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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 [ 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]
In the citing paper
“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].”
In the citing paper
“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].”
In the citing paper
“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
In the citing paper
“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
Citations
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“Missing data for outcomes at 6 weeks, 6 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.”
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Huque, M. H., Moreno‐Betancur, M., Quartagno, M., Simpson, J. A., Carlin, J. B., & Lee, K. J. (2020) · Biometrical Journal · doi:10.1002/bimj.201900051 · Cited by 61
“…at equal intervals, standard cross-sectional implementations of MVNI and FCS can be employed to impute missing values by treating the time-dependent longitudinal measurements as distinct variables (Schafer, 1997; van Buuren et al., 2006) ; we denote these as JM-MVN and FCS-Standard, respectively.”
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Conscientiousness, Extraversion, College Education, and Longevity of High-Ability Individuals
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Adriaensens, S., Beyers, W., & Struyf, E. (2015) · Journal of Communication Disorders · doi:10.1016/j.jcomdis.2015.10.003 · Cited by 50
Kuppermann, M., Kaimal, A. J., Blat, C., Gonzalez, J., Thiet, M.-P., Bermingham, Y., Altshuler, A. L., Bryant, A. S., Bacchetti, P., & Grobman, W. A. (2020) · JAMA · doi:10.1001/jama.2020.5952 · Cited by 50
Spahn, G., Choi, K.-E., Kennemann, C., Lüdtke, R., Franken, U., Langhorst, J., Paul, A., & Dobos, G. J. (2013) · Integrative Cancer Therapies · doi:10.1177/1534735413492727 · Cited by 49
“Missing data in the questionnaires were multiply imputed by Markov chain Monte Carlo methods.(29,30) This gave a total of 5 complete data sets, each of which was analyzed separately as described below.”
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Bellón, J. Á., Rodríguez-Bayón, A., de Dios Luna, J., & Torres-González, F. (2008) · British Journal of General Practice · doi:10.3399/bjgp08x280182 · Cited by 49
“The ‘multiple imputation by chained equations’ (MICE) method was used to represent the outcomes in participants whose outcome was unobserved.(36,37) In these analyses, an imputation model was developed which relates the primary outcome (consultations over 1 year follow-up) to covariates with complete data using chaine…”
MethodWitte, S. S., Altantsetseg, B., Aira, T., Riedel, M., Chen, J., Potocnik, K., El-Bassel, N., Wu, E., Gilbert, L., Carlson, C., et al. (2011) · AIDS and Behavior · doi:10.1007/s10461-011-9984-0 · Cited by 48
MacNeil Vroomen, J., Bosmans, J. E., Eekhout, I., Joling, K. J., van Mierlo, L. D., Meiland, F. J. M., van Hout, H. P. J., & de Rooij, S. E. (2016) · PLOS ONE · doi:10.1371/journal.pone.0160908 · Cited by 48
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Van Ginkel, J. R., Andries Van der Ark, L., Sijtsma, K., & Vermunt, J. K. (2007) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2006.12.022 · Cited by 46
“Schafer (1997, p. 105) defined a multiple-imputation method to be Bayesianly proper if the imputed values are independent realizations of P( Xmis|Xobs), given some complete-data model and a prior distribution of a set of model parameters, denoted by . If this condition is met, the distribution of P( Xmis|Xobs) equals”
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Shin, S., Livchits, V., Connery, H. S., Shields, A., Yanov, S., Yanova, G., Fitzmaurice, G. M., Nelson, A. K., Greenfield, S. F., & Tomsk Tuberculosis Alcohol Working Group (2013) · Addiction · doi:10.1111/add.12148 · Cited by 44
Ma, J., Yank, V., Lv, N., Goldhaber-Fiebert, J. D., Lewis, M. A., Kramer, M. K., Snowden, M. B., Rosas, L. G., Xiao, L., & Blonstein, A. C. (2015) · Contemporary Clinical Trials · doi:10.1016/j.cct.2015.06.010 · Cited by 45
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Ciarrochi, J., Sahdra, B., Marshall, S., Parker, P., & Horwath, C. (2014) · Journal of Contextual Behavioral Science · doi:10.1016/j.jcbs.2014.07.002 · Cited by 42
Unconstraining methods in revenue management systems: research overview and prospects
Guo, P., Xiao, B., & Li, J. (2012) · Advances in Operations Research · doi:10.1155/2012/270910 · Cited by 40
Miragall, M., Baños, R. M., Cebolla, A., & Botella, C. (2015) · Frontiers in Psychology · doi:10.3389/fpsyg.2015.01531 · Cited by 29
Pascal de Raykeer, R., Hoertel, N., Blanco, C., Lavaud, P., Kaladjian, A., Blumenstock, Y., Cuervo-Lombard, C.-V., Peyre, H., Lemogne, C., Limosin, F., et al. (2019) · Journal of Affective Disorders · doi:10.1016/j.jad.2019.05.063 · Cited by 35
El-Bassel, N., Gilbert, L., Vinocur, D., Chang, M., & Wu, E. (2011) · American Journal of Public Health · doi:10.2105/ajph.2009.181842 · Cited by 30
Niles, A. N., Wolitzky-Taylor, K. B., Arch, J. J., & Craske, M. G. (2017) · Behaviour Research and Therapy · doi:10.1016/j.brat.2017.01.001 · Cited by 26
“In Stata 13, we used multiple imputation with multivariate normal regression and three imputations (Rubin, 2004; Schafer, 1997).”
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The Cost of Relapse for Patients with a Manic/Mixed Episode of Bipolar Disorder in the EMBLEM Study
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Wages, N. P., Simon, J. E., Clark, L. A., Amano, S., Russ, D. W., Manini, T. M., & Clark, B. C. (2020) · BMC Geriatrics · doi:10.1186/s12877-020-01656-y · Cited by 27
Labelling effects and adolescent responses to peers with depression: an experimental investigation
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Does Model Matter? Examining Change Across Time for Youth in Group Homes
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Sensitivity analysis in multiple imputation in effectiveness studies of psychotherapy
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“Although MI is a procedure for MAR, it may, following Schafer (1997), perform well in cases when MNAR mechanisms are suspected, provided the richness of the available multivariate data and the complexity of the imputation model allow a good prediction of the variable affected by missing values.”
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“Descriptive Statistics, Preliminary Analyses, and Group Characteristics Missing data for particular items were imputed using the multiple-imputation method for each scale separately (e.g., Schafer 1997).”
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Zgierska, A. E., Burzinski, C. A., Garland, E. L., Lennon, R. P., Jamison, R., Nakamura, Y., Barrett, B., Sehgal, N., Mirgain, S. A., Singles, J. M., et al. (2021) · Contemporary Clinical Trials · doi:10.1016/j.cct.2021.106548 · Cited by 18
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Comparison of methods of estimating missing values in time series
Iwueze, I. S., Nwogu, E. C., Nlebedim, V. U., Nwosu, U. I., & Chinyem, U. E. (2018) · Open Journal of Statistics · doi:10.4236/ojs.2018.82025 · Cited by 12
“Indeed, procedures have been developed by statisticians to mitigate problems caused by missing data and various estimation methods have been reportedly used by different researchers to replace missing values [19] [20].”
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Schmidt, E. M., Gupta, S., Bowe, T., Ellerbe, L. S., Phelps, T. E., Finney, J. W., Asch, S. M., Humphreys, K., Trafton, J., Vanneman, M., et al. (2017) · Substance Abuse · doi:10.1080/08897077.2016.1212779 · Cited by 12
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“Under most conditions, this approach produces less biased parameter estimations than alternatives in the presence of missing data [26].”
BackgroundA Fast, Easy, and Efficient Estimator for Multiparty Electoral Data
Honaker, J., Katz, J. N., & King, G. (2002) · Political Analysis · doi:10.1093/pan/10.1.84 · Cited by 5
ACT with Pain : measurement, efficacy and mechanisms of Acceptance & Commitment Therapy
Trompetter, H. (2014) · doi:10.3990/1.9789036537087 · Cited by 4
Neidhart (2000) · Pure Amsterdam UMC · doi:10.1002/1529-0131(200012)43:12 · Cited by 1
Multiple imputation of missing blood pressure covariates in survival analysis
van Buuren, S., Boshuizen, H. C., & Knook, D. L. (1999) · Statistics in Medicine · doi:10.1002/(sici)1097-0258(19990330)18:6
Metadata
- 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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