Analysis of Incomplete Multivariate Data
Abstract
The last two decades have seen enormous developments in statistical methods for incomplete data. The EM algorithm and its extensions, multiple imputation, and Markov Chain Monte Carlo provide a set of flexible and reliable tools from inference in large classes of missing-data problems. Yet, in practical terms, those developments have had surprisingly little impact on the way most data analysts handle missing values on a routine basis. Analysis of Incomplete Multivariate Data helps bridge the gap between theory and practice, making these missing-data tools accessible to a broad audience. It presents a unified, Bayesian approach to the analysis of incomplete multivariate data, covering datasets in which the variables are continuous, categorical, or both. The focus is applied, where necessary, to help readers thoroughly understand the statistical properties of those methods, and the behavior of the accompanying algorithms.All techniques are illustrated with real data examples, with extended discussion and practical advice. All of the algorithms described in this book have been implemented by the author for general use in the statistical languages S and S Plus. The software is available free of charge on the Internet.
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Methods to deal with data missing at random (MAR) fall into three broad classes: likelihood-based approaches, weighted estimation and multiple imputation.
“Methods to deal with data missing at random (MAR) fall into three broad classes [ 108 , 111 ]: likelihood-based approaches [ 112 ], weighted estimation [ 113 ] and multiple imputation [ 111 , 114 ].”
From Strengthening the Reporting of Observational Studies in Epidemiology (STROBE): Explanation and Elaboration (2007) · cites this paper as [111]
Missing values in the WAI-TECH-SF, PHQ-9, and CEQ ranged from 0% to 1.6%, were missing at random according to Little's MCAR test (p > 0.05), and were imputed using the Expectation-Maximization Algorithm method.
“The percentage of missing values in the WAI-TECH-SF, PHQ-9, and CEQ scores ranged from 0% to 1.6%. After testing that the values were missing at random using Little’s MCAR test (p > 0.05), they were imputed using the Expectation–Maximization Algorithm method [40].”
Parallel Analysis showed that one factor had to be retained because only one factor had an eigenvalue of 9.08, greater than the 95th percentile eigenvalue of 1.53 for randomly generated data.
“Parallel Analysis [40] showed that one factor had to be retained because only one factor had an eigenvalue (raw data eigenvalue = 9.08) greater than the eigenvalue at the 95th percentile for randomly generated data (95th percentile eigenvalue = 1.53) [48].”
The study used a planned missing data design or matrix sampling to keep participant burden to a minimum.
“For the remaining part of the survey, we utilized a planned missing data design or “matrix sampling” (Schafer, 1997; Graham et al., 2006) to keep the burden on participants to a minimum, and used multiple imputations to deal with the uncertainty related to missing data (as described in detail in Section Multiple Imputation Procedure below).”
From Using Genetic Algorithms in a Large Nationally Representative American Sample to Abbreviate the Multidimensional Experiential Avoidance Questionnaire (2016) · cites this paper as Schafer, 1997
Missing data for outcomes at six weeks, six months and 12 months were imputed using multiple imputation based on Markov chain Monte Carlo and maximum likelihood imputation based on the Expectation-Maximization algorithm.
“Missing data for outcomes at six weeks, six months and 12 months were imputed using multiple imputation based upon the Markov chain Monte Carlo method (Schafer, 1997) and maximum likelihood imputation based upon the Expectation-Maximization (EM) algorithm.”
From A randomised clinical trial to evaluate the acceptability and efficacy of an early phase, online, guided augmentation of outpatient care for adults with anorexia nervosa (2020) · cites this paper as Schafer, 1997
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Methods for Handling Missing Data
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Jerez, J. M., Molina, I., García-Laencina, P. J., Alba, E., Ribelles, N., Martín, M., & Franco, L. (2010) · Artificial Intelligence in Medicine · doi:10.1016/j.artmed.2010.05.002 · Cited by 608 · see citing works
What Improves with Increased Missing Data Imputations?
Bodner, T. E. (2008) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1080/10705510802339072 · Cited by 695 · see citing works
“Several sources document the practical and technical issues one encounters when using the multiple imputation technique (e.g., Allison, 2002; Longford, 2005; Rubin, 1987; Schafer, 1997a).”
BackgroundMethodResultInfluentialMarsh, H. W., Lüdtke, O., Nagengast, B., Trautwein, U., Morin, A. J. S., Abduljabbar, A. S., & Köller, O. (2012) · Educational Psychologist · doi:10.1080/00461520.2012.670488 · Cited by 612 · see citing works
Duckitt, J. H., & Sibley, C. G. (2006) · European Journal of Personality · doi:10.1002/per.614 · Cited by 593 · see citing works
Missing covariate data in medical research: To impute is better than to ignore
Janssen, K. J. M., Donders, A. R. T., Harrell, F. E., Vergouwe, Y., Chen, Q., Grobbee, D. E., & Moons, K. G. M. (2010) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2009.12.008 · Cited by 593 · see citing works
King, L. A., King, D. W., Vogt, D. S., Knight, J., & Samper, R. E. (2006) · Military Psychology · doi:10.1207/s15327876mp1802_1 · Cited by 599 · see citing works
Duckitt, J. H., Bizumić, B., Krauss, S. W., & Heled, E. (2010) · Political Psychology · doi:10.1111/j.1467-9221.2010.00781.x · Cited by 580 · see citing works
Englund, M. M., Luckner, A. E., Whaley, G. J. L., & Egeland, B. R. (2004) · Journal of Educational Psychology · doi:10.1037/0022-0663.96.4.723 · Cited by 585 · see citing works
Defection detection: measuring and understanding the predictive accuracy of customer churn models
Neslin, S. A., Gupta, S., Kamakura, W., Lu, J., & Mason, C. H. (2006) · Journal of Marketing Research · doi:10.1509/jmkr.43.2.204 · Cited by 574 · see citing works
Collins, L. M. (2006) · Annual Review of Psychology · doi:10.1146/annurev.psych.57.102904.190146 · Cited by 565 · see citing works
“A much better option is to use modern missing data procedures (Schafer 1997, Schafer & Graham 2002), such as multiple imputation and maximum likelihood, which are based in statistical theory.”
Twisk, J., de Boer, M., de Vente, W., & Heymans, M. (2013) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2013.03.017 · Cited by 568 · see citing works
Are expressive suppression and cognitive reappraisal associated with stress-related symptoms?
Moore, S. A., Zoellner, L. A., & Mollenholt, N. (2008) · Behaviour Research and Therapy · doi:10.1016/j.brat.2008.05.001 · Cited by 575 · see citing works
- Authors
- Schafer, J. L.
- Corresponding author
- Schafer, J.L.
- Venue
- Chapman & Hall/CRC eBooks
- Publisher
- Chapman and Hall/CRC
- Published
- 1997-08-01
- Volume
- 1997
- Issue
- 1997
- Pages
- 1-99
- OpenAlex
- W4300187280
- Semantic Scholar
- d9b083d7366dc17f2a71d7799f417707c86a37d3
- Cited by
- 6,012
- References
- 0
- Citations by year
- 2026: 92 · 2025: 128 · 2024: 161 · 2023: 143 · 2022: 197 · 2021: 277 · 2020: 222 · 2019: 288 · 2018: 261 · 2017: 264 · 2016: 308 · 2015: 323 · 2014: 322 · 2013: 292 · 2012: 320
- FWCI
- 17.88
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Chapman & Hall/CRC eBooks (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Methods and Mixture Models
- Keywords
- missing data, Bayesian analysis, expectation-maximization algorithm, multiple imputation, Markov chain Monte Carlo
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