Standard procedures for drawing inferences from complex samples do not apply when the variable of interest θ cannot be observed directly, but must be inferred from the values of secondary random variables that depend on θ stochastically. Examples are proficiency variables in item response models and class memberships in latent class models. Rubin's “multiple imputation” techniques yield approximations of sample statistics that would have been obtained, had θ been observable, and associated variance estimates that account for uncertainty due to both the sampling of respondents and the latent nature of θ. The approach is illustrated with data from the National Assessment for Educational Progress.

Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis. In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR). These tests of MCAR require large sample sizes n and/or large group sample sizes ni, and they usually fail when applied to nonnormal data. Hawkins (Technometrics 23:105–110, 1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when ni are small. This paper proposes a modification of this test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete. Moreover, it is shown that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and nonnormal data. It is explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality. Simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates. Also, a power study of the proposed tests indicates good power. The proposed methods use appropriate missing data imputations to impute missing data. Methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods. Examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.

The U.S. National Assessment of Educational Progress (NAEP), the Third International Mathematics and Science Study (TIMSS), and the U.S. Adult Literacy Survey collect probability samples of students (or adults) who are administered brief examinations in subject areas such as mathematics and reading (cognitive variables), along with background demographic (primary) and educational environment (secondary) questions. The demographic questions are used in the primary reporting, while the numerous “explanatory” secondary variables, or “covariates”, are only directly utilized in subsequent secondary analyses. The covariates are also used indirectly to create the plausible values (multiple imputations) that are an integral part of analyses because of the use of sparse matrix sampling of cognitive items. The improvement in the precision of the primary reporting due to the inclusion of the covariates is assessed here and contrasted with the precision of reporting using plausible values created using only the primary demographic variables.

The results demonstrate that the improvement in precision depends on the matrix sampling designs for the cognitive assessments. The improvements range from essentially none for the most common designs, to moderate for some less common designs. Consequently, two potential changes in the reporting procedures that could improve the statistical and operational efficiency of primary reporting are (a) eliminate or reduce the collection of covariates and increase the number of cognitive items, (b) to avoid delays, eliminate the covariates from the creation of plausible values used for the primary reports, but include them later when creating public-use files for secondary analyses. The potential improvements in statistical and operational efficiency must be weighed against the intrinsic interest in the covariates, and the potential for small discrepancies in the primary and secondary reporting.

Whenever statistical analyses are applied to multiply imputed datasets, specific formulas are needed to combine the results into one overall analysis, also called combination rules. In the context of regression analysis, combination rules for the unstandardized regression coefficients, the t-tests of the regression coefficients, and the F-tests for testing $R^2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R^{2}$$\end{document} for significance have long been established. However, there is still no general agreement on how to combine the point estimators of $R^2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R^{2}$$\end{document} in multiple regression applied to multiply imputed datasets. Additionally, no combination rules for standardized regression coefficients and their confidence intervals seem to have been developed at all. In the current article, two sets of combination rules for the standardized regression coefficients and their confidence intervals are proposed, and their statistical properties are discussed. Additionally, two improved point estimators of $R^2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R^{2}$$\end{document} in multiply imputed data are proposed, which in their computation use the pooled standardized regression coefficients. Simulations show that the proposed pooled standardized coefficients produce only small bias and that their 95% confidence intervals produce coverage close to the theoretical 95%. Furthermore, the simulations show that the newly proposed pooled estimates for $R^2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R^{2}$$\end{document} are less biased than two earlier proposed pooled estimates.

Rubin's “multiple imputation” approach to missing data creates synthetic data sets, in which each missing variable is replaced by a draw from its predictive distribution, conditional on the observed data. By construction, analyses of such filled-in data sets as if the imputations were true values have the correct expectations for population parameters. In a recent paper, Mislevy showed how this approach can be applied to estimate the distributions of latent variables from complex samples. Multiple imputations for a latent variable bear a surface similarity to classical “multiple indicators” of a latent variable, as might be addressed in structural equation modelling or hierarchical modelling of successive stages of random sampling. This note demonstrates with a simple example why analyzing “multiple imputations” as if they were “multiple indicators” does not generally yield correct results; they must instead be analyzed by means concordant with their construction.

The basic theme of the EM algorithm, to repeatedly use complete-data methods to solve incomplete data problems, is also a theme of several more recent statistical techniques. These techniques—multiple imputation, data augmentation, stochastic relaxation, and sampling importance resampling—combine simulation techniques with complete-data methods to attack problems that are difficult or impossible for EM.

This study considers data from 5 waves of the English Longitudinal Study of Ageing (ELSA). We aim to study the impact of demographic and self-rated health variables including disability and diseases on the survival of the population aged 50+. The disability variables that we consider are mobility impairment, difficulties in performing Activities of Daily Living (ADL) and Instrumental Activities of Daily Living (IADL). One of the problems with the survey study is missing observations. This may happen due to different reasons, such as errors, nonresponse and temporary withdrawals. We address this problem by applying single and multiple imputation methods. We then fit a Generalized Linear model (GLM) and Generalized Linear Mixed model (GLMM) to our data and show that a GLMM performs better than a GLM in terms of information criteria. We also look at the predictability of our models in terms of the time-dependent receiver operating characteristic (ROC) and the area of ROC, i.e. AUC. We conclude that among the disability factors, IADL and among the diseases, cancer significantly affect the survival of the English population aged 50 and older.

Food insecurity is an important public health concern; however, research into this phenomenon within the Netherlands is limited. Food insecurity is not solely related to individual factors, but can also be influenced by various factors in the social and physical environment. Therefore, this study aimed to identify determinants of food insecurity within the personal, social and physical environment, based on the social ecological model (SEM), and to identify their relative importance for experiencing food insecurity. The study population consisted of 307 participants living in disadvantaged neighbourhoods of the Dutch city The Hague, of which approximately one-quarter were food insecure. Participant characteristics showing bivariate associations P < 0⋅20 were placed in a predetermined level of the SEM, after which a multivariate logistic regression was performed for each level and the Nagelkerke pseudo R2 was presented. Determinants of food insecurity were BMI, gross monthly income, highest educational attainment, smoking status, diet quality, employment status, marital status and religion (P < 0⋅05). The results showed that 29⋅7 % of the total variance in food insecurity status was explained by all included determinants together. The personal, social and physical environment explained 20⋅6, 14⋅0 and 2⋅4 % of the total variance, respectively. Our findings suggest that determinants within the personal environment are most important for explaining differences in experienced food insecurity. The present study contributes to furthering the knowledge about the relative importance of the personal, social and physical environment, indicating that determinants within the personal environment may be most promising for developing targeted interventions to reduce food insecurity.

Mounting evidence suggests that the first few months of life are critical for the development of obesity. The relationships between the timing of solid food introduction and the risk of childhood obesity have been examined previously; however, evidence for the association of timing of infant formula introduction remains scarce. This study aimed to examine whether the timing of infant formula introduction is associated with growth z-scores and overweight at ages 1 and 3 years. This study included 5733 full-term (≥ 37 gestational weeks) and normal birth weight (≥ 2500 and < 4000 g) children in the Born in Guangzhou Cohort Study, a prospective cohort study with data collected at 6 weeks, 6, 12 and 36 months. Compared with infant formula introduction at 0–3 months, introduction at 4–6 months was associated with the lower BMI, weight-for-age and weight-for-length z-scores at 1 and 3 years old. Also, introduction at 4–6 months was associated with the lower odds of at-risk of overweight at age 1 (adjusted OR 0·72, 95 % CI 0·55, 0·94) and 3 years (adjusted OR 0·50, 95 % CI 0·30, 0·85). Introduction at 4–6 months also decreased the odds of overweight at age 1 year (adjusted OR 0·42, 95 % CI 0·21, 0·84) but not at age 3 years. Based on our findings, compared with introduction within the first 3 months, introduction at 4–6 months has a reduction on later high BMI risk and at-risk of overweight. However, these results need to be replicated in other well-designed studies before more firm recommendations can be made.

The use of a Kaplan–Meier (K–M) survival time approach is generally considered appropriate to report antimalarial efficacy trials. However, when a treatment arm has 100% efficacy, confidence intervals may not be computed. Furthermore, methods that use probability rules to handle missing data for instance by multiple imputation, encounter perfect prediction problem when a treatment arm has full efficacy, in which case all imputed values are either treatment success or all imputed values are failures. The use of a survival K–M method addresses this imputation problem in estimating the efficacy estimates also referred to as cure rates. We discuss the statistical challenges and propose a potential way forward.

The proposed approach includes the use of K–M estimates as the main measure of efficacy. Confidence intervals could be computed using the binomial exact method. p-Values for comparison of difference in efficacy between treatments can be estimated using Fisher’s exact test. We emphasize that when efficacy rates are not 100% in both groups, the K–M approach remains the main strategy of analysis considering its statistical robustness in handling missing data and confidence intervals can be computed under such scenarios.

Principled methods for analyzing missing values, based chiefly on multiple imputation, have become increasingly popular yet can struggle to handle the kinds of large and complex data that are also becoming common. We propose an accurate, fast, and scalable approach to multiple imputation, which we call MIDAS (Multiple Imputation with Denoising Autoencoders). MIDAS employs a class of unsupervised neural networks known as denoising autoencoders, which are designed to reduce dimensionality by corrupting and attempting to reconstruct a subset of data. We repurpose denoising autoencoders for multiple imputation by treating missing values as an additional portion of corrupted data and drawing imputations from a model trained to minimize the reconstruction error on the originally observed portion. Systematic tests on simulated as well as real social science data, together with an applied example involving a large-scale electoral survey, illustrate MIDAS’s accuracy and efficiency across a range of settings. We provide open-source software for implementing MIDAS.

Although modern lines for dealing with missing data are well established from the 1970s, today there is a challenge when researchers encounter this problem in multilevel models. First, there is a variety of existing software to handle missing data based on multiple imputation (MI), currently pointed out by experts as the most promising strategy. Second, the two principal paradigms of MI are joint modelling (JM) and fully conditional specification (FCS), one more complication because they are not equally useful depending on the combination of multilevel model and the estimated parameters affected by missing data. Technical literature do not contribute to ease the number of decisions that researcher has to do. Given these inconveniences, the present paper has three objectives. (1) To present a thorough revision of the most recently developed software and functions about multiple imputation in multilevel models. (2) We derive a set of suggestions, recommendations, and guides for helping researchers to handle missing data. We list a number of key questions to consider when analyzing multilevel models. (3) Finally, based on the previous relevant questions, we present two detailed examples using the recommended R packages to be easy for the researcher applying multiple imputation in multilevel models.