Estimation for zero-inflated over-dispersed count data model with missing response
In this paper, we develop estimation procedure for the parameters of a zero‐inflated over‐dispersed/under‐dispersed count model in the presence of missing responses. In particular, we deal with a zero‐inflated extended negative binomial model in the presence of missing responses. A weighted expectat...
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| Abstract | In this paper, we develop estimation procedure for the parameters of a zero‐inflated over‐dispersed/under‐dispersed count model in the presence of missing responses. In particular, we deal with a zero‐inflated extended negative binomial model in the presence of missing responses. A weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved. Some simulations are conducted to study the properties of the estimators. Robustness of the procedure is shown when count data follow other over‐dispersed models, such as the log‐normal mixture of the Poisson distribution or even from a zero‐inflated Poisson model. An illustrative example and a discussion leading to some conclusions are given. Copyright © 2016 John Wiley & Sons, Ltd. |
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| AbstractList | In this paper, we develop estimation procedure for the parameters of a zero-inflated over-dispersed/under-dispersed count model in the presence of missing responses. In particular, we deal with a zero-inflated extended negative binomial model in the presence of missing responses. A weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved. Some simulations are conducted to study the properties of the estimators. Robustness of the procedure is shown when count data follow other over-dispersed models, such as the log-normal mixture of the Poisson distribution or even from a zero-inflated Poisson model. An illustrative example and a discussion leading to some conclusions are given. Copyright © 2016 John Wiley & Sons, Ltd. In this paper, we develop estimation procedure for the parameters of a zero-inflated over-dispersed/under-dispersed count model in the presence of missing responses. In particular, we deal with a zero-inflated extended negative binomial model in the presence of missing responses. A weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved. Some simulations are conducted to study the properties of the estimators. Robustness of the procedure is shown when count data follow other over-dispersed models, such as the log-normal mixture of the Poisson distribution or even from a zero-inflated Poisson model. An illustrative example and a discussion leading to some conclusions are given. |
| Author | Mian, Rajibul Paul, Sudhir |
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| References | Ibrahim JG, Chen MH, Lipsitz SR. Monte Carlo EM for missing covariates in parametric regression models. Biometrics 1999; 55:591-596. Paul SR, Plackett RL. Inference sensitivity for Poisson mixtures. Biometrika 1978; 65:591-602. Ibrahim JG, Chen MH, Lipsitz SR. Missing responses in generalized linear mixed models when the missing data mechanism is nonignorable. Biometrika 2001; 88:551-556. Paul SR, Banergee T. Analysis of two-way layout of count data involving multiple counts in each cell. Journal of the American Statistical Association 1998; 93:1419-1429. Anderson TW, Taylor JB. Strong consistency of least squares estimates in normal linear regression. The Annals of Statistics 1976; 4:788-790. Prentice RL. Binary regression using an extended beta-binomial distribution, with discussion of correlation induced by covariate measurement errors. Journal of the American Statistical Association 1986; 81:321-327. Raftery AE, Madigan D, Hoeting JA. Bayesian model averaging for linear regression models. Journal of the American Statistical Association 1997; 92:179-191. Rubin DB. Formalizing subjective notions about the effect of nonrespondents in sample surveys. Journal of the American Statistical Association 1977; 72:538-543. Sahu SK, Roberts GO. On convergence of the EM algorithm and the Gibbs sampler. Statistics and Computing 1999; 9:55-64. Lipsitz SR, Ibrahim JG. A conditional model for incomplete covariates in parametric regression models. Biometrika 1996; 83:916-922. Ibrahim JG, Chen MH, Lipsitz SR, Herring AH. Missing-data methods for generalized linear models. Journal of the American Statistical Association 2005; 100:332-346. Casella G, George EL. Explaining the Gibbs sampler. The American Statistician 1992; 46:167-174. Maiti T, Pradhan V. Bias reduction and a solution for separation of logistic regression with missing covariates. Biometrics 2009; 65:1262-1269. Ibrahim JG, Lipsitz SR. Parameter estimation from incomplete data in binomial regression when the missing data mechanism is nonignorable. Biometrics 1996; 52:1071-1078. Lawless JF. Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics 1987; 15:209-225. Cameron AC, Trivedi PK. Regression Analysis of Count Data. Cambridge University Press: New York, 2013. Geweke J. Inference in the inequality constrained normal linear regression model. Journal of Applied Econometrics 1986; 1:117-141. Deng D, Paul SR. Score tests for zero inflation and over dispersion in generalized linear models. Statistica Sinica 2005; 15:257-276. Efron B, Hinkley DV. Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika 1978; 65:457-487. Sinha S, Maiti T. Analysis of matched case-control data in presence of nonignorable missing exposure. Biometrics 2007; 64:106-114. Ibrahim JG. Incomplete data in generalized linear model. Journal of the American Statistical Association 1990; 85:765-769. Dempster AP, Larid NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 1977; 39:1-38. Jiang X, Paul SR. Analysis of covariance of zero-inflated paired count data using a zero-inflated bivariate Poisson regression model. Calcutta Statistical Bulletin (Special Volume) 2009; 61:113-124. Little RJA, Rubin DB. (1987, 2002, 2014). Statistical Analysis with Missing Data (2nd ed). Wiley: New York. Zhang CH, Huang J. The sparsity and bias of the Lasso selection in high-dimensional linear regression. The Annals of Statistics 2008; 36:1567-1594. Bohning D, Dietz E, Schlattmann P, Mendonca L, Kirchner U. The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology. Journal of the Royal Statistical Society A 1999; 162:195-209. Chen J, Hubbard S, Rubin Y. Estimating the hydraulic conductivity at the South Oyster site from geophysical tomographic data using Bayesian techniques based on the normal linear regression model. Water Resources Research 2001; 37:1603-1613. Deng D, Paul SR. Score tests for zero inflation in generalized linear models. The Canadian Journal of Statistics 2000; 87:451-457. Barnwal RK, Paul SR. Analysis of one-way layout of count data with negative binomial variation. Biometrika 1988; 75:215-22. Piegorsch WW. Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics 1990; 46:863-867. Kelly BC. Some aspects of measurement error in linear regression of astronomical data. The Astrophysical Journal 2007; 665:1489-1506. Dean CB. Testing for overdispersion in Poisson and binomial regression models. Journal of the American Statistical Association 1992; 87:451-457. 2007; 665 2009; 65 2009; 61 2000; 87 2008; 36 1999; 162 1996; 52 1988; 75 2001; 88 1976; 4 1987; 15 1999; 9 1990; 85 1986; 1 1986; 81 1990; 46 1997; 92 2005; 100 1978; 65 1977; 39 1996; 83 1977; 72 1999; 55 2001; 37 1992; 46 1998; 93 2013 2005; 15 2007; 64 1992; 87 |
| References_xml | – reference: Barnwal RK, Paul SR. Analysis of one-way layout of count data with negative binomial variation. Biometrika 1988; 75:215-22. – reference: Cameron AC, Trivedi PK. Regression Analysis of Count Data. Cambridge University Press: New York, 2013. – reference: Zhang CH, Huang J. The sparsity and bias of the Lasso selection in high-dimensional linear regression. The Annals of Statistics 2008; 36:1567-1594. – reference: Rubin DB. Formalizing subjective notions about the effect of nonrespondents in sample surveys. Journal of the American Statistical Association 1977; 72:538-543. – reference: Prentice RL. Binary regression using an extended beta-binomial distribution, with discussion of correlation induced by covariate measurement errors. Journal of the American Statistical Association 1986; 81:321-327. – reference: Sahu SK, Roberts GO. On convergence of the EM algorithm and the Gibbs sampler. Statistics and Computing 1999; 9:55-64. – reference: Bohning D, Dietz E, Schlattmann P, Mendonca L, Kirchner U. The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology. Journal of the Royal Statistical Society A 1999; 162:195-209. – reference: Ibrahim JG. Incomplete data in generalized linear model. Journal of the American Statistical Association 1990; 85:765-769. – reference: Lawless JF. Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics 1987; 15:209-225. – reference: Anderson TW, Taylor JB. Strong consistency of least squares estimates in normal linear regression. The Annals of Statistics 1976; 4:788-790. – reference: Ibrahim JG, Chen MH, Lipsitz SR, Herring AH. Missing-data methods for generalized linear models. Journal of the American Statistical Association 2005; 100:332-346. – reference: Deng D, Paul SR. Score tests for zero inflation in generalized linear models. The Canadian Journal of Statistics 2000; 87:451-457. – reference: Deng D, Paul SR. Score tests for zero inflation and over dispersion in generalized linear models. Statistica Sinica 2005; 15:257-276. – reference: Dempster AP, Larid NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 1977; 39:1-38. – reference: Maiti T, Pradhan V. Bias reduction and a solution for separation of logistic regression with missing covariates. Biometrics 2009; 65:1262-1269. – reference: Paul SR, Banergee T. Analysis of two-way layout of count data involving multiple counts in each cell. Journal of the American Statistical Association 1998; 93:1419-1429. – reference: Geweke J. Inference in the inequality constrained normal linear regression model. Journal of Applied Econometrics 1986; 1:117-141. – reference: Paul SR, Plackett RL. Inference sensitivity for Poisson mixtures. Biometrika 1978; 65:591-602. – reference: Little RJA, Rubin DB. (1987, 2002, 2014). Statistical Analysis with Missing Data (2nd ed). Wiley: New York. – reference: Jiang X, Paul SR. Analysis of covariance of zero-inflated paired count data using a zero-inflated bivariate Poisson regression model. Calcutta Statistical Bulletin (Special Volume) 2009; 61:113-124. – reference: Dean CB. Testing for overdispersion in Poisson and binomial regression models. Journal of the American Statistical Association 1992; 87:451-457. – reference: Ibrahim JG, Chen MH, Lipsitz SR. Monte Carlo EM for missing covariates in parametric regression models. Biometrics 1999; 55:591-596. – reference: Ibrahim JG, Chen MH, Lipsitz SR. Missing responses in generalized linear mixed models when the missing data mechanism is nonignorable. Biometrika 2001; 88:551-556. – reference: Kelly BC. Some aspects of measurement error in linear regression of astronomical data. The Astrophysical Journal 2007; 665:1489-1506. – reference: Casella G, George EL. Explaining the Gibbs sampler. The American Statistician 1992; 46:167-174. – reference: Ibrahim JG, Lipsitz SR. Parameter estimation from incomplete data in binomial regression when the missing data mechanism is nonignorable. Biometrics 1996; 52:1071-1078. – reference: Sinha S, Maiti T. Analysis of matched case-control data in presence of nonignorable missing exposure. Biometrics 2007; 64:106-114. – reference: Raftery AE, Madigan D, Hoeting JA. Bayesian model averaging for linear regression models. Journal of the American Statistical Association 1997; 92:179-191. – reference: Chen J, Hubbard S, Rubin Y. Estimating the hydraulic conductivity at the South Oyster site from geophysical tomographic data using Bayesian techniques based on the normal linear regression model. Water Resources Research 2001; 37:1603-1613. – reference: Efron B, Hinkley DV. Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika 1978; 65:457-487. – reference: Piegorsch WW. Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics 1990; 46:863-867. – reference: Lipsitz SR, Ibrahim JG. A conditional model for incomplete covariates in parametric regression models. Biometrika 1996; 83:916-922. – volume: 65 start-page: 591 year: 1978 end-page: 602 article-title: Inference sensitivity for Poisson mixtures publication-title: Biometrika – volume: 87 start-page: 451 year: 1992 end-page: 457 article-title: Testing for overdispersion in Poisson and binomial regression models publication-title: Journal of the American Statistical Association – volume: 65 start-page: 457 year: 1978 end-page: 487 article-title: Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information publication-title: Biometrika – volume: 162 start-page: 195 year: 1999 end-page: 209 article-title: The zero‐inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology publication-title: Journal of the Royal Statistical Society A – volume: 52 start-page: 1071 year: 1996 end-page: 1078 article-title: Parameter estimation from incomplete data in binomial regression when the missing data mechanism is nonignorable publication-title: Biometrics – volume: 4 start-page: 788 year: 1976 end-page: 790 article-title: Strong consistency of least squares estimates in normal linear regression publication-title: The Annals of Statistics – volume: 100 start-page: 332 year: 2005 end-page: 346 article-title: Missing‐data methods for generalized linear models publication-title: Journal of the American Statistical Association – volume: 46 start-page: 167 year: 1992 end-page: 174 article-title: Explaining the Gibbs sampler publication-title: The American Statistician – volume: 36 start-page: 1567 year: 2008 end-page: 1594 article-title: The sparsity and bias of the Lasso selection in high‐dimensional linear regression publication-title: The Annals of Statistics – volume: 65 start-page: 1262 year: 2009 end-page: 1269 article-title: Bias reduction and a solution for separation of logistic regression with missing covariates publication-title: Biometrics – volume: 9 start-page: 55 year: 1999 end-page: 64 article-title: On convergence of the EM algorithm and the Gibbs sampler publication-title: Statistics and Computing – volume: 39 start-page: 1 year: 1977 end-page: 38 article-title: Maximum likelihood from incomplete data via the EM algorithm publication-title: Journal of the Royal Statistical Society B – volume: 1 start-page: 117 year: 1986 end-page: 141 article-title: Inference in the inequality constrained normal linear regression model publication-title: Journal of Applied Econometrics – volume: 85 start-page: 765 year: 1990 end-page: 769 article-title: Incomplete data in generalized linear model publication-title: Journal of the American Statistical Association – volume: 87 start-page: 451 year: 2000 end-page: 457 article-title: Score tests for zero inflation in generalized linear models publication-title: The Canadian Journal of Statistics – volume: 15 start-page: 257 year: 2005 end-page: 276 article-title: Score tests for zero inflation and over dispersion in generalized linear models publication-title: Statistica Sinica – volume: 64 start-page: 106 year: 2007 end-page: 114 article-title: Analysis of matched case–control data in presence of nonignorable missing exposure publication-title: Biometrics – volume: 81 start-page: 321 year: 1986 end-page: 327 article-title: Binary regression using an extended beta‐binomial distribution, with discussion of correlation induced by covariate measurement errors publication-title: Journal of the American Statistical Association – volume: 75 start-page: 215 year: 1988 end-page: 22 article-title: Analysis of one‐way layout of count data with negative binomial variation publication-title: Biometrika – volume: 61 start-page: 113 year: 2009 end-page: 124 article-title: Analysis of covariance of zero‐inflated paired count data using a zero‐inflated bivariate Poisson regression model publication-title: Calcutta Statistical Bulletin (Special Volume) – volume: 88 start-page: 551 year: 2001 end-page: 556 article-title: Missing responses in generalized linear mixed models when the missing data mechanism is nonignorable publication-title: Biometrika – volume: 93 start-page: 1419 year: 1998 end-page: 1429 article-title: Analysis of two‐way layout of count data involving multiple counts in each cell publication-title: Journal of the American Statistical Association – volume: 15 start-page: 209 year: 1987 end-page: 225 article-title: Negative binomial and mixed Poisson regression publication-title: The Canadian Journal of Statistics – volume: 92 start-page: 179 year: 1997 end-page: 191 article-title: Bayesian model averaging for linear regression models publication-title: Journal of the American Statistical Association – volume: 37 start-page: 1603 year: 2001 end-page: 1613 article-title: Estimating the hydraulic conductivity at the South Oyster site from geophysical tomographic data using Bayesian techniques based on the normal linear regression model publication-title: Water Resources Research – volume: 83 start-page: 916 year: 1996 end-page: 922 article-title: A conditional model for incomplete covariates in parametric regression models publication-title: Biometrika – volume: 72 start-page: 538 year: 1977 end-page: 543 article-title: Formalizing subjective notions about the effect of nonrespondents in sample surveys publication-title: Journal of the American Statistical Association – volume: 46 start-page: 863 year: 1990 end-page: 867 article-title: Maximum likelihood estimation for the negative binomial dispersion parameter publication-title: Biometrics – volume: 55 start-page: 591 year: 1999 end-page: 596 article-title: Monte Carlo EM for missing covariates in parametric regression models publication-title: Biometrics – volume: 665 start-page: 1489 year: 2007 end-page: 1506 article-title: Some aspects of measurement error in linear regression of astronomical data publication-title: The Astrophysical Journal – year: 2013 |
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| SubjectTerms | Algorithms count data EM algorithm Estimating techniques Humans Likelihood Functions Medical statistics Models, Statistical over dispersion Poisson Distribution regression model Simulation zero inflation |
| Title | Estimation for zero-inflated over-dispersed count data model with missing response |
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