In this paper, we introduce a transformation of the truncated Weibull distribution, originally defined in the interval $ (0,2) $ (0,2), to facilitate the analysis of differences between rates or proportions. The transformed distribution is defined on the interval $ (-1,1) $ (-1,1) and parameterized in terms of its mode and shape parameters. We refer to this new distribution as the modal truncated Weibull (MTW) distribution. Building on the MTW distribution, we propose a novel modal regression model in which the parameters are estimated using the maximum likelihood method. To assess the performance of the maximum likelihood estimators, we conducted a Monte Carlo simulation study under various scenarios and regression structures. Additionally, we derive analytical expressions for the score vector and the Fisher information matrix, providing a foundation for parameter estimation and inference. We also briefly discuss diagnostic tools for evaluating the fit and performance of the model. Finally, we present and analyze an empirical application to demonstrate the practical utility and relevance of the proposed MTW regression model.

We propose a zero-adjusted promotion cure-rate regression model that jointly accommodates instantaneous failures (times at or near-zero), a non-susceptible cured fraction, competing risks, and latent heterogeneity modeled with a Gamma frailty. The number of latent causes follows a Poisson distribution and their activation times follow a Weibull distribution. We derive closed-form survival and density functions under three activation schemes-minimum, maximum, and random-and allow covariates to act on the zero inflation and cure proportions as well as on the baseline time-to-event distribution. Parameters are estimated by maximum likelihood, and finite-sample performance is assessed via Monte Carlo experiments. We illustrate the method with data on time to insulin initiation among 390 pregnant women with gestational diabetes, where the random activation scheme provides the best fit by Akaike Information Criterion and Bayesian Information Criterion and yields interpretable covariate effects. The proposed framework unifies and extends existing cure-rate and zero-adjusted survival models, providing a flexible tool for complex settings with cure fractions, competing risks, and frailty.

Count data are common in medical research. When these data have more zeros than expected by the most used count distributions, it is common to employ a zero-inflated regression model. However, the interpretability of these models is much lower than the most used count regression models. We present a more interpretable regression model that estimates the mean event rate and models covariate-dependent dispersion directly. Additionally, the dispersion parameter can be interpreted as an index of clumping, a control parameter for overdispersion. We discuss inferential and diagnostic tools and perform a Monte Carlo simulation study to evaluate the performance of the maximum likelihood estimator. Finally, the usefulness of the proposed regression model is illustrated through an application on antenatal care visits.

In this article, we propose a slashed lognormal regression model based on a reparameterization defined in terms of the mean, precision, and shape parameters. The model is suitable for situations in which the response variable is continuous, positively skewed, exhibits a flexible shape, and presents high kurtosis and heteroskedasticity. The proposed methodology includes parameter estimation, hypothesis testing for the precision parameter, residual analysis, and influence diagnostics. A Monte Carlo simulation study is carried out to evaluate the performance of the estimators and residuals in finite samples. Finally, the applicability of the model is illustrated using a mineral concentration dataset.

This research investigates the properties of the posterior distribution of the gamma distribution, especially in the context of right-censored data. We establish necessary and sufficient conditions for determining when improper priors lead to proper posteriors. Additionally, we derive conditions to ascertain the finiteness of the posterior moments. The study addresses the challenges posed by censoring and delves into the application of various objective priors. We introduce a novel estimator for censored data, enhancing the efficiency of the Markov Chain Monte Carlo (MCMC) algorithm. Through a simulation study, we evaluate the performance of Bayesian estimators under different priors. Our methodology is applied to a dataset from the Cancer Genome Atlas, focussing on lung adenocarcinoma in patients over 70, offering valuable insights into disease progression and mortality patterns.

We followed the COVID-19 pandemic in Manaus, one of the epicenters of COVID-19 in Brazil, using an epidemiological mathematical model and made five main conclusions. First, in early 2022, the actual cases exceed officially reported data by up to 8 times. Second, despite vaccination campaigns, the collective immunity threshold necessary was insufficient to contain severe cases of COVID-19. Next, the low observed mortality demonstrated the effectiveness of vaccination. Next, the drop in the vaccination rate combined with immune escape by the Omicron sub-variants (BA.2.12.1, BA.4, and BA.5) resulted in new wave after November 2022. Finally, to minimize severe cases of COVID-19, we need to raise vaccination thresholds above 90–95

In this paper, we discuss some theoretical results and properties of a discrete version of the Birnbaum-Saunders distribution. We present a proof of the unimodality of this model. Moreover, results on moments, quantile function, reliability and order statistics are also presented. In addition, we propose a regression model based on the discrete Birnbaum-Saunders distribution. The model parameters are estimated by the maximum likelihood method and a Monte Carlo study is performed to evaluate the performance of the estimators. Finally, we illustrate the proposed methodology with the use of real data sets.

The two-parameter weighted Lindley distribution has become much popular due to its simplicity, attractive properties, and flexibility to fit data when compared with similar generalizations of the exponential model, such as gamma and Weibull, among others. In this paper, we introduce a regression model based on a weighted Lindley distribution, which is reparameterized in terms of mean and precision parameters. In this model, both the mean and precision parameters vary with the explanatory variable values and general link functions are used in order to account for these relationships. We developed and implemented local influence diagnostics to identify potential influential observations. Hessian and Fisher information matrices are computed on the closed-form as well as their inverses. Classical inference based on the maximum likelihood method is presented. Extensive Monte Carlo simulation studies are carried out for a special case of the regression model in order to verify the asymptotic properties of the maximum likelihood estimators. Finally, the usefulness of the proposed model is illustrated through an empirical analysis.

We introduce a new modelling for long-term survival models, assuming that the number of competing causes follows a mixture of Poisson and the Birnbaum-Saunders distribution. In this context, we present some statistical properties of our model and demonstrate that the promotion time model emerges as a limiting case. We delve into detailed discussions of specific models within this class. Notably, we examine the expected number of competing causes, which depends on covariates. This allows for direct modeling of the cure rate as a function of covariates. We present an Expectation-Maximization (EM) algorithm for parameter estimation, to discuss the estimation via maximum likelihood (ML) and provide insights into parameter inference for this model. Additionally, we outline sufficient conditions for ensuring the consistency and asymptotic normal distribution of ML estimators. To evaluate the performance of our estimation method, we conduct a Monte Carlo simulation to provide asymptotic properties and a power study of LR test by contrasting our methodology against the promotion time model. To demonstrate the practical applicability of our model, we apply it to a real medical dataset from a population-based study of incidence of breast cancer in São Paulo, Brazil. Our results illustrate that the proposed model can outperform traditional approaches in terms of model fitting, highlighting its potential utility in real-world scenarios.

The class of models known as Arithmetic Reduction of Age (ARA) has found widespread use in modeling equipment maintenance data, assuming that the system continuously degrades between failures, as indicated by beta>1 in the Power Law Process (PLP), commonly employed for fitting such data. However, there are situations where system improvement between failures may occur, as signaled by beta<1 in the PLP. Systems or equipment that enhance over time to a certain degree are less common, as degradation often proves to be a more typical feature. Nevertheless, some systems may exhibit this initial improvement due to various factors such as adaptation, maturation, or updates. In this context, the study introduces the modified ARA1 model (ARAM1), enabling the modeling of systems in a state of reduction or degradation, thereby overcoming this limitation. Additionally, it proposes a new reparameterization of the PLP as a time truncation while preserving the original interpretation of the PLP parameters. An example with a real-world dataset illustrates the potential applications of the proposed model. From the results, it is evident that the ARAM1 model serves as a valuable tool for enhancing the understanding of the reliability of systems subject to imperfect repairs.

Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. In this type of model, the number of concurrent causes is assumed to be a random variable. However, in practice, it is natural to admit that the distribution of the number of competing causes is different from individual to individual. Our proposal is to assume that the number of competing causes belongs to a class of a finite mixture of competing causes distributions. We assume the number of malignant cells follow a mixture of two power series distributions and suppose that the time to the event of interest follows a Weibull distribution. We consider the proportion of the cured number of competing causes depending on covariates, allowing direct modeling of the cure rate. The proposed model includes several well-known models as special cases and defines many new special models. An expectation-maximization algorithm is proposed for parameter estimation, where the expectation step involves the computation of the expected number of concurrent causes for each individual. A Monte Carlo simulation is performed to assess the behavior of the estimation method. In order to show the potential for the practice of our model, we apply it to the real medical data set from a population-based study of incident cases of cutaneous melanoma diagnosed in the state of São Paulo, Brazil, illustrating that the model proposed can outperform traditional models in terms of model fitting.

Tendentious projections about COVID-19 in Brazil provided an appealing excuse for individuals and decision-makers to justify poor choices during a critical phase of the pandemic. The erroneous results likely contributed to premature resumption of in-person school classes and easing of restrictions on social contact, favoring the resurgence of COVID-19. In Manaus, the largest city in the Amazon region, the COVID-19 pandemic did not end in 2020 of its own accord, but rather rebounded in a disastrous second wave of the disease.

The foundry industry involves pouring liquid metal or metal alloy into molds to create parts of specific shapes and measurements. The Tecumseh foundry is a notable player in this field, producing roughly 42,000 tons of metal each year. However, like many foundries in Brazil, Tecumseh faces challenges in accurately converting its productive volume into the number of pieces produced. Without a reliable and effective system for this conversion, the only option is manual counting. To address this issue, the group sought to estimate the number of pieces to be sampled while minimizing errors.

In this paper, we propose a new cure rate frailty regression model based on a two-parameter weighted Lindley distribution. The weighted Lindley distribution has attractive properties such as flexibility on its probability density function, Laplace transform function on closed-form, among others. An advantage of proposed model is the possibility to jointly model the heterogeneity among patients by their frailties and the presence of a cured fraction of them. To make the model parameters identifiable, we consider a reparameterized version of the weighted Lindley distribution with unit mean as frailty distribution. The proposed model is very flexible in sense that has some traditional cure rate models as special cases. The statistical inference for the model’s parameters is discussed in detail using the maximum likelihood estimation under random right-censoring. Further, we present a Monte Carlo simulation study to verify the maximum likelihood estimators’ behavior assuming different sample sizes and censoring proportions. Finally, the new model describes the lifetime of 22,148 patients with stomach cancer, obtained from the Fundação Oncocentro de São Paulo, Brazil.

Extreme-value distributions are important when modeling weather events, such as temperature and rainfall. These distributions are also important for modeling air pollution events. Particularly, the extreme-value Birnbaum-Saunders regression is a helpful tool in the modeling of extreme events. However, this model is implemented by adding covariates to the location parameter. Given the importance of quantile regression to estimate the effects of covariates along the wide spectrum of a response variable, we introduce a quantile extreme-value Birnbaum-Saunders distribution and its corresponding quantile regression model. We implement a likelihood-based approach for parameter estimation and consider two types of statistical residuals. A Monte Carlo simulation is performed to assess the behavior of the estimation method and the empirical distribution of the residuals. We illustrate the introduced methodology with unpublished real air pollution data.

The city of Manaus (the capital of Brazil's state of Amazonas) has become a key location for understanding the dynamics of the global pandemic of COVID-19. Different groups of scientists have foreseen different scenarios, such as the second wave or that Manaus could escape such a wave by having reached herd immunity. Here we test five hypotheses that explain the second wave of COVID-19 in Manaus: 1) The greater transmissibility of the Amazonian (gamma or P.1) variant is responsible for the second wave; 2) SARS-CoV-2 infection levels during the first wave were overestimated by those foreseeing herd immunity, and the population remained below this threshold when the second wave began at the beginning of December 2020; 3) Antibodies acquired from infection by one lineage do not confer immunity against other lineages; 4) Loss of immunity has generated a feedback phenomenon among infected people, which could generate future waves, and 5) A combination of the foregoing hypotheses. We also evaluated the possibility of a third wave in Manaus despite advances in vaccination, the new wave being due to the introduction of the delta variant in the region and the loss of immunity from natural contact with the virus. We developed a multi-strain SEIRS (Susceptible-Exposed-Infected-Removed-Susceptible) model and fed it with data for Manaus on mobility, COVID-19 hospitalizations, numbers of cases and deaths. Our model contemplated the current vaccination rates for all vaccines applied in Manaus and the individual protection rates already known for each vaccine. Our results indicate that the SARS-CoV-2 gamma (P.1) strain that originated in the Amazon region is not the cause of the second wave of COVID-19 in Manaus, but rather this strain originated during the second wave and became predominant in January 2021. Our multi-strain SEIRS model indicates that neither the doubled transmission rate of the gamma variant nor the loss of immunity alone is sufficient to explain the sudden rise of hospitalizations in late December 2020. Our results also indicate that the most plausible explanation for the current second wave is a SARS-CoV-2 infection level at around 50% of the population in early December 2020, together with loss of population immunity and early relaxation of restrictive measures. The most-plausible model indicates that contact with one strain does not provide protection against other strains and that the gamma variant has a transmissibility rate twice that of the original SARS-CoV-2 strain. Our model also shows that, despite the advance of vaccination, and even if future vaccination advances at a steady pace, the introduction of the delta variant or other new variants could cause a new wave of COVID-19.

We present a proposal to deal with the non-normality issue in the context of regression models with measurement errors when both the response and the explanatory variable are observed with error. We extend the normal model by jointly modelling the unobserved covariate and the random errors by a finite mixture of scale mixture of skew-normal distributions. This approach allows us to model data with great flexibility, accommodating skewness, heavy tails, and multi-modality. The main virtue of considering measurement error models under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows easy implementation of inference. In order to illustrate the usefulness of the proposed method some simulation studies are presented and a real dataset (Systemic lupus erythematosus) is analyzed.

Background: Manaus, the capital of Brazil’s state of Amazonas, became the center of the world's attention when the health system collapsed causing thousands of deaths after politicians orchestrated an attempt to induce herd immunity in the population through natural contact with SARS- CoV-2. Three waves of COVID-19 hit Manaus causing more than 9600 deaths in this city of 2.2 million. Given that local data have shown that natural contact with the virus does not provide lasting immunity, it is relevant to estimate when the population could reach herd immunity through vaccination so that severe forms of COVID 19 can be eradicated.Methods: We estimated the time when herd immunity via vaccination would be reached in Manaus using the same SEIRS model (Susceptible – Exposed – Infected – Removed- Susceptible) that was able to predict the second and third waves of COVID-19 in Manaus (four and twelve months in advance, respectively).Findings: Our results point to a new wave of COVID-19 in Manaus caused by the Omicron BA.2 variant and indicate that herd immunity will only be achieved when 90 to 95% of the entire population, including children, is vaccinated against COVID-19.Interpretation: Our results support the need for at least an annual or biannual booster dose of vaccine for protection against new variants, thus eradicating the severe forms of COVID-19. Our results indicate that the pandemic is far from over, given the current level of vaccination, and do not support the current government’s decision to relax requirements for the use of masks.Funding: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) and Fundação de Amparo à Pesquisa do Estado do Amazonas (FAPEAM). LF thanks PROEX and POSGRAD 2020- Biologia (Ecologia) Edital: Resolução N. 006/2020.Declaration of Interest: Authors declare no competing interests.

In this article, we discuss an extension of the classical negative binomial cure rate model with piecewise exponential distribution of the time to event for concurrent causes, which enables the modeling of monotonic and non‐monotonic hazard functions (ie, the shape of the hazard function is not assumed as in traditional parametric models). This approach produces a flexible cure rate model, depending on the choice of time partition. We discuss local influence on this negative binomial power piecewise exponential model. We report on Monte Carlo simulation studies and application of the model to real melanoma and leukemia datasets.