Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model

Multiple imputation is a practical, principled approach to handling missing data. When used to impute missing values in covariates of regression models, imputation models may be misspecified if they are not compatible with the substantive model of interest for the outcome. In this article, we introduce the smcfcs command, which imputes covariates by substantive-model compatible fully conditional specification. This modifies the popular fully conditional specification or chained-equations approach to multiple imputation by imputing each covariate compatibly with a user-specified substantive model. We compare the smcfcs command with standard fully conditional specification imputation using mi impute chained in a simulation study and illustrative analysis of data from a study investigating time to tumor recurrence in breast cancer.

In patient follow‐up studies, events of interest may take place between periodic clinical assessments and so the exact time of onset is not observed. Such events are known as “bounded” or “interval‐censored.” Methods for handling such events can be categorized as either (i) applying multiple imputation (MI) strategies or (ii) taking a full likelihood‐based (LB) approach. We focused on MI strategies, rather than LB methods, because of their flexibility. We evaluated MI strategies for bounded event times in a competing risks analysis, examining the extent to which interval boundaries, features of the data distribution and substantive analysis model are accounted for in the imputation model. Candidate imputation models were predictive mean matching (PMM); log‐normal regression with postimputation back‐transformation; normal regression with and without restrictions on the imputed values and Delord and Genin's method based on sampling from the cumulative incidence function. We used a simulation study to compare MI methods and one LB method when data were missing at random and missing not at random, also varying the proportion of missing data, and then applied the methods to a hematopoietic stem cell transplantation dataset. We found that cumulative incidence and median event time estimation were sensitive to model misspecification. In a competing risks analysis, we found that it is more important to account for features of the data distribution than to restrict imputed values based on interval boundaries or to ensure compatibility with the substantive analysis by sampling from the cumulative incidence function. We recommend MI by type 1 PMM.

Background:The National Trauma Data Bank (NTDB) is plagued by the problem of missing physiological data. The Glasgow Coma Scale score, Respiratory Rate and Systolic Blood Pressure are an essential part of risk adjustment strategies for trauma system evaluation and clinical research. Missing data on these variables may compromise the feasibility and the validity of trauma group comparisons.Aims:To evaluate the validity of Multiple Imputation (MI) for completing missing physiological data in the National Trauma Data Bank (NTDB), by assessing the impact of MI on 1) frequency distributions, 2) associations with mortality, and 3) risk adjustment.Methods:Analyses were based on 170,956 NTDB observations with complete physiological data (observed data set). Missing physiological data were artificially imposed on this data set and then imputed using MI (MI data set). To assess the impact of MI on risk adjustment, 100 pairs of hospitals were randomly selected with replacement and compared using adjusted Odds Ratios (OR) of mortality. OR generated by the observed data set were then compared to those generated by the MI data set.Results:Frequency distributions and associations with mortality were preserved following MI. The median absolute difference between adjusted OR of mortality generated by the observed data set and by the MI data set was 3.6% (inter-quartile range: 2.4%-6.1%).Conclusions:This study suggests that, provided it is implemented with care, MI of missing physiological data in the NTDB leads to valid frequency distributions, preserves associations with mortality, and does not compromise risk adjustment in inter-hospital comparisons of mortality.

Multiple imputation is now well established as a practical and flexible method for analyzing partially observed data, particularly under the missing at random assumption. However, when the substantive model is a weighted analysis, there is concern about the empirical performance of Rubin’s rules and also about how to appropriately incorporate possible interaction between the weights and the distribution of the study variables. One approach that has been suggested is to include the weights in the imputation model, potentially also allowing for interactions with the other variables. We show that the theoretical criterion justifying this approach can be approximately satisfied if we stratify the weights to define level-two units in our data set and include random intercepts in the imputation model. Further, if we let the covariance matrix of the variables have a random distribution across the level-two units, we also allow imputation to reflect any interaction between weight strata and the distribution of the variables. We evaluate our proposal in a number of simulation scenarios, showing it has promising performance both in terms of coverage levels of the model parameters and bias of the associated Rubin’s variance estimates. We illustrate its application to a weighted analysis of factors predicting reception-year readiness in children in the UK Millennium Cohort Study.

Multiple imputation techniques are commonly used when data are missing, however, there are many options one can consider. Multivariate imputation by chained equations is a popular method for generating imputations but relies on specifying models when imputing missing values. In this work, we introduce multiple imputation by super learning, an update to the multivariate imputation by chained equations method to generate imputations with ensemble learning. Ensemble methodologies have recently gained attention for use in inference and prediction as they optimally combine a variety of user-specified parametric and non-parametric models and perform well when estimating complex functions, including those with interaction terms. Through two simulations we compare inferences made using the multiple imputation by super learning approach to those made with other commonly used multiple imputation methods and demonstrate multiple imputation by super learning as a superior option when considering characteristics such as bias, confidence interval coverage rate, and confidence interval width.

Cox's proportional hazards (PH) model assumes constant-over-time covariate effects. Furthermore, most applications assume linear effects of continuous covariates on the logarithm of the hazard. Yet, many prognostic factors have time-dependent (TD) and/or nonlinear (NL) effects, that is, violate these conventional assumptions. Detection of such complex effects could affect prognosis and clinical decisions. However, assessing the effects of each of the multiple, often correlated, covariates in flexible multivariable analyses is challenging. In simulations, we investigated the impact of the approach used to build the flexible multivariable model on inference about the TD and NL covariate effects. Results demonstrate that the conclusions regarding the statistical significance of the TD/NL effects depend heavily on the strategy used to decide which effects of the other covariates should be adjusted for. Both a failure to adjust for true TD and NL effects of relevant covariates and inclusion of spurious effects of covariates that conform to the PH and linearity assumptions increase the risk of incorrect conclusions regarding other covariates. In this context, iterative backward elimination of nonsignificant NL and TD effects from the multivariable model, which initially includes all these effects, may help discriminate between true and spurious effects. The practical importance of these issues was illustrated in an example that reassessed the predictive ability of selected biomarkers for survival in advanced non-small-cell lung cancer. In conclusion, a careful model-building strategy and flexible modeling of multivariable survival data can yield new insights about predictors' roles and improve the validity of analyses.

Prognostic Value of Late Heart Rate Recovery After Treadmill Exercise

Nonadherence to assigned treatment is common in randomized controlled trials (RCTs). Recently, there has been increased interest in estimating causal effects of treatment received, for example, the so-called local average treatment effect (LATE). Instrumental variables (IV) methods can be used for identification, with estimation proceeding either via fully parametric mixture models or two-stage least squares (TSLS). TSLS is popular but can be problematic for binary outcomes where the estimand of interest is a causal odds ratio. Mixture models are rarely used in practice, perhaps because of their perceived complexity and need for specialist software. Here, we propose using multiple imputation (MI) to impute the latent compliance class appearing in the mixture models. Since such models include an interaction term between the latent compliance class and randomized treatment, we use "substantive model compatible" MI (SMC MIC), which can additionally handle missing data in outcomes and other variables in the model, before fitting the mixture models via maximum likelihood to the MI data sets and combining results via Rubin's rules. We use simulations to compare the performance of SMC MIC to existing approaches and also illustrate the methods by reanalyzing an RCT in UK primary health. We show that SMC MIC can be more efficient than full Bayesian estimation when auxiliary variables are incorporated, and is superior to two-stage methods, especially for binaryoutcomes.

Missing data are frequently encountered in biomedical, epidemiologic and social research. It is well known that a naive analysis without adequate handling of missing data may lead to bias and/or loss of efficiency. Partly due to its ease of use, multiple imputation has become increasingly popular in practice for handling missing data. However, it is unclear what is the best strategy to conduct multiple imputation in the presence of high-dimensional data. To answer this question, we investigate several approaches of using regularized regression and Bayesian lasso regression to impute missing values in the presence of high-dimensional data. We compare the performance of these methods through numerical studies, in which we also evaluate the impact of the dimension of the data, the size of the true active set for imputation, and the strength of correlation. Our numerical studies show that in the presence of high-dimensional data the standard multiple imputation approach performs poorly and the imputation approach using Bayesian lasso regression achieves, in most cases, better performance than the other imputation methods including the standard imputation approach using the correctly specified imputation model. Our results suggest that Bayesian lasso regression and its extensions are better suited for multiple imputation in the presence of high-dimensional data than the other regression methods.

The usual methods for analyzing case-cohort studies rely on sometimes not fully efficient weighted estimators. Multiple imputation might be a good alternative because it uses all the data available and approximates the maximum partial likelihood estimator. This method is based on the generation of several plausible complete data sets, taking into account uncertainty about missing values. When the imputation model is correctly defined, the multiple imputation estimator is asymptotically unbiased and its variance is correctly estimated. We show that a correct imputation model must be estimated from the fully observed data (cases and controls), using the case status among the explanatory variable. To validate the approach, we analyzed case-cohort studies first with completely simulated data and then with case-cohort data sampled from two real cohorts. The analyses of simulated data showed that, when the imputation model was correct, the multiple imputation estimator was unbiased and efficient. The observed gain in precision ranged from 8 to 37 per cent for phase-1 variables and from 5 to 19 per cent for the phase-2 variable. When the imputation model was misspecified, the multiple imputation estimator was still more efficient than the weighted estimators but it was also slightly biased. The analyses of case-cohort data sampled from complete cohorts showed that even when no strong predictor of the phase-2 variable was available, the multiple imputation was unbiased, as precised as the weighted estimator for the phase-2 variable and slightly more precise than the weighted estimators for the phase-1 variables. However, the multiple imputation estimator was found to be biased when, because of interaction terms, some coefficients of the imputation model had to be estimated from small samples. Multiple imputation is an efficient technique for analyzing case-cohort data. Practically, we suggest building the analysis model using only the case-cohort data and weighted estimators. Multiple imputation can eventually be used to reanalyze the data using the selected model in order to improve the precision of the results.

Gaining more flexibility in Cox proportional hazards regression models with cubic spline functions

This study aimed to document the ways by which missing data were handled in clinical pharmacy research to provide an insight into the amount of attention paid to the importance of missing data in this field of research. Our cross-sectional descriptive report evaluated 10 journals affiliated with pharmacy organizations in the United States, Canada, the United Kingdom, and Australia. Randomized controlled trials, cohort studies, case-control studies, and cross-sectional studies published in 2018 were included. The primary outcome measure was the proportion of studies that reported the handling of missing data in their methods or results. A total of 178 studies were included in the analysis. Of these, 19.7% (n = 35) mentioned missing data either in their methods (3.4%, n = 6), results (15.2%, n = 27), or in both sections (1.1%, n = 2). Only 4.5% (n = 8) of the studies mentioned how they handled missing data, the most common method being multiple imputation (n = 3), followed by indicator (n = 2), complete case analysis (n = 2), and simple imputation (n = 1). One study using multiple imputation and both studies using an indicator method also combined other strategies to account for missing data. One study only used complete case analysis for subgroup analysis, and the other study only used this method if a specific baseline variable was missing. Very few studies in clinical pharmacy literature report any handling of missing data. This has the potential to lead to biased results. We advocate that researchers should report how missing data were handled to increase the transparency of findings and minimize bias.

Missing data occur in many types of studies and typically complicate the analysis. Multiple imputation, either using joint modeling or the more flexible fully conditional specification approach, are popular and work well in standard settings. In settings involving nonlinear associations or interactions, however, incompatibility of the imputation model with the analysis model is an issue often resulting in bias. Similarly, complex outcomes such as longitudinal or survival outcomes cannot be adequately handled by standard implementations. In this paper, we introduce the R package JointAI, which utilizes the Bayesian framework to perform simultaneous analysis and imputation in regression models with incomplete covariates. Using a fully Bayesian joint modeling approach it overcomes the issue of uncongeniality while retaining the attractive flexibility of fully conditional specification multiple imputation by specifying the joint distribution of analysis and imputation models as a sequence of univariate models that can be adapted to the type of variable. JointAI provides functions for Bayesian inference with generalized linear and generalized linear mixed models and extensions thereof as well as survival models and joint models for longitudinal and survival data, that take arguments analogous to the corresponding well known functions for the analysis of complete data from base R and other packages. Usage and features of JointAI are described and illustrated using various examples and the theoretical background is outlined.

We propose a flexible and scalable approximate Bayesian inference methodology for the Cox Proportional Hazards model with partial likelihood. The model we consider includes nonlinear covariate effects and correlated survival times. The proposed method is based on nested approximations and adaptive quadrature, and the computational burden of working with the log-partial likelihood is mitigated through automatic differentiation and Laplace approximation. We provide two simulation studies to show the accuracy of the proposed approach, compared with the existing methods. We demonstrate the practical utility of our method and its computational advantages over Markov Chain Monte Carlo methods through the analysis of Kidney infection times, which are paired, and the analysis of Leukemia survival times with a semi-parametric covariate effect and spatial variation.

PurposeHypospadias is one of the most frequent male congenital malformations. It remains controversial whether maternal lifestyle during pregnancy may affects the risk of having a son with hypospadias, especially for smoking with many suggesting lower risk. We assessed the individual and joint associations between maternal cigarette smoking, pre-pregnancy body mass index (BMI), alcohol consumption, binge drinking, and caffeine consumption and occurrence of hypospadias in sons.Patients and MethodsThis cohort study utilized the Danish National Birth Cohort and the Aarhus Birth Cohort, holding detailed information on lifestyle factors in early pregnancy between 1989 and 2012. The Danish health registers were used to identify boys with hypospadias, according to International Classification of Diseases. Potential confounders and covariates were identified by literature search and use of directed acyclic graphs. Missing data were handled by multiple imputation and Cox proportional hazards models were applied to analyse data.ResultsIn total, 85,923 live-born singleton boys were included in the study of whom 502 (0.6%) were diagnosed with hypospadias. Maternal smoking in early pregnancy was associated with lower occurrence of hypospadias. An increase of one cigarette smoked per day was associated with lower risk of having a son with hypospadias (adjusted hazard ratio (HR) 0.97 (95% confidence interval (CI) 0.94, 1.00)). However, sub-analyses suggested that the results may be prone to unadjusted confounding. We found no association between pre-pregnancy BMI, alcohol consumption, binge drinking, or caffeine consumption and hypospadias.ConclusionMaternal smoking during pregnancy was associated with lower occurrence of hypospadias but we cannot exclude uncontrolled confounding. The other investigated maternal lifestyle factors were not associated with hypospadias in sons.