Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

A tractable method to account for high-dimensional nonignorable missing data in intensive longitudinal data

Despite the need for sensitivity analysis to nonignorable missingness in intensive longitudinal data (ILD), such analysis is greatly hindered by novel ILD features, such as large data volume and complex nonmonotonic missing-data patterns. Likelihood of alternative models permitting nonignorable missingness often involves very high-dimensional integrals, causing curse of dimensionality and rendering solutions computationally prohibitive to obtain. We aim to overcome this challenge by developing a computationally feasible method, nonlinear indexes of local sensitivity to nonignorability (NISNI). We use linear mixed effects models for the incomplete outcome and covariates. We use Markov multinomial models to describe complex missing-data patterns and mechanisms in ILD, thereby permitting missingness probabilities to depend directly on missing data. Using a second-order Taylor series to approximate likelihood under nonignorability, we develop formulas and closed-form expressions for NISNI. Our approach permits the outcome and covariates to be missing simultaneously, as is often the case in ILD, and can capture U-shaped impact of nonignorability in the neighborhood of the missing at random model without fitting alternative models or evaluating integrals. We evaluate performance of this method using simulated data and real ILD collected by the ecological momentary assessment method.     

Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

Analysis of incomplete quality of life data in advanced stage cancer: A practical application of multiple imputation

This paper presents a practical approach to analyzing incomplete quality of life (QOL) data that contains non-ignorable dropouts in patients with advanced non-small-cell lung cancer (NSCLC). QOL scores for the physical domain at baseline and at the end of the first and second courses of chemotherapy were compared between two treatment groups in a phase III trial. One hundred and 103 eligible patients were randomized to receive cisplatin and irinotecan (CPT-P) or cisplatin and vindesine, respectively; of those two groups, 83 and 85, respectively, completed a QOL questionnaire at least at baseline. A multiple imputation incorporating auxiliary QOL variables was implemented as one of alternatives of sensitivity analyses; these were complete case, available case, and pattern mixture analyses. Although larger sensitivity to missing data was found for CPT-P treatment, none of the alternative analyses demonstrated a significant difference in estimated slopes over time between the groups. This study presents an analytical approach for dealing with the complex problem of missing QOL data. It must be noted, however, that the validity of the multiple imputation method we present is not certain unless we can specify sufficiently informative auxiliary variables to ensure the conversion of non-ignorable missingness to ignorable.     

Latent trait shared-parameter mixed models for missing ecological momentary assessment data

Latent trait shared-parameter mixed models for ecological momentary assessment (EMA) data containing missing values are developed in which data are collected in an intermittent manner. In such studies, data are often missing due to unanswered prompts. Using item response theory models, a latent trait is used to represent the missing prompts and modeled jointly with a mixed model for bivariate longitudinal outcomes. Both one- and two-parameter latent trait shared-parameter mixed models are presented. These new models offer a unique way to analyze missing EMA data with many response patterns. Here, the proposed models represent missingness via a latent trait that corresponds to the students' “ability” to respond to the prompting device. Data containing more than 10 300 observations from an EMA study involving high school students' positive and negative affects are presented. The latent trait representing missingness was a significant predictor of both positive affect and negative affect outcomes. The models are compared to a missing at random mixed model. A simulation study indicates that the proposed models can provide lower bias and increased efficiency compared to the standard missing at random approach commonly used with intermittent missing longitudinal data.     

EMLasso: logistic lasso with missing data

In clinical settings, missing data in the covariates occur frequently. For example, some markers are expensive or hard to measure. When this sort of data is used for model selection, the missingness is often resolved through a complete case analysis or a form of single imputation. An alternative sometimes comes in the form of leaving the most damaged covariates out. All these strategies jeopardise the goal of model selection. In earlier work, we have applied the logistic Lasso in combination with multiple imputation to obtain results in such settings, but we only provided heuristic arguments to advocate the method. In this paper, we propose an improved method that builds on firm statistical arguments and that is developed along the lines of the stochastic expectation–maximisation algorithm. We show that our method can be used to handle missing data in both categorical and continuous predictors, as well as in a nonpenalised regression. We demonstrate the method by applying it to data of 273 lung cancer patients. The objective is to select a model for the prediction of acute dysphagia, starting from a large set of potential predictors, including clinical and treatment covariates as well as a set of single‐nucleotide polymorphisms. Copyright © 2013 John Wiley & Sons, Ltd.     

Quality of life in advanced non-small-cell lung cancer: Results of a Southwest Oncology Group randomized trial

Purpose: The main purpose of this paper is to present the results of a randomized trial comparing the effects of two chemotherapy regimens on the Quality of life (QOL) of patients with advanced non-small-cell lung cancer (NSCLC). Trials in advanced stage disease represent an important treatment context for QOL assessment. A second purpose of this paper is to examine methods for handling the level of missing data commonly observed in the advanced stage disease context. Methods: Patients were randomized to receive cisplatin plus vinorelbine or carboplatin plus paclitaxel. The QOL of 222 patients was assessed with the Functional Assessment of Cancer Therapy – Lung (FACT-L) prior to randomization; follow-up assessments occurred at 13 and 25 weeks. Three methods were used to analyze the QOL data: (1) cross-sectional analysis of four patient categories (improved, stable, missing, and declined) based on changes in the FACT-L score, (2) a mixed linear model, and (3) a pattern mixture model. The longitudinal analyses addressed two potential data biases. Results: Questionnaire submission rates were 91% at baseline, 68% at 13 weeks, and 47% at 25 weeks. The cross-sectional and mixed linear model analyses did not show significant differences by treatment arm in patient-reported QOL. The pattern mixture model analysis, more appropriate given non-ignorable missing data, also found no statistically significant effect of treatment on patient QOL. Conclusion: We present a sensitivity analysis approach with multiple methods for analyzing treatment effects on patient QOL in the presence of substantial, non-ignorable missing data in an advanced stage disease clinical trial. We conclude that the two treatment arms did not differ statistically in their effects on patient QOL over a 25-week treatment period.     

An approach to joint analysis of longitudinal measurements and competing risks failure time data

Joint analysis of longitudinal measurements and survival data has received much attention in recent years. However, previous work has primarily focused on a single failure type for the event time. In this paper we consider joint modelling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint which occurs frequently in clinical trials. Our model uses latent random variables and common covariates to link together the sub-models for the longitudinal measurements and competing risks failure time data, respectively. An EM-based algorithm is derived to obtain the parameter estimates, and a profile likelihood method is proposed to estimate their standard errors. Our method enables one to make joint inference on multiple outcomes which is often necessary in analyses of clinical trials. Furthermore, joint analysis has several advantages compared with separate analysis of either the longitudinal data or competing risks survival data. By modelling the event time, the analysis of longitudinal measurements is adjusted to allow for non-ignorable missing data due to informative dropout, which cannot be appropriately handled by the standard linear mixed effects models alone. In addition, the joint model utilizes information from both outcomes, and could be substantially more efficient than the separate analysis of the competing risk survival data as shown in our simulation study. The performance of our method is evaluated and compared with separate analyses using both simulated data and a clinical trial for the scleroderma lung disease.     

The impact of missing data on estimation of health-related quality of life outcomes: an analysis of a randomized longitudinal clinical trial

Missing responses for health-related quality of life (HRQL) outcomes are common in clinical trials and may introduce bias as such data are often not missing at random. To evaluate the missingness (dropout) effect when comparing two treatment groups in a longitudinal randomized trial, we analyzed the Functional Assessment of Cancer Therapy Trial Outcome Index (TOI) change over 12 months for newly diagnosed patients with chronic myeloid leukemia. HRQL assessment was expected at baseline and months 1, 2, 3, 4, 5, 6, 9 and 12. We defined completers as those with baseline and month 12 TOI, and dropouts as all others as long as they had a baseline score. We defined censoring time as the time interval between baseline and the scheduled month 12 visit dates and approximate time-to-dropout as the time interval from baseline to the midpoint between date of the last reported TOI and the scheduled next visit date. A mixed-effects model was first built to assess treatment effect; a pattern-mixture model and a joint model were then built to account for non-ignorable dropout. Intermittent missing data were assumed to be missing at random. A square root transformation of TOI scores was taken to fulfill the normality and homogeneity assumption at each time point in all the models. The mixed-effects model revealed significant (P < 0.001) between-group differences at each visit except for baseline. The joint model generated similar parameter estimates as the separate longitudinal and survival sub-models with a significant association parameter (P = 0.039) indicating negative association between slope of TOI and hazard of dropout and thus non-ignorable dropout. The pattern-mixture model parameter estimates were fairly similar to those generated from the joint model. When non-ignorable missing data exist in longitudinal studies, a joint model is useful to quantify the relationship between dropout and outcome. In addition, it is important to examine underlying assumptions and utilize multiple missing data models including the pattern mixture model to assess sensitivity of model based inference to assumptions about missing mechanisms.     

Pattern mixture models for longitudinal quality of life studies in advanced stage disease

Analyses of longitudinal quality of life (QOL) for patients with advanced stage disease are frequently plagued by problems of non-random drop-out attributable to deteriorating health and/or death. As an example, Moinpour et al. cite specific challenges which limited their report and assessment of QOL for patients treated for advanced stage colorectal cancer in a clinical trial of several chemotherapeutic regimes performed by the Southwest Oncology Group. A particular source of confusion that arises in studies of advanced stage disease is whether or not to differentiate loss of follow-up due to death from drop-out where the patient is still alive but has dropped from the study. In this paper we examine exploratory data techniques for longitudinal QOL data with non-random missingness due to drop-out and censorship by death. We propose a pattern mixture model for longitudinal QOL, time of drop-out and survival, which allows for straightforward implementation of sensitivity analyses and explicit comparisons to the raw data. Our method is illustrated in the context of analysing the data and addressing the challenges posed by Moinpour et al.     

Addressing missing data in confounders when estimating propensity scores for continuous exposures

Propensity score models are frequently used to estimate causal effects in observational studies. One unresolved issue in fitting these models is handling missing values in the propensity score model covariates. As these models usually contain a large set of covariates, using only individuals with complete data significantly decreases the sample size and statistical power. Several missing data imputation approaches have been proposed, including multiple imputation (MI), MI with missingness pattern (MIMP), and treatment mean imputation. Generalized boosted modeling (GBM), which is a nonparametric approach to estimate propensity scores, can automatically handle missingness in the covariates. Although the performance of MI, MIMP, and treatment mean imputation have previously been compared for binary treatments, they have not been compared for continuous exposures or with single imputation and GBM. We compared these approaches in estimating the generalized propensity score (GPS) for a continuous exposure in both a simulation study and in empirical data. Using GBM with the incomplete data to estimate the GPS did not perform well in the simulation. Missing values should be imputed before estimating propensity scores using GBM or any other approach for estimating the GPS.     

Analysing the effects of anaemia on local recurrence of head and neck cancer when covariate values are missing

We present a case study in the analysis of the prognostic effects of anaemia and other covariates on the local recurrence of head and neck cancer in patients who have been treated with radiation therapy. Because it is believed that a large fraction of the patients are cured by the therapy, we use a failure time mixture model for the outcomes, which simultaneously models both the relationship of the covariates to cure and the relationship of the covariates to local recurrence times for subjects who are not cured. A problematic feature of the data is that two covariates of interest having missing values, so that only 75 per cent of the subjects have complete data. We handle the missing-data problem by jointly modelling the covariates and the outcomes, and then fitting the model to all of the data, including the incomplete cases. We compare our approach to two traditional methods for handling missingness, that is, complete-case analysis and the use of an indicator variable for missingness. The comparison with complete-case analysis demonstrates gains in efficiency for joint modelling as well as sensitivity of some results to the method used to handle missing data. The use of an indicator variable yields results that are very similar to those from joint modelling for our data. We also compare the results obtained for the mixture model with results obtained for a standard (non-mixture) survival model. It is seen that the mixture model separates out effects in a way that is not possible with a standard survival model. In particular, conditional on other covariates, we find strong evidence of an association between anaemia and cure, whereas the evidence of an association between anaemia and time to local recurrence for patients who are not cured is weaker.     

Bias due to missing exposure data using complete-case analysis in the proportional hazards regression model

We studied bias due to missing exposure data in the proportional hazards regression model when using complete-case analysis (CCA). Eleven missing data scenarios were considered: one with missing completely at random (MCAR), four missing at random (MAR), and six non-ignorable missingness scenarios, with a variety of hazard ratios, censoring fractions, missingness fractions and sample sizes. When missingness was MCAR or dependent only on the exposure, there was negligible bias (2-3 per cent) that was similar to the difference between the estimate in the full data set with no missing data and the true parameter. In contrast, substantial bias occurred when missingness was dependent on outcome or both outcome and exposure. For models with hazard ratio of 3.5, a sample size of 400, 20 per cent censoring and 40 per cent missing data, the relative bias for the hazard ratio ranged between 7 per cent and 64 per cent. We observed important differences in the direction and magnitude of biases under the various missing data mechanisms. For example, in scenarios where missingness was associated with longer or shorter follow-up, the biases were notably different, although both mechanisms are MAR. The hazard ratio was underestimated (with larger bias) when missingness was associated with longer follow-up and overestimated (with smaller bias) when associated with shorter follow-up. If it is known that missingness is associated with a less frequently observed outcome or with both the outcome and exposure, CCA may result in an invalid inference and other methods for handling missing data should be considered.     

Longitudinal latent variable models given incompletely observed biomarkers and covariates

In this paper, we analyze a two‐level latent variable model for longitudinal data from the National Growth and Health Study where surrogate outcomes or biomarkers and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to re‐express the desired model as a joint distribution of variables, including the biomarkers, that are subject to missingness conditional on all of the covariates that are completely observed, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. We show that the over‐identified joint model produces biased estimation of the latent variable model and describe how to impose constraints on the joint model so that it has a one‐to‐one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation‐maximization algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

An autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose-modification

The assessment of the dose-response relationship is important but not straightforward when the therapeutic agent is administered repeatedly with dose-modification in each patient and a continuous response is measured repeatedly. We recently proposed an autoregressive linear mixed effects model for such data in which the current response is regressed on the previous response, fixed effects, and random effects. The model represents profiles approaching each patient's asymptote, takes into account the past dose history, and provides a dose-response relationship of the asymptote as a summary measure. In an autoregressive model, intermittent missing data mean the missing values in previous responses as covariates. We previously provided the marginal (unconditional on the previous response) form of the proposed model to deal with intermittent missing data. Irregular timings of dose-modification or measurement can also be treated as equally spaced data with intermittent missing values by selecting an adequately small unit of time. The likelihood is, however, expressed by matrices whose sizes depend on the number of observations for a patient, and the computational burden is large. In this study, we propose a state space form of the autoregressive linear mixed effects model to calculate the marginal likelihood without using large matrices. The regression coefficients of the fixed effects can be concentrated out of the likelihood in this model by the same way of a linear mixed effects model. As an illustration of the approach, we analyzed immunologic data from a clinical trial for multiple sclerosis patients and estimated the dose-response curves for each patient and the population mean.     

Modelling the rate of change in a longitudinal study with missing data, adjusting for contact attempts

The Collaborative Ankle Support Trial (CAST) is a longitudinal trial of treatments for severe ankle sprains in which interest lies in the rate of improvement, the effectiveness of reminders and potentially informative missingness. A model is proposed for continuous longitudinal data with non-ignorable or informative missingness, taking into account the nature of attempts made to contact initial non-responders. The model combines a non-linear mixed model for the outcome model with logistic regression models for the reminder processes. A sensitivity analysis is used to contrast this model with the traditional selection model, where we adjust for missingness by modelling the missingness process. The conclusions that recovery is slower, and less satisfactory with age and more rapid with below knee cast than with a tubular bandage do not alter materially across all models investigated. The results also suggest that phone calls are most effective in retrieving questionnaires.     

NON-RESPONSE MODELS FOR THE ANALYSIS OF NON-MONOTONE IGNORABLE MISSING DATA

We discuss a new class of ignorable non-monotone missing data models – the randomized monotone missingness (RMM) models. We argue that the RMM models represent the most general plausible physical mechanism for generating non-monotone ignorable data. We show that there exists ignorable missing data processes that are not RMM. We argue that it may therefore be inappropriate to analyse non-monotone missing data under the assumption that the missingness mechanism is ignorable, if a statistical test has rejected the hypothesis that the missing data process is RMM representable. We use RMM models to analyse data from a case-control study of the effects of radiation on breast cancer. © 1997 by John Wiley & Sons, Ltd.     

HIV viral dynamic models with dropouts and missing covariates

In recent years HIV viral dynamic models have received great attention in AIDS studies. Often, subjects in these studies may drop out for various reasons such as drug intolerance or drug resistance, and covariates may also contain missing data. Statistical analyses ignoring informative dropouts and missing covariates may lead to misleading results. We consider appropriate methods for HIV viral dynamic models with informative dropouts and missing covariates and evaluate these methods via simulations. A real data set is analysed, and the results show that the initial viral decay rate, which may reflect the efficacy of the anti-HIV treatment, may be over-estimated if dropout patients are ignored. We also find that the current or immediate previous viral load values may be most predictive for patients' dropout. These results may be important for HIV/AIDS studies.     

A Bayesian proportional hazards regression model with non-ignorably missing time-varying covariates

Missing covariate data are common in observational studies of time to an event, especially when covariates are repeatedly measured over time. Failure to account for the missing data can lead to bias or loss of efficiency, especially when the data are non-ignorably missing. Previous work has focused on the case of fixed covariates rather than those that are repeatedly measured over the follow-up period, hence, here we present a selection model that allows for proportional hazards regression with time-varying covariates when some covariates may be non-ignorably missing. We develop a fully Bayesian model and obtain posterior estimates of the parameters via the Gibbs sampler in WinBUGS. We illustrate our model with an analysis of post-diagnosis weight change and survival after breast cancer diagnosis in the Long Island Breast Cancer Study Project follow-up study. Our results indicate that post-diagnosis weight gain is associated with lower all-cause and breast cancer-specific survival among women diagnosed with new primary breast cancer. Our sensitivity analysis showed only slight differences between models with different assumptions on the missing data mechanism yet the complete-case analysis yielded markedly different results.     

Explicating the Conditions Under Which Multilevel Multiple Imputation Mitigates Bias Resulting from Random Coefficient-Dependent Missing Longitudinal Data

Random coefficient-dependent (RCD) missingness is a non-ignorable mechanism through which missing data can arise in longitudinal designs. RCD, for which we cannot test, is a problematic form of missingness that occurs if subject-specific random effects correlate with propensity for missingness or dropout. Particularly when covariate missingness is a problem, investigators typically handle missing longitudinal data by using single-level multiple imputation procedures implemented with long-format data, which ignores within-person dependency entirely, or implemented with wide-format (i.e., multivariate) data, which ignores some aspects of within-person dependency. When either of these standard approaches to handling missing longitudinal data is used, RCD missingness leads to parameter bias and incorrect inference. We explain why multilevel multiple imputation (MMI) should alleviate bias induced by a RCD missing data mechanism under conditions that contribute to stronger determinacy of random coefficients. We evaluate our hypothesis with a simulation study. Three design factors are considered: intraclass correlation (ICC; ranging from .25 to .75), number of waves (ranging from 4 to 8), and percent of missing data (ranging from 20 to 50%). We find that MMI greatly outperforms the single-level wide-format (multivariate) method for imputation under a RCD mechanism. For the MMI analyses, bias was most alleviated when the ICC is high, there were more waves of data, and when there was less missing data. Practical recommendations for handling longitudinal missing data are suggested.