The use of complete-case and multiple imputation-based analyses in molecular epidemiology studies that assess interaction effects

Background

In molecular epidemiology studies biospecimen data are collected, often with the purpose of evaluating the synergistic role between a biomarker and another feature on an outcome. Typically, biomarker data are collected on only a proportion of subjects eligible for study, leading to a missing data problem. Missing data methods, however, are not customarily incorporated into analyses. Instead, complete-case (CC) analyses are performed, which can result in biased and inefficient estimates.

Methods

Through simulations, we characterized the performance of CC methods when interaction effects are estimated. We also investigated whether standard multiple imputation (MI) could improve estimation over CC methods when the data are not missing at random (NMAR) and auxiliary information may or may not exist.

Results

CC analyses were shown to result in considerable bias and efficiency loss. While MI reduced bias and increased efficiency over CC methods under specific conditions, it too resulted in biased estimates depending on the strength of the auxiliary data available and the nature of the missingness. In particular, CC performed better than MI when extreme values of the covariate were more likely to be missing, while MI outperformed CC when missingness of the covariate related to both the covariate and outcome. MI always improved performance when strong auxiliary data were available. In a real study, MI estimates of interaction effects were attenuated relative to those from a CC approach.

Conclusions

Our findings suggest the importance of incorporating missing data methods into the analysis. If the data are MAR, standard MI is a reasonable method. Auxiliary variables may make this assumption more reasonable even if the data are NMAR. Under NMAR we emphasize caution when using standard MI and recommend it over CC only when strong auxiliary data are available. MI, with the missing data mechanism specified, is an alternative when the data are NMAR. In all cases, it is recommended to take advantage of MI's ability to account for the uncertainty of these assumptions.     

Joint Longitudinal Models for Dealing With Missing at Random Data in Trial-Based Economic Evaluations

ObjectivesIn trial-based economic evaluation, some individuals are typically associated with missing data at some time point, so that their corresponding aggregated outcomes (eg, quality-adjusted life-years) cannot be evaluated. Restricting the analysis to the complete cases is inefficient and can result in biased estimates, while imputation methods are often implemented under a missing at random (MAR) assumption. We propose the use of joint longitudinal models to extend standard approaches by taking into account the longitudinal structure to improve the estimation of the targeted quantities under MAR.MethodsWe compare the results from methods that handle missingness at an aggregated (case deletion, baseline imputation, and joint aggregated models) and disaggregated (joint longitudinal models) level under MAR. The methods are compared using a simulation study and applied to data from 2 real case studies.ResultsSimulations show that, according to which data affect the missingness process, aggregated methods may lead to biased results, while joint longitudinal models lead to valid inferences under MAR. The analysis of the 2 case studies support these results as both parameter estimates and cost-effectiveness results vary based on the amount of data incorporated into the model.ConclusionsOur analyses suggest that methods implemented at the aggregated level are potentially biased under MAR as they ignore the information from the partially observed follow-up data. This limitation can be overcome by extending the analysis to a longitudinal framework using joint models, which can incorporate all the available evidence.     

A multiple imputation approach for MNAR mechanisms compatible with Heckman's model

Standard implementations of multiple imputation (MI) approaches provide unbiased inferences based on an assumption of underlying missing at random (MAR) mechanisms. However, in the presence of missing data generated by missing not at random (MNAR) mechanisms, MI is not satisfactory. Originating in an econometric statistical context, Heckman's model, also called the sample selection method, deals with selected samples using two joined linear equations, termed the selection equation and the outcome equation. It has been successfully applied to MNAR outcomes. Nevertheless, such a method only addresses missing outcomes, and this is a strong limitation in clinical epidemiology settings, where covariates are also often missing. We propose to extend the validity of MI to some MNAR mechanisms through the use of the Heckman's model as imputation model and a two‐step estimation process. This approach will provide a solution that can be used in an MI by chained equation framework to impute missing (either outcomes or covariates) data resulting either from a MAR or an MNAR mechanism when the MNAR mechanism is compatible with a Heckman's model. The approach is illustrated on a real dataset from a randomised trial in patients with seasonal influenza. Copyright © 2016 John Wiley & Sons, Ltd.     

Population-calibrated multiple imputation for a binary/categorical covariate in categorical regression models

Multiple imputation (MI) has become popular for analyses with missing data in medical research. The standard implementation of MI is based on the assumption of data being missing at random (MAR). However, for missing data generated by missing not at random mechanisms, MI performed assuming MAR might not be satisfactory. For an incomplete variable in a given data set, its corresponding population marginal distribution might also be available in an external data source. We show how this information can be readily utilised in the imputation model to calibrate inference to the population by incorporating an appropriately calculated offset termed the “calibrated-δ adjustment.” We describe the derivation of this offset from the population distribution of the incomplete variable and show how, in applications, it can be used to closely (and often exactly) match the post-imputation distribution to the population level. Through analytic and simulation studies, we show that our proposed calibrated-δ adjustment MI method can give the same inference as standard MI when data are MAR, and can produce more accurate inference under two general missing not at random missingness mechanisms. The method is used to impute missing ethnicity data in a type 2 diabetes prevalence case study using UK primary care electronic health records, where it results in scientifically relevant changes in inference for non-White ethnic groups compared with standard MI. Calibrated-δ adjustment MI represents a pragmatic approach for utilising available population-level information in a sensitivity analysis to explore potential departures from the MAR assumption.     

A general method for elicitation,imputation, and sensitivity analysis for incomplete repeated binary data

We develop and demonstrate methods to perform sensitivity analyses to assess sensitivity to plausible departures from missing at random in incomplete repeated binary outcome data. We use multiple imputation in the not at random fully conditional specification framework, which includes one or more sensitivity parameters (SPs) for each incomplete variable. The use of an online elicitation questionnaire is demonstrated to obtain expert opinion on the SPs, and highest prior density regions are used alongside opinion pooling methods to display credible regions for SPs. We demonstrate that substantive conclusions can be far more sensitive to departures from the missing at random assumption (MAR) when control and intervention nonresponders depart from MAR differently, and show that the correlation of arm specific SPs in expert opinion is particularly important. We illustrate these methods on the iQuit in Practice smoking cessation trial, which compared the impact of a tailored text messaging system versus standard care on smoking cessation. We show that conclusions about the effect of intervention on smoking cessation outcomes at 8 week and 6 months are broadly insensitive to departures from MAR, with conclusions significantly affected only when the differences in behavior between the nonresponders in the two trial arms is larger than expert opinion judges to be realistic.     

Doubly robust generalized estimating equations for longitudinal data

A popular method for analysing repeated‐measures data is generalized estimating equations (GEE). When response data are missing at random (MAR), two modifications of GEE use inverse‐probability weighting and imputation. The weighted GEE (WGEE) method involves weighting observations by their inverse probability of being observed, according to some assumed missingness model. Imputation methods involve filling in missing observations with values predicted by an assumed imputation model. WGEE are consistent when the data are MAR and the dropout model is correctly specified. Imputation methods are consistent when the data are MAR and the imputation model is correctly specified. Recently, doubly robust (DR) methods have been developed. These involve both a model for probability of missingness and an imputation model for the expectation of each missing observation, and are consistent when either is correct. We describe DR GEE, and illustrate their use on simulated data. We also analyse the INITIO randomized clinical trial of HIV therapy allowing for MAR dropout. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluation of two‐fold fully conditional specification multiple imputation for longitudinal electronic health record data

Most implementations of multiple imputation (MI) of missing data are designed for simple rectangular data structures ignoring temporal ordering of data. Therefore, when applying MI to longitudinal data with intermittent patterns of missing data, some alternative strategies must be considered. One approach is to divide data into time blocks and implement MI independently at each block. An alternative approach is to include all time blocks in the same MI model. With increasing numbers of time blocks, this approach is likely to break down because of co‐linearity and over‐fitting. The new two‐fold fully conditional specification (FCS) MI algorithm addresses these issues, by only conditioning on measurements, which are local in time. We describe and report the results of a novel simulation study to critically evaluate the two‐fold FCS algorithm and its suitability for imputation of longitudinal electronic health records. After generating a full data set, approximately 70% of selected continuous and categorical variables were made missing completely at random in each of ten time blocks. Subsequently, we applied a simple time‐to‐event model. We compared efficiency of estimated coefficients from a complete records analysis, MI of data in the baseline time block and the two‐fold FCS algorithm. The results show that the two‐fold FCS algorithm maximises the use of data available, with the gain relative to baseline MI depending on the strength of correlations within and between variables. Using this approach also increases plausibility of the missing at random assumption by using repeated measures over time of variables whose baseline values may be missing. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Using the outcome for imputation of missing predictor values was preferred

BACKGROUND AND OBJECTIVE: Epidemiologic studies commonly estimate associations between predictors (risk factors) and outcome. Most software automatically exclude subjects with missing values. This commonly causes bias because missing values seldom occur completely at random (MCAR) but rather selectively based on other (observed) variables, missing at random (MAR). Multiple imputation (MI) of missing predictor values using all observed information including outcome is advocated to deal with selective missing values. This seems a self-fulfilling prophecy. METHODS: We tested this hypothesis using data from a study on diagnosis of pulmonary embolism. We selected five predictors of pulmonary embolism without missing values. Their regression coefficients and standard errors (SEs) estimated from the original sample were considered as "true" values. We assigned missing values to these predictors--both MCAR and MAR--and repeated this 1,000 times using simulations. Per simulation we multiple imputed the missing values without and with the outcome, and compared the regression coefficients and SEs to the truth. RESULTS: Regression coefficients based on MI including outcome were close to the truth. MI without outcome yielded very biased--underestimated--coefficients. SEs and coverage of the 90% confidence intervals were not different between MI with and without outcome. Results were the same for MCAR and MAR. CONCLUSION: For all types of missing values, imputation of missing predictor values using the outcome is preferred over imputation without outcome and is no self-fulfilling prophecy.     

Bias and efficiency of multiple imputation compared with complete‐case analysis for missing covariate values

When missing data occur in one or more covariates in a regression model, multiple imputation (MI) is widely advocated as an improvement over complete‐case analysis (CC). We use theoretical arguments and simulation studies to compare these methods with MI implemented under a missing at random assumption. When data are missing completely at random, both methods have negligible bias, and MI is more efficient than CC across a wide range of scenarios. For other missing data mechanisms, bias arises in one or both methods. In our simulation setting, CC is biased towards the null when data are missing at random. However, when missingness is independent of the outcome given the covariates, CC has negligible bias and MI is biased away from the null. With more general missing data mechanisms, bias tends to be smaller for MI than for CC. Since MI is not always better than CC for missing covariate problems, the choice of method should take into account what is known about the missing data mechanism in a particular substantive application. Importantly, the choice of method should not be based on comparison of standard errors. We propose new ways to understand empirical differences between MI and CC, which may provide insights into the appropriateness of the assumptions underlying each method, and we propose a new index for assessing the likely gain in precision from MI: the fraction of incomplete cases among the observed values of a covariate (FICO). Copyright © 2010 John Wiley & Sons, Ltd.     

Meta‐analysis with missing study‐level sample variance data

We consider a study‐level meta‐analysis with a normally distributed outcome variable and possibly unequal study‐level variances, where the object of inference is the difference in means between a treatment and control group. A common complication in such an analysis is missing sample variances for some studies. A frequently used approach is to impute the weighted (by sample size) mean of the observed variances (mean imputation). Another approach is to include only those studies with variances reported (complete case analysis). Both mean imputation and complete case analysis are only valid under the missing‐completely‐at‐random assumption, and even then the inverse variance weights produced are not necessarily optimal. We propose a multiple imputation method employing gamma meta‐regression to impute the missing sample variances. Our method takes advantage of study‐level covariates that may be used to provide information about the missing data. Through simulation studies, we show that multiple imputation, when the imputation model is correctly specified, is superior to competing methods in terms of confidence interval coverage probability and type I error probability when testing a specified group difference. Finally, we describe a similar approach to handling missing variances in cross‐over studies. Copyright © 2016 John Wiley & Sons, Ltd.     

The impact of missing data on analyses of a time-dependent exposure in a longitudinal cohort: a simulation study

Background

Missing data often cause problems in longitudinal cohort studies with repeated follow-up waves. Research in this area has focussed on analyses with missing data in repeated measures of the outcome, from which participants with missing exposure data are typically excluded. We performed a simulation study to compare complete-case analysis with Multiple imputation (MI) for dealing with missing data in an analysis of the association of waist circumference, measured at two waves, and the risk of colorectal cancer (a completely observed outcome).

Methods

We generated 1,000 datasets of 41,476 individuals with values of waist circumference at waves 1 and 2 and times to the events of colorectal cancer and death to resemble the distributions of the data from the Melbourne Collaborative Cohort Study. Three proportions of missing data (15, 30 and 50%) were imposed on waist circumference at wave 2 using three missing data mechanisms: Missing Completely at Random (MCAR), and a realistic and a more extreme covariate-dependent Missing at Random (MAR) scenarios. We assessed the impact of missing data on two epidemiological analyses: 1) the association between change in waist circumference between waves 1 and 2 and the risk of colorectal cancer, adjusted for waist circumference at wave 1; and 2) the association between waist circumference at wave 2 and the risk of colorectal cancer, not adjusted for waist circumference at wave 1.

Results

We observed very little bias for complete-case analysis or MI under all missing data scenarios, and the resulting coverage of interval estimates was near the nominal 95% level. MI showed gains in precision when waist circumference was included as a strong auxiliary variable in the imputation model.

Conclusions

This simulation study, based on data from a longitudinal cohort study, demonstrates that there is little gain in performing MI compared to a complete-case analysis in the presence of up to 50% missing data for the exposure of interest when the data are MCAR, or missing dependent on covariates. MI will result in some gain in precision if a strong auxiliary variable that is not in the analysis model is included in the imputation model.

     

Multiple imputation using linked proxy outcome data resulted in important bias reduction and efficiency gains: a simulation study

Background

When an outcome variable is missing not at random (MNAR: probability of missingness depends on outcome values), estimates of the effect of an exposure on this outcome are often biased. We investigated the extent of this bias and examined whether the bias can be reduced through incorporating proxy outcomes obtained through linkage to administrative data as auxiliary variables in multiple imputation (MI).

Methods

Using data from the Avon Longitudinal Study of Parents and Children (ALSPAC) we estimated the association between breastfeeding and IQ (continuous outcome), incorporating linked attainment data (proxies for IQ) as auxiliary variables in MI models. Simulation studies explored the impact of varying the proportion of missing data (from 20 to 80%), the correlation between the outcome and its proxy (0.1–0.9), the strength of the missing data mechanism, and having a proxy variable that was incomplete.

Results

Incorporating a linked proxy for the missing outcome as an auxiliary variable reduced bias and increased efficiency in all scenarios, even when 80% of the outcome was missing. Using an incomplete proxy was similarly beneficial. High correlations (> 0.5) between the outcome and its proxy substantially reduced the missing information. Consistent with this, ALSPAC analysis showed inclusion of a proxy reduced bias and improved efficiency. Gains with additional proxies were modest.

Conclusions

In longitudinal studies with loss to follow-up, incorporating proxies for this study outcome obtained via linkage to external sources of data as auxiliary variables in MI models can give practically important bias reduction and efficiency gains when the study outcome is MNAR.

     

Bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function

In long‐term follow‐up studies, irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points. Modeling the covariance structure for this type of data is challenging, as it requires specification of a covariance function that is positive definite. Moreover, in certain settings, careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure. Two common settings where this occurs are studies with ‘outcome‐dependent follow‐up’ and studies with ‘ignorable missing data’. ‘Outcome‐dependent follow‐up’ occurs when individuals with a history of poor health outcomes had more follow‐up measurements, and the intervals between the repeated measurements were shorter. When the follow‐up time process only depends on previous outcomes, likelihood‐based methods can still provide consistent estimates of the regression parameters, given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow‐up time process is required. For ‘ignorable missing data’, the missing data mechanism does not need to be specified, but valid likelihood‐based inference requires correct specification of the covariance structure. In both cases, flexible modeling approaches for the covariance structure are essential. In this paper, we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function. In particular, we propose semiparametric non‐stationary partial autocorrelation function models, which do not suffer from complex positive definiteness restrictions like the autocorrelation function. We describe a Bayesian approach, discuss computational issues, and apply the proposed methods to CD4 count data from a pediatric AIDS clinical trial. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Missing data strategies for time-varying confounders in comparative effectiveness studies of non-missing time-varying exposures and right-censored outcomes

The treatment of missing data in comparative effectiveness studies with right-censored outcomes and time-varying covariates is challenging because of the multilevel structure of the data. In particular, the performance of an accessible method like multiple imputation (MI) under an imputation model that ignores the multilevel structure is unknown and has not been compared to complete-case (CC) and single imputation methods that are most commonly applied in this context. Through an extensive simulation study, we compared statistical properties among CC analysis, last value carried forward, mean imputation, the use of missing indicators, and MI-based approaches with and without auxiliary variables under an extended Cox model when the interest lies in characterizing relationships between non-missing time-varying exposures and right-censored outcomes. MI demonstrated favorable properties under a moderate missing-at-random condition (absolute bias <0.1) and outperformed CC and single imputation methods, even when the MI method did not account for correlated observations in the imputation model. The performance of MI decreased with increasing complexity such as when the missing data mechanism involved the exposure of interest, but was still preferred over other methods considered and performed well in the presence of strong auxiliary variables. We recommend considering MI that ignores the multilevel structure in the imputation model when data are missing in a time-varying confounder, incorporating variables associated with missingness in the MI models as well as conducting sensitivity analyses across plausible assumptions.     

A bias‐corrected estimator in multiple imputation for missing data

Multiple imputation (MI) is one of the most popular methods to deal with missing data, and its use has been rapidly increasing in medical studies. Although MI is rather appealing in practice since it is possible to use ordinary statistical methods for a complete data set once the missing values are fully imputed, the method of imputation is still problematic. If the missing values are imputed from some parametric model, the validity of imputation is not necessarily ensured, and the final estimate for a parameter of interest can be biased unless the parametric model is correctly specified. Nonparametric methods have been also proposed for MI, but it is not so straightforward as to produce imputation values from nonparametrically estimated distributions. In this paper, we propose a new method for MI to obtain a consistent (or asymptotically unbiased) final estimate even if the imputation model is misspecified. The key idea is to use an imputation model from which the imputation values are easily produced and to make a proper correction in the likelihood function after the imputation by using the density ratio between the imputation model and the true conditional density function for the missing variable as a weight. Although the conditional density must be nonparametrically estimated, it is not used for the imputation. The performance of our method is evaluated by both theory and simulation studies. A real data analysis is also conducted to illustrate our method by using the Duke Cardiac Catheterization Coronary Artery Disease Diagnostic Dataset.     

L’imputation multiple des données manquantes aléatoirement : concepts généraux et présentation d’une méthode Monte-Carlo

BackgroundStatistical analysis of a data set with missing data is a frequent problem to deal with in epidemiology. Methods are available to manage incomplete observations, avoiding biased estimates and improving their precision, compared to more traditional methods, such as the analysis of the sub-sample of complete observations.MethodsOne of these approaches is multiple imputation, which consists in imputing successively several values for each missing data item. Several completed data sets having the same distribution characteristics as the observed data (variability and correlations) are thus generated. Standard analyses are done separately on each completed dataset then combined to obtain a global result. In this paper, we discuss the various assumptions made on the origin of missing data (at random or not), and we present in a pragmatic way the process of multiple imputation. A recent method, Multiple Imputation by Chained Equations (MICE), based on a Monte-Carlo Markov Chain algorithm under missing at random data (MAR) hypothesis, is described. An illustrative example of the MICE method is detailed for the analysis of the relation between a dichotomous variable and two covariates presenting MAR data with no particular structure, through multivariate logistic regression.ResultsCompared with the original dataset without missing data, the results show a substantial improvement of the regression coefficient estimates with the MICE method, relatively to those obtained on the dataset with complete observations.ConclusionThis method does not require any direct assumption on joint distribution of the variables and it is presently implemented in standard statistical software (Splus, Stata). It can be used for multiple imputation of missing data of several variables with no particular structure.     

Statistical analysis with missing exposure data measured by proxy respondents: a misclassification problem within a missing‐data problem

In studies of older adults, researchers often recruit proxy respondents, such as relatives or caregivers, when study participants cannot provide self‐reports (e.g., because of illness). Proxies are usually only sought to report on behalf of participants with missing self‐reports; thus, either a participant self‐report or proxy report, but not both, is available for each participant. Furthermore, the missing‐data mechanism for participant self‐reports is not identifiable and may be nonignorable. When exposures are binary and participant self‐reports are conceptualized as the gold standard, substituting error‐prone proxy reports for missing participant self‐reports may produce biased estimates of outcome means. Researchers can handle this data structure by treating the problem as one of misclassification within the stratum of participants with missing self‐reports. Most methods for addressing exposure misclassification require validation data, replicate data, or an assumption of nondifferential misclassification; other methods may result in an exposure misclassification model that is incompatible with the analysis model. We propose a model that makes none of the aforementioned requirements and still preserves model compatibility. Two user‐specified tuning parameters encode the exposure misclassification model. Two proposed approaches estimate outcome means standardized for (potentially) high‐dimensional covariates using multiple imputation followed by propensity score methods. The first method is parametric and uses maximum likelihood to estimate the exposure misclassification model (i.e., the imputation model) and the propensity score model (i.e., the analysis model); the second method is nonparametric and uses boosted classification and regression trees to estimate both models. We apply both methods to a study of elderly hip fracture patients. Copyright © 2014 John Wiley & Sons, Ltd.     

The handling of missing data in trial-based economic evaluations: should data be multiply imputed prior to longitudinal linear mixed-model analyses?

Introduction

For the analysis of clinical effects, multiple imputation (MI) of missing data were shown to be unnecessary when using longitudinal linear mixed-models (LLM). It remains unclear whether this also applies to trial-based economic evaluations. Therefore, this study aimed to assess whether MI is required prior to LLM when analyzing longitudinal cost and effect data.

Methods

Two-thousand complete datasets were simulated containing five time points. Incomplete datasets were generated with 10, 25, and 50% missing data in follow-up costs and effects, assuming a Missing At Random (MAR) mechanism. Six different strategies were compared using empirical bias (EB), root-mean-squared error (RMSE), and coverage rate (CR). These strategies were: LLM alone (LLM) and MI with LLM (MI-LLM), and, as reference strategies, mean imputation with LLM (M-LLM), seemingly unrelated regression alone (SUR-CCA), MI with SUR (MI-SUR), and mean imputation with SUR (M-SUR).

Results

For costs and effects, LLM, MI-LLM, and MI-SUR performed better than M-LLM, SUR-CCA, and M-SUR, with smaller EBs and RMSEs as well as CRs closers to nominal levels. However, even though LLM, MI-LLM and MI-SUR performed equally well for effects, MI-LLM and MI-SUR were found to perform better than LLM for costs at 10 and 25% missing data. At 50% missing data, all strategies resulted in relatively high EBs and RMSEs for costs.

Conclusion

LLM should be combined with MI when analyzing trial-based economic evaluation data. MI-SUR is more efficient and can also be used, but then an average intervention effect over time cannot be estimated.

     

Joint analysis of left‐censored longitudinal biomarker and binary outcome via latent class modeling

Joint latent class modeling is an appealing approach for evaluating the association between a longitudinal biomarker and clinical outcome when the study population is heterogeneous. The link between the biomarker trajectory and the risk of event is reflected by the latent classes, which accommodate the underlying population heterogeneity. The estimation of joint latent class models may be complicated by the censored data in the biomarker measurements due to detection limits. We propose a modified likelihood function under the parametric assumption of biomarker distribution and develop a Monte Carlo expectation‐maximization algorithm for joint analysis of a biomarker and a binary outcome. We conduct simulation studies to demonstrate the satisfactory performance of our Monte Carlo expectation‐maximization algorithm and the superiority of our method to the naive imputation method for handling censored biomarker data. In addition, we apply our method to the Genetic and Inflammatory Markers of Sepsis study to investigate the role of inflammatory biomarker profile in predicting 90‐day mortality for patients hospitalized with community‐acquired pneumonia.     

Analyzing clinical trial outcomes based on incomplete daily diary reports

A case study is presented assessing the impact of missing data on the analysis of daily diary data from a study evaluating the effect of a drug for the treatment of insomnia. The primary analysis averaged daily diary values for each patient into a weekly variable. Following the commonly used approach, missing daily values within a week were ignored provided there was a minimum number of diary reports (i.e., at least 4). A longitudinal model was then fit with treatment, time, and patient‐specific effects. A treatment effect at a pre‐specified landmark time was obtained from the model. Weekly values following dropout were regarded as missing, but intermittent daily missing values were obscured. Graphical summaries and tables are presented to characterize the complex missing data patterns. We use multiple imputation for daily diary data to create completed data sets so that exactly 7 daily diary values contribute to each weekly patient average. Standard analysis methods are then applied for landmark analysis of the completed data sets, and the resulting estimates are combined using the standard multiple imputation approach. The observed data are subject to digit heaping and patterned responses (e.g., identical values for several consecutive days), which makes accurate modeling of the response data difficult. Sensitivity analyses under different modeling assumptions for the data were performed, along with pattern mixture models assessing the sensitivity to the missing at random assumption. The emphasis is on graphical displays and computational methods that can be implemented with general‐purpose software. Copyright © 2016 John Wiley & Sons, Ltd.