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A practical guide to analysing partially observed data. Collecting, analysing and drawing inferences from data is central to research in the medical and social sciences. Unfortunately, it is rarely possible to collect all the intended data. The literature on inference from the resulting incomplete  data is now huge, and continues to grow both as methods are developed for large and complex data structures, and as increasing computer power and suitable software enable researchers to apply these methods. This book focuses on a particular statistical method for analysing and drawing inferences from incomplete data, called Multiple Imputation (MI). MI is attractive because it is both practical and widely applicable. The authors aim is to clarify the issues raised by missing data, describing the rationale for MI, the relationship between the various imputation models and associated algorithms and its application to increasingly complex data structures. Multiple Imputation and its Application: * Discusses the issues raised by the analysis of partially observed data, and the assumptions on which analyses rest. * Presents a practical guide to the issues to consider when analysing incomplete data from both observational studies and randomized trials. * Provides a detailed discussion of the practical use of MI with real-world examples drawn from medical and social statistics. * Explores handling non-linear relationships and interactions with multiple imputation, survival analysis, multilevel multiple imputation, sensitivity analysis via multiple imputation, using non-response weights with multiple imputation and doubly robust multiple imputation. Multiple Imputation and its Application is aimed at quantitative researchers and students in the medical and social sciences with the aim of clarifying the issues raised by the analysis of incomplete data data, outlining the rationale for MI and describing how to consider and address the issues that arise in its application.

Background Missing data are common in statistical analyses, and imputation methods based on random forests (RF) are becoming popular for handling missing data especially in biomedical research. Unlike standard imputation approaches, RF-based imputation methods do not assume normality or require specification of parametric models. However, it is still inconclusive how they perform for non-normally distributed data or when there are non-linear relationships or interactions. Methods To examine the effects of these three factors, a variety of datasets were simulated with outcome-dependent missing at random (MAR) covariates, and the performances of the RF-based imputation methods missForest and CALIBERrfimpute were evaluated in comparison with predictive mean matching (PMM). Results Both missForest and CALIBERrfimpute have high predictive accuracy but missForest can produce severely biased regression coefficient estimates and downward biased confidence interval coverages, especially for highly skewed variables in nonlinear models. CALIBERrfimpute typically outperforms missForest when estimating regression coefficients, although its biases are still substantial and can be worse than PMM for logistic regression relationships with interaction. Conclusions RF-based imputation, in particular missForest, should not be indiscriminately recommended as a panacea for imputing missing data, especially when data are highly skewed and/or outcome-dependent MAR. A correct analysis requires a careful critique of the missing data mechanism and the inter-relationships between the variables in the data.

Background Multiple imputation (MI) is now widely used to handle missing data in longitudinal studies. Several MI techniques have been proposed to impute incomplete longitudinal covariates, including standard fully conditional specification (FCS-Standard) and joint multivariate normal imputation (JM-MVN), which treat repeated measurements as distinct variables, and various extensions based on generalized linear mixed models. Although these MI approaches have been implemented in various software packages, there has not been a comprehensive evaluation of the relative performance of these methods in the context of longitudinal data. Method Using both empirical data and a simulation study based on data from the six waves of the Longitudinal Study of Australian Children ( N  = 4661), we investigated the performance of a wide range of MI methods available in standard software packages for investigating the association between child body mass index (BMI) and quality of life using both a linear regression and a linear mixed-effects model. Results In this paper, we have identified and compared 12 different MI methods for imputing missing data in longitudinal studies. Analysis of simulated data under missing at random (MAR) mechanisms showed that the generally available MI methods provided less biased estimates with better coverage for the linear regression model and around half of these methods performed well for the estimation of regression parameters for a linear mixed model with random intercept. With the observed data, we observed an inverse association between child BMI and quality of life, with available data as well as multiple imputation. Conclusion Both FCS-Standard and JM-MVN performed well for the estimation of regression parameters in both analysis models. More complex methods that explicitly reflect the longitudinal structure for these analysis models may only be needed in specific circumstances such as irregularly spaced data.

Missing data are ubiquitous in clinical epidemiological research. Individuals with missing data may differ from those with no missing data in terms of the outcome of interest and prognosis in general. Missing data are often categorized into the following three types: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). In clinical epidemiological research, missing data are seldom MCAR. Missing data can constitute considerable challenges in the analyses and interpretation of results and can potentially weaken the validity of results and conclusions. A number of methods have been developed for dealing with missing data. These include complete-case analyses, missing indicator method, single value imputation, and sensitivity analyses incorporating worst-case and best-case scenarios. If applied under the MCAR assumption, some of these methods can provide unbiased but often less precise estimates. Multiple imputation is an alternative method to deal with missing data, which accounts for the uncertainty associated with missing data. Multiple imputation is implemented in most statistical software under the MAR assumption and provides unbiased and valid estimates of associations based on information from the available data. The method affects not only the coefficient estimates for variables with missing data but also the estimates for other variables with no missing data.

Background Multiple imputation is frequently used to deal with missing data in healthcare research. Although it is known that the outcome should be included in the imputation model when imputing missing covariate values, it is not known whether it should be imputed. Similarly no clear recommendations exist on: the utility of incorporating a secondary outcome, if available, in the imputation model; the level of protection offered when data are missing not-at-random; the implications of the dataset size and missingness levels. Methods We used realistic assumptions to generate thousands of datasets across a broad spectrum of contexts: three mechanisms of missingness (completely at random; at random; not at random); varying extents of missingness (20–80% missing data); and different sample sizes (1,000 or 10,000 cases). For each context we quantified the performance of a complete case analysis and seven multiple imputation methods which deleted cases with missing outcome before imputation, after imputation or not at all; included or did not include the outcome in the imputation models; and included or did not include a secondary outcome in the imputation models. Methods were compared on mean absolute error, bias, coverage and power over 1,000 datasets for each scenario. Results Overall, there was very little to separate multiple imputation methods which included the outcome in the imputation model. Even when missingness was quite extensive, all multiple imputation approaches performed well. Incorporating a secondary outcome, moderately correlated with the outcome of interest, made very little difference. The dataset size and the extent of missingness affected performance, as expected. Multiple imputation methods protected less well against missingness not at random, but did offer some protection. Conclusions As long as the outcome is included in the imputation model, there are very small performance differences between the possible multiple imputation approaches: no outcome imputation, imputation or imputation and deletion. All informative covariates, even with very high levels of missingness, should be included in the multiple imputation model. Multiple imputation offers some protection against a simple missing not at random mechanism.

Background Incomplete data are of particular important influence in mental measurement questionnaires. Most experts, however, mostly focus on clinical trials and cohort studies and generally pay less attention to this deficiency. We aim is to compare the accuracy of four common methods for handling items missing from different psychology questionnaires according to the items non-response rates. Method All data were drawn from the previous studies including the self-acceptance scale (SAQ), the activities of daily living scale (ADL) and self-esteem scale (RSES). SAQ and ADL dataset, simulation group, were used to compare and assess the ability of four imputation methods which are direct deletion, mode imputation, Hot-deck (HD) imputation and multiple imputation (MI) by absolute deviation, the root mean square error and average relative error in missing proportions of 5, 10, 15 and 20%. RSES dataset, validation group, was used to test the application of imputation methods. All analyses were finished by SAS 9.4. Results The biases obtained by MI are the smallest under various missing proportions. HD imputation approach performed the lowest absolute deviation of standard deviation values. But they got the similar results and the performances of them are obviously better than direct deletion and mode imputation. In a real world situation, the respondents’ average score in complete data set was 28.22 ± 4.63, which are not much different from imputed datasets. The direction of the influence of the five factors on self-esteem was consistent, although there were some differences in the size and range of OR values in logistic regression model. Conclusion MI shows the best performance while it demands slightly more data analytic capacity and skills of programming. And HD could be considered to impute missing values in psychological investigation when MI cannot be performed due to limited circumstances.

Background Missing data are common in randomised controlled trials (RCTs) and can bias results if not handled appropriately. A statistically valid analysis under the primary missing-data assumptions should be conducted, followed by sensitivity analysis under alternative justified assumptions to assess the robustness of results. Controlled Multiple Imputation (MI) procedures, including delta-based and reference-based approaches, have been developed for analysis under missing-not-at-random assumptions. However, it is unclear how often these methods are used, how they are reported, and what their impact is on trial results. This review evaluates the current use and reporting of MI and controlled MI in RCTs. Methods A targeted review of phase II-IV RCTs (non-cluster randomised) published in two leading general medical journals (The Lancet and New England Journal of Medicine) between January 2014 and December 2019 using MI. Data was extracted on imputation methods, analysis status, and reporting of results. Results of primary and sensitivity analyses for trials using controlled MI analyses were compared. Results A total of 118 RCTs (9% of published RCTs) used some form of MI. MI under missing-at-random was used in 110 trials; this was for primary analysis in 43/118 (36%), and in sensitivity analysis for 70/118 (59%) (3 used in both). Sixteen studies performed controlled MI (1.3% of published RCTs), either with a delta-based ( n  = 9) or reference-based approach ( n  = 7). Controlled MI was mostly used in sensitivity analysis ( n  = 14/16). Two trials used controlled MI for primary analysis, including one reporting no sensitivity analysis whilst the other reported similar results without imputation. Of the 14 trials using controlled MI in sensitivity analysis, 12 yielded comparable results to the primary analysis whereas 2 demonstrated contradicting results. Only 5/110 (5%) trials using missing-at-random MI and 5/16 (31%) trials using controlled MI reported complete details on MI methods. Conclusions Controlled MI enabled the impact of accessible contextually relevant missing data assumptions to be examined on trial results. The use of controlled MI is increasing but is still infrequent and poorly reported where used. There is a need for improved reporting on the implementation of MI analyses and choice of controlled MI parameters.

Background Missing observations within the univariate time series are common in real-life and cause analytical problems in the flow of the analysis. Imputation of missing values is an inevitable step in every incomplete univariate time series. Most of the existing studies focus on comparing the distributions of imputed data. There is a gap of knowledge on how different imputation methods for univariate time series affect the forecasting performance of time series models. We evaluated the prediction performance of autoregressive integrated moving average (ARIMA) and long short-term memory (LSTM) network models on imputed time series data using ten different imputation techniques. Methods Missing values were generated under missing completely at random (MCAR) mechanism at 10%, 15%, 25%, and 35% rates of missingness using complete data of 24-h ambulatory diastolic blood pressure readings. The performance of the mean, Kalman filtering, linear, spline, and Stineman interpolations, exponentially weighted moving average (EWMA), simple moving average (SMA), k-nearest neighborhood (KNN), and last-observation-carried-forward (LOCF) imputation techniques on the time series structure and the prediction performance of the LSTM and ARIMA models were compared on imputed and original data. Results All imputation techniques either increased or decreased the data autocorrelation and with this affected the forecasting performance of the ARIMA and LSTM algorithms. The best imputation technique did not guarantee better predictions obtained on the imputed data. The mean imputation, LOCF, KNN, Stineman, and cubic spline interpolations methods performed better for a small rate of missingness. Interpolation with EWMA and Kalman filtering yielded consistent performances across all scenarios of missingness. Disregarding the imputation methods, the LSTM resulted with a slightly better predictive accuracy among the best performing ARIMA and LSTM models; otherwise, the results varied. In our small sample, ARIMA tended to perform better on data with higher autocorrelation. Conclusions We recommend to the researchers that they consider Kalman smoothing techniques, interpolation techniques (linear, spline, and Stineman), moving average techniques (SMA and EWMA) for imputing univariate time series data as they perform well on both data distribution and forecasting with ARIMA and LSTM models. The LSTM slightly outperforms ARIMA models, however, for small samples, ARIMA is simpler and faster to execute.

Background The multiple imputation by chained equations (MICE) is a widely used approach for handling missing data. However, its robustness, especially for high missing proportions in health indicators, is under-researched. The study aimed to provide a preliminary guideline for the choice of the extent of missing proportion to impute longitudinal health-related data using the MICE method. Methods The study obtained complete data on five mortality-related health indicators of 100 countries (2015–2019) from the Global Health Observatory. Nine incomplete datasets with missing rates from 10 to 90% were generated and imputed using MICE. The robustness of MICE was assessed through three approaches: comparison of means using the Repeated Measures- Analysis of variance, estimation of evaluation metrics (Root mean square error, mean absolute deviation, Bias, and proportionate variance), and visual inspection of box plots of imputed and non-imputed data. Results The Repeated Measures- Analysis of variance revealed significant differences between complete and imputed data, primarily in imputed data with over 50% missing proportions. Evaluation metrics exhibited ‘high performance’ for the dataset with a 50% missing proportion for various health indicators However, with missing proportions exceeding 70%, the majority of indicators demonstrated a ‘low’ performance level in terms of most evaluation metrics. The visual inspection of the box plot revealed severe variance shrinkage in imputed datasets with missing proportions beyond 70%, corroborating the findings from the evaluation metrics. Conclusion It demonstrates high robustness up to 50% missing values, with marginal deviations from complete datasets. Caution is warranted for missing proportions between 50 and 70%, as moderate alterations are observed. Proportions beyond 70% lead to significant variance shrinkage and compromised data reliability, emphasizing the importance of acknowledging imputation limitations for practical decision-making.

Genotype imputation methods are now being widely used in the analysis of genome-wide association studies. Most imputation analyses to date have used the HapMap as a reference dataset, but new reference panels (such as controls genotyped on multiple SNP chips and densely typed samples from the 1,000 Genomes Project) will soon allow a broader range of SNPs to be imputed with higher accuracy, thereby increasing power. We describe a genotype imputation method (IMPUTE version 2) that is designed to address the challenges presented by these new datasets. The main innovation of our approach is a flexible modelling framework that increases accuracy and combines information across multiple reference panels while remaining computationally feasible. We find that IMPUTE v2 attains higher accuracy than other methods when the HapMap provides the sole reference panel, but that the size of the panel constrains the improvements that can be made. We also find that imputation accuracy can be greatly enhanced by expanding the reference panel to contain thousands of chromosomes and that IMPUTE v2 outperforms other methods in this setting at both rare and common SNPs, with overall error rates that are 15%-20% lower than those of the closest competing method. One particularly challenging aspect of next-generation association studies is to integrate information across multiple reference panels genotyped on different sets of SNPs; we show that our approach to this problem has practical advantages over other suggested solutions.