Estimating treatment effects under untestable assumptions with nonignorable missing data.

Nonignorable missing data poses key challenges for estimating treatment effects because the substantive model may not be identifiable without imposing further assumptions. For example, the Heckman selection model has been widely used for handling nonignorable missing data but requires the study to make correct assumptions, both about the joint distribution of the missingness and outcome and that there is a valid exclusion restriction. Recent studies have revisited how alternative selection model approaches, for example estimated by multiple imputation (MI) and maximum likelihood, relate to Heckman-type approaches in addressing the first hurdle. However, the extent to which these different selection models rely on the exclusion restriction assumption with nonignorable missing data is unclear. Motivated by an interventional study (REFLUX) with nonignorable missing outcome data in half of the sample, this article critically examines the role of the exclusion restriction in Heckman, MI, and full-likelihood selection models when addressing nonignorability. We explore the implications of the different methodological choices concerning the exclusion restriction for relative bias and root-mean-squared error in estimating treatment effects. We find that the relative performance of the methods differs in practically important ways according to the relevance and strength of the exclusion restriction. The full-likelihood approach is less sensitive to alternative assumptions about the exclusion restriction than Heckman-type models and appears an appropriate method for handling nonignorable missing data. We illustrate the implications of method choice for inference in the REFLUX study, which evaluates the effect of laparoscopic surgery on long-term quality of life for patients with gastro-oseophageal reflux disease.

Sensitivity analysis for clinical trials with missing continuous outcome data using controlled multiple imputation: A practical guide.

Missing data due to loss to follow-up or intercurrent events are unintended, but unfortunately inevitable in clinical trials. Since the true values of missing data are never known, it is necessary to assess the impact of untestable and unavoidable assumptions about any unobserved data in sensitivity analysis. This tutorial provides an overview of controlled multiple imputation (MI) techniques and a practical guide to their use for sensitivity analysis of trials with missing continuous outcome data. These include δ- and reference-based MI procedures. In δ-based imputation, an offset term, δ, is typically added to the expected value of the missing data to assess the impact of unobserved participants having a worse or better response than those observed. Reference-based imputation draws imputed values with some reference to observed data in other groups of the trial, typically in other treatment arms. We illustrate the accessibility of these methods using data from a pediatric eczema trial and a chronic headache trial and provide Stata code to facilitate adoption. We discuss issues surrounding the choice of δ in δ-based sensitivity analysis. We also review the debate on variance estimation within reference-based analysis and justify the use of Rubin's variance estimator in this setting, since as we further elaborate on within, it provides information anchored inference.

Reference-based sensitivity analysis for time-to-event data.

The analysis of time-to-event data typically makes the censoring at random assumption, ie, that-conditional on covariates in the model-the distribution of event times is the same, whether they are observed or unobserved (ie, right censored). When patients who remain in follow-up stay on their assigned treatment, then analysis under this assumption broadly addresses the de jure, or "while on treatment strategy" estimand. In such cases, we may well wish to explore the robustness of our inference to more pragmatic, de facto or "treatment policy strategy," assumptions about the behaviour of patients post-censoring. This is particularly the case when censoring occurs because patients change, or revert, to the usual (ie, reference) standard of care. Recent work has shown how such questions can be addressed for trials with continuous outcome data and longitudinal follow-up, using reference-based multiple imputation. For example, patients in the active arm may have their missing data imputed assuming they reverted to the control (ie, reference) intervention on withdrawal. Reference-based imputation has two advantages: (a) it avoids the user specifying numerous parameters describing the distribution of patients' postwithdrawal data and (b) it is, to a good approximation, information anchored, so that the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. In this article, we build on recent work in the survival context, proposing a class of reference-based assumptions appropriate for time-to-event data. We report a simulation study exploring the extent to which the multiple imputation estimator (using Rubin's variance formula) is information anchored in this setting and then illustrate the approach by reanalysing data from a randomized trial, which compared medical therapy with angioplasty for patients presenting with angina.

Meta-analysis of Gaussian individual patient data: Two-stage or not two-stage?

Quantitative evidence synthesis through meta-analysis is central to evidence-based medicine. For well-documented reasons, the meta-analysis of individual patient data is held in higher regard than aggregate data. With access to individual patient data, the analysis is not restricted to a "two-stage" approach (combining estimates and standard errors) but can estimate parameters of interest by fitting a single model to all of the data, a so-called "one-stage" analysis. There has been debate about the merits of one- and two-stage analysis. Arguments for one-stage analysis have typically noted that a wider range of models can be fitted and overall estimates may be more precise. The two-stage side has emphasised that the models that can be fitted in two stages are sufficient to answer the relevant questions, with less scope for mistakes because there are fewer modelling choices to be made in the two-stage approach. For Gaussian data, we consider the statistical arguments for flexibility and precision in small-sample settings. Regarding flexibility, several of the models that can be fitted only in one stage may not be of serious interest to most meta-analysis practitioners. Regarding precision, we consider fixed- and random-effects meta-analysis and see that, for a model making certain assumptions, the number of stages used to fit this model is irrelevant; the precision will be approximately equal. Meta-analysts should choose modelling assumptions carefully. Sometimes relevant models can only be fitted in one stage. Otherwise, meta-analysts are free to use whichever procedure is most convenient to fit the identified model.

A penalized framework for distributed lag non-linear models.

Distributed lag non-linear models (DLNMs) are a modelling tool for describing potentially non-linear and delayed dependencies. Here, we illustrate an extension of the DLNM framework through the use of penalized splines within generalized additive models (GAM). This extension offers built-in model selection procedures and the possibility of accommodating assumptions on the shape of the lag structure through specific penalties. In addition, this framework includes, as special cases, simpler models previously proposed for linear relationships (DLMs). Alternative versions of penalized DLNMs are compared with each other and with the standard unpenalized version in a simulation study. Results show that this penalized extension to the DLNM class provides greater flexibility and improved inferential properties. The framework exploits recent theoretical developments of GAMs and is implemented using efficient routines within freely available software. Real-data applications are illustrated through two reproducible examples in time series and survival analysis.

Estimation of the linear mixed integrated Ornstein-Uhlenbeck model.

The linear mixed model with an added integrated Ornstein-Uhlenbeck (IOU) process (linear mixed IOU model) allows for serial correlation and estimation of the degree of derivative tracking. It is rarely used, partly due to the lack of available software. We implemented the linear mixed IOU model in Stata and using simulations we assessed the feasibility of fitting the model by restricted maximum likelihood when applied to balanced and unbalanced data. We compared different (1) optimization algorithms, (2) parameterizations of the IOU process, (3) data structures and (4) random-effects structures. Fitting the model was practical and feasible when applied to large and moderately sized balanced datasets (20,000 and 500 observations), and large unbalanced datasets with (non-informative) dropout and intermittent missingness. Analysis of a real dataset showed that the linear mixed IOU model was a better fit to the data than the standard linear mixed model (i.e. independent within-subject errors with constant variance).

Comparisons between mild and severe cases of hand, foot and mouth disease in temporal trends: a comparative time series study from mainland China.

BACKGROUND: Over recent decades, hand, foot and mouth disease (HFMD) has emerged as a serious public health threat in the Asia-Pacific region because of its high rates of severe complications. Understanding the differences and similarities between mild and severe cases can be helpful in the control of HFMD. In this study, we compared the two types of HFMD cases in their temporal trends. METHODS: We retrieved the daily series of disease counts of mild and severe HFMD cases reported in mainland China in the period of 2009-2014. We applied a quasi-Poisson regression model to decompose each series into the long-term linear trend, periodic variations, and short-term fluctuations, and then we compared each component between two series separately. RESULTS: A total of 11,101,860 clinical HFMD cases together with 115,596 severe cases were included into this analysis. We found a biennial increase of 24.46 % (95 % CI: 22.80-26.14 %) for the baseline of disease incidence of mild cases, whereas a biennial decrease of 8.80 % (95 % CI: 7.26-10.31 %) was seen for that of severe cases. The periodic variations of both two series could be characterized by a mixture of biennial, annual, semi-annual and eight-monthly cycles. However, compared to the mild cases, we found the severe cases vary more widely for the biennial and annual cycle, and started its annual epidemic earlier. We also found the short-term fluctuations between two series were still significantly correlated at the current day with a correlation coefficient of 0.46 (95 % CI: 0.43-0.49). CONCLUSIONS: We found some noticeable differences and also similarities between the daily series of mild and severe HFMD cases at different time scales. Our findings can help us to deepen the understanding of the transmission of different types of HFMD cases, and also provide evidences for the planning of the associated disease control strategies.

Is ethnic density associated with risk of child pedestrian injury? A comparison of inter-census changes in ethnic populations and injury rates.

OBJECTIVES: Research on inequalities in child pedestrian injury risk has identified some puzzling trends: although, in general, living in more affluent areas protects children from injury, this is not true for those in some minority ethnic groups. This study aimed to identify whether 'group density' effects are associated with injury risk, and whether taking these into account alters the relationship between area deprivation and injury risk. 'Group density' effects exist when ethnic minorities living in an area with a higher proportion of people from a similar ethnic group enjoy better health than those who live in areas with a lower proportion, even though areas with dense minority ethnic populations can be relatively more materially disadvantaged. DESIGN: This study utilised variation in minority ethnic densities in London between two census periods to identify any associations between group density and injury risk. Using police data on road traffic injury and population census data from 2001 to 2011, the numbers of 'White,' 'Asian' and 'Black' child pedestrian injuries in an area were modelled as a function of the percentage of the population in that area that are 'White,' 'Asian' and 'Black,' controlling for socio-economic disadvantage and characteristics of the road environment. RESULTS: There was strong evidence (p < 0.001) of a negative association between 'Black' population density and 'Black' child pedestrian injury risk [incidence (of injury) rate ratios (IRR) 0.575, 95% CI 0.515-0.642]. There was weak evidence (p = 0.083) of a negative association between 'Asian' density and 'Asian' child pedestrian injury risk (IRR 0.901, 95% CI 0.801-1.014) and no evidence (p = 0.412) of an association between 'White' density and 'White' child pedestrian injury risk (IRR 1.075, 95% CI 0.904-1.279). When group density effects are taken into account, area deprivation is associated with injury risk of all ethnic groups. CONCLUSIONS: Group density appears to protect 'Black' children living in London against pedestrian injury risk. These findings suggest that future research should focus on structural properties of societies to explain the relationships between minority ethnicity and risk.

Reference-based sensitivity analysis via multiple imputation for longitudinal trials with protocol deviation.

Randomized controlled trials provide essential evidence for the evaluation of new and existing medical treatments. Unfortunately, the statistical analysis is often complicated by the occurrence of protocol deviations, which mean we cannot always measure the intended outcomes for individuals who deviate, resulting in a missing-data problem. In such settings, however one approaches the analysis, an untestable assumption about the distribution of the unobserved data must be made. To understand how far the results depend on these assumptions, the primary analysis should be supplemented by a range of sensitivity analyses, which explore how the conclusions vary over a range of different credible assumptions for the missing data. In this article, we describe a new command, mimix, that can be used to perform reference-based sensitivity analyses for randomized controlled trials with longitudinal quantitative outcome data, using the approach proposed by Carpenter, Roger, and Kenward (2013, Journal of Biopharmaceutical Statistics 23: 1352-1371). Under this approach, we make qualitative assumptions about how individuals' missing outcomes relate to those observed in relevant groups in the trial, based on plausible clinical scenarios. Statistical analysis then proceeds using the method of multiple imputation.

Estimating Disease Duration in Cross-sectional Surveys.

BACKGROUND: In some common episodic conditions, such as diarrhea, respiratory infections, or fever, episode duration can reflect disease severity. The mean episode duration in a population can be estimated if both the incidence and prevalence of the condition are known. In this article, we discuss how an estimator of the average episode duration may be obtained based on prevalence alone if data are collected for two consecutive units of time (usually days) in the same person. METHODS: We derive a maximum likelihood estimator of episode duration, explore its behavior through a simulation study, and illustrate its use through a real example. RESULTS: We show that for two consecutive days, the estimator of the mean episode duration in a population equals one plus twice the ratio of the number of subjects with the condition on both days to the number of subjects with only 1 day ill. The estimator can be extended to account for 3 or 4 consecutive days. The estimator assumes nonoverlapping episodes and a time-constant incidence rate and is more precise for shorter than for longer average episode durations. CONCLUSION: The proposed method allows estimating the mean duration of disease episodes in cross-sectional studies and is applicable to large demographic and health surveys in low-income settings that routinely collect data on diarrhea and respiratory illness. The method may further be used for the calculation of the duration of infectiousness if test results are available for two consecutive days, such as paired throat swabs for influenza.

Maternal dietary fatty acid intake during pregnancy and the risk of preclinical and clinical type 1 diabetes in the offspring.

The aim of the present study was to examine the associations between the maternal intake of fatty acids during pregnancy and the risk of preclinical and clinical type 1 diabetes in the offspring. The study included 4887 children with human leucocyte antigen (HLA)-conferred type 1 diabetes susceptibility born during the years 1997-2004 from the Finnish Type 1 Diabetes Prediction and Prevention Study. Maternal diet was assessed with a validated FFQ. The offspring were observed at 3- to 12-month intervals for the appearance of type 1 diabetes-associated autoantibodies and development of clinical type 1 diabetes (average follow-up period: 4·6 years (range 0·5-11·5 years)). Altogether, 240 children developed preclinical type 1 diabetes and 112 children developed clinical type 1 diabetes. Piecewise linear log-hazard survival model and Cox proportional-hazards regression were used for statistical analyses. The maternal intake of palmitic acid (hazard ratio (HR) 0·82, 95 % CI 0·67, 0·99) and high consumption of cheese during pregnancy (highest quarter v. intermediate half HR 0·52, 95 % CI 0·31, 0·87) were associated with a decreased risk of clinical type 1 diabetes. The consumption of sour milk products (HR 1·14, 95 % CI 1·02, 1·28), intake of protein from sour milk (HR 1·15, 95 % CI 1·02, 1·29) and intake of fat from fresh milk (HR 1·43, 95 % CI 1·04, 1·96) were associated with an increased risk of preclinical type 1 diabetes, and the intake of low-fat margarines (HR 0·67, 95 % CI 0·49, 0·92) was associated with a decreased risk. No conclusive associations between maternal fatty acid intake or food consumption during pregnancy and the development of type 1 diabetes in the offspring were detected.

Are missing data adequately handled in cluster randomised trials? A systematic review and guidelines.

BACKGROUND: Missing data are a potential source of bias, and their handling in the statistical analysis can have an important impact on both the likelihood and degree of such bias. Inadequate handling of the missing data may also result in invalid variance estimation. The handling of missing values is more complex in cluster randomised trials, but there are no reviews of practice in this field. OBJECTIVES: A systematic review of published trials was conducted to examine how missing data are reported and handled in cluster randomised trials. METHODS: We systematically identified cluster randomised trials, published in English in 2011, using the National Library of Medicine (MEDLINE) via PubMed. Non-randomised and pilot/feasibility trials were excluded, as were reports of secondary analyses, interim analyses, and economic evaluations and those where no data were at the individual level. We extracted information on missing data and the statistical methods used to deal with them from a random sample of the identified studies. RESULTS: We included 132 trials. There was evidence of missing data in 95 (72%). Only 32 trials reported handling missing data, 22 of them using a variety of single imputation techniques, 8 using multiple imputation without accommodating the clustering and 2 stating that their likelihood-based complete case analysis accounted for missing values because the data were assumed Missing-at-Random. LIMITATIONS: The results presented in this study are based on a large random sample of cluster randomised trials published in 2011, identified in electronic searches and therefore possibly missing some trials, most likely of poorer quality. Also, our results are based on information in the main publication for each trial. These reports may omit some important information on the presence of, and reasons for, missing data and on the statistical methods used to handle them. Our extraction methods, based on published reports, could not distinguish between missing data in outcomes and missing data in covariates. This distinction may be important in determining the assumptions about the missing data mechanism necessary for complete case analyses to be valid. CONCLUSIONS: Missing data are present in the majority of cluster randomised trials. However, they are poorly reported, and most authors give little consideration to the assumptions under which their analysis will be valid. The majority of the methods currently used are valid under very strong assumptions about the missing data, whose plausibility is rarely discussed in the corresponding reports. This may have important consequences for the validity of inferences in some trials. Methods which result in valid inferences under general Missing-at-Random assumptions are available and should be made more accessible.

Using multi-level data to estimate the effect of social capital on hazardous alcohol consumption in the former Soviet Union.

BACKGROUND: Hazardous alcohol consumption is a leading cause of mortality in the former Soviet Union (fSU), but little is known about the social factors associated with this behaviour. We set out to estimate the association between individual- and community-level social capital and hazardous alcohol consumption in the fSU. METHODS: Data were obtained from Health in Times of Transition 2010, a household survey of nine fSU countries (n = 18 000 within 2027 communities). Individual-level indicators of social isolation, civic participation, help in a crisis and interpersonal trust were aggregated to the community level. Adjusting for demographic factors, the association of individual- and community-level indicators with problem drinking (CAGE) and episodic heavy drinking was estimated using a population average model for the analysis of multi-level data. RESULTS: Among men, individual social isolation [odds ratio (OR) = 1.20], community social isolation (OR = 1.18) and community civic participation (OR = 4.08) were associated with increased odds of CAGE. Community civic participation (OR = 2.91) increased the odds of episodic heavy drinking, while community interpersonal trust (OR = 0.89) decreased these odds. Among women, individual social isolation (OR = 1.30) and community civic participation (OR = 2.94) increased odds of CAGE. CONCLUSION: Our results provide evidence of the role of some elements of social capital in problem drinking in the fSU, and highlight the importance of community effects. The nature of civic organizations in the fSU, and the communities in which civic participation is high, should be further investigated to inform alcohol policy in the region.

The impact of alcohol consumption on patterns of union formation in Russia 1998-2010: an assessment using longitudinal data.

Using data from the Russian Longitudinal Monitoring Survey, 1998-2010, we investigated the extent to which patterns of alcohol consumption in Russia are associated with the subsequent likelihood of entry into cohabitation and marriage. Using discrete-time event history analysis we estimated for 16-50 year olds the extent to which the probabilities of entry into the two types of union were affected by the amount of alcohol drunk and the pattern of drinking, adjusted to allow for social and demographic factors including income, employment, and health. The results show that individuals who did not drink alcohol were less likely to embark on either cohabitation or marriage, that frequent consumption of alcohol was associated with a greater chance of entering unmarried cohabitation than of entering into a marriage, and that heavy drinkers were less likely to convert their relationship from cohabitation to marriage.

Missing data sensitivity analysis for recurrent event data using controlled imputation.

Statistical analyses of recurrent event data have typically been based on the missing at random assumption. One implication of this is that, if data are collected only when patients are on their randomized treatment, the resulting de jure estimator of treatment effect corresponds to the situation in which the patients adhere to this regime throughout the study. For confirmatory analysis of clinical trials, sensitivity analyses are required to investigate alternative de facto estimands that depart from this assumption. Recent publications have described the use of multiple imputation methods based on pattern mixture models for continuous outcomes, where imputation for the missing data for one treatment arm (e.g. the active arm) is based on the statistical behaviour of outcomes in another arm (e.g. the placebo arm). This has been referred to as controlled imputation or reference-based imputation. In this paper, we use the negative multinomial distribution to apply this approach to analyses of recurrent events and other similar outcomes. The methods are illustrated by a trial in severe asthma where the primary endpoint was rate of exacerbations and the primary analysis was based on the negative binomial model.

Timing of infant feeding in relation to childhood asthma and allergic diseases.

BACKGROUND: Emerging evidence questions current recommendations on the timing of infant feeding for the prevention of childhood allergies. The evidence for asthma is inconclusive. OBJECTIVE: We sought to investigate the associations between the duration of breast-feeding and timing of introduction of complementary foods and the development of asthma and allergies by the age of 5 years. METHODS: Data were analyzed for 3781 consecutively born children. The dietary exposures were categorized into thirds and analyzed as time-dependent variables. Asthma, allergic rhinitis, and atopic eczema end points were assessed by using the International Study of Asthma and Allergies in Childhood questionnaire, whereas IgE antibodies were analyzed from serum samples at the age of 5 years. Cox proportional hazard and logistic regressions were used for the analyses. RESULTS: The median duration of exclusive and total breast-feeding was 1.4 months (interquartile range, 0.2-3.5 months) and 7.0 months (interquartile range, 4.0-11.0 months), respectively. Total breast-feeding of 9.5 months or less was associated with an increased risk of nonatopic asthma. Introduction of wheat, rye, oats, or barley at 5 to 5.5 months was inversely associated with asthma and allergic rhinitis, whereas introduction of other cereals at less than 4.5 months increased the risk of atopic eczema. Introduction of egg at 11 months or less was inversely associated with asthma, allergic rhinitis, and atopic sensitization, whereas introduction of fish at 9 months or less was inversely associated with allergic rhinitis and atopic sensitization. CONCLUSION: Early introduction of wheat, rye, oats, and barley cereals; fish; and egg (respective to the timing of introduction of each food) seems to decrease the risk of asthma, allergic rhinitis, and atopic sensitization in childhood. Longer duration of total breast-feeding, rather than its exclusivity, was protective against the development of nonatopic but not atopic asthma, suggesting a potential differing effect of breast-feeding on different asthma phenotypes.

A characterization of missingness at random in a generalized shared-parameter joint modeling framework for longitudinal and time-to-event data, and sensitivity analysis.

We consider a conceptual correspondence between the missing data setting, and joint modeling of longitudinal and time-to-event outcomes. Based on this, we formulate an extended shared random effects joint model. Based on this, we provide a characterization of missing at random, which is in line with that in the missing data setting. The ideas are illustrated using data from a study on liver cirrhosis, contrasting the new framework with conventional joint models.

Some consequences of assuming simple patterns for the treatment effect over time in a linear mixed model.

Linear mixed models are often used for the analysis of data from clinical trials with repeated quantitative outcomes. This paper considers linear mixed models where a particular form is assumed for the treatment effect, in particular constant over time or proportional to time. For simplicity, we assume no baseline covariates and complete post-baseline measures, and we model arbitrary mean responses for the control group at each time. For the variance-covariance matrix, we consider an unstructured model, a random intercepts model and a random intercepts and slopes model. We show that the treatment effect estimator can be expressed as a weighted average of the observed time-specific treatment effects, with weights depending on the covariance structure and the magnitude of the estimated variance components. For an assumed constant treatment effect, under the random intercepts model, all weights are equal, but in the random intercepts and slopes and the unstructured models, we show that some weights can be negative: thus, the estimated treatment effect can be negative, even if all time-specific treatment effects are positive. Our results suggest that particular models for the treatment effect combined with particular covariance structures may result in estimated treatment effects of unexpected magnitude and/or direction. Methods are illustrated using a Parkinson's disease trial.

Preventing smoking relapse using text messages: analysis of data from the txt2stop trial.

INTRODUCTION: Interactive text message-based technologies which operate in real time have the potential to be especially effective for delivery of relapse prevention interventions. We examined predictors of use of a text message system for providing support for lapses and cravings, describe the natural history of requests for support, and predictors of time to requests for support. METHODS: Data were collected prospectively from participants in the intervention arm of txt2stop, a large randomized controlled trial of an automated, text message-based smoking cessation intervention. Txt2stop included 2,915 men and women aged 16-78, recruited from London, United Kingdom from 2009 to 2010. Participants could text "crave" or "lapse" when they experienced either; an automated system registered the time of the text message to the nearest second. RESULTS: One thousand one hundred and twenty one (38.5%) participants sent a lapse or crave message to request support. Women were more likely to lapse at some point during the trial. Of those who lapsed, being female, younger age, and setting a Saturday quit date were predictors of sending a lapse text requesting support. Half of all crave texts arrived within 106 hr of quitting. Half of all lapse texts arrived between 4 and 17 days after the quit date. Sending a crave text, being female, younger, and setting a quit date on a Saturday were associated with shorter time to sending a first lapse text. CONCLUSIONS: Text-based lapse support should be developed and evaluated, especially for women. Smokers may benefit from additional support to prevent lapses on days 4-17 postquit attempt.