Maximum Likelihood Imputation

Maximum likelihood (ML) estimation is widely used in statistics. The h-likelihood has been proposed as an extension of Fisher’s likelihood to statistical models in-cluding unobserved latent variables of recent interest. Its advantage is that the joint maximization gives ML estimators (MLEs) of both fixed and random parameters with their standard error estimates. However, the current h-likelihood approach does not allow MLEs of variance components as Henderson’s joint likelihood does not in linear mixed models. In this paper, we show how to form the h-likelihood in order to facilitate joint maximization for MLEs of whole parameters. We also show the role of the Jacobian term which allows MLEs in the presence of unobserved latent variables. To obtain MLEs for fixed parameters, intractable integration is not necessary. As an illustration, we show one-shot ML imputation for missing data by treating them as realized but unobserved random parameters. We show that the h-likelihood bypasses the expectation step in the expectation-maximization (EM) algorithm and allows single ML imputation instead of multiple imputations. We also discuss the difference in predictions in random effects and missing data.

[1]  Youngjo Lee,et al.  Enhanced Laplace Approximation , 2022, 2207.09871.

[2]  Gwangsun Kim,et al.  Properties of h‐Likelihood Estimators in Clustered Data , 2019, International Statistical Review.

[3]  Youngjo Lee,et al.  H‐likelihood Predictive Intervals for Unobservables , 2016 .

[4]  Anders Nielsen,et al.  TMB: Automatic Differentiation and Laplace Approximation , 2015, 1509.00660.

[5]  Shu Yang,et al.  A note on multiple imputation for method of moments estimation , 2015, 1508.06977.

[6]  Jae Kwang Kim,et al.  Statistical Methods for Handling Incomplete Data , 2013 .

[7]  Jae Kwang Kim.  Parametric fractional imputation for missing data analysis , 2011 .

[8]  J. Nelder,et al.  Rejoinder: Likelihood Inference for Models with Unobservables Another View , 2009, 1010.0814.

[9]  Xiao-Li Meng,et al.  Decoding the H-likelihood , 2009, 1010.0810.

[10]  M. Kenward,et al.  Every missingness not at random model has a missingness at random counterpart with equal fit , 2008 .

[11]  J. Nelder,et al.  Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood , 2006 .

[12]  John A. Nelder,et al.  Fitting via alternative random-effect models , 2006, Stat. Comput..

[13]  Nicole A. Lazar,et al.  Statistical Analysis With Missing Data , 2003, Technometrics.

[14]  J. Nelder,et al.  Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions , 2001 .

[15]  J. Robins,et al.  Inference for imputation estimators , 2000 .

[16]  S. Raudenbush,et al.  Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation , 2000 .

[17]  James M. Robins,et al.  Large-sample theory for parametric multiple imputation procedures , 1998 .

[18]  N. Breslow,et al.  Bias correction in generalised linear mixed models with a single component of dispersion , 1995 .

[19]  Xiao-Li Meng,et al.  Multiple-Imputation Inferences with Uncongenial Sources of Input , 1994 .

[20]  R. Butler.  Predictive Likelihood Inference with Applications , 1986 .

[21]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[22]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[23]  D. Rubin.  INFERENCE AND MISSING DATA , 1975 .

[24]  Lena Osterhagen,et al.  Multiple Imputation For Nonresponse In Surveys , 2016 .

[25]  Youngjo Lee,et al.  Frequentist Inference on Random Effects Based on Summarizability , 2015 .

[26]  J. F. Bjørnstad.  ----On the Generalization of the Likelihood Function and the Likelihood Principle , 2008 .

[27]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[28]  J. Neyman,et al.  Consistent Estimates Based on Partially Consistent Observations , 1948 .

[29]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .