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Multiple Imputation to Estimate Hierarchical Models from Data Missing at Random: Latent Covariates, Random Coefficients, and Statistical Interactions

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Title: Multiple Imputation to Estimate Hierarchical Models from Data Missing at Random: Latent Covariates, Random Coefficients, and Statistical Interactions
Language: English
Authors: Yongyun Shin (ORCID 0000-0003-0654-2620), Stephen W. Raudenbush
Source: Grantee Submission. 2025.
Peer Reviewed: Y
Page Count: 52
Publication Date: 2025
Sponsoring Agency: Institute of Education Sciences (ED)
Contract Number: R305D210022
Document Type: Reports - Research
Descriptors: Hierarchical Linear Modeling, Maximum Likelihood Statistics, Sampling, Error of Measurement, Algorithms, Equations (Mathematics), Computation, Bayesian Statistics
DOI: 10.3102/10769986251385210
Abstract: Consider the conventional multilevel model Y=C[gamma]+Zu+e where [gamma] represents fixed effects and (u,e) are multivariate normal random effects. The continuous outcomes Y and covariates C are fully observed with a subset Z of C. The parameters are [theta]=([gamma],var(u),var(e)). Dempster, Rubin and Tsutakawa (1981) framed the estimation as a missing data problem, where (Y,u) are the complete data and the random effects u are conceived as missing data. Viewed in this way, the Expectation-Maximization (EM) algorithm has proven to be a natural and popular approach to estimation. However, when C is partially observed or subject to measurement error, it is natural to formulate a multilevel model for C that includes random effects, [nu]. In this article, we extend this thinking to allow estimation of the joint distribution of data Y=(Y,C)=(U[subscript o],U[subscript m]) and random effects b=(u,v) from observed data Y[subscript o]=(Y[subscript o],C[subscript o]) and to generate multiple imputations of missing data (Y[subscript m],b) based on the estimated distribution under the assumption that the data Y are missing at random. This approach contributes to the literature on multiple imputation in three ways: (a) it allows random effects [nu] to be conceived as latent covariates, thus addressing measurement errors of C; (b) it allows non-linearities, including random coefficients, interaction effects, and other polynomial effects involving partially observed covariates; (c) it imputes (Y[subscript m],b) using two-step importance sampling. In these cases, the joint distribution of Y is not analytically tractable even if the analytic multilevel model of interest to the analyst follows a multivariate normal distribution. We prove that our method of maximizing the likelihood and imputing missing data ensures compatibility of the non-normal joint distribution with the analytic normal theory multilevel model via provisionally known random effects. We present and evaluate a sufficient condition under which the produced imputations are compatible with the analytic model. [This paper will be published in the "Journal of Educational and Behavioral Statistics."]
Abstractor: As Provided
IES Funded: Yes
Entry Date: 2025
Accession Number: ED677391
Database: ERIC
FullText Text:
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PubType: Report
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  Data: Multiple Imputation to Estimate Hierarchical Models from Data Missing at Random: Latent Covariates, Random Coefficients, and Statistical Interactions
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  Data: <searchLink fieldCode="AR" term="%22Yongyun+Shin%22">Yongyun Shin</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0654-2620">0000-0003-0654-2620</externalLink>)<br /><searchLink fieldCode="AR" term="%22Stephen+W%2E+Raudenbush%22">Stephen W. Raudenbush</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Grantee+Submission%22"><i>Grantee Submission</i></searchLink>. 2025.
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  Data: 52
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  Data: 10.3102/10769986251385210
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  Data: Consider the conventional multilevel model Y=C[gamma]+Zu+e where [gamma] represents fixed effects and (u,e) are multivariate normal random effects. The continuous outcomes Y and covariates C are fully observed with a subset Z of C. The parameters are [theta]=([gamma],var(u),var(e)). Dempster, Rubin and Tsutakawa (1981) framed the estimation as a missing data problem, where (Y,u) are the complete data and the random effects u are conceived as missing data. Viewed in this way, the Expectation-Maximization (EM) algorithm has proven to be a natural and popular approach to estimation. However, when C is partially observed or subject to measurement error, it is natural to formulate a multilevel model for C that includes random effects, [nu]. In this article, we extend this thinking to allow estimation of the joint distribution of data Y=(Y,C)=(U[subscript o],U[subscript m]) and random effects b=(u,v) from observed data Y[subscript o]=(Y[subscript o],C[subscript o]) and to generate multiple imputations of missing data (Y[subscript m],b) based on the estimated distribution under the assumption that the data Y are missing at random. This approach contributes to the literature on multiple imputation in three ways: (a) it allows random effects [nu] to be conceived as latent covariates, thus addressing measurement errors of C; (b) it allows non-linearities, including random coefficients, interaction effects, and other polynomial effects involving partially observed covariates; (c) it imputes (Y[subscript m],b) using two-step importance sampling. In these cases, the joint distribution of Y is not analytically tractable even if the analytic multilevel model of interest to the analyst follows a multivariate normal distribution. We prove that our method of maximizing the likelihood and imputing missing data ensures compatibility of the non-normal joint distribution with the analytic normal theory multilevel model via provisionally known random effects. We present and evaluate a sufficient condition under which the produced imputations are compatible with the analytic model. [This paper will be published in the "Journal of Educational and Behavioral Statistics."]
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        Value: 10.3102/10769986251385210
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 52
    Subjects:
      – SubjectFull: Hierarchical Linear Modeling
        Type: general
      – SubjectFull: Maximum Likelihood Statistics
        Type: general
      – SubjectFull: Sampling
        Type: general
      – SubjectFull: Error of Measurement
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      – SubjectFull: Algorithms
        Type: general
      – SubjectFull: Equations (Mathematics)
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      – SubjectFull: Computation
        Type: general
      – SubjectFull: Bayesian Statistics
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      – TitleFull: Multiple Imputation to Estimate Hierarchical Models from Data Missing at Random: Latent Covariates, Random Coefficients, and Statistical Interactions
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