A Parametric Regression Model using Power Chris-Jerry Distribution with Application to Censored Data

Conclusion

    A great deal of effort has been made to investigate regression modeling using Power Chris-Jerry distribution in this article. The log-transformed Power Chris-Jerry regression model was developed and its survival function, and other functional forms were studied. The model parameters were estimated using maximum likelihood method under censored sample consequent upon the nature of the data deployed. The log-transformed Power Chris-Jerry regression model (LPCJ) designed in this paper was compared with those of the Log-Power Lomax (LPLO) distribution, Log-Power Zeghdoudi (LPZe) distribution, Log-Power Suja (LPSuj) distribution, Log-Exponentiated Power Ishita (LEPI), Log-Power Ishita (LPI), Log-Exponentiated Weibull (LEWe), Log-Power Rama (LPR), Log-Exponentiated Frechet (LEF), Log-Power Lindley (LPLi), Log-Exponentiated Power Akash (LEPA), Log-Exponentiated Power Lindley (LEPLi) distributions. The LPCJ outperformed the rest of the models and proves to fit better any skewed data.

Acknowledgment

The authors are very grateful to the anonymous reviewers and the editors for their comments which improved the quality of this article.

Conflict of interest

The author declare that there is  no conflict of interest

  A good number of authors have investigated parametric regression model from the probability scenario using various approaches. Cordeiro, Biazatti, and Santana (2023) introduced a four-parameter Weibull extended Weibull (WEW) distribution that presents greater flexibility and can model data with bathtub-shape and unimodal failure rate taking the Extended Weibull PDF as baseline to form the new distribution. The new support for the new distribution is  with properties such as quantile function, kurtosis, skewness, moments were discussed. Estimation of the parameters was done using method of maximum likelihood. Carlo monte simulation study was carried out to show the new distribution WEW has consistent MLE’s with lower AIC, BIC and Global Deviance GD from generated data. Also regression model  was constructed on Log-Weilbull Extended Weibull (LWEW) distribution for  when X has WEW pdf where has LWEW Pdf, is the vector coefficients and is the vector covariates for the ith response which models the location parameter . Two data set was used to show the applicability and superiority of the proposed model over other existing ones compared.

Biazatti, Cordeiro, Rodrigues, Ortega and De Santana (2022) introduced Weibull-Beta Prime (WBP) distribution from Beta Prime BP distribution due to wide use of the BP and to provide better fit to complex real data. Some structural properties of the new distribution such as quantile function, linear representation and moment were obtained. Method of maximum likelihood estimation was used for parameter estimation. The simulation study carried out shows that all estimators improve as n increases. Furthermore, the WBP regression model was constructed for censored samples. Since censored samples are commonly considered as systematic component for the shape parameter . Considering systematic component   where  is the vector of covariates and  is the vector of unknown parameters. Real data set were used to show the importance and superiority of the proposed model when compared with some known competing models. Rodrigues, Ortega, Cordeiro and Vila (2022) proposed Odd Log-Logistic Weibull (OLLW) regression model for censored data to identify factors that increase the risk of death of hospitalized patients diagnosed with Covid’19.

The properties of the distribution; mode, stochastic representation, closure under changes of scale and of power, identifiability, tail behavior and moment were discussed. OLLW regression model was defined by two systematic component for  and for(i=1,..,n) follows  and  where   are vector length for unknown coefficients functionally independent, is the number of explanatory variables related to the jth parameter,  are the linear predictors and  are observation on  and  known repressors. From the data set, older age, asthma, diabetes obesity and chronic neurological diseases were identified as risk factors associated with death of diagnosed Covid’19 patients in the city of Campinas, Brazil.

Segovia, Gomez and Gallardo (2021) introduced an Extension of Power Maxwell (EPW) distribution called Exponentiated Power Maxwell and proposed Re-parameterized Exponentiated Power Maxwell (REPM) distribution. The properties of the new distribution which includes, moment, quantile function, median were discussed. They also introduced regression framework for applying the model to any quantile of the distribution, where quantile of such variables is related to a set of covariates say  through the logarithmic link  i=1,2,…n where  are the regression coefficient. The maximum likelihood estimation for REPM regression model under classical approach was discussed and simulation study to assess the performance of the ML estimators for REPM regression model was conducted. Two real data set used to show the applicability of the model and was compared with other proposed in literature shows the value of AIC and K-S of REPM lower than the compared ones.

Wongrin and Bodhisuwan (2017) used the generalized linear model to create a new linear regression model called Generalized Poisson Linear model which was based on a Generalized Poisson-Lindley (GPLi) distribution for seven parameters. The conditional distribution  for  , the GPL mean, the probability mass function  were discussed. the maximum likelihood estimation for the model parameter estimation was derived. The applicability of the new model was seen in the analyzing a real dataset on the corona virus-infected patients.

Reis (2023) proposed a new distribution called Pezeta distribution that has support on the interval . It was obtained after transforming the random variable  with exponential distribution . Its properties such as; mode, moment, quantile function, random number generation, proof of exponential family, MLE and MLE bias correction when n is small were discussed. Also, regression model was introduced for the dependent variable with support at . The model has a regression structure on the median of the distribution  where is the k-vector of unknown parameters is the vector of k explanatory variables  which are assumed fixed and known  is the linear predictor. Simulation study was conducted to show the performance of the MLE’s for the proposed regression model  Pezeta regression model was compared with the unit Lindley UL regression model using a dataset. Discriminate between the two regression models was assessed using AIC, BIC and Hannan Information Criterion (HQIC). The new model presents the smallest value of these statistics which shows its superiority over the compared one.

Badmus, Akinyemi and Onyeka-Ubaka (2021) introduced a location-scale regression model based on the logarithm of an extended Raleigh Lomax distribution which has the ability to model survival data than classical regression model called Log-Beta Rayleigh Lomax (LBRL) regression model. They presented two important classes of the distribution, firstly Beta Rayleigh Lomax (BRL)distribution using Logit Beta function. Secondly, Log –Beta Rayleigh Lomax LBRL distribution. Hazard function, reliability function, moment, moment generating function linear combination and other properties of the new distribution were derived. Based on the LBRL distribution, a linear regression model linking the response variables explanatory variables  is defined as  where the random error has LBRL density function with the unknown parameters and the explanatory variable vector modeling the linear predictor . The linear predictor vector of LBRL regression model is written as  where  is the known model matrix. The MLE was used for parameter estimation. Applicability of the new model was shown using breast cancer dataset referring to time spent and explanatory variables age , occupation , martial status , event statues  and type of treatment for n=623 observations. Fitting the above dataset with proposed model and 5 other existing regression model and using model selection criteria AIC, BIC and CAIC the proposed model outperformed the compared ones.

Eliwa, Attun, Alhussian, Ahmed, Salah, Ahamed and El-Morshedy (2021) deployed the Odd Lindley-G family Oli-G to proposed a new generalization of Half Logistic HL distribution with only one parameter called Odd Lindley Half Logistic (OLiHL) distribution. The statistical properties of the new distribution such as raw and central moment, incomplete moment, moment generating functions, quantile function were discussed. the estimation method used are MLE, LS, Weighted Least Square and Cramer-Von Mises. Simulation study shows the relative performance of the used estimation methods. A log-location-scale regression model called log-OLiHL regression model was introduced based on the  transformation and a suitable re-parameterization on the baseline distribution OLiHL considering  where the response variable has the Log-OLiHL density, the covariates are linked to location of  with identity link function  where is the model matrix consists of the observation and the independent variables and is the unknown regression coefficients. Two datasets were considered to show the flexibility of the OLiHL distributions against the several one-parameter competitive model and it showed better modeling ability.

Nasiru, Abubakari, Chesneau (2022) proposed the Bounded Truncated Cauchy Power Exponential (BTCPE) distribution for modeling dataset on the unit interval

Relevant properties of the BTCPE distribution which includes the distribution of inequalities, quantile function, moment, moment generating function and order statistics were discussed. The bivariate extension of the new model was shown. The parameter estimation method used are; MLE, OLS, WLS, Cramer-Von Mises, Percentile estimation, Anderson-Darling method and maximum and minimum product spacing method. Simulation study was conducted to compare the estimation methods using bias, RMSE of the estimates. Using 3 dataset, application of the BTCPE distribution was illustrated and its performance was compared to other competitive distributions defined in the unit interval based on AIC, BIC criterion. They also define BTCPE quantile regression as  where is the vector of unknown parameter,  is the ith quantile parameter and are the known ith vector covariates. The log-likelihood for estimating the parameter of the BTCPE quantile regression was given. Monte Carlo simulation was carried out to examine the performance of the ML estimates of the parameter of the model using Absolute Bias and RMSEs. It shows that the regression parameter are consistent. 

Nwankwo, Nwankwo and Obulezi (2023) proposed a new three exponentiated Power Akash (EPA) distribution. The properties of the distribution such as moment, incomplete moment were discussed. Maximum likelihood estimation was used for the parameter estimation. Letting  where and defining and , the log-Exponentiated Power Akash (LEPA) density for y  R was derived and a parametric regression model for response variable and vector of explanatory variables constructed as  for i=1,2,…,n where , and is the vector of unknown regression coefficient and z is the random error, also likelihood of  was derived. Using a Covid’19 censored data, the applicability and performance of the new distribution when compared with other competitive ones in the literature were shown using Bayesian Information Criterion (BIC), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC) and  Hannan-Quinn Information Criterion (HQIC) measure criteria.

Methods:        Log-Transformed Regression from Power Chris-Jerry Distribution

Ezeilo, Umeh, Osuagwu and Onyekwere (2023) introduced the Power Chris-Jerry (PCJ) distribution with PDF given as

                                                                                                              (1)

and CDF

                                                                                         (2)                               

Where  and  are the shape and scale parameters respectively. Essentially, the goal here is to create a new reparameterized regression model using log-transformation of the PCJ distribution.         The baseline distribution being the one-parameter Chris-Jerry distribution proposed by Onyekwere and Obulezi (2022) and its extensions namely Oramulu et al. (2023a, 2023b), Chukwuma et al. (2024), Chinedu et al. (2023a, 2023b), Etaga et al. (2023a, 2023b, 2023c), Tolba et al. (2023), Musa et al. (2023a, 2023b), Anabike et al. (2023), Obulezi et al. (2023a, 2023b, 2023c, 2023d, 2023e), Oha et al. (2024), Onyekwere et al. (2022), Nwankwo B. C. et al. (2023), and Nwankwo, M. P. et al. (2023).     

Let  where  defined in equation (2.1). Assume and  , the log-PCJ density for  using  is

                                                                                 (3)

Where  and . If , the . Similarly, we deploy the same technique in equation (2.3) to derive the survival and density function of  which are

                                                                                                    (4)

                                                                    (5)

         Where

Using equation (2.5), we construct a parametric regression model for the response variable and a vector of explanatory variables  as

                  i = 1, 2, …, n                                                                                          (6) 

Where  ,  is the vector of unknown regression coefficients and z is the random error with density in equation (2.6), define the survival and density function of  as

                                                                                                    (7)

  and

                                                                                                              (8)

Where  and

2.1       Maximum Likelihood Estimation of Log-PCJ Parameters under Censored Sample

To estimate the parameters in equation (2.6) for right censored data, we defined and as the lifetime and non-information censoring time (assuming independence) and . Then, the log-likelihood function for  is

=                      (9)

Where F and C are the sets of uncensored and censored observations respectively and d is the number of failures. The MLE  of the unknown parameter vector can be obtained by maximizing equation (9).

3.0       Simulation Study

    The simulation conditions deployed by Ferreira and Cordeiro (2023) are used in this article due to the compatibility of the two distributions. For the LPCJ distribution under different scenarios, the accuracy of the MLEs is examined. For 1000 repetitions, the acceptance and rejection method is adopted to generate random samples of sizes n=50, 100, 300, and 600 from the LPCJ distribution. The Average estimates (AEs) of the parameters, Biases, and mean squared error (MSEs) are calculated. The algorithm for generating random samples uses the acceptance-rejection method. Note that LPCJ means Log-PCJ.

Table 1: Simulation Measures from the Log-PCJ Regression Model

The statistics in Table 1, indicate that the AEs converge to the true parameters and that the biases and MSEs tend to zero when n increases, which proves the consistency of the LPCJ estimators. Overall, the simulation results suggest that larger sample sizes and the appropriate choice of ξ are crucial for accurate parameter estimation of the LPCJ distribution.

Fig. 1: Empirical cdf and estimated cdf for                 Fig 2: Estimated PDF and histogram for generated samples using the scenario (9, 3.5)                                                                                  samples using the scenario (9, 3.5).

Figures 1 and 2 reveal the approximation of the acceptance-rejection method. The estimated PDF and CDF of the PCJ distribution are very close to the histogram and empirical CDF of the generated samples, indicating a good performance of the method.

4.0       Application to COVID-19 Censored Data

   The dataset comprises the lifetime (in days) of 322 individuals diagnosed with COVID-19 through RT-PCR screening in Campinas, Brazil. These data were previously studied by Ferreira and Cordeiro (2023), and Nwankwo, Nwankwo and Obulezi (2023). The response variable  represents the time elapsed from the onset of symptoms until death due to COVID-19 (failure). Ferreira and Cordeiro (2023), observed that about 66.45% of the observations are censored. The variables considered, for  include : censoring indicator; 0 for censored and 1 for observed lifetime;  : age (in years), and  : diabetes mellitus   yes,  no or uninformed. The suggested regression model for these COVID-19 data is written as

where  the PDF in equation (2.8).

The Power Lomax (PLO) distribution by Rady, El-Houssainy, Hassanein and Elhaddad (2016), Power Zeghdoudi (PZe) distribution (new), Power Suja (PSuj) distribution (new), exponentiated Power Ishita (EPI) by Ferreira and Cordeiro (2023), Power Ishita (PI) by Shukla and Shanker (2018), exponentiated Weibull (EWe) by Pal, Ali and Woo (2006), Power Rama (PR) by Abebe, Tesfay, Eyob and Shanker (2019), exponentiated Frechet (EF) by Nadarajah and Kotz (2003), Power Lindley (PLi) by Ghitany, Al-Mutairi, Balakrishnan and Al-Enezi (2013), exponentiated Power Akash (EPA) by Nwankwo, Nwankwo and Obulezi (2023), exponentiated Power Lindley (EPLi) by Ashour and Eltehiwy (2015) are used to compare with the proposed Power Chris-Jerry (PCJ) distribution. Note, that the log- of each distribution is derived following the procedure in section 2 to obtain LPLO, LPZe, LPSuj, LEPI, LPI, LEWe, LPR, LEF, LPLi, LEPA and LEPLi respectively. 

The result from Table 3 shows that the explanatory variables age and diabetes mellitus are significant at the 5% significance level. Note that in table 3,  is the exponentiated parameter for some of the fitted distributions. The negative signs of  and  mean that older individuals or those with diabetes tend to have shorter failure times. This result is in agreement with that obtained from earlier study by Ferreira and Cordeiro (2023. From Table 2, the LPCJ regression has the lowest criterion values hence confirming that the LPCJ model provides a better fit for the COVID-19 data than the competing regression model using Bayesian Information Criterion (BIC), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC) and  Hannan-Quinn Information Criterion (HQIC) measure criteria.

 Table 2: Metrics of Model Performance for the Log-transformed Regression Models using the COVID-19 Censored data

Table 3: Parameter Estimates of the Log-transformed Regression Models using COVID-19 censored data

            Fig 4.3: QQ plot                       Fig. 4.4: Histogram                 Fig. 4.5: Quantile residual plot

Figures 4.3, 4.4 and 4.5 are the non-parametric plots of the data set. The potential of the log-PCJ regression modeling skewed data is one interesting feature explored using the COVID-19 data. The properties of this distribution were derived and the log-transformation has been used to create a parametric regression model called the Power Chris-Jerry distribution regression model. The maximum likelihood estimation aided the estimation process for uncensored samples while the procedure for the estimation of the unknown parameters when data is censored was also shown. Essentially, the censored COVID-19 data set with the age of patients and diabetic mellitus index was deployed to justify the importance of the distribution. Furthermore, the distribution was fitted to the data on infant mortality rate (below age 5 years) reported for some countries by the World Health Organization in 2021. The distribution performs pretty well in both instances of application.