Subjective prior distributions for modeling longitudinal continuous outcomes with non-ignorable dropout

Nonstandard conditionally specified models for nonignorable missing data

We consider data-analysis settings where data are missing not at random. In these cases, the two basic modeling approaches are 1) pattern-mixture models, with separate distributions for missing data and observed data, and 2) selection models, with a distribution for the data preobservation and a missing-data mechanism that selects which data are observed. These two modeling approaches lead to distinct factorizations of the joint distribution of the observed-data and missing-data indicators. In this paper, we explore a third approach, apparently originally proposed by J. W. Tukey as a remark in a discussion between Rubin and Hartigan, and reported by Holland in a two-page note, which has been so far neglected.

Data analyses typically rely upon assumptions about the missingness mechanisms that lead to observed versus missing data, assumptions that are typically unassessable. We explore an approach where the joint distribution of observed data and missing data are specified in a nonstandard way. In this formulation, which traces back to a representation of the joint distribution of the data and missingness mechanism, apparently first proposed by J. W. Tukey, the modeling assumptions about the distributions are either assessable or are designed to allow relatively easy incorporation of substantive knowledge about the problem at hand, thereby offering a possibly realistic portrayal of the data, both observed and missing. We develop Tukey’s representation for exponential-family models, propose a computationally tractable approach to inference in this class of models, and offer some general theoretical comments. We then illustrate the utility of this approach with an example in systems biology.

Binary variable multiple-model multiple imputation to address missing data mechanism uncertainty: application to a smoking cessation trial

The true missing data mechanism is never known in practice. We present a method for generating multiple imputations for binary variables, which formally incorporates missing data mechanism uncertainty. Imputations are generated from a distribution of imputation models rather than a single model, with the distribution reflecting subjective notions of missing data mechanism uncertainty. Parameter estimates and standard errors are obtained using rules for nested multiple imputation. Using simulation, we investigate the impact of missing data mechanism uncertainty on post-imputation inferences and show that incorporating this uncertainty can increase the coverage of parameter estimates. We apply our method to a longitudinal smoking cessation trial where nonignorably missing data were a concern. Our method provides a simple approach for formalizing subjective notions regarding nonresponse and can be implemented using existing imputation software.

Impact of plain packaging of tobacco products on smoking in adults and children: an elicitation of international experts' estimates

BACKGROUND

Governments sometimes face important decisions in the absence of direct evidence. In these cases, expert elicitation methods can be used to quantify uncertainty. We report the results of an expert elicitation study regarding the likely impact on smoking rates in adults and children of plain packaging of tobacco products.

METHODS

Thirty-three tobacco control experts were recruited from the UK (n = 14), Australasia (n = 12) and North America (n = 7). Experts' estimates were individually elicited via telephone interviews, and then linearly pooled. Elicited estimates consisted of (1) the most likely, (2) the highest possible, and (3) the lowest possible value for the percentage of (a) adult smokers and (b) children trying smoking, two years after the introduction of plain packaging (all other things being constant) in a target country in the expert's region of residence.

RESULTS

The median estimate for the impact on adult smoking prevalence was a 1 percentage point decline (99% range 2.25 to 0), and for the percentage of children trying smoking was a 3 percentage point decline (99% range 6.1 to 0), the latter estimated impact being larger than the former (P < 0.001, sign test). There were no differences in either estimate by region (I2: Adults: 0; Children: 0) but there was considerable variability between experts' estimates within regions (I2: Adults: 0.91; Children: 0.89).

CONCLUSIONS

In the absence of direct evidence for the impact of introducing plain packaging on smoking rates in adults and children, this study shows that tobacco control experts felt the most likely outcomes would be a reduction in smoking prevalence in adults, and a greater reduction in the numbers of children trying smoking, although there was substantial variability in the estimated size of these impacts. No experts judged an increase in smoking as a likely outcome.

Addressing Missing Data Mechanism Uncertainty using Multiple-Model Multiple Imputation: Application to a Longitudinal Clinical Trial

We present a framework for generating multiple imputations for continuous data when the missing data mechanism is unknown. Imputations are generated from more than one imputation model in order to incorporate uncertainty regarding the missing data mechanism. Parameter estimates based on the different imputation models are combined using rules for nested multiple imputation. Through the use of simulation, we investigate the impact of missing data mechanism uncertainty on post-imputation inferences and show that incorporating this uncertainty can increase the coverage of parameter estimates. We apply our method to a longitudinal clinical trial of low-income women with depression where nonignorably missing data were a concern. We show that different assumptions regarding the missing data mechanism can have a substantial impact on inferences. Our method provides a simple approach for formalizing subjective notions regarding nonresponse so that they can be easily stated, communicated, and compared.

Missing value imputation in longitudinal measures of alcohol consumption

Attrition in longitudinal studies can lead to biased results. The study is motivated by the unexpected observation that alcohol consumption decreased despite increased availability, which may be due to sample attrition of heavy drinkers. Several imputation methods have been proposed, but rarely compared in longitudinal studies of alcohol consumption. The imputation of consumption level measurements is computationally particularly challenging due to alcohol consumption being a semi‐continuous variable (dichotomous drinking status and continuous volume among drinkers), and the non‐normality of data in the continuous part. Data come from a longitudinal study in Denmark with four waves (2003–2006) and 1771 individuals at baseline. Five techniques for missing data are compared: Last value carried forward (LVCF) was used as a single, and Hotdeck, Heckman modelling, multivariate imputation by chained equations (MICE), and a Bayesian approach as multiple imputation methods. Predictive mean matching was used to account for non‐normality, where instead of imputing regression estimates, "real" observed values from similar cases are imputed. Methods were also compared by means of a simulated dataset. The simulation showed that the Bayesian approach yielded the most unbiased estimates for imputation. The finding of no increase in consumption levels despite a higher availability remained unaltered.