Negative binomial regression model (NBR) is a popular approach for modelling over-dispersed count data with covariates. Several parameterizations have been performed for NBR, and the two well-known models, negative binomial-1 regression model (NBR-1) and negative binomial-2 regression model (NBR-2), have been applied. Another parameterization of NBR is negative binomial-P regression model (NBR-P), which has an additional parameter and the ability to nest both NBR-1 and NBR-2. This paper introduces several forms of bivariate negative binomial regression model (BNBR) which can be fitted to bivariate count data with covariates. The main advantages of having several forms of BNBR are that they are nested and allow likelihood ratio test to be performed for choosing the best model, they have flexible forms of mean-variance relationship, they can be fitted to bivariate count data with positive, zero or negative correlations, and they allow over-dispersion of the two dependent variables. Applications of several forms of BNBR have been illustrated on two sets of count data; Australian health care and Malaysian motor insurance.

Background & Aim: Mixed Poisson and mixed negative binomial distributions have been considered as alternatives for fitting count data with over-dispersion. This study introduces a new discrete distribution which is a weighted version of Poisson-Lindley distribution. Methods & Materials: The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p. m. f., p. g. f. and simulation procedure of the new weighted distribution, namely weighted negative binomial-Poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameters estimation is also presented. Results: The WNBPL distribution is fitted to several datasets, related to genetics and compared with the Poison distribution. The goodness of fit test shows that the WNBPL can be a useful tool for modeling genetics datasets. Conclusion: This paper introduces a new weighted Poisson-Lindley distribution which is obtained using negative binomial weight function and can be used for fitting over-dispersed count data. The p. m. f., p. g. f. and simulation procedure are provided for the new weighted distribution, namely the weighted negative binomial-Poisson Lindley (WNBPL) to better inform parents from possible time of occurrence reflux and treatment strategies.

Background & Aim: Statistical modeling is one of the most suitable methods for analyzing the relationship between health and medical issues. In the situation of analysis of zero-inflated data, there are different methods. In this study, the models Poisson, Poisson gamma, and Poisson lognormal regression were compared.Methods & Materials: This cross-sectional study was conducted to determine the influential factors on decay-missing-filled (DMF) index by the three mentioned models using the data of 808 first-grade children of the primary school in Kerman, Iran. The command PROC NLMIXED in SAS software was applied for fitting the models on data. For comparing the models, we applied the Akaike’s criterion (AIC), mean square error (MSE) criterion and confidence interval (CI).Results: The AIC and CI showed that the Poisson lognormal model was better than the others due to a level of significance. The variables of the students’ place of living, mothers’ jobs, fathers’ jobs, the region, sex, optic problems, and behavioral problems had a significant effect on DMF index.Conclusion: Poisson lognormal was better than the other models in dental health data.

so many natural phenomena of determining relationship and the effect of input variables on response variable in statistical studies may be different from the suggested model that the researcher selects for his study due to the occupant exists in the structure of data. It may be so influential on different distributions considered for response variables. The optimal properties of estimators evaluated and studied for two statistical variables considered for response variable and input variables in the suggested model. It has been simulated for study and real data has been also investigated. The results confirmed the superiority of a model which is close to the structure of the data.