Efficient estimation in a regression model with missing responses |
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| Institution: | 1. Centre for Green Process Engineering, School of Engineering, London South Bank University, 103 Borough Road, London SE1 0AA, UK;2. Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK;3. WestCHEM, Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow G1 1XL, UK;1. University of North Carolina Wilmington, United States;2. The University of York, United Kingdom;1. Institute of Biological Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia;2. Department of Chemical and Environmental Engineering, Faculty of Engineering, University of Nottingham Malaysia Campus, Jalan Broga, Semenyih 43500, Selangor Darul Ehsan, Malaysia;3. Manufacturing and Industrial Processes Division, Faculty of Engineering, Centre for Food and Bioproduct Processing, University of Nottingham Malaysia Campus, Jalan Broga, Semenyih 43500, Selangor Darul Ehsan, Malaysia;4. Nanotechnology & Catalysis Research Centre (NANOCAT), University of Malaya, 50603 Kuala Lumpur, Malaysia;5. University Center for Bioscience and Biotechnology, National Cheng Kung University, Tainan 701, Taiwan;6. Department of Chemical Engineering, National Cheng Kung University, Tainan 701, Taiwan;7. Research Center for Energy Technology and Strategy, National Cheng Kung University, Tainan 701, Taiwan;1. Laboratoire de Mathématiques Appliquées de Compiègne-L.M.A.C., Université de Technologie de Compiègne, B.P. 529, 60205 Compiègne Cedex, France;2. L.S.T.A., Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France;1. Department of Statistics and Biostatistics, Rutgers University, Piscataway, NJ 08854, USA;2. Department of Biostatistics, Yale University, School of Public Health, New Haven, CT 06511, USA;3. Independent Consultant, Sudbury, MA 01776, USA |
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| Abstract: | This article examines methods to efficiently estimate the mean response in a linear model with an unknown error distribution under the assumption that the responses are missing at random. We show how the asymptotic variance is affected by the estimator of the regression parameter, and by the imputation method. To estimate the regression parameter, the ordinary least squares is efficient only if the error distribution happens to be normal. If the errors are not normal, then we propose a one step improvement estimator or a maximum empirical likelihood estimator to efficiently estimate the parameter.To investigate the imputation’s impact on the estimation of the mean response, we compare the listwise deletion method and the propensity score method (which do not use imputation at all), and two imputation methods. We demonstrate that listwise deletion and the propensity score method are inefficient. Partial imputation, where only the missing responses are imputed, is compared to full imputation, where both missing and non-missing responses are imputed. Our results reveal that, in general, full imputation is better than partial imputation. However, when the regression parameter is estimated very poorly, the partial imputation will outperform full imputation. The efficient estimator for the mean response is the full imputation estimator that utilizes an efficient estimator of the parameter. |
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| Keywords: | Efficiency Missing at random Full imputation |
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