A Bayesian transition model for missing longitudinal binary outcomes and an application to a smoking cessation study

Abstract

Smoking cessation intervention studies often produce data on smoking status at discrete follow-up assessments, often with missing data in different amounts at each assessment. Smoking status in these studies is a dynamic process with individuals transitioning from smoking to abstinent, as well as abstinent to smoking, at different times during the intervention. Directly assessing transitions provides an opportunity to answer important questions like ‘Does the proposed intervention help smokers remain abstinent or quit smoking more effectively than other interventions?’ In this article, we model changes in smoking status and examine how interventions and other covariates affect the transitions. We propose a Bayesian approach for fitting the transition model to the observed data and impute missing outcomes based on a logistic model, which accounts for both missing at random (MAR) and missing not at random (MNAR) mechanisms. The proposed Bayesian approach treats missing data as additional unknown quantities and samples them from their posterior distributions. The performance of the proposed method is investigated through simulation studies and illustrated by data from a randomized controlled trial of smoking cessation interventions. Finally, posterior predictive checking and log pseudo marginal likelihood (LPML) are used to assess model assumptions and perform model comparisons, respectively.

1. Introduction

Tobacco smoking continues to be a pressing healthcare problem because it significantly shortens the lifespan of smokers (Jha and Peto, 2014). According to Papadakis et al. (2010), there is still a need for low-cost, easily implemented and effective interventions. Smoking cessation studies are commonly used to investigate interventions which may help smokers to quit smoking, for example, see the meta-analyses on self-help interventions by Fiore et al. (2008) and Hartmann-Boyce (2014). In a recent randomized controlled trial by Brandon et al. (2016) (see Section 4 for more details), smoking status (‘smoker’ versus ‘abstinent’ using 7-day-point-prevalence) was assessed every 6 months for 2 years for participants randomly assigned to one of three treatment conditions. Longitudinal data analysis of binary smoking status at multiple times needs to adequately account for the correlations of the responses from each, and deal with missing responses that often arise in such studies.

There is a vast literature for analysing repeated binary data with missing values. We refer to Daniels and Hogan (2008) for a comprehensive review. Most modelling approaches focus on understanding covariates’ effects on the binary response, where they typically equate the expected value of the response variable to a systematic component, which specifies covariates in a linear predictor function. Developed methodologies of this category include weighted generalized estimation equation (GEE), multiple imputation combined with GEE (Robins et al., 1995; Carpenter et al., 2006; Beunckens et al., 2008), multiple imputation coupled with generalized linear mixed effects model (GLMM; Diggle et al., 2002; Jansen et al., 2006), and marginalized transition models (Azzalini, 1994; Heagerty, 2002; Kurland and Heagerty, 2004). The marginalized transition approach combines a mean model for the response with a conditional mean model that describes a serial dependence and treats the serial dependence as a nuisance quantity.

Another class of transition models (Steele et al., 2004; Steele, 2011; Yeh et al., 2012) focus on quantifying the variations in the expected value of the response variable in response to covariates as well as past responses. We refer to this class of models as multilevel transition models and summarize it in the following:

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(P\left(Y_{ti}=1|Y_{t-1,i}=0\right)\right)={\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{0i},$$

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(P\left(Y_{ti}=0|Y_{t-1,i}=1\right)\right)={\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i},$$

where logit(·) = exp(·)/(1 + exp(·)), Y_ti is a binary variable indicating the state for individual i at time t, x_0ti and x_1ti are covariates, β⁰¹ and β¹⁰ are regression coefficients, and u_0i and u_1i are time-invariant random effects. Probabilities P(Y_ti =1| Y_t−1,i 0) and P(Y_ti = 0 Y_t−1,i = 1) are referred as transition probabilities. This class of models has potential to better understand the dynamics of smoking status using transition probabilities rather than only the means of responses. Figure 1 presents an example of smoking status transitions with all participants smoking at the start of the two-group comparison study. In both groups, 80% continue to smoke and 20% transition to abstinent at the 6-month assessment. The two groups diverge between the 6-month and 12-month assessments. In Group 1, 25% of smokers at 6 months transition to abstinent at 12 months and 20% of those abstinent at 6 months transition to smokers at 12 months. In Group 2, 40% of smokers at 6 months transition to abstinent at 12 months and 80% of those abstinent at 6 months transition to smokers at 12 months. At 12 months, both groups will have 64% smokers and 36% abstinent. However, there are group differences in the composition of the smokers. In Group 1, 60% of smokers at 12 months were continuous smokers (i.e., also smoked at 6 months). In Group 2, only 48% of smokers at 12 were continuous smokers. In terms of those abstinent at 12 months, Group 2 has a larger proportion who quit smoking later in the study. Without missing responses, the joint likelihood of observing {y_i,i = 1,..., n} under a multilevel transition model is equivalent to that under a GLMM where

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(P\left(Y_{ti}=1\right)\right)=\left({\mathbf{x}}_{0ti}^T-y_{t-1}{{}_{,}}_i{\mathbf{x}}_{0ti}^T\right){\mathbf{\beta }}^{01}-y_{t-1,i}{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+\left(1-y_{t-1,i}\right)u_{0i}-y_{t-1,i}{u}_{1i}.$$

However, when responses are partially missing, the likelihood marginalized over missing responses under a multilevel transition event model can no longer find an equivalence under the GLMM framework. Hence multiple imputations coupled with GLMM can no longer apply here. To our knowledge, extensions of the multilevel transition models that accommodate missing responses are limited.

Figure 1.

Smoking status transition map for Group 1 and 2

Notes: S–smoking; A—abstinent; arrows are the transitions; the numeric values above the arrows are transition percentages; the numeric values after smoking statues are the overall percentages of smoking and abstinent.

In this article, we propose a Bayesian approach for extending the multilevel transition model to account for missing responses. Typically, Bayesian methods for missing longitudinal responses require a specification of the joint distribution of the responses and the missing indicators. Once the missing model is specified, missing responses can be augmented as additional unknown parameters and be sampled together with other parameters. We refer to Daniels and Hogan (2008) for a review of Bayesian missing imputation which has become a highly useful paradigm for handling missing values in many settings as it accounts for the uncertainty in imputation naturally. We further refer to Ibrahim and Molenberghs (2009) for a comprehensive review of missing data modelling in longitudinal studies. Notably, there are three important missing data mechanisms: missing completely at random (MCAR) when missingness is unrelated to the data, missing at random (MAR) when missingness depends on the observed data only, and missing not at random (MNAR) when missingness depends on the unobserved data, perhaps in addition to the observed data. A favourite type of model that accommodates the aforementioned three missing mechanisms is the selection model (Diggle and Kenward, 1994; Ibrahim et al., 2001), which specifies a conditional distribution, for example, logistic distribution, for the indicators of missing responses, given hypothetical complete data. In this work, we combine the multilevel transition models for the responses with logistic regressions for the missing indicators.

For the proposed joint modelling of transition events and missing responses, we developed a new Bayesian approach for fitting the models while employing Bayesian multiple imputations for missing values. In our proposed Bayesian approach, we utilize an approximation of a particular class of t-distributions to a class of logistic distributions (Kinney and Dunson, 2007), which devises conjugate conditional distributions for most of the parameters. The resulting MCMC mixes quickly after a short burn-in, which is appealing for imputing a large number of missing responses. The posterior samples allow us to compute the posterior mean probabilities of each smoking patterns, together with their credible intervals. Using the posterior samples, we are also able to assess models’ goodness of fit based on a posterior predictive checking method (Gelman et al., 2014; Xu et al., 2016) and perform model comparisons by log pseudo-marginal likelihood (LPML) (Geisser and Eddy, 1979; Linero and Daniels, 2015) and Watanabe-Akaike information criterion (WAIC) (Watanabe, 2010). We present simulation results to demonstrate the performance of the proposed algorithm and model selection criteria. We also demonstrate a real data analysis using data from the randomized controlled trial by Brandon et al. (2016).

The rest of the article is organized as follows: Section 2 presents the details of the model development, which include model base, prior construction, posterior inference and model assessment; Section 3 evaluates the proposed method using simulation studies; Section 4 illustrates the data analyses for a recent randomized controlled intervention trial for smoking cessation; we conclude the article with a summary in Section 5.

2. Model development

2.1. Model base

Two states of smoking behaviour are considered in our study: abstinent (A) and smoking (S). Let Y_ti be a binary variable indicating the state for individual i at time t; Y_ti = 0 for abstinent and Y_ti = 1 for smoking. The probabilities of transitioning from a state at time t − 1 to either smoking or abstinent are:

$$p_{ti}^{01}=P\left(Y_{ti}=1|Y_{t-1,i}=0\right),p_{ti}^{00}=P\left(Y_{ti}=0|Y_{t-1,i}=0\right),$$ (2.1)

$$p_{ti}^{10}=P\left(Y_{ti}=0|Y_{t-1,i}=1\right),p_{ti}^{11}=P\left(Y_{ti}=1|Y_{t-1,i}=1\right),$$ (2.2)

where $p_{ti}^{00}=1-p_{ti}^{01}$, $p_{ti}^{11}=1-p_{ti}^{10}$,t = 1...,m and i = 1,..., n. Suppose all initial states y_0i are known. If all initial states are ‘smoking’, using the definition of transition probabilities in equations (2.1) and (2.2), the probability of the pattern S → A → A → A → A for the first four states is $p_{1i}^{10}{p}_{2i}^{00}{p}_{3i}^{00}{p}_{4i}^{00}$ and the probability of the pattern S → S → S → S → S is $p_{1i}^{11}{p}_{2i}^{11}{p}_{3i}^{11}{p}_{4i}^{11}$. All other types of patterns are feasible following these notations. The class of multilevel transition models we consider here assumes logistic regression models for the transition probabilities,

$$\log \left(\frac{p_{ti}^{01}}{1-p_{ti}^{01}}\right)={\mathbf{x}}_{oti}^T{\mathbf{\beta }}^{01}+u_{0i}$$ (2.3)

$$\log \left(\frac{p_{ti}^{10}}{1-p_{ti}^{10}}\right)={\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i}$$ (2.4)

where x_0ti and x_1ti (including intercepts) are covariates that may or may not differ from each other, β⁰¹ and β¹⁰ are regression coefficients, u_0i and u_1i are time-invariant random effects, allowing unobserved heterogeneities among individuals in their probabilities of transitioning from state 0 to 1 and 1 to 0, respectively. Random vectors u_i = (u_0i, u_1i),i = 1,...,n are assumed to follow a bivariate normal distribution independently, that is, ${\mathbf{u}}_i\stackrel{ind.}{~}N\left(\mathbf{0},\sum \right)$, where Σ is a two-dimensional covariance matrix.

As we mentioned in the introduction, the observed dataset has a large portion of missing values of the response variable, among which most are non-monotonic. Let δ_ti be a binary indicator which equals 1 if the state variable y_ti is observed and equals 0 if y_ti is missing. Define π_ti = P(δ_ti = 1). We consider a logistic regression model for the missing indicators,

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ti}\right)={\mathbf{\alpha }}_1^T{\mathbf{z}}_{ti}+{\mathbf{\alpha }}_2^T{\mathbf{y}}_i^{*},$$ (2.5)

where z_ti (z_t,i,1, ... , z_t,i,p) is a covariate vector and ${\mathbf{y}}_i^{*}=\left(y_{ti},y_{t-1,i}\right)$. The aforementioned model implies MCAR when all components in α₁ and α₂ except the intercept equal zero, MAR when α₂ equals zero and MNAR otherwise. We refer to Ibrahim et al. (2001) for other flexible missing data models and our method can be extended to consider those missing models. An important issue related to complicated missing data mechanism is model assumption validation. In Section 2.4, we propose a posterior predictive checking for validating the transition model and the missing data model. Let β = {β⁰¹, β¹⁰} and α = {α₁, α₂}. Denote y_obs,i and y_mis,i as the observed and missing responses for individual i, respectively. Denote δ_i = (δ_1i,..., δ_mi) as missing indicators for individual i at m time points, x_i = (x_1i, ..., x_mi) and z_i = (z_1i,..., z_mi) as the corresponding set of covariates. We write the observed dataset as $\mathcal{D}=\left\{\left({\mathbf{y}}_{obs,i},{\mathbf{x}}_i,{\mathbf{z}}_i,{\delta }_i\right);i=1,\dots ,n\right\}$.

2.2. Prior specifications

Fully Bayesian methods require specifying priors for all the model parameters, which include β = (β⁰¹, β¹⁰), α and Σ. We assume independent multivariate Normal priors for β⁰¹, β¹⁰, and α, that is, ${\mathbf{\beta }}^{01}~N\left(\mathbf{0},c_{\mathbf{\beta }}^2\mathbf{I}\right)$, ${\mathbf{\beta }}^{10}~N\left(\mathbf{0},c_{\mathbf{\beta }}^2\mathbf{I}\right)$ and $\mathbf{\alpha }~N\left(\mathbf{0},c_{\mathbf{\alpha }}^2\mathbf{I}\right)$, where $c_{\beta }^2$ and $c_{\alpha }^2$ are scalars. Larger values of $c_{\beta }^2$ and $c_{\alpha }^2$ lead to less informative priors on the corresponding parameters. Our simulations in Section 3 show that setting $c_{\beta }^2=c_{\alpha }^2=5^2$ allows accurate estimations of the parameters for all scenarios.

For the covariance matrix Σ, we adopt the hierarchical half-t prior in Huang and Wand (2013), which is a multivariate extension of the half-t prior in Gelman (2006). An important advantage of the hierarchical prior over the commonly used inverse Wishart prior is that it allows weakly informative priors on the standard deviation terms and imposes less restrictions on the relationships between standard deviations and correlations. A hierarchical half-t prior for two-dimensional Σ is written as follows:

$$\sum ~IW\left(a+1,2a\mathrm{\Lambda }\right),$$

$$\mathrm{\Lambda }=diag\left({\lambda }_1,{\lambda }_2\right),$$

$${\lambda }_j\stackrel{ind}{~}\mathrm{\Gamma }\left(\frac{1}{2},\frac{1}{{\varsigma }_j^2}\right),$$

where a is the degree of freedom, $1/{\varsigma }_j^2$ is the rate parameter of the Gamma distribution $\mathrm{\Gamma }\left(\cdot ,\cdot \right)$, and IW(·, ·) represents inverse Wishart distribution. Denote the standard deviation terms as ${\sigma }_j=\sqrt{{\sum }_{jj}}$, j = 1,2 and the correlation term as $\rho ={\sum }_{12}/\sqrt{{\sum }_{11}{\sum }_{22}}$

According to Huang and Wand (2013), the joint distribution of σ₁, σ₂, and ρ is

$$\begin{gathered} p\left({\sigma }_1,{\sigma }_2,\rho \right)\propto {\left(1-{\rho }^2\right)}^{-\frac{a}{2}-2}{\sigma }_1^{-a-2}{\sigma }_2^{-a-2}{\left[-\frac{a}{{\left(1-\rho \right)}^2{\sigma }_1^2}+\frac{1}{{\varsigma }_1^2}\right]}^{-\left(a+2\right)/2} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[-\frac{a}{{\left(1-\rho \right)}^2{\sigma }_2^2}+\frac{1}{{\varsigma }_2^2}\right]}^{-\left(a+2\right)/2},{\sigma }_1,{\sigma }_2>0,-1<\rho <1. \end{gathered}$$

The marginal distribution of σ_j is Half-t(a, ς_j), which has density

$$p\left({\sigma }_j\right)\propto {\{1+{\left({\sigma }_j/{\varsigma }_j\right)}^2/a\}}^{-\left(a+1\right)/2},$$

and

$$\mathrm{E}\left({\sigma }_{jj}\right)=2{\varsigma }_j\sqrt{\frac{a}{\pi }}\frac{\mathrm{\Gamma }\left(\frac{a+1}{2}\right)}{\mathrm{\Gamma }\left(\frac{a}{2}\right)\left(a-1\right)},\mathrm{V}\mathrm{a}\mathrm{r}\left({\sigma }_{jj}\right)={\varsigma }_j^2\frac{a}{a-2}\frac{4a}{\pi {\left(a-1\right)}^2}{\left(\frac{\mathrm{\Gamma }\left(\frac{a+1}{2}\right)}{\mathrm{\Gamma }\left(\frac{a}{2}\right)}\right)}^2.$$

Therefore, larger values of ${\varsigma }_j^2$ lead to less informative priors on the corresponding standard deviation term. Huang and Wand (2013) recommend setting ${\varsigma }_j^2={10}^{\mathit{5}}$, which leads to a weakly informative prior on the standard deviation term. The marginal distribution of the correlation parameter ρ has density

$$p\left(\rho \right)\propto {\left(1-{\rho }^2\right)}^{\frac{a}{2}-1}.$$

When a = 2, the marginal distributions for the correlations are uniform distributions over [−1, 1]. Since the conditional distribution of ρ given σ₁ and σ₂ is symmetric around 0, E(ρ|σ₁, σ₂) = 0. Therefore, Σ1₂ = ρσ₁σ₂ also has prior mean at 0. We set a = 2 throughout simulations and data analyses.

In summary, the joint prior of the parameters in $\mathrm{\Theta }=\left\{\mathbf{\beta },\sum ,\mathbf{\alpha }\right\}$ is

$$\begin{gathered} N\left({\mathbf{\beta }}^{01};0,c_{\mathbf{\beta }}^2\mathbf{I}\right)N\left({\mathbf{\beta }}^{10};0,c_{\mathbf{\beta }}^2\mathbf{I}\right)N\left(\mathbf{\alpha };0,c_{\mathbf{\alpha }}^2\mathbf{I}\right) \\ \times IW\left(\sum ;a+1,2a\mathrm{\Lambda }\right)\prod _{j=1}^2\mathrm{\Gamma }\left({\lambda }_j;\frac{1}{2},\frac{1}{{\zeta }_1^2}\right). \end{gathered}$$ (2.6)

2.3. Bayesian multiple imputations and posterior inference

Denote $\mathcal{M}$ as the set of indices ti for all the missing responses. Let u = {u_i,i = 1,..., n} be the set of random effects. The joint likelihood $L\left(\mathbf{u},y_{ti},\left(t,i\right)\in \mathcal{M},\mathrm{\Theta }|\mathcal{D}\right)$ is

$$\begin{gathered} \prod _{i=1}^n\prod _{t=1}^m\left[{\left(p_{ti}^{01}\right)}^{y_{ti}\left(1-y_{t-1,i}\right)}{\left(p_{ti}^{00}\right)}^{\left(1-y_{ti}\right)\left(1-y_{t-1,i}\right)}{\left(p_{ti}^{11}\right)}^{y_{ti}{y}_{t-1,i}}{\left(p_{ti}^{10}\right)}^{\left(1-y_{ti}\right)y_{t-1,i}}\right] \\ \times {\pi }_{ti}^{{\delta }_{ti}}{\left(1-{\pi }_{ti}\right)}^{1-{\delta }_{ti}}N\left({\mathbf{u}}_i;\mathbf{0},\sum \right) \end{gathered}$$ (2.7)

where ${\pi }_{ti}=P\left({\delta }_{ti}=1\right)$. To further exploit conjugacy in the exponential families for MCMC, we introduce auxiliary variables ω_ti,t = 1,..., m; i = 1,..., n, which follow logistic distributions with location parameters as $\left[{\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{01}\right]I\left(y_{t-1,i}=0\right)+\left[{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i}\right]I\left(y_{t-1,i}=1\right)$, where I(·) is an indicator function taking values 0 and 1. The distribution of ω_ti can be approximated by $N\left({\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{01},{\tilde{\sigma }}^2/{\varphi }_{ti}\right)l\left(y_{t-1,i}=0\right)+N\left({\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{11},{\tilde{\sigma }}^2/{\varphi }_{ti}\right)l\left(y_{t-1,i}=1\right)$, ${\tilde{\sigma }}^2={\pi }^2\left(c-2\right)/3c$, ${\varphi }_{ti}~\mathrm{\Gamma }\left(c/2,c/2\right)$, π = 3.1416, and c = 7.3, for which the approximation error is negligible and can be corrected by importance sampling (Kinney and Dunson, 2007). Probabilities $p_{ti}^{01}$ and $p_{ti}^{10}$ are then approximated by $\mathrm{\Phi }\left({\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{0i}\right)$ and $\mathrm{\Phi }\left({\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i}\right)$, where Ф(·) is the standard normal cumulative distribution function. Utilizing the auxiliary variables ϕ = {ϕ_ti,t = 1, ... , m; i = 1, ... , n} and ω = {ω_ti,t = 1, ..., m; i = 1, ... , n}, the augmented likelihood $L\left(\mathbf{\omega },\mathbf{u},\mathbf{\varphi },y_{ti},\left(t,i\right)\in \mathcal{M},\mathrm{\Theta }|\mathcal{D}\right)$ is

$$\begin{gathered} \prod _{i=1}^n\prod _{t=1}^m\{N\left({\omega }_{ti};{\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{0i},\frac{{\tilde{\sigma }}^2}{{\varphi }_{ti}}\right)\left(1-y_{t-1,i}\right)\left[I\left({\omega }_{ti}>0\right)y_{ti}+I\left({\omega }_{ti}\le 0\right)\left(1-y_{ti}\right)\right] \\ +N\left({\omega }_{ti};{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i},\frac{{\tilde{\sigma }}^2}{{\varphi }_{ti}}\right)y_{t-1,i}\left[I\left({\omega }_{ti}\le 0\right)y_{ti}+I\left({\omega }_{ti}>0\right)\left(1-y_{ti}\right)\right]\} \\ \times \mathrm{\Gamma }\left({\varphi }_{ti};\frac{c}{2},\frac{c}{2}\right){\pi }_{ti}^{\delta ti}{\left(1-{\pi }_{ti}\right)}^{1-{\delta }_{ti}}\prod _{i=1}^{n}N\left({\mathbf{u}}_i;\mathbf{0},\sum \right). \end{gathered}$$ (2.8)

We draw posterior samples of each variable in the set $\{\mathbf{\omega },\mathbf{u},\mathbf{\varphi },y_{ti},\left(t,i\right)\in \mathcal{M},\mathrm{\Theta },\mathrm{\Lambda }\}$ from its corresponding conditional posterior distribution, that is, conditional distribution given observed data and the rest of the variables in the set. The following steps are repeated to obtain a desired number of samples.

Sample missing y_lis: For $\left(l,i\right)\in \mathcal{M}$, We sample y_li from a Bernoulli distribution with P(y_li = 1) and P(y_li = 0) proportional to

$$\prod _{t=1}^m\left\{{\left[{\left(p_{ti}^{01}\right)}^{yti}{\left(p_{ti}^{00}\right)}^{1-y_{ti}}\right]}^{\left(1-y_{t-1,i}\right)}{\left[{\left(p_{ti}^{10}\right)}^{1-y_{ti}}{\left(p_{ti}^{00}\right)}^{y_{ti}}\right]}^{y_{t-1,i}}{\pi }_{ti}^{{\delta }_{ti}}{\left(1-{\pi }_{ti}\right)}^{1-{\delta }_{ti}}\right\}$$ (2.9)

with y_li evaluated at 1 and 0 respectively.

Sampleω_tis: We sample ω_ti from its conditional posterior distribution that is proportional to

$$\{\begin{array}{c}N\left({\omega }_{ti};{\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{0i},\frac{{\tilde{\sigma }}^2}{{\varphi }_{ti}}\right)\left[I\left({\omega }_{ti}>0\right)y_{ti}+I\left({\omega }_{ti}<0\right)\left(1-y_{ti}\right)\right];if\hspace{0.17em}{y}_{t-1,i}=0,\\ N\left({\omega }_{ti};{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i},\frac{{\tilde{\sigma }}^2}{{\varphi }_{ti}}\right)\left[I\left({\omega }_{ti}\le 0\right)y_{ti}+I\left({\omega }_{ti}>0\right)\left(1-y_{ti}\right)\right];if\hspace{0.17em}{y}_{t-1,i}=1.\end{array}$$

Sample u_is: Define ${\zeta }_{0i}={\sum }_{t=1}^m\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}\left(1-y_{t-1,i}\right)\left({\omega }_{ti}-{\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}\right)$, ${\zeta }_{1i}={\sum }_{t=1}^m\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}{y}_{t-1,i}\left({\omega }_{ti}-{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}\right)$, ${\kappa }_{0i}={\sum }_{t=1}^m\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}\left(1-y_{t-1,i}\right)$ and ${\kappa }_{1i}={\sum }_{t=1}^m\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}{y}_{t-1,i}$. Sample u_i from a bivariate normal distribution N(µi,K_i) where ${\mu }_i={\Omega }_i{\left({\zeta }_{0i},{\zeta }_{1i}\right)}^T$ and

$${\Omega }_i={\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\kappa }_{0i},{\kappa }_{1i}\right)+{\sum }^{-1}\right]}^{-1}.$$

Sample ϕ_tis: Sample ϕ_ti from $\mathrm{\Gamma }=\left(\frac{c}{2}+0.5,\frac{A_{ti}}{2{\tilde{\sigma }}^2}+\frac{c}{2}\right)$ where $A_{ti}={\left({\omega }_{ti}-{\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}-u_{0i}\right)}^{2}I\left(y_{t-1,i}=0\right)+{\left({\omega }_{ti}-{\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}-u_{1i}\right)}^{2}I\left(y_{t-1,i}=1\right)$ and c = 7.3.

Sample Σ: Sample Σ from

$$IW(n+a+1,\sum _{i=1}^n{\mathbf{u}}_i{\mathbf{u}}_i^T+2a\mathrm{\Lambda }),$$

where a is the degree of freedom in the hierarchical half-t prior.

Sample Λ: Sample diagonal value ʎ_j from $\mathrm{\Gamma }\left(\left(a+2\right)/2,\frac{1}{{\varsigma }_j^2}+a{\sum }_{jj}^{-1}\right)$, where ${\sum }_{jj}^{-1}$ is the jth diagonal value of Σ⁻¹.

Sample β: Sample β⁰¹ from N(ѱ₀B₀, ѱ₀) where

$${\mathrm{\Psi }}_0^{-1}=\sum _{t=1}^m\sum _{i\in S_{0t}}\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}{\mathbf{x}}_{0ti}{\mathbf{x}}_{0ti}^T\frac{1}{c_{\mathbf{\beta }}^2}\mathbf{I}+\hspace{0.17em}{B}_0=\sum _{t=1}^m\sum _{i\in S_{0t}}\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}\left({\omega }_{ti}-u_{0i}\right){\mathbf{x}}_{0ti},$$

and S_0t is the set of indices for individuals whose states are 0 at time t − 1. Similarly, define S_1t is the set of indices for individuals whose states are 1 at time t − 1. We sample β¹⁰ from N(ѱ₁B₁, ѱ₁) where

$${\mathrm{\Psi }}_1^{-1}=\sum _{t=1}^m\sum _{i\in S_{1t}}\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}{\mathbf{x}}_{1ti}{\mathbf{x}}_{1ti}^T\frac{1}{c_{\mathbf{\beta }}^2}\mathbf{I},\hspace{0.17em}{B}_1=\sum _{t=1}^m\sum _{i\in S_{1t}}\frac{{\varphi }_{ti}}{{\tilde{\sigma }}^2}\left({\omega }_{ti}-u_{1i}\right){\mathbf{x}}_{1ti}.$$

Sample α: We propose a Metropolis–Hasting step for updating α. The conditional distribution of α given y, u, $\mathrm{\Theta }$, and $\mathcal{D}$, denoted as p(α|·), is proportional to

$${\prod _{i=1}^n\prod _{t=1}^m{\pi }_{ti}^{{\delta }_{ti}}\left(1-{\pi }_{ti}\right)}^{1-{\delta }_{ti}}N\left(\mathbf{\alpha };0;c_{\mathbf{\alpha }}^2\mathbf{I}\right).$$

Let α^∗ be the latest accepted value for α. Sample α′ from proposal distribution N(α^∗, V) and accept α′ with probability min{p(α′|.)/p(α*|.), 1}. The automatic tuning of V in Haario et al. (2005) is used in this article to get acceptance rate in the 20%–50% range. Specifically, let the sequence α⁽¹⁾, α⁽²⁾,... be the states of the Markov chain for α. When deciding the k-th state α, we sample α^j ~ N(α^(k−1), V(k)) with

$${\mathbf{V}}^{\left(k\right)}=\{\begin{array}{ll}{\mathbf{V}}^{\left(0\right)}\hfill & k<k_0,\hfill \\ sVar\left\{{\mathbf{\alpha }}^{\left(1\right)},\dots ,{\mathbf{\alpha }}^{\left(k-1\right)}\right\}+s_0{\mathbf{I}}_q,\hfill & k>k_0,\hfill \end{array}$$

where q is the dimension of α, s is recommended to be 2.4²/q, k₀ is the number of burn-in iterations, s₀ is a small constant, V⁽⁰⁾ is the initial variance of the proposal distribution and I_q is an identity matrix.

We point out that a recently developed Polya–Gamma data augmentation approach (Polson et al., 2013) can also yield a Gibbs sampler for the Bayesian logistic-regression type model. It is our future interest to compare the two augmentation approaches in extensions of the transition event model.

2.4. Posterior predictive checking and model comparison

Posterior predictive checking:

To assess the goodness of fit of our models, we adopt the posterior predictive checking approach in Gelman et al. (2014, Chapter 6) and Xu et al. (2016). The idea is that the replicated data generated under the model should look similar to observed data if the model fits. In our scenario, observed data include y_obs and δ. Define ${\mathbf{y}}_{obs}^{rep}$ and δ^rep as the replicated data that could have been observed, or we would see if the data generating process for y_obs and δ were replicated with the same model (including model covariates and examination times) and the same value of model parameters. As an illustration, we define a test quantity $G\left({\mathbf{y}}_{t,obs}\right)={\sum }_{i\in {\mathcal{O}}_t}\hspace{0.17em}{y}_{ti}/n$ as the percentage of individuals smoking at time t among the all responses, where ${\mathcal{O}}_t$ as the set of indices (ti) for all the responses that are not missing at time t. Posterior predictive distribution of the test quantity can be approximated by MCMC samples $G\left({\mathbf{y}}_{t,obs}^{rep,l}\right)={\sum }_{i\in {\mathcal{O}}_t^{rep,l}}\hspace{0.17em}{y}_{t,obs}^{rep,l}/n$, l = 1, …, L. We summarize the posterior predictive distribution by the sample mean, 2.5% quantile and 97.5% quantile of the samples. Since we are interested in smoking status transitions and missing model estimation, we also compare observed transition percentages to simulated transition percentages and observed missing percentages to simulated missing percentages.

Model comparison:

We compare models using LPML (Geisser and Eddy, 1979) and WAIC (Watanabe, 2010). LPML is a measure of a model’s predictive ability and WAIC is a fully Bayesian approach for estimating point wise out-of-sample prediction accuracy from a fitted Bayesian model. More recently, Linero and Daniels (2015) used LPML to compare models for longitudinal studies with missing data. Denote y_obs and y_mis as the observed and missing responses respectively, y_{obs, i} and δ_−i as the observed responses and missing indicators for those with individual i removed.

By definition,

$$\mathrm{L}\mathrm{P}\mathrm{M}\mathrm{L}=\sum _{i=1}^n\log \left(p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{y}}_{obs,-i},{\mathbf{\delta }}_{-i}\right)\right),$$

where $p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{y}}_{obs,-i},{\mathbf{\delta }}_{-i}\right)$ is the predictive probability of y_obs,i and δ_i given remaining observed data y_obs,−i and δ_−i. Using the representation in Gelfand and Dey (1994),

$$p{\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{y}}_{obs,-i},{\mathbf{\delta }}_{-i}\right)}^{-1}=\int \frac{1}{p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)}p\left(\mathrm{\Theta },\mathbf{u}|{\mathbf{y}}_{obs,}\mathbf{\delta }\right)d\mathbf{u}d\mathrm{\Theta }.$$

Therefore, LPML can be approximated by

$$\frac{1}{L}\sum _{l=1}^L\frac{1}{p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{u}}_i^{\left(l\right)},{\mathrm{\Theta }}^{\left(l\right)}\right)}$$ (2.10)

where $\{u_i^{\left(l\right)},{\mathrm{\Theta }}^{\left(l\right)},l=1,\dots ,L\}$ are MCMC iterates.

By definition,

$$\mathrm{W}\mathrm{A}\mathrm{I}\mathrm{C}=\sum _{i=1}^n\log \int p\left({\mathbf{y}}_{obs,i,},{\mathbf{\delta }}_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)p\left(\mathrm{\Theta },\mathbf{u}|{\mathbf{y}}_{obs,}\mathbf{\delta }\right)d\mathbf{u}d\mathrm{\Theta }-p_{\mathrm{w}\mathrm{a}\mathrm{i}\mathrm{c}}$$

where p_waic is the effective number of parameters. Two approaches of formulating p_waic have been proposed in literature (Gelman et al., 2014). In this work, we use the variance of individual terms in the log predictive density summed over the n data points, that is,$p_{\mathrm{w}\mathrm{a}\mathrm{i}\mathrm{c}}={\sum }_{i=1}^n\mathrm{V}\mathrm{a}{\mathrm{r}}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}\left(\log \left(p\left({\mathbf{y}}_{obs,i,}{\delta }_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)\right)\right)$. The WAIC value can be approximated by

$$\sum _{i=1}^n\log \left(\frac{1}{L}\sum _{l=1}^{L}p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{u}}_i^{\left(l\right)},{\mathrm{\Theta }}^{\left(l\right)}\right)\right)-\sum _{i=1}^n{V}_{l=1}^L\left(\log \left(p({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{u}}_i^{\left(l\right)},{\mathrm{\Theta }}^{\left(l\right)}\right)\right)$$ (2.11)

where $V_{l=1}^L{a}_1=\frac{1}{L-1}{{\sum }_{l=1}^L\left(a_1-\overline{a}\right)}^2.$

Note that for the missing model we propose,

$$p\left({\mathbf{y}}_{obs,i,}{\mathbf{\delta }}_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)=\sum _{y_{mis,i}}p\left({\mathbf{y}}_{obs,i,}{\mathbf{y}}_{mis,i},{\mathbf{\delta }}_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)$$

and

$$p\left({\mathbf{y}}_i,{\mathbf{\delta }}_i|{\mathbf{u}}_i,\mathrm{\Theta }\right)=\prod _{t=1}^m\left\{{\left[{\left(p_{ti}^{01}\right)}^{y_{ti}}{\left(p_{ti}^{00}\right)}^{1-y_{ti}}\right]}^{\left(1-y_{t-1,i}\right)}{\left[{\left(p_{ti}^{10}\right)}^{1-y_{ti}}{\left(p_{ti}^{00}\right)}^{y_{ti}}\right]}^{y_{t-1,i}}{\pi }_{ti}^{{\mathbf{\delta }}_{ti}}{\left(1-{\pi }_{ti}\right)}^{1-{\delta }_{ti}}\right\}.$$

When comparing two models, a greater LPML or WAIC indicates a better fit of the data.

3. Simulations

In this section, we aim to examine parameter estimation under a correctly specified or partially mis-specified joint model of multilevel transitions and missing imputations. Specifically, we consider (a) both the multilevel transition model and the missing model are correctly specified; (b) the missing model is mis-specified and (c) the random effects distribution is mis-specified. Also, we perform sensitivity analyses on the hyper-parameters and conduct evaluations on the performance of model selection criteria LPML and WAIC for the proposed models.

Simulation I:

In this simulation, we examine parameter estimation under a correctly specified multilevel transition model and missing model. For each simulated dataset, consider n individuals and m follow-up times. Covariate vectors x_sti; s = 0, 1; t = 1, ... , m; i = 1, ... ,n were simulated independently, with their first elements being one, their second elements sampled from N(0, 1) and their third elements sampled from Bernoulli (0.5). Initial states {y_0i,i = 1, ... , n} were simulated from Bernoulli(0.5). We set true regression vectors β⁰¹ = (1.0,−0.5, 0.5) and β¹⁰ = (−1.0, 0.5,−0.5). Random effects u_i,i = 1, ... ,n were sampled independently from a bivariate normal distribution N₂(0, Σ), where the diagonal elements of Σ are 0.6 and 0.4, and the off-diagonal elements are −0.3. Conditional on the simulated u_i and x_sti, we simulated responses from the transition event model:

$$\log \left(\frac{p_{ti}^{01}}{1-p_{ti}^{01}}\right)={\mathbf{x}}_{0ti}^T{\mathbf{\beta }}^{01}+u_{0i}$$

$$\log \left(\frac{p_{ti}^{10}}{1-p_{ti}^{10}}\right)={\mathbf{x}}_{1ti}^T{\mathbf{\beta }}^{10}+u_{1i}$$

where $P\left(Y_{ti}=1|Y_{t-1,i}=0\right)=p_{ti}^{01}$ and $P\left(Y_{ti}=1|Y_{t-1,i}=1\right)=1-p_{ti}^{10}$. Conditional on α and the sampled responses, we simulated the missing indicators from model (2.5),

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ti}=1\right)={\alpha }_1+{\alpha }_2{z}_{ti}+{\alpha }_3{y}_{ti}+{\alpha }_4{y}_{t-1,i},$$

where z_ti was sampled from N(0, 0.5²) and α = (2.0, 0.5, −1.0, −1.0). We consider two follow-up times for m—6 and 10, and two sample sizes for n—200 and 300. The simulated datasets have approximately 30% missing values. The starting values are zeros for β⁰¹, β¹⁰, and random effects, (1, 0, 0) for α and 0.2, 0, 0.2 for Σ₁₁, Σ₁₂, and Σ₂₂, respectively. Hyper-parameters $c_{\mathbf{\beta }}^2$, $c_{\alpha }^2$ and ${\varsigma }_j^2$ were fixed at 5², 5² and 10⁵, respectively. We obtained a chain of 6 000 iterates after a burn-in of 20 000 and thinning of every other 5 iterates. Each simulation scenario was repeated 500 times using the MCMC steps outlined in Section 2.3. For each parameter in β⁰¹, β¹⁰, α and Σ, we present the numerical summaries of posterior mean, bias, mean squared error (MSE) and 95% coverage probability.

Under ‘missing model correctly specified’ in Table 1, we represent the numerical summaries of Simulation I for β⁰¹, β¹⁰, Σ and α. We observe significant reductions in biases and MSEs when sample size or follow-up times increase. We also see the same patterns under a different set-up of α (simulations not shown here). The results suggest that the posterior means under correct model specifications provide a consistent estimation of all parameters. Under the scenario n = 300,m = 10, all parameters except Σ₁₂ have estimated biases ranging from 0.01% to 5% of their true values. For Σ₁₂, the bias is 29% of its true value. In fact, when n = 200, the percentages of bias for Σ₁₂ are 58% and 42% for m = 6 and 10, respectively; when n = 300, they are 55% and 29% for m = 6 and 10, respectively. As we increased m to 20 and kept n at 200, the percentage of bias is 26% (based on simulations not shown in the table). Random effects distributions are generally hard to estimate. The correlation parameter of u_i1 and u_i2 is even more difficult to estimate as it relies on the correct estimations of u_i1 and u_i2 simultaneously for each individual. Without much information about the correlation, the posterior mean of Σ₁₂ is skewed towards its prior mean, which is zero under the hierarchical half-t prior with a = 2. Using a simulated dataset under the setting n = 200,m = 6, we observe that the empirical posterior distributions of Σ₁₁ and Σ₂₂ are skewed toward positive infinity, but the empirical posterior distribution of Σ₁₂ is skewed towards 0. Finally, the coverage probabilities of all parameters except Σ₁₂ are close to 95% and above 93%. Both the bias and skewness of the posterior distribution for Σ₁₂ have impacts on its 95% credible interval and results in their comparatively lower coverage probability. Greater sample sizes or follow-up times (n and m) draw the coverage probabilities of all parameters closer to 95% slightly. In addition to the effect of sample size and follow-up times, prior distributions with less prior variances may lead to greater biases and changes in coverage probabilities, which we will discuss in the section of Simulation III.

Table 1.

Simulated data: True value and Monte Carlo mean, bias, and mean square error (MSE) and 95% coverage probability of the transition event model parameters where α=(2.0, 0.5, −1.0, −1.0)

				Missing model correctly specified				Missing model mis-specified
n	m	Parameter	True Value	Mean	Bias	MSE	95% CP	Mean	Bias	MSE	95% CP
200	6	${\beta }_1^{01}$	1.0	0.999	0.001	0.120	0.936	0.867	0.133	0.147	0.912
		${\beta }_2^{01}$	−0.5	−0.532	0.032	0.066	0.940	−0.544	0.044	0.073	0.932
		${\beta }_3^{01}$	0.5	0.532	0.032	0.216	0.942	0.534	0.034	0.234	0.940
		${\beta }_1^{10}$	−1.0	−1.044	0.044	0.064	0.938	−0.641	0.359	0.196	0.680
		${\beta }_2^{10}$	0.5	0.529	0.029	0.034	0.934	0.567	0.067	0.042	0.924
		${\beta }_3^{10}$	−0.5	−0.497	0.003	0.097	0.944	−0.528	0.028	0.121	0.944
		${\sum }_{11}$	0.6	0.752	0.152	0.207	0.942	1.051	0.451	0.499	0.864
		${\sum }_{12}$	−0.3	−0.126	0.174	0.051	0.860	−0.117	0.183	0.062	0.872
		${\sum }_{22}$	0.4	0.503	0.103	0.098	0.960	0.496	0.096	0.145	0.964
		α1	2.0	2.038	0.038	0.097	0.944	0.689	1.311	1.723	0.000
		α2	0.5	0.500	0.000	0.017	0.952	0.452	0.048	0.017	0.932
		α3	−1.0	−1.014	0.014	0.139	0.934	*	*	*	*
		α4	−1.0	−1.017	0.017	0.028	0.948	*	*	*	*
200	10	${\beta }_1^{01}$	1.0	1.007	0.007	0.084	0.938	0.856	0.144	0.103	0.894
		${\beta }_2^{01}$	−0.5	−0.528	0.028	0.038	0.944	−0.531	0.031	0.040	0.944
		${\beta }_3^{01}$	0.5	0.528	0.028	0.140	0.934	0.527	0.027	0.139	0.944
		${\beta }_1^{10}$	−1.0	−1.042	0.042	0.042	0.934	−0.627	0.373	0.181	0.526
		${\beta }_2^{10}$	0.5	0.516	0.016	0.018	0.942	0.556	0.056	0.024	0.926
		${\beta }_3^{10}$	−0.5	−0.492	0.008	0.063	0.942	−0.526	0.026	0.077	0.950
		${\sum }_{11}$	0.6	0.667	0.067	0.075	0.952	0.868	0.268	0.161	0.864
		${\sum }_{12}$	−0.3	−0.175	0.125	0.028	0.892	−0.17	0.13	0.035	0.904
		${\sum }_{22}$	0.4	0.425	0.025	0.036	0.956	0.45	0.05	0.072	0.950
		α1	2.0	2.020	0.020	0.046	0.956	0.679	1.321	1.748	0.000
		α2	0.5	0.502	0.002	0.010	0.952	0.451	0.049	0.011	0.926
		α3	−1.0	−1.005	0.005	0.067	0.948	*	*	*	*
		α4	−1.0	−1.012	0.012	0.016	0.940	*	*	*	*
300	6	${\beta }_1^{01}$	1.0	0.993	0.007	0.078	0.942	0.853	0.147	0.102	0.904
		${\beta }_2^{01}$	−0.5	−0.540	0.040	0.039	0.958	−0.553	0.053	0.044	0.952
		${\beta }_3^{01}$	0.5	0.546	0.046	0.144	0.952	0.561	0.061	0.157	0.948
		${\beta }_1^{10}$	−1.0	−1.025	0.025	0.039	0.940	−0.614	0.386	0.191	0.498
		${\beta }_2^{10}$	0.5	0.507	0.007	0.019	0.942	0.535	0.035	0.024	0.938
		${\beta }_3^{10}$	−0.5	−0.515	0.015	0.071	0.948	−0.546	0.046	0.086	0.936
		${\sum }_{11}$	0.6	0.668	0.068	0.100	0.954	0.961	0.361	0.269	0.856
		${\sum }_{12}$	−0.3	−0.134	0.166	0.045	0.850	−0.117	0.183	0.057	0.840
		${\sum }_{22}$	0.4	0.457	0.057	0.061	0.976	0.403	0.003	0.080	0.948
		α1	2.0	2.017	0.017	0.051	0.954	0.688	1.312	1.725	0.000
		α2	0.5	0.505	0.005	0.011	0.960	0.459	0.041	0.011	0.932
		α3	−1.0	−1.012	0.012	0.073	0.938	*	*	*	*
		α4	−1.0	−0.999	0.001	0.018	0.948	*	*	*	*
300	10	${\beta }_1^{01}$	1.0	1.011	0.011	0.046	0.947	0.849	0.151	0.065	0.923
		${\beta }_2^{01}$	−0.5	−0.521	0.021	0.024	0.937	−0.526	0.026	0.024	0.943
		${\beta }_3^{01}$	0.5	0.477	0.023	0.079	0.953	0.479	0.021	0.081	0.957
		${\beta }_1^{10}$	−1.0	−1.021	0.021	0.024	0.950	−0.603	0.397	0.182	0.283
		${\beta }_2^{10}$	0.5	0.503	0.003	0.011	0.947	0.538	0.038	0.015	0.920
		${\beta }_3^{10}$	−0.5	−0.511	0.011	0.038	0.933	−0.546	0.046	0.050	0.947
		${\sum }_{11}$	0.6	0.646	0.046	0.045	0.960	0.838	0.238	0.111	0.853
		${\sum }_{12}$	−0.3	−0.214	0.086	0.018	0.897	−0.209	0.091	0.025	0.890
		${\sum }_{22}$	0.4	0.402	0.002	0.023	0.973	0.407	0.007	0.041	0.947
		α1	2.0	2.024	0.024	0.031	0.957	0.686	1.314	1.730	0.000
		α2	0.5	0.505	0.005	0.007	0.947	0.458	0.042	0.008	0.907
		α3	−1.0	−1.014	0.014	0.045	0.960	*	*	*	*
		α4	−1.0	−1.006	0.006	0.011	0.943	*	*	*	*

Note:

^*indicates that the corresponding parameter is not estimated.

Simulation II:

In this simulation, we fitted the simulated datasets from Simulation I again but assumes the MAR assumption for the missing model:

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ti}=1\right)={\alpha }_1+{\alpha }_2{z}_{ti}.$$

Comparing the estimated biases under ‘missing model correctly specified’ (assuming MNAR) and ‘missing model mis-specified’ (assuming MAR) in Table 1, mis-specifying the missing model leads to greater estimated biases for all parameters and the differences are especially significant for the intercept terms of β⁰¹ and β¹⁰, and Σ₁₁. Under the scenario of n = 300,m = 10, estimated biases for the intercept terms of β⁰¹, β¹⁰, and Σ₁₁ are 17%, 18%, and 40% of the true values respectively, which are much higher than the range of 0.01% to 5% from Simulation I. Therefore, we conclude that mis-specifying the missing model can lead to substantial biases in estimation.

Simulation III:

In this simulation, we fitted the simulated datasets from Simulation I to three additional set-ups of hyper-parameters: (a)${\varsigma }_j^2=10$, $c_{\alpha }^2=c_{\mathbf{\beta }}^2=5^2$, (b)${\varsigma }_j^2={10}^2$, $c_{\alpha }^2=c_{\mathbf{\beta }}^2=5^2$ and (c)${\varsigma }_j^2={10}^5$,$c_{\alpha }^2=c_{\mathbf{\beta }}^2=3^2$. Smaller ${\varsigma }_j^2$, $c_{\alpha }^2$ and $c_{\mathbf{\beta }}^2$ correspond to priors with smaller variances. The results are presented in Table 2. Comparing the results in Table 1 under ‘missing model correctly specified’$\left({\varsigma }_j^2={10}^5,c_{\alpha }^2=c_{\mathbf{\beta }}^2=5^2\right)$, prior settings (a), (b) and (c) have smaller MSEs for all parameters but have greater biases and changes in coverage probabilities for some parameters. For example, under ${\zeta }_j^2=10$ and $c_{\alpha }^2=c_{\mathbf{\beta }}^2=5^2$, β⁰¹, β¹⁰ and α have greater biases in general, while Σ₁₁ and Σ₁₂ have slightly smaller biases. Coverage probabilities for β⁰¹, β¹⁰ and α are less close to 95%, compared to Table 1. Similar comparisons are observed for ${\varsigma }_j^2=10$ and 10². Changing $c_{\alpha }^2=c_{\mathbf{\beta }}^2$ from 5² to 3² have small impacts in biases and coverage probabilities. We therefore recommend setting ${\varsigma }_j^2={10}^5$, $c_{\alpha }^2=c_{\mathbf{\beta }}^2=5^2$ and performing sensitivity analysis on hyper-parameters.’

Table 2.

Simulated data: True value and Monte Carlo mean, bias and mean square error (MSE) and 95% coverage probability of the transition event model parameters where α=(2.0, 0.5, −1.0, −1.0)

				${\varsigma }_j^2=10,$$c_{\alpha }^2=c_{\beta }^2=5^2$				${\varsigma }_j^2={10}^2,$$c_{\alpha }^2=c_{\beta }^2=5^2$				${\varsigma }_j^2={10}^5,$$c_{\alpha }^2=c_{\beta }^2=3^2$
n	m	Parameter	True Value	Mean	Bias	MSE	95% CP	Mean	Bias	MSE	95% CP	Mean	Bias	MSE	95% CP
200	6	${\beta }_1^{01}$	1.0	1.015	0.015	0.120	0.952	1.006	0.006	0.121	0.942	0.987	0.013	0.114	0.942
		${\beta }_2^{01}$	−0.5	−0.530	0.030	0.065	0.932	−0.545	0.045	0.066	0.950	−0.530	0.030	0.063	0.946
		${\beta }_3^{01}$	0.5	0.528	0.028	0.189	0.962	0.526	0.026	0.206	0.952	0.540	0.040	0.176	0.960
		${\beta }_1^{10}$	−1.0	−1.035	0.035	0.062	0.952	−1.015	0.015	0.070	0.938	−1.043	0.043	0.070	0.920
		${\beta }_2^{10}$	0.5	0.527	0.027	0.031	0.952	0.534	0.034	0.032	0.938	0.512	0.012	0.030	0.944
		${\beta }_3^{10}$	−0.5	−0.536	0.036	0.114	0.938	−0.550	0.050	0.114	0.926	−0.508	0.008	0.104	0.946
		${\sum }_{11}$	0.6	0.714	0.114	0.165	0.952	0.755	0.155	0.187	0.952	0.704	0.104	0.175	0.946
		${\sum }_{12}$	−0.3	−0.120	0.180	0.053	0.856	−0.127	0.173	0.052	0.870	−0.112	0.188	0.056	0.820
		${\sum }_{22}$	0.4	0.489	0.089	0.094	0.964	0.526	0.126	0.117	0.962	0.509	0.109	0.106	0.968
		α1	2.0	2.054	0.054	0.080	0.954	2.039	0.039	0.095	0.930	2.021	0.021	0.080	0.948
		α2	0.5	0.502	0.002	0.017	0.960	0.511	0.011	0.021	0.942	0.517	0.017	0.018	0.950
		α3	−1.0	−1.036	0.036	0.108	0.958	−1.012	0.012	0.131	0.938	−0.999	0.001	0.109	0.946
		α4	−1.0	−1.017	0.017	0.024	0.950	−1.014	0.014	0.024	0.952	−1.010	0.010	0.026	0.962
200	10	${\beta }_1^{01}$	1.0	0.974	0.026	0.092	0.910	1.000	0.000	0.082	0.946	1.002	0.002	0.069	0.958
		${\beta }_2^{01}$	−0.5	−0.540	0.040	0.041	0.910	−0.534	0.034	0.040	0.934	−0.517	0.017	0.034	0.946
		${\beta }_3^{01}$	0.5	0.519	0.019	0.142	0.960	0.539	0.039	0.143	0.938	0.524	0.024	0.129	0.944
		${\beta }_1^{10}$	−1.0	−1.024	0.024	0.033	0.980	−1.034	0.034	0.040	0.938	−1.038	0.038	0.036	0.950
		${\beta }_2^{10}$	0.5	0.505	0.005	0.018	0.950	0.514	0.014	0.016	0.956	0.507	0.007	0.016	0.950
		${\beta }_3^{10}$	−0.5	−0.520	0.020	0.061	0.950	−0.503	0.003	0.061	0.942	−0.487	0.013	0.061	0.946
		${\sum }_{11}$	0.6	0.701	0.101	0.081	0.970	0.668	0.068	0.081	0.948	0.655	0.055	0.071	0.938
		${\sum }_{12}$	−0.3	−0.174	0.126	0.030	0.880	−0.171	0.129	0.030	0.872	−0.181	0.119	0.030	0.868
		${\sum }_{22}$	0.4	0.407	0.007	0.028	0.990	0.414	0.014	0.034	0.958	0.428	0.028	0.038	0.962
		α1	2.0	2.018	0.018	0.037	0.980	2.020	0.020	0.053	0.944	2.011	0.011	0.050	0.942
		α2	0.5	0.501	0.001	0.013	0.910	0.511	0.011	0.010	0.954	0.507	0.007	0.010	0.962
		α3	−1.0	−0.986	0.014	0.057	0.950	−1.005	0.005	0.074	0.944	−1.002	0.002	0.070	0.942
		α4	−1.0	−1.022	0.022	0.013	0.970	−1.008	0.008	0.015	0.954	−0.999	0.001	0.015	0.948
300	6	${\beta }_1^{01}$	1.0	0.992	0.008	0.081	0.934	0.985	0.015	0.070	0.954	0.973	0.027	0.074	0.942
		${\beta }_2^{01}$	−0.5	−0.526	0.026	0.037	0.940	−0.521	0.021	0.041	0.946	−0.507	0.007	0.038	0.942
		${\beta }_3^{01}$	0.5	0.517	0.017	0.142	0.936	0.531	0.031	0.134	0.956	0.506	0.006	0.122	0.956
		${\beta }_1^{10}$	−1.0	−1.042	0.042	0.044	0.934	−1.027	0.027	0.037	0.960	−1.014	0.014	0.038	0.954
		${\beta }_2^{10}$	0.5	0.513	0.013	0.021	0.940	0.518	0.018	0.019	0.956	0.518	0.018	0.020	0.954
		${\beta }_3^{10}$	−0.5	−0.500	0.000	0.068	0.958	−0.513	0.013	0.065	0.958	−0.538	0.038	0.078	0.934
		${\sum }_{11}$	0.6	0.660	0.060	0.101	0.950	0.655	0.055	0.105	0.942	0.675	0.075	0.104	0.964
		${\sum }_{12}$	−0.3	−0.134	0.166	0.046	0.818	−0.136	0.164	0.044	0.826	−0.140	0.160	0.042	0.832
		${\sum }_{22}$	0.4	0.452	0.052	0.070	0.940	0.456	0.056	0.063	0.972	0.456	0.056	0.063	0.960
		α1	2.0	2.040	0.040	0.062	0.942	2.012	0.012	0.049	0.964	2.009	0.009	0.046	0.958
		α2	0.5	0.496	0.004	0.012	0.942	0.500	0.000	0.012	0.952	0.512	0.012	0.011	0.940
		α3	−1.0	−1.028	0.028	0.090	0.920	−0.997	0.003	0.071	0.954	−0.996	0.004	0.068	0.946
		α4	−1.0	−1.006	0.006	0.016	0.952	−1.011	0.011	0.018	0.936	−1.003	0.003	0.017	0.940
300	10	${\beta }_1^{01}$	1.0	0.999	0.001	0.050	0.953	0.967	0.033	0.051	0.940	0.981	0.019	0.049	0.943
		${\beta }_2^{01}$	−0.5	−0.531	0.031	0.024	0.947	−0.524	0.024	0.024	0.947	−0.512	0.012	0.022	0.953
		${\beta }_3^{01}$	0.5	0.498	0.002	0.075	0.973	0.548	0.048	0.087	0.957	0.528	0.028	0.084	0.957
		${\beta }_1^{10}$	−1.0	−1.036	0.036	0.028	0.933	−1.033	0.033	0.024	0.933	−1.005	0.005	0.026	0.930
		${\beta }_2^{10}$	0.5	0.500	0.000	0.010	0.923	0.511	0.011	0.010	0.963	0.511	0.011	0.012	0.947
		${\beta }_3^{10}$	−0.5	−0.496	0.004	0.040	0.950	−0.500	0.000	0.038	0.950	−0.531	0.031	0.043	0.940
		${\sum }_{11}$	0.6	0.633	0.033	0.045	0.943	0.628	0.028	0.043	0.947	0.663	0.063	0.053	0.937
		${\sum }_{12}$	−0.3	−0.206	0.094	0.023	0.877	−0.197	0.103	0.022	0.850	−0.204	0.096	0.019	0.913
		${\sum }_{22}$	0.4	0.411	0.011	0.026	0.940	0.397	0.003	0.020	0.970	0.401	0.001	0.022	0.967
		α1	2.0	2.017	0.017	0.028	0.960	2.009	0.009	0.030	0.943	1.996	0.004	0.030	0.950
		α2	0.5	0.498	0.002	0.006	0.960	0.492	0.008	0.008	0.913	0.501	0.001	0.008	0.950
		α3	−1.0	−1.003	0.003	0.041	0.957	−0.995	0.005	0.038	0.967	−0.985	0.015	0.043	0.960
		α4	−1.0	−1.010	0.010	0.010	0.953	−1.010	0.010	0.009	0.963	−1.005	0.005	0.013	0.917

Simulation IV:

In this simulation, true random effects were simulated from a normal mixture distribution 0.5N(µ₁,Σ) + 0.5N(µ₂,Σ) where µ₁ = (−0.5, 0.5), µ₂ (0.5,−0.5), and Σ has diagonal elements 0.6 and 0.4 and off-diagonal elements −0.3. The marginal distribution of each random effect term has zero mean and two modes. Results in Table 4 show small estimated biases for β⁰¹ and β¹⁰ but large estimated biases for the intercept term and the coefficient of y_ti in α. Under the scenario of n = 300 and m = 10, estimated biases range from 0.4% to 6% for β⁰¹ and β¹⁰ and 1.4% to 18% for α. The small estimated biases for β⁰¹ and β¹⁰ suggest a degree of robustness of the proposed model to random effect distribution mis-specification. It agrees with McCulloch and Neuhaus (2011) who demonstrate a large degree of robustness of estimated covariates effects in generalized linear mixed models. Nevertheless, the missing model parameters are sensitive to random effect distribution mis-specification. A Bayesian non-parametric modelling of the random effects distribution (Antonelli et al., 2016) may reduce bias in parameter estimation and provide a framework to assess the Normality assumption. It is our interest to explore this extension in the future.

Table 4.

Simulated data: True value and Monte Carlo mean, bias and mean square error (MSE) and 95% coverage probability of the transition event model and missing model parameters where random effects were simulated from 0.5N(µ₁,Σ) + 0.5N(µ₂,Σ), µ₁ = (−0.5, 0.5), µ₂ = (0.5, −0.5) and Σ has diagonal elements 0.6 and 0.4, and off-diagonal elements −0.3

n	m	Parameter	True Value	Mean	Bias	MSE	95% CP
200	6	${\beta }_1^{01}$	1.0	0.963	0.037	0.090	0.958
		${\beta }_2^{01}$	−0.5	−0.520	0.020	0.031	0.942
		${\beta }_3^{01}$	0.5	0.559	0.059	0.109	0.954
		${\beta }_1^{10}$	−1.0	−1.043	0.043	0.058	0.944
		${\beta }_2^{10}$	0.5	0.501	0.001	0.016	0.950
		${\beta }_3^{10}$	−0.5	−0.521	0.021	0.059	0.940
		α1	2.0	2.400	0.400	0.748	0.936
		α2	0.5	0.502	0.002	0.016	0.954
		α3	−1.0	−1.373	0.373	0.843	0.942
		α4	−1.0	−1.017	0.017	0.030	0.954
200	10	${\beta }_1^{01}$	1.0	0.986	0.014	0.059	0.944
		${\beta }_2^{01}$	−0.5	−0.512	0.012	0.017	0.948
		${\beta }_3^{01}$	0.5	0.537	0.037	0.069	0.936
		${\beta }_1^{10}$	−1.0	−1.070	0.070	0.042	0.928
		${\beta }_2^{10}$	0.5	0.496	0.004	0.008	0.960
		${\beta }_3^{10}$	−0.5	−0.484	0.016	0.035	0.932
		α1	2.0	2.279	0.279	0.441	0.928
		α2	0.5	0.497	0.003	0.010	0.952
		α3	−1.0	−1.272	0.272	0.520	0.920
		α4	−1.0	−0.999	0.001	0.018	0.944
300	6	${\beta }_1^{01}$	1.0	0.994	0.006	0.066	0.936
		${\beta }_2^{01}$	−0.5	−0.528	0.028	0.022	0.948
		${\beta }_3^{01}$	0.5	0.521	0.021	0.078	0.944
		${\beta }_1^{10}$	−1.0	−1.049	0.049	0.040	0.948
		${\beta }_2^{10}$	0.5	0.509	0.009	0.010	0.96
		${\beta }_3^{10}$	−0.5	−0.497	0.003	0.035	0.948
		α1	2.0	2.283	0.283	0.492	0.948
		α2	0.5	0.511	0.011	0.012	0.940
		α3	−1.0	−1.286	0.286	0.564	0.950
		α4	−1.0	−0.993	0.007	0.019	0.946
300	10	${\beta }_1^{01}$	1.0	1.023	0.023	0.040	0.942
		${\beta }_2^{01}$	−0.5	−0.522	0.022	0.014	0.932
		${\beta }_3^{01}$	0.5	0.512	0.012	0.040	0.954
		${\beta }_1^{10}$	−1.0	−1.056	0.056	0.028	0.918
		${\beta }_2^{10}$	0.5	0.498	0.002	0.005	0.962
		${\beta }_3^{10}$	−0.5	−0.494	0.006	0.022	0.954
		α1	2.0	2.178	0.178	0.229	0.960
		α2	0.5	0.504	0.004	0.007	0.942
		α3	−1.0	−1.181	0.181	0.268	0.962
		α4	−1.0	−0.997	0.003	0.011	0.952

Simulation V:

In this simulation, we evaluate the performance of model selection criteria LPML and WAIC. We simulated data from two additional set-ups of α and computed the proportions of fits choosing MNAR over MAR for the missing model based on LPML and WAIC. Null set considers α =(1, 0.5, 0.0, 0.0), for which the true missing model assumes MAR; Sets 1 and 2 assume α =(1, 0.5, −0.5, −0.5) and (2, 0.5, −1, −1) respectively. Both Sets 1 and 2 assume MNAR for the true missing model. Greater LPML (WAIC) indicates a better model fit. Based on a cutoff of 0 (Table 3), the proportions of selecting the true models increase for both LPML and WAIC as sample size n or follow-up times m increases. Under the scenario n = 300 and m = 10, the proportions of choosing the true models are all close to 1. WAIC has lower proportions of selecting MNAR than LPML does under the Null set and Set 1, suggesting a more conservative measure favouring the simpler model (MAR). Both criteria appear to be reasonable model selection tools for our proposed model.

Table 3.

Simulated data: Proportions of greater LPML (WAIC) for models assuming MNAR than those assuming MAR. Null set: α=(1, 0.5, 0.0, 0.0), for which the true model assumes MAR; Set 1: α=(1, 0.5, −0.5, −0.5); Set 2: α=(2, 0.5, −1, −1). Both Sets 1 and 2 assume MNAR for the true model

		LPML			WAIC
n	m	Null set	Set 1	Set 2	Null set	Set 1	Set 2
100	6	0.16	0.87	1.00	0.10	0.75	1.00
100	10	0.17	0.93	1.00	0.11	0.87	1.00
200	6	0.13	0.99	1.00	0.09	0.95	1.00
200	10	0.13	1.00	1.00	0.04	0.98	1.00
300	6	0.11	1.00	1.00	0.05	0.98	1.00
300	10	0.07	1.00	1.00	0.01	1.00	1.00

4. Data analysis

We apply the proposed Bayesian transition method to the data from a randomized controlled trial evaluating extended self-help intervention for smoking cessation (Brandon et al., 2016). In this trial, participants (smokers who wanted to quit smoking) were randomly assigned to one of the three interventions to help them quit smoking: traditional self-help (TSH), standard repeated mailing (SRM) and intensive repeated mailing (IRM). TSH received an existing self-help booklet for quitting smoking. SRM received eight different cessation booklets mailed over a 12-month period. IRM received monthly mailings of 10 booklets and additional material designed to enhance social support over 18 months. The authors hypothesized that IRM would produce the greatest smoking abstinence over time, followed by SRM and then TSH, which was included as a ‘usual care’ comparison condition. The primary outcomes were the self-reported 7-day-point-prevalence (i.e., no tobacco cigarettes smoked in previous seven days) smoking status collected at 6, 12, 18 and 24 months, and were coded as ‘1’ for smoking and ‘0’ for abstinent. All participants were reported to be smoking at the baseline, that is, y_0t = 1 for all individuals. Their findings indicated that the self-help interventions with increased intensity and duration resulted in significantly improved abstinence rates, using a logistic regression model at each time point and the GEE approach over time. To manage missing data, they applied multiple imputations under the MAR assumption. A post hoc approach was then used to address the influence of MNAR on smoking status. For our analysis, baseline characteristics expected to be related to smoking status at follow-up assessments were age, sex, marital status and income level. After excluding observations with missing values of the baseline features and another 308 participants who missed all of the follow-up measurements, we have n = 1 530 participants as the analytical data. Figure 2 displays the smoking status transitions, and the numeric values (outside the parentheses) are the observed transition percentages.

Figure 2.

Smoking status transition map based on MNAR assumption

Notes: S—smoking; A—abstinent; M—missing; arrows are the transitions; predictive summaries of the transitions are placed in the same vertical orders of the corresponding arrows; the numeric values outside the brackets are observed transition percentages; the numeric values inside the brackets are 95% sample intervals of transition percentages of the replicated data.

In the following analyses below, age was standardized; sex=‘1’ for male and ‘0’ for female; marital status =‘1’ for married, and ‘0’ otherwise; income level was standardized. Reference group refers to female participants at the sample mean age (48 years old), not married, with income US $30 000–40 000 annually, and treated with IRM intervention. More detailed information about the study design and covariates can be found in Brandon et al. (2016). Hyper-parameters for the priors are initially fixed at ${\varsigma }_j^2={10}^3$, c_β = 5² and c_α = 4². Starting values for the parameters are same as those in the simulation studies. All summaries were based on 30 000 posterior samples for the parameters, which were obtained after a burn-in of 200 000 iterations and a thinning of every other 10 iterates.

We first considered a full MNAR model that assesses all prospective moderators (i.e., sex, age, marital status and income level) of the treatment effect via interaction terms. We found that marital status interacts with treatment significantly. Therefore, we refitted a reduced MNAR model which includes the main effects of all baseline covariates plus an interaction term: treatment × marital status. LPML and WAIC for the full model are −5 014 and −4 806, and for the reduced model are −5 020 and −4 827. The full model is slightly preferred. For the reduced model, Table 5 provides estimates of the posterior means and the associated 95% credible intervals for the transition parameters β⁰¹ and β¹⁰ (01: abstinent to smoking and 10: smoking to abstinent) and the missing model parameters α, respectively.

Table 5.

Smoking cessation data: Posterior mean (95% credible interval) for model parameters with treatments interacting with marital status based on the reduced MNAR and MAR model

	MNAR			MAR
	β01	β10	α	β01	β10	α
Intercept (IRM)	0.53 (−0.90, 1.92)	−2.28 (−2.60, −1.97)	4.18 ( 2.61, 7.10)	−1.42 (−3.21, 0.23)	−2.00 (−2.34, −1.68)	1.52 ( 1.39, 1.66)
Intercept (TSH)	1.53 (−0.08, 3.32)	−3.09 (−3.46, −2.74)	4.37 ( 2.76, 7.30)	−0.20 (−2.32, 1.86)	−2.88 (−3.26, −2.52)	1.58 ( 1.46, 1.72)
Intercept (SRM)	0.28 (−1.42, 1.89)	−2.74 (−3.10, −2.40)	4.41 ( 2.83, 7.34)	−1.96 (−4.23, 0.07)	−2.55 (−2.93, −2.21)	1.69 ( 1.55, 1.82)
Age	−0.47 (−0.91, −0.06)	−0.07 (−0.06, 0.21)	0.17 ( 0.11, 0.24)	−0.53 (−1.15, 0.02)	0.04 (−0.11, 0.18)	0.18 ( 0.12, 0.24)
Sex	−0.34 (−1.29, 0.50)	0.27 (−0.02, 0.55)	−0.36 (−0.50, −0.23)	−0.95 (−2.19, 0.17)	0.34 ( 0.05, 0.64)	−0.30 (−0.43, −0.17)
Marital status (IRM)	−0.01 (−1.33, 1.29)	−0.18 (−0.66, 0.30)	0.04 (−0.12, 0.19)	−0.41 (−2.18, 1.30)	−0.24 (−0.75, 0.25)	0.02 (−0.13, 0.17)
Marital status (TSH)	−0.00 (−1.60, 1.64)	0.11 (−0.41, 0.63)	*	−0.66 (−2.81, 1.41)	0.09 (−0.47, 0.64)	*
Marital status (SRM)	2.07 ( 0.46, 3.94)	0.23 (−0.30, 0.78)	*	2.38 ( 0.35, 4.68)	0.27 (−0.28, 0.82)	*
Income	0.03 (−0.43, 0.50)	0.11 (−0.04, 0.26)	−0.16 (−0.23, −0.09)	−0.01 (−0.63, 0.63)	0.16 ( 0.00, 0.32)	−0.13 (−0.20, −0.07)
Time t	*	*	−0.35 (−0.41, −0.30)	*	*	−0.32 (−0.37, −0.26)
I (yti = 1)	*	*	−3.58 (−6.05, −1.77)	*	*	*
I (yt−1,i = 1)	*	*	0.68 ( 0.38, 0.98)	*	*	*

Note:

^*means that the corresponding parameter is not estimated. Blank indicates that values are not defined. I (·) is an indicator function.

Based on the summaries for β⁰¹, P(A → S) ≈ exp(0.53)/(1 + exp(0.53)) ≈ 0.629 for the reference group, which indicates that participants in the reference group have a probability of 62.9% to transition from abstinence to smoking. Age has a negative effect on P(A → S). Sex and income have no significant effect on resuming smoking or remaining abstinent. Marital status interacts with treatment. Married participants in the SRM group are more likely to resume smoking than those in the IRM group. Based on the 95% credible intervals of the contrasts in Table 6, P(A → S) of the SRM group and the IRM group for married participants are also significantly different.

Table 6.

Smoking cessation data: Posterior mean (95% credible interval) for contrasts of the coefficients for intercept and marital status between TSH and IRM, and SRM and IRM

	MNAR		MAR
	β01	β10	β01	β10
Intercept (TSH − IRM)	1.00 (−0.19, 2.32)	−0.82 (−1.24, −0.41)	1.23 (−0.31, 2.90)	−0.88 (−1.31, −0.46)
Intercept (SRM −IRM)	−0.25 (−1.48, 0.94)	−0.46 (−0.88, −0.06)	−0.53 (−2.17, 1.03)	−0.56 (−0.98, −0.14)
Marital status (TSH − IRM)	−0.00 (−1.94, 1.89)	0.29 (−0.39, 0.97)	−0.26 (−2.77, 2.21)	0.33 (−0.37, 1.04)
Marital status (SRM − IRM)	2.06 ( 0.16, 4.24)	0.41 (−0.27, 1.10)	2.79 ( 0.36, 5.50)	0.51 (−0.19, 1.23)

Based on the summaries for β¹⁰, P(S → A) ≈ 0.093 for the reference group. If we modify the treatment for the reference group to TSH or SRM, the probability of quitting smoking decreases by 5.0% or 3.3%, respectively, that is, (P(S → A) ≈ 0.043 or 0.060). The differences are also significant according to the 95% credible intervals of the contrasts in Table 6. Age and marital status have no significant effect on cessation. Male participants are more likely to quit in any of the three groups than female participants, as the probability of the coefficient for sex being greater than zero is 97%. Finally, participants with higher income are more likely to quit as the probability of the coefficient of income being greater than zero is 92%.

Based on the summaries for α in Table 5 under MNAR, the coefficient for current smoking status is significantly negative, indicating that participants who are currently smoking are more likely to have missing responses. Sex, income and time t (months 6, 12, 18, and 24 were coded as t = −1, 0, 1, 2) all have negative effects, indicating that male participants are more likely to have missing responses than women; higher income participants are more likely to have missing responses than lower income participants; more responses are missing as time increases.

Lastly, the correlation between u_0i and u_1i $\left({\sum }_{12}/\sqrt{{\sum }_{11}{\sum }_{22}}\right)$ has an estimated value −0.536 and 95% credible interval (−0.827, −0.156). The correlation is significantly less than zero, which implies that entering abstinent status is negatively correlated with entering smoking status. The standard deviation of ${\mathbf{u}}_{0i}\left(\sqrt{{\sum }_{11}}\right)$ has an estimated value 3.248 and 95% credible interval (2.241, 4.617). The standard deviation of ${\mathbf{u}}_{1i}\left(\sqrt{{\sum }_{22}}\right)$ has an estimated value 1.652 and 95% credible interval (1.347, 1.966).

Using the MCMC samples for the reduced MNAR model, we can compute the population averaged probabilities of smoking patterns for every subgroup. To illustrate, we focus on patterns ‘S → A → A → A → A’ and ‘S → S → S → S → S’ for each treatment group, while other variables are held at reference values. Given β⁰¹, β¹⁰ and Σ, the probability of observing pattern S → A → A → A → A is

$$\int \frac{\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{10}+u_1\right)}{1+\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{10}+u_1\right)}{\left[\frac{1}{1+\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{01}+u_0\right)}\right]}^{3}N\left(\mathbf{u};\mathbf{0};\sum \right)d\mathbf{u},$$

where u = (u₀, u₁). We approximate the integration above by

$$\frac{1}{L}\sum _{l=1}^L\left\{\frac{\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{10}+u_1^{\left(l\right)}\right)}{1+\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{10}+u_1^{\left(l\right)}\right)}{\left[\frac{1}{1+\exp \left({\mathbf{x}}^T{\mathbf{\beta }}^{01}+u_0^{\left(l\right)}\right)}\right]}^3\right\}$$

where u¹, ... , u^(L) are independent samples drawn from N(0, Σ). Using the MCMC samples of β⁰¹, β¹⁰ and Σ, we obtain the posterior mean and the 95% credible interval of the probability for each pattern. Table 7 displays the results for each smoking cessation group. The IRM group has the highest estimated probability of ‘S → A → A → A → A’ pattern and lowest estimated probability for ‘S → S → S → S → S’.

Table 7.

Smoking cessation data: Posterior means and 95% credible intervals for the probability of observing patterns ‘S→A→A→A→A’ and ‘S→S→S→S→S’ based on the reduced MNAR model . The probabilities are for a female population with age 48 and annual income US $30 000–40 000

Marital status	Smoking pattern	IRM	SRM	TSH	TSH − IRM	SRM − IRM
Not married	S → A → A → A → A	0.08 (0.05, 0.11)	0.07 (0.04, 0.09)	0.04 (0.02, 0.06)	−0.04 (−0.07, −0.02)	−0.01 (−0.04, 0.01)
	S → S → S → S → S	0.60 (0.53, 0.66)	0.67 (0.62, 0.73)	0.73 (0.68, 0.78)	0.14 ( 0.07, 0.20)	0.08 ( 0.01, 0.15)
Married	S → A → A → A → A	0.07 (0.05, 0.11)	0.04 (0.02, 0.07)	0.04 (0.02, 0.07)	−0.03 (−0.06, 0.00)	−0.03 (−0.07, 0.00)
	S → S → S → S → S	0.63 (0.55, 0.70)	0.63 (0.55 , 0.71)	0.71 (0.64 , 0.78)	0.09 ( 0.00, 0.18)	0.01 (−0.09, 0.10)

To assess the goodness of fit of the reduced MNAR model, we use the proposed posterior predictive checking method in Section 2.4 and 6 000 replicated datasets. Table 8 displays the summaries of smoking percentages and response rates (percentages of participants who responded smoking status) for both the observed and replicated data. We find a minimal discrepancy between the observed and replicated data. We also see no discrepancy between the observed and replicated data in response rates. In addition, we displayed in Figure 2 the summaries of transition probabilities among smoking, abstinent and missing. The transition probabilities between smoking and abstinent are well matched by the replicated data. The transition probabilities between missing and smoking statuses are not fully predicted by the replicated data.

Table 8.

Values in columns ‘observed’ are smoking percentages and response rates by questionnaire time of the smoking cessation data. Values in columns ‘MNAR’ and ‘MAR’ are sample means and 95% sample intervals of smoking percentages and response rates of the replicated data

	Smoking percentage			Response percentage
time in month	Observed	MNAR	MAR	Observed	MNAR	MAR
6	0.733	0.719 (0.690, 0.748)	0.616 (0.584, 0.646)	0.871	0.859 (0.837, 0.880)	0.742 (0.715, 0.769)
12	0.620	0.638 (0.609, 0.667)	0.582 (0.551, 0.612)	0.796	0.812 (0.790, 0.834)	0.753 (0.726, 0.779)
18	0.563	0.565 (0.536, 0.595)	0.550 (0.519, 0.581)	0.758	0.763 (0.738, 0.786)	0.756 (0.730, 0.782)
24	0.505	0.496 (0.462, 0.528)	0.533 (0.500, 0.564)	0.710	0.703 (0.675, 0.733)	0.759 (0.733, 0.784)

For sensitivity analysis, we first compare the reduced MNAR model to a MAR version of the model. The LPML and WAIC of the MAR model are −5 056 and −4 866. Therefore the reduced MNAR model’s predictive ability is better than that of reduced MAR model. The posterior predictive checking of the reduced MAR model in Table 8 and the transition map in Figure 3 also suggest that the MNAR model fits the observed data slightly better than the MAR model. We present the parameter estimation results of the reduced MAR model in Tables 5 and 6.

Figure 3.

Smoking status transition map based on MAR assumption

Notes: S—smoking; A—abstinent; M—missing; arrows are the transitions; predictive summaries of the transitions are placed in the same vertical orders of the corresponding arrows; the numeric values outside the brackets are observed transition percentages; the numeric values inside the brackets are 95% sample intervals of transition percentages of the replicated data.

We also perform sensitivity analyses on hyper-parameters under the reduced MNAR model. The resulting LPML and WAIC of each set-up is represented in Table 9. We noticed very small changes in the LPML and WAIC values. Model parameter estimates also show no change under different set-ups of $c_{\mathbf{\beta }}^2$ and ${\varsigma }_j^2$. When we increase or decrease $c_{\alpha }^2$, we noticed small changes in the estimates. Results of the reduced MNAR model under $c_{\alpha }^2=3^2\hspace{0.17em}and\hspace{0.17em}{5}^2$ are represented in Table 10.

Table 9.

Smoking cessation data: Reduced MNAR model performance under different set-ups of hyper-parameters

			MNAR
${\varsigma }_j^2$	cβ	cα	LPML	WAIC
103	52	42	−5020	−4827
102	52	42	−5019	−4827
104	52	42	−5019	−4827
103	42	42	−5019	−4828
103	62	42	−5021	−4827
103	52	32	−5021	−4831
103	52	52	−5019	−4825

Table 10.

Smoking cessation data: Posterior mean (95% credible interval) for model parameters with treatments interacting with marital status based on the reduced MNAR model under c_α = 3² and 5²

	cα = 32			cα = 52
	β01	β10	α	β01	β10	α
Intercept (IRM)	0.42 (−1.13, 1.87)	−2.27 (−2.60, −1.95)	3.68 ( 2.34, 5.60)	0.58 (−0.84, 1.93)	−2.28 (−2.61, −1.97)	4.58 ( 2.64, 8.53)
Intercept (TSH)	1.46 (−0.30, 3.26)	−3.09 (−3.47, −2.73)	3.87 ( 2.51, 5.81)	1.55 ( 0.06, 3.26)	−3.10 (−3.48, −2.74)	4.77 ( 2.80, 8.73)
Intercept (SRM)	0.15 (−1.76, 1.86)	−2.74 (−3.10, −2.40)	3.91 ( 2.56, 5.85)	0.35 (−1.39, 1.94)	−2.74 (−3.10, −2.40)	4.81 ( 2.85, 8.77)
Age	−0.47 (−0.96, −0.06)	−0.07 (−0.06, 0.21)	0.17 ( 0.10, 0.24)	−0.47 (−0.90, −0.07)	−0.07 (−0.06, 0.21)	0.17 ( 0.11, 0.24)
Sex	−0.40 (−1.39, 0.48)	0.27 (−0.01, 0.56)	−0.36 (−0.50, −0.23)	−0.31 (−1.22, 0.51)	0.27 (−0.01, 0.56)	−0.36 (−0.50, −0.23)
Marital status (IRM)	−0.02 (−1.39, 1.34)	−0.18 (−0.66, 0.30)	0.04 (−0.12, 0.19)	−0.00 (−1.29, 1.27)	−0.18 (−0.66, 0.30)	0.04 (−0.12, 0.19)
Marital status (TSH)	−0.05 (−1.77, 1.60)	0.11 (−0.42, 0.64)	*	−0.01 (−1.58, 1.57)	0.12 (−0.41, 0.64)	*
Marital status (SRM)	2.15 ( 0.51, 4.11)	0.24 (−0.30, 0.78)	*	2.00 ( 0.44, 3.84)	0.23 (−0.31, 0.77)	*
Income	0.03 (−0.43, 0.52)	0.11 (−0.04, 0.26)	−0.15 (−0.23, −0.08)	0.03 (−0.40, 0.47)	0.11 (−0.04, 0.26)	−0.16 (−0.23, −0.09)
Time t	*	*	−0.35 (−0.41, −0.29)	*	*	−0.35 (−0.41, −0.29)
I (yti = 1)	*	*	−3.03 (−5.11, −1.42)	*	*	−4.00 (−8.04, −1.81)
I (yt−1,i = 1)	*	*	0.64 ( 0.30, 0.94)	*	*	0.70 ( 0.38, 0.98)

Note: Blank indicates that values are not defined. I (·) is an indicator function.

^*means that the corresponding parameter is not estimated.

Finally, we assess the MCMC convergence of the samples for the reduced MNAR model. Geweke’ z-statistics (Geweke, 1992) for the parameters are displayed in Figure 4 which suggests convergence for all parameters. The z-statistic of β⁰¹ (marital status SRM) is a little larger than two but the MCMC chain shows no obvious poor mixing.

Figure 4.

Smoking cessation data: Geweke’ z-statistic for each parameter in the reduced MNAR model using the first 10% and last 50% of the MCMC chain

5. Conclusion

We proposed an extension of a two-state transition model to manage missing smoking status and applied the model to a smoking cessation intervention study. We further proposed an MCMC algorithm for obtaining accurate Bayesian inferences under MAR and MNAR missing model assumption. We developed a posterior predictive checking to assess model assumption and evaluated the performances of LPML and WAIC in model comparison. The proposed model can be used to evaluate the effectiveness of smoking interventions by comparing the estimated transitioning probabilities of smoking behaviours and probabilities of longitudinal smoking behaviour patterns. The application of this method to a randomized controlled intervention trial for smoking cessation provides information on how the interventions influence the transition of smoking behaviours. The results are consistent with the hypothesis and conclusions from the previous study (Brandon et al., 2016) in which the IRM group (intensive intervention) produced the best outcomes, followed by the SRM group (standard intervention) and then the TSH group (usual care). But unlike the original analyses, our analyses provide a further understanding of the transitions of smoking statuses between each follow-up point. For example, the pair-wise comparisons at each time point in the original analyses reported that outcome of the 7-day-point-prevalence abstinence rate of the IRM group is significantly better than that of the SRM group at 18 months. Our analyses suggest that the difference is mostly due to the significant difference in the transition probability P(S → A), instead of the difference in the transition probability P(A → S) or P(A → A). In other words, IRM is superior to SRM in helping people quit smoking, but the intervention is not significantly better at helping people to remain abstinent. The original study has many other covariates in addition to those used in our models. One of the challenges of including many covariates is dealing with missing observations. In a Bayesian framework, the challenge lies in the fact that we need to specify a flexible joint distribution for incomplete categorical and continuous variables that will be used for imputation. Our future research will focus on extending the current model to allow missing covariates. We are also interested in extending the current missing model of the response to a flexible class of missing data models proposed in Ibrahim et al. (2001). Instead of assuming independent binary distribution of missing indicators δ_1i, ... , δ_mi, one can assume a joint distribution of the missing indicators, for example, p(δ_1i, ... , δ_mi | y_i, z_i) = p(δ_1i|y_i, z_1i, γ₁)p(δ_2i|δ_1i, y_i, z_1i, γ₂) ... p(δ_mi|δ_1i, ... , δ_{m 1,i}, y_i, z_mi, γ_m) where each conditional distribution is modeled using a logistic regression.