abstract

Response variables in medical sciences are often bounded, e.g. proportions, rates or fractions of incidence of some disease. In this work, we are interested to study if some characteristics of the population, e.g. sex and race which can explain the incidence rate of colorectal cancer cases. To accommodate such responses, we propose a new class of regression models for bounded response by considering a new distribution in the open unit interval which includes a new parameter to make a more flexible distribution. The proposal is to obtain compound power normal distribution as a base distribution with a quantile transformation of another family of distributions with the same support and then is to study some properties of the new family. In addition, the new family is extended to regression models as an alternative to the regression model with a unit interval response. We also present inferential procedures based on the Bayesian methodology, specifically a Metropolis–Hastings algorithm is used to obtain the Bayesian estimates of parameters. An application to real data to illustrate the use of the new family is considered.

Multivariate generalized linear mixed models for continuous bounded outcomes: Analyzing the body fat percentage data

“…In this article, we used the R package TMB to apply the Laplace approximation method in (12) and to compute the required derivatives of the marginal log-likelihood by automatic differentiation. 47,48 TMB offers fast and general tools for fitting complex random effects models that are based on C++ templates and high performing libraries.…”

“…Examples include the proportion of diseased tooth sites, 10 health-related quality of life scores, 11 and the incidence rate of colorectal cancer. 12 For the analysis of such data, several regression models have been proposed over the last two decades. Ferrari and Cribari-Neto 13 proposed the beta regression model, which allows us to model the mean of the response variable as a function of a set of covariates.…”

Parametric quantile regression models for fitting double bounded response with application to COVID-19 mortality rate data

“…Unlike the beta regression, the median in the re-parameterized Johnson S B distribution is related to a linear predictor. Cancho et al (2020) generalized the Johnson S B model to a general class of distributions. The authors introduced an extra parameter to model the shape of the Johnson S B distribution, and studied a quantile regression model for limited range response variables.…”

“…In this paper, we formulate two rich classes of parametric quantile regression models for a bounded response, where the response variable is power Johnson S B distributed (Cancho et al, 2020) using a new parametrization of this distribution that is indexed by quantile (not only for median regression) and shape parameters. The estimation and inference for the proposed quantile regression models can be carried out based on the likelihood paradigm (parametric approach).…”

“…Therefore, a regression structure on ξ and δ is studied by the authors, providing a rich class of median regression model with varying dispersion. Cancho et al (2020) considered g(u) = φ(u) (where φ(•) denotes the pdf of the standard normal model) and the power model transformation (Lehmann, 1953;Durrans, 1992) to extend this class of models (named as PJSB), which cdf is given by…”