Bayesian methods for nonignorable dropout in joint models in smoking cessation studies

Abstract

Inference on data with missingness can be challenging, particularly if the knowledge that a measurement was unobserved provides information about its distribution. Our work is motivated by the Commit to Quit II study, a smoking cessation trial that measured smoking status and weight change as weekly outcomes. It is expected that dropout in this study was informative and that patients with missed measurements are more likely to be smoking, even after conditioning on their observed smoking and weight history. We jointly model the categorical smoking status and continuous weight change outcomes by assuming normal latent variables for cessation and by extending the usual pattern mixture model to the bivariate case. The model includes a novel approach to sharing information across patterns through a Bayesian shrinkage framework to improve estimation stability for sparsely observed patterns. To accommodate the presumed informativeness of the missing data in a parsimonious manner, we model the unidentified components of the model under a non-future dependence assumption and specify departures from missing at random through sensitivity parameters, whose distributions are elicited from a subject-matter expert.

1. Introduction

One of the more challenging aspects in analyzing repeatedly measured data on human subjects is handling missing observations. This is particularly true when the knowledge that an observation is missing may itself provide information about the distribution of the unobserved data. Methodology to deal with this informative missingness is both challenging and potentially controversial, as any analysis will require the statistician to make untestable assumptions about the unobserved data. Despite this difficulty missingness in experimental data is ubiquitous, and the analyst risks mischaracterizing the results or drawing faulty conclusions by failing to deal with it appropriately.

We recall some key results from the missing data literature (e.g., Little and Rubin, 2002). Let Y_i = (Y_i1, …, Y_iT) denote the vector of (potentially observed) responses, and let R_it be an indicator that response Y_it is observed, i.e., R_it = I(Y_it is observed). Define the (full) data response model as the probability model for Y, p(y|θ₁), parametrized by θ₁. We then define the missing data mechanism (MDM) as p(r|y, θ₂) with parameter θ₂. Finally, y_obs (respectively, y_mis) is the set of Y_its that are observed (missing). The missing data mechanism is ignorable if the following three conditions hold: 1) p(r|y, θ₂) = p(r|y_obs, θ₂), i.e., the missingness only depends on the observed responses, so it is missing at random (MAR); 2) the parameter θ of the full model p(y, r|θ) can be decomposed as (θ₁, θ₂) with p(y|θ₁) and p(r|y, θ₂); 3) the parameters of the response model and the missing data mechanism are a priori independent, i.e. p(θ) = p(θ₁)p(θ₂). Models with ignorable missingness comprise a wide and commonly used class of techniques. In particular, standard Bayesian analysis using the observed data likelihood for the responses implicitly makes the assumption that missingness is ignorable (Schafer, 1997; Little and Rubin, 2002).

When any of the three conditions above are not satisfied, the missingness is referred to as non-ignorable. Often, this is due to the first condition failing, i.e., p(r|y, θ₂) ≠ p(r|y_obs, θ₂), known as missing not at random (MNAR). It is then necessary to specify the MDM as a function of both y_obs and y_mis. The joint model for (y, r) will require untestable assumptions about the missing data, as can be seen from the factorization

$$p(\mathbf{y},\mathbf{r}|\mathit{\theta })=p({\mathbf{y}}_{\text{mis}}|{\mathbf{y}}_{\text{obs}},\mathbf{r},{\mathit{\theta }}_E)p({\mathbf{y}}_{\text{obs}},\mathbf{r}|{\mathit{\theta }}_O).$$ (1)

While the parameters of the observed data model θ_O are identifiable, it is clear that the observed data provide no information about θ_E. We call the leading term on the right the extrapolation distribution (Daniels and Hogan, 2008).

There are three main classes of models based on different factorizations of the full data distribution p(y, r|θ): selection models, shared parameter models, and pattern mixture models. Selection models involve the factorization p(y, r|θ) = p(y|θ₁)p(r|y, θ₂) (Diggle and Kenward, 1994). If data are MNAR, structural assumptions about the MDM lead to a fully identified model, including the extrapolation distribution for which the observed data give no information. Shared parameter models assume a latent variable β_i, and typically set y_i and r_i to be independent given β_i by modeling p(y_i|β_i) and p(r_i|β_i) (Wu and Carroll, 1988; Cowles et al., 1996; Dunson and Perreault, 2001). A drawback of shared parameter models is that formulas for the MDM are generally difficult to obtain (β must be integrated out), making it challenging to specify MAR models.

Alternatively, pattern mixture models (PMM) factor p(y, r|θ) as p(r|π)p(y|r, ψ), first drawing the missingness indicators R, referred to as the missingness pattern, then the responses conditional on the pattern (Little, 1993, 1994). The data response model is recovered by marginalizing over R, p(y|θ₁) = Σ_r p(r|π)p(y|r, ψ). A key advantage to PMMs with missingness due to dropout is that the extrapolation distribution appears explicitly in the model specification. Hence, PMMs often lead to a straightforward understanding of the role the missing data assumptions play and provide easier specification of θ_E through sensitivity parameters, as opposed to selection and shared parameter models (Daniels and Hogan, 2000).

While many of these methods are well understood when the response Y_it is univariate, additional considerations arise if a bivariate (or more generally, multivariate) response is repeatedly observed and subject to non-ignorable missingness. Our work is partially motivated by the analysis of a smoking cessation trial (Marcus et al., 2005). In this study patients were measured weekly for whether they had smoked during the previous week and for the percentage weight change from baseline. In addition to study dropout, the analysis is further complicated by the joint modeling of the discrete and continuous longitudinal outcomes.

In the next section we provide details on the motivating data. Section 3 introduces our proposed bivariate pattern mixture model and the role of the extrapolation distribution. In particular, we propose new methodology to share information across patterns to gain stability in the parameter estimates for patterns with few patients. Section 4 contains a simulation study to evaluate the performance of our proposed model, and we then apply the methods to the smoking cessation data in Section 5. Next, we discuss issues arising from non-ignorability including the role of sensitivity parameters, elicitation of their distribution, and the estimation of treatment effects. We provide some concluding remarks in Section 7.

2. The Commit to Quit II smoking cessation trial

The Commit to Quit II study (CTQ2; Marcus et al., 2003, 2005) was a 4-year randomized trial undertaken to test the efficacy of moderate-intensity physical activity as an aid for smoking cessation in women. Study enrollees were healthy women aged 18–65 who had regularly smoked five or more cigarettes per day for at least a year and who routinely exercise for less than 90 minutes a week. Patients were randomized into one of two treatments, a moderate-intensity exercise condition (denoted as exercise) and a contact condition (denoted as wellness). The outcomes of interest, measured weekly, were quit status (a longitudinal binary outcome) and weight change (a longitudinal continuous outcome). As it is believed that many cessation attempts fail due to weight gain, the goal of the study was to test whether an exercise treatment may lead to higher quit rates by better managing weight changes.

Two hundred, seventeen women were enrolled into the study, and measurements were intended to be taken for each of T = 8 weeks. For our analysis we exclude patients who were missing a baseline weight or who missed all eight measurement times. This left 208 patients with 104 in each treatment arm. As is common in studies of this type, there was substantial missingness with only a third of patients having an observed smoking status for all time points. Figure 1 displays smoking and missingness statuses for each patient by treatment; black, gray, and white represent smoking, not smoking, and missing for a given week. Patients are not asked to quit until week 3 (bottom two rows are mostly black), and the amount of missingness (white) increases by week. It is believed that patients who miss an appointment are more likely to be smoking than those who are observed, so a careful handling of the missingness is necessary to make appropriate conclusions from this study. In addition to comparing cessation success and weight change between the two treatments, we also wish to determine the sensitivity of these conclusions to our assumptions about the role of the missingness.

Figure 1.

Depiction of smoking status and missingness values by treatment group. Each column represents a patient’s status at each of the T = 8 measurement occasions. Black represents an observed value with smoking in the given week, gray represents an observed value of no smoking in the given week, and white represents the patient was missing for the week.

We let Q_it denote weekly smoking quit status, equal to 1 (0) if patient i abstains (smokes) during week t. The percentage weight change from baseline is W_it, and a_i denotes the indicator of whether subject i was randomized to the exercise treatment. The corresponding vectors of the binary and continuous outcomes are Q_i = (Q_i1, ⋯, Q_iT)^⊤ and W_i = (W_i1, ⋯, W_iT)^⊤, with Q_i,obs and W_i,obs representing the observed parts of Q_i and W_i respectively. We assume a normally-distributed, latent variable Z_it whose sign determines the weekly cessation status through Q_it = I(Z_it ≥ 0), and let Z_i = (Z_i1, ⋯, Z_iT)^⊤.

Liu et al. (2009) analyze the CTQ2 data by modeling the joint distribution of the binary and continuous variables by a multivariate normal specification on the latent quit propensity and weight change through ${({\mathbf{Z}}_i^{\top },{\mathbf{W}}_i^{\top })}^{\top }~{\mathrm{N}}_{2T}({\mathbf{X}}_i{\mathit{\beta }}_{a_i},{\mathbf{\sum }}_{a_i})$. An important complication is that the T × T block of Σ_ai corresponding to Z_i must be a correlation matrix for identifiability (Chib and Greenberg, 1998; Gueorguieva and Agresti, 2001). However, their analysis is made under the assumption of ignorable missingness, whereas the goal of this work to develop models appropriate for analysis under MNAR. To that end we introduce a pattern mixture model for this data.

There are alternatives to using normal latent variables for the binary responses to specify a joint distribution of categorical and continuous outcomes. One could potentially use the general location model (Olkin and Tate, 1961; Liu and Rubin, 1998) or Gaussian copulas (Nelsen, 1999). An overview of Bayesian methods for mixed data can be found in the chapter Daniels and Gaskins (2013). As the latent variable choice yields a model that can more easily be extended to non-ignorability, we do not explore other choices here. In the next section we introduce a bivariate pattern mixture model for this type of data.

3. Bivariate pattern mixture model

3.1. Partial ignorability and the model for the missingness

The pattern mixture model is specified as p(y, r|θ) = p(r|π)p(y|r, ψ), where r is the missingness pattern. The CTQ2 data has 2^T − 1 = 255 potential patterns, far too many to handle efficiently. Hence, we model this missingness through a dropout process. We say patient i drops out at time t if their final observed value occurs at t (R_it = 1 and R_ij = 0 for all j > t). We denote this by D_i = d(R_i) = t, and D_i = T indicates that patient i completed the study. (Note that sometimes dropout is defined by D_i + 1 and ranges from 2 to T + 1.) We assume that any missed values before dropout (called intermittent missingness) are MAR conditional on D_i, while those after dropout may be MNAR. This is called partial ignorability (Harel and Schafer, 2009) and requires an MDM of the form p(r|y, θ₂) = p(d|y, θ_2A)p(r|y_obs, d(r), θ_2B) with θ₂ = (θ_2A, θ_2B). Note R depends only on the observed data and the dropout time D, whereas D may depend on both the observed and missing data. If p(d|y, θ_2A) = p(d|y_obs, θ_2A), then the model is (fully) ignorable and MAR.

Under the partially ignorable mechanism described above, we base inference on the joint distribution p(y, d) instead of p(y, r) as partial ignorability implies that the only relevant information about Y from R is found in D = d(R) (Harel and Schafer, 2009, Proposition 2). See Section A.1 of the Web Appendix for details. We model p(y|θ₁)p(d|y, θ_2A) using the PMM factorization p(d|π)p(y|d, ψ) where p(d|π) is the model for dropout and p(y|d, ψ) is the model for the response. The distribution of dropout D_i is multinomial on {1, …, T} with probabilities π_ai,d = P (D_i = d) depending on the treatment assignment a_i. A convenient conjugate prior for π_a = (π_a,1, …, π_a,T) is Dirichlet(1,…,1) for a = 0, 1. The complexity lies in specifying the response model p(y|d, ψ) as a function of the pattern d. We assume different sets of model parameters (π_a, ψ_a) for the two treatments, but to simplify notation we suppress the dependence on a in the following.

At each measurement time we have a pair of observations, and in all but a few instances, smoking status and weight change are either both observed or both missing at a particular week. When only one is observed, we treat this as observed to assign pattern membership and assume the missing measurement is partially ignorable since it occurs before dropout. Table 1 provides the number of patients in each pattern by treatment. As stated previously, our model is defined through the weight change W_it and the cessation latent variable Z_it not the actual smoking status Q_it. Likewise, partial ignorability assumes R depends on dropout time, the observed weight changes, and the “observed” Z_its (Z_it corresponding to an observed Q_it). We return to the role of partial ignorability in the discussion of Section 7.

Table 1.

Dropout time d_i by treatment group.

Treatment	Number of patients with drop out at week d
	1	2	3	4	5	6	7	8
Exercise	7	2	7	7	1	5	5	70
Wellness	9	6	8	4	5	3	3	66

3.2. Model for the observed data response

We now introduce the response model conditional on the dropout time. To that end, let Y_it = (Z_it, W_it)^⊤ be the quit status latent variable and weight change pair at time t, and the time-ordered arrangement of the full response is Y_i = (Z_i1, W_i1, Z_i2, W_i2, …, Z_iT, W_iT)^⊤. The history at week t (t > 1) is denoted by ${\overline{\mathbf{Y}}}_{\mathit{\text{it}}}={({\mathbf{Y}}_{i1}^{\top },\dots ,{\mathbf{Y}}_{i,t-1}^{\top })}^{\top }$. We construct the distribution for each pattern d sequentially through the factorization

$$p({\mathbf{y}}_i|D_i=d)=f_d({\mathbf{y}}_i)=f_{d;1}({\mathbf{y}}_{i1})\prod _{t=2}^T{f}_{d;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}}),$$

where f_d;t(y_it|ȳ_it) is the density of the pair of measurements Y_it at time t for subjects who leave the study at time d, conditional on the history Ȳ_it. Only those distributions f_d;t(y_it|ȳ_it) with d ≥ t are identified by the observed data. The remaining distributions, which taken together form the extrapolation distribution in equation (1), need to be specified through a combination of modeling choices and sensitivity parameters.

The identifiable distributions are modeled as follows. The week one distributions f_d;1(y_i1) have the form N₂(ζ_d;1, Ω_d;1) (d = 1, …, T), where ζ_d;1 is a 2-vector and Ω_d;1 is 2 × 2 positive definite. Recall that the first component of Y_i1 is Z_i1, the latent quit propensity. As its scale is unidentified, the variance is constrained to be 1. For the identified distributions at time t > 1, f_d;t(y_it|ȳ_it) (d ≥ t), we choose N₂ (ζ_d;t + (Φ_d;1t, …, Φ_d;t−1,t)ȳ_it, Ω_d;t); ζ_d;t is a 2-vector, each Φ_d;jt is a 2 × 2 matrix, and Ω_d;t is 2 × 2 positive definite. The Φ_d;jt matrix contains regression coefficients for the patient’s history at time j, controlling the longitudinal dependence on the earlier observations. ζ_d;t is called the conditional intercept, the intercept for the condition regression of Ȳ_it onto Y_it. Ω_d;t is the covariance matrix for this conditional regression, and as with t = 1 the leading variance is constrained to 1. We parametrize this matrix as ${\mathbf{\Omega }}_{d;t}[1,2]={\rho }_{d;t}\sqrt{{\omega }_{d;t}}$ and Ω_d;t[2, 2] = ω_d;t. Positive definiteness is guaranteed by ρ_d;t ∈ (−1, 1) and ω_d;t > 0. We refer to the elements of the Φ_d;jt matrices as generalized auto-regressive parameters (GARPs) and Ω_d;t as the innovation covariance matrix, treating our model as a multivariate extension of the modified Cholesky parametrization of the covariance matrix (Pourahmadi, 1999). This model is also related to the vector autoregressive model (VAR) from time series analysis (Lütkepohl, 1991), but we do not require stationarity. Allowing Ω_d;t to be constant across t, ζ_d;t to be zero for t > 1, and Φ_d;jt to be constant in t − j would lead to the stationary VAR model.

For a particular pattern d, let ${\mathit{\zeta }}_d={({\mathit{\zeta }}_{d;1}^{\top },\dots ,{\mathit{\zeta }}_{d;d}^{\top })}^{\top }$ be the vector of identified conditional intercepts and Ω_d be the block diagonal matrix of Ω_d;1, …, Ω_d;d. Further define Φ_d to be the 2d × 2d block lower triangular matrix with (t, t) block as the 2 × 2 identity matrix, I₂, and the (t, j) block −Φ_d;jt for j < t. The joint distribution of the observed data ${\overline{\mathbf{Y}}}_{i,D_i+1}={({\mathbf{Y}}_{i1}^{\top },\dots ,{\mathbf{Y}}_{i,D_i}^{\top })}^{\top }$ given the pattern D_i = d is ${\mathrm{N}}_{2d}({\mathbf{\Phi }}_d^{-1}{\mathit{\zeta }}_d,{\mathbf{\Phi }}_d^{-1}{\mathbf{\Omega }}_d{\mathbf{\Phi }}_d^{-\top })$ (see the Web Appendix for derivation).

Under partial ignorability the distribution p(ȳ_i,d+1|D_i = d) is the identifiable piece of the response model p(y_i|D_i = d) from factorization (1), and if d ≠ T the distribution of the unobserved variables p(y_i,d+1, …, y_iT|ȳ_i,d+1, D_i = d) is the unidentified extrapolation distribution. As mentioned previously, this transparency of the extrapolation is an important benefit to using PMMs. Furthermore, it provides intuitive choices for specifying the unidentifiable distributions, which we explore in Section 6. For now we consider the bivariate PMM in the context of the MAR assumption, so that we have a fully defined model. The MAR restriction uniquely specifies the unidentified distributions (d < t) to be

$$f_{d;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})=p({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D\ge t)=\sum _{s=t}^T\left\{\frac{{\pi }_s{f}_{s;1}({\mathbf{y}}_{i1})\cdots f_{s;t-1}({\mathbf{y}}_{i,t-1}|{\overline{\mathbf{y}}}_{i,t-1})}{{\sum }_{j=t}^T{\pi }_j{f}_{j;1}({\mathbf{y}}_{i1})\cdots f_{j;t-1}({\mathbf{y}}_{i,t-1}|{\overline{\mathbf{y}}}_{i,t-1})}\right\}{f}_{s;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})$$ (2)

(Molenberghs et al., 1998), which is a mixture over the distributions at time t for the identified patterns s = t, …, T. The terms in braces are the mixing coefficients, which are equal to P (D = s|ȳ_it, D ≥ t). Note that this distribution f_d;t(y_it|ȳ_it) is constant in d (d < t), and hence, f_d;t(y_it|ȳ_it) = f_d′;t(y_it|ȳ_it) for all d, d′ < t at each time t.

We remind the reader that we have specified our model in terms of the full data Y which contains both Z_obs, the unobserved latent variables corresponding to the observed smoking statuses Q_obs, and (Q_mis, W_mis), the values that are missing either intermittently or due to dropout. We obtain the observed data likelihood by integrating the identified model p(d, ȳ_d+1) with respect to the latent variables and the intermittently missed responses:

$$p(d_i,{\mathbf{q}}_{i,\text{obs}},{\mathbf{w}}_{i,\text{obs}})={\pi }_{d_i}\int I({\mathbf{z}}_i\in 𝒵({\mathbf{q}}_{i,\text{obs}}))f_{d_i;1}({\mathbf{y}}_{i1})\prod _{t=2}^{d_i}{f}_{d_i;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})\mathrm{d}{\mathbf{z}}_i\mathrm{d}{\mathbf{w}}_{i,\text{int}},$$ (3)

where z has length d_i, w_int = {w_it : t ≤ d_i, r_it = 0} denotes the intermittently missed weight changes, and 𝒵(q_i,obs) represents the set of latent variables consistent with the sign restrictions of the observed smoking statuses. We obtain parameter samples from our model using MCMC with data augmentation that draws (z_i, w_i,int) given (d_i, q_i,obs, w_i,obs) during each iteration.

The identified distributions from p(y_i|D_i = d) are $f_{d;1}({\mathbf{y}}_{i1}){\prod }_{t=2}^d{f}_{d;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})$ which contain a large number of parameters. In the next two sections, we formulate the prior distributions in such a way as to effectively reduce the number of parameters that must be estimated from the data.

3.2.1. Priors to induce sharing information across patterns

A common issue when using a PMM is that some of the patterns contain relatively few observations, leading to instability in the parameter estimates of the pattern-specific distributions. Our model as presented thus far estimates 2 conditional intercepts, 4(t − 1) GARPs, and 2 covariance parameters for each d and t with d ≥ t. In the CTQ2 data only the completer pattern (d = T) has more than ten patients (Table 1), so we must develop methodology to handle these sparsely observed patterns.

An obvious solution is to set the parameters equal across patterns, yielding f_d;t(y_it|ȳ_it) = f_d′;t(y_it|ȳ_it) for all d, d′ ≥ t. However, together with (2) this will imply p(y|d, ψ) = p(y|ψ), which is the missing completely at random (MCAR) assumption. While requiring fewer parameters, assuming y and d are independent seems unlikely to hold in most practical settings, particularly in the context of smoking cessation. Other parameter reduction choices include grouping the dropout times into a smaller number of patterns (Hogan et al., 2004) or assuming the distribution of Y_i differs across a small number of latent classes C_i whose distribution depends on dropout D_i (Roy, 2003; Roy and Daniels, 2008).

Rather than reducing the number of patterns, Wang and Daniels (2011) consider equality constraints on subsets of model parameters across patterns. They show that for the full response distribution p(y_i|D_i = d) under MAR (2) to be multivariate normal for each pattern, the distribution of Y_it given Ȳ_it must be the same across all patterns d for t > 1, that is, for t > 1, f_d;t(y_it|ȳ_it) = f_t(y_it|ȳ_it) for all d. Hence, the conditional intercepts, GARPs, and innovation covariances are equal across d for t > 1. The t = 1 means ζ_d;1 differ across patterns, and the covariances Ω_d;1 may also differ across d but are generally assumed equal. This constraint results in a MDM that depends only on Y_i1 (Wang and Daniels, 2011, Corollary 1).

A somewhat more flexible model would be to assume the dependence parameters are equal across identifiable patterns, ${\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star }={\mathbf{\Phi }}_{d;\mathit{\text{jt}}}$ and ${\mathbf{\Omega }}_t^{\star }={\mathbf{\Omega }}_{d;t}(j<t\le d)$ and to allow the conditional intercepts ζ_d;t to differ. Unlike the previous model where the extrapolation distributions f_d;t(y_it|ȳ_it) (d > t) are multivariate normal, the extrapolation distributions here will be a mixture of T − t + 1 normals as defined in (2).

While these types of equality assumptions can provide stable estimation, the resulting models may be a poor representations of the true response distribution. Hence, a middle ground between equality and independence through sharing information across patterns would be welcome. There is a growing literature on Bayesian estimation for multiple, potentially similar covariance matrices (e.g., Daniels, 2006; Pourahmadi et al., 2007; Hoff, 2009; Gaskins and Daniels, 2013, 2015), but due to the constraint in the (1, 1) component of each Ω_d;t and the unidentifiability of the extrapolation parameters, these methods cannot be directly implemented. Hence, we introduce a method that borrows strength in estimating the identifiable parameters across patterns.

To that end, we propose shrinking the pattern-specific parameters toward a global value for the distributions identified by the observed data. Let ${\mathit{\zeta }}_t^{\star },{\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star },{\rho }_t^{\star },{\omega }_t^{\star }$ be these global parameters, which are the shrinkage targets of the identified parameters ζ_d;t, Φ_d;jt, ρ_d;t, ω_d;t (d ≥ t). While we suppress the notation, we use a distinct set of ${\mathit{\zeta }}_t^{\star },{\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star },{\rho }_t^{\star },{\omega }_t^{\star }$ for each treatment. These are connected through the following distributions for d ≥ t:

$${\mathit{\zeta }}_{d;t}|{\mathit{\zeta }}_t^{\star }~{\mathrm{N}}_2({\mathit{\zeta }}_t^{\star },{\tau }_{\zeta }^2{\mathbf{I}}_2),$$ (4)

$$\text{vec}({\mathbf{\Phi }}_{d;\mathit{\text{jt}}})|{\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star }~{\mathrm{N}}_4(\text{vec}({\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star }),{\tau }_{\varphi }^2{\mathbf{I}}_4)(j<t),$$ (5)

$$\text{log}\left[\frac{1+{\rho }_{d;t}}{1-{\rho }_{d;t}}\right]|{\rho }_t^{\star }~\mathrm{N}(\text{log}\left[\frac{1+{\rho }_t^{\star }}{1-{\rho }_t^{\star }}\right],{\tau }_{\rho }^2),$$ (6)

$$\text{log}({\omega }_{d;t})|{\omega }_t^{\star }~\mathrm{N}(\text{log}({\omega }_t^{\star }),{\tau }_{\omega }^2).$$ (7)

Here, vec(·) is the standard vector operator that stacks the columns of a matrix. This model assumes the identifiable parameters in each pattern are exchangeable and makes no use of the temporal ordering of the patterns. Extensions along these lines are possible but not pursued here.

For each of the four sets of parameters, the shrinkage variance τ² governs the amount of information shared across patterns. Large values of τ² allow large differences between the parameters of different patterns, while the identified f_d;t(y_it|ȳ_it) (d ≥ t) will be similar under small τ². Note that in the special case where ${\tau }_{\zeta }^2,{\tau }_{\varphi }^2,{\tau }_{\rho }^2,{\tau }_{\omega }^2\to 0$, this shrinkage model goes to MCAR (assuming MAR for the extrapolation terms). Further with ${\tau }_{\varphi }^2,{\tau }_{\rho }^2,{\tau }_{\omega }^2\to 0$ and ${\tau }_{\zeta }^2$ not too small, this will be equivalent to the model that sets the dependence parameters equal and leaves the mean parameters flexible. Hence, our model allows the data to inform the appropriate level of information sharing.

To fully define our Bayesian model, we must choose prior distributions for the remaining parameters. For the (treatment-specific) shrinkage targets, we use

$${\mathit{\zeta }}_t^{\star }~{\mathrm{N}}_2({\mathbf{0}}_2,{\sigma }_{\zeta }^2{\mathbf{I}}_2)$$ (8)

$$\text{vec}({\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star })~p_{\mathbf{\Phi }}({\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star })(j<t)$$ (9)

$${\rho }_t^{\star }~\text{Unif}(-1,1)$$ (10)

$${\omega }_t^{\star }~\text{InvGamma}({\lambda }_1,{\lambda }_2),$$ (11)

where p_Φ(·) will be defined in the following section and 0_k represents the k-vector of zeros. We use a half-Cauchy prior for the shrinkage standard deviations τ_ζ, τ_ϕ, τ_ρ, τ_ω. That is, for k ∈ {ζ, ϕ, ρ, ω}, $p({\tau }_k)=\frac{2}{\pi }{\gamma }_k{({\tau }_k^2+{\gamma }_k^2)}^{-1}$, τ_k > 0. This is the Cauchy distribution with location zero and scale γ_k, restricted to positive half-line. Under the support restriction γ_k is both the scale and median of τ_k. The half-Cauchy distribution has found an important role in Bayesian variable shrinkage since it has non-zero density at 0 and heavy tails (Carvalho et al., 2010). Here, we choose γ_ζ = γ_ω = 0.1 and γ_ϕ = γ_ρ = 0.05 as the hyperparameter values to represent reasonable guesses at the prior median of the τ s. For the other hyperparameters the prior choices are ${\sigma }_{\zeta }^2~\text{ InvGamma}(0.1,0.1)$ and λ₁, λ₂ ~ Gamma(1, 1). With the exception of the shrinkage targets, the other hyperparameters ( ${\tau }_{\zeta }^2,{\tau }_{\varphi }^2,{\tau }_{\rho }^2,{\tau }_{\omega }^2,{\sigma }_{\zeta }^2$, λ₁, λ₂) are common across the two treatments.

3.2.2. Sparse prior models for GARP matrices

A well known issue in the modeling of a covariance matrix is the quadratic number of parameters. In our model this is manifested in the 2(d − 1)(d − 2) GARPs defining Φ_d. One Bayesian solution is to use as the prior for the GARP shrinkage target in (9) a mixture of a point mass at zero and a normal distribution, known as the “spike-and-slab” prior (Smith and Kohn, 2002). A computationally faster alternative uses a prior that shrinks the GARPs toward zero. We explore this using a longitudinal extension of the normal-gamma model of Griffin and Brown (2010).

We define the prior p_Φ(·) for ${\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star }$ in (9) through the following hierarchical model, where ${\varphi }_{\mathit{\text{jt}};k}^2$ is the kth component of $\text{vec}({\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star })$ (k = 1, …, 4),

$${\varphi }_{\mathit{\text{jt}};k}^{\star }|{\sigma }_{\mathit{\text{jt}};k}^2~\mathrm{N}(0,{\sigma }_{\mathit{\text{jt}};k}^2),$$ (12)

$${\sigma }_{\mathit{\text{jt}};k}^2~\text{Gamma }(\lambda ,2{\gamma }_0{\xi }^{t-j}),$$ (13)

for λ, γ₀ > 0 and ξ ∈ (0, 1). Here, ${\sigma }_{\mathit{\text{jt}};k}^2$ represents a GARP-specific shrinkage factor, showing that this model falls in the global-local shrinkage framework (Polson and Scott, 2010). Marginally, ${\varphi }_{\mathit{\text{jt}};k}^{\star }$ has mean zero, variance 2λγ₀ξ^t−j, and excess kurtosis 3/λ (Griffin and Brown, 2010). The constraint on ξ implies the variance of ${\varphi }_{\mathit{\text{jt}};k}^{\star }$ decreases in the lag t − j, that is, the regression coefficient of the responses at time j onto the time t measurement is more aggressively shrunk for j further back in time. This is consistent with the longitudinal nature of the history, as less recent responses will generally be less relevant for predicting the current measurement. Smaller values of λ give the distribution heavier tails, providing protection from over-shrinking large GARPs. As a special case, setting λ = 1 implies that (13) is an exponential distribution as in the Bayesian Lasso (Park and Casella, 2008) and a GARP-shrinkage model proposed in Gaskins and Daniels (2015). For hyperpriors we choose λ ~ Exp(1), γ₀ ~ InvGamma(1, 1), and ξ ~ Unif(0, 1). The GARP shrinkage target ${\varphi }_{\mathit{\text{jt}};k}^{\star }$ and its variance ${\sigma }_{\mathit{\text{jt}};k}^2$ are treatment specific, while the hyperparameters λ, γ₀, ξ are common across treatments.

4. Simulation study to evaluate model performance

To assess the performance of our proposed methodology, we consider a simulation study based on the Commit to Quit II study. We compare our model to frequently-used choices. For the model on the conditional intercepts, let mvn-mar denote the constraint ${\mathit{\zeta }}_t^{\star }={\mathit{\zeta }}_{d;t}$ for d ≥ t and t ≠ 1, which provides a multivariate normal distribution for the full response p(y_i|D_i = d) under MAR (assuming equality of GARPs and innovation covariance matrix) (Wang and Daniels, 2011). For modeling the intercepts distinctly across patterns, each ζ_d;t (d ≥ t) is drawn independently from the distribution in (8), and we call this model pattern. Our proposed model that shrinks ζ_d;t toward ${\mathit{\zeta }}_t^{\star }$ given by (4) and (8) is shrink.

With each of the three intercept models, we assume equality in the dependence structure across patterns ( ${\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star }={\mathbf{\Phi }}_{d;\mathit{\text{jt}}}$ and ${\mathbf{\Omega }}_t^{\star }={\mathbf{\Omega }}_{d;t}$ for d ≥ t). With this equal structure, the GARPs have the sparse prior in (12)–(13). We also consider the shrink dependence model from equations (5)–(7) and (9)–(11) with the sparse GARP model and the shrink intercept choice. Finally, we pair the pattern mean model with a pattern dependence model with no information sharing across dependence parameters and a non-sparse (normal) prior for the GARPs. As a shorthand, we denote these five models by the triple that defines the mean structure, the dependence structure, and the prior on the GARPs.

We consider four data generating mechanism. Model (A) draws the data using the parameter estimates from shrink/equal/sparse model fit to the CTQ2 data. Data generating model (B) is consistent with mvn-mar assumption for the intercepts, and in model (C) the ζ_d;t differ more substantially across patterns than model (A). Model (C) should favor the pattern mean model or shrink with a large value of τ_ζ. While choices (A)–(C) all assume common (equal) dependence structures across patterns, model (D) allows the intercepts, GARPs, correlation, and innovation variances to each vary across patterns. Model (D) is consistent with shrink/shrink and pattern/pattern mean/dependence models. Details about selection of parameter values can be found in the Section A.3 of the Web Appendix.

For each model specification considered, we run a Markov chain Monte Carlo (MCMC) algorithm to obtain a sample of the parameters from the posterior distribution. We run the chain for 75,000 iterations after a burn-in of 15,000 and retain every 50th iteration for inference. We use the data augmentation algorithm in Liu et al. (2009, Proposition 1) to sample the constrained latent variables Z, which is more efficient that one-at-a-time conditioning (Robert, 1995). Many of the model parameters are updated conjugately (π_a, Z_it for t ≤ d_i, any missing W_it for t ≤ d_i, ζ_d;t, ${\mathit{\zeta }}_t^{\star }$, Φ_d;jt, ${\mathbf{\Phi }}_{\mathit{\text{jt}}}^{\star },{\sigma }_{\zeta }^2$, λ₂, ${\sigma }_{\mathit{\text{jt}};k}^2$, γ₀). For those parameters whose distributions are non-conjugate (ρ_d;t, ${\rho }_t^{\star }$, ω_d;t, ${\omega }_t^{\star },{\tau }_{\zeta }^2,{\tau }_{\varphi }^2,{\tau }_{\rho }^2,{\tau }_{\omega }^2$, λ₁, λ, ξ), we update using the slice sampler (Neal, 2003). See the Web Appendix, Section A.2 for the form of the full conditional (sampling) distributions. Depending on the complexity of the model involved, the MCMC algorithm takes between 10 and 18 hours to run on a desktop computer per data set. R code is available at the website of the second author, *****.

For each of the true models, we generate 100 data sets, maintaining the same dropout and intermittent missingness patterns from the original CTQ2 data. For each of data sets, we run MCMC chains to compare five model specifications. To evaluate the estimation accuracy, we compute the risk for estimating the conditional intercepts, the mean functions, and the covariance matrices using the following loss functions: $L(\hat{\mathit{\zeta }};\mathit{\zeta })={\sum }_{\mathit{\text{trt}}}{\sum }_{d=1}^T{({\hat{\mathit{\zeta }}}_d-{\mathit{\zeta }}_d)}^{\top }({\hat{\mathit{\zeta }}}_d-{\mathit{\zeta }}_d),L(\hat{\mathit{\mu }};\mathit{\mu })={\sum }_{\mathit{\text{trt}}}{\sum }_{d=1}^T{({\hat{\mathit{\mu }}}_d-{\mathit{\mu }}_d)}^{\top }({\hat{\mathit{\mu }}}_d-{\mathit{\mu }}_d),L(\hat{\mathbf{\sum }},\mathbf{\sum })={\sum }_{\mathit{\text{trt}}}{\sum }_{d=1}^T(\text{trace}\{{\hat{\mathbf{\sum }}}_d{\mathbf{\sum }}_d^{-1}\}-\text{log }|{\hat{\mathbf{\sum }}}_d{\mathbf{\sum }}_d^{-1}|-2d)$, where ${\mathit{\mu }}_d={\mathbf{\Phi }}_d^{-1}{\mathit{\zeta }}_d$ is the marginal mean vector and ${\mathbf{\sum }}_d={\mathbf{\Phi }}_d^{-1}{\mathbf{\Omega }}_d{\mathbf{\Phi }}_d^{-\top }$ is the (2d) × (2d) marginal covariance matrix. Results are in Table 2.

Table 2.

Estimated risk for each parameter set (conditional intercepts, marginal mean, covariance matrix) by estimation model under each of the four data generating mechanisms.

Loss Function	Estimation Model
Mean: Dependence: GARP:	mvn-mar equal sparse	pattern equal sparse	shrink equal sparse	shrink shrink sparse	pattern pattern non-sparse
Data Generating Model (A)
Zeta	33.9	68.6	8.2	34.9	56.7
Mu	53.5	130.6	9.2	54.4	120.7
Sigma	22.8	33.6	16.7	49.8	54.2
Data Generating Model (B)
Zeta	29.8	65.5	16.8	46.9	56.9
Mu	47.3	120.3	32.1	57.3	110.4
Sigma	17.6	30.8	15.0	49.8	53.2
Data Generating Model (C)
Zeta	310.8	67.9	61.9	140.9	95.1
Mu	191.7	111.8	90.8	97.2	123.2
Sigma	30.3	32.8	20.6	51.1	52.9
Data Generating Model (D)
Zeta	243.6	245.7	190.5	31.8	85.1
Mu	250.7	120.5	106.9	92.3	119.2
Sigma	388.4	377.7	367.6	29.9	45.6

Generating models (A)–(C) have the same dependence structure for each pattern, and we find that the shrink/equal/sparse is found to be the best in each case. It is perhaps surprising that using the shrinkage framework produces better estimation than the correct mvn-mar choice in model (B). This is, in part, a consequence of the unbalanced pattern memberships. All of the non-completer patterns have fewer than 10 subjects, implying that there is very little information from the data about ζ_d;1 for d < T. This leads to large sampling variability (and higher risk) in these intercept estimates and the resulting marginal means. Even though the shrink models are biased toward the population—not pattern—mean, the stability it imposes yields better estimation. For data generating model (C), there are large differences in the conditional intercept across patterns that should favor the pattern or shrink with large τ_ζ, but by introducing a small amount of shrinkage, shrink/equal/sparse has slightly lower risk. In the shrink-mean model the average value of τ̂_ζ is 1.18 (80% of τ̂_ζ ’s are between 1.10 and 1.26) compared to 0.09 (0.06, 0.15) and 0.13 (0.07, 0.24) in scenarios (A) and (B), respectively. Clearly, our shrinkage framework is flexible enough to adapt to the situation when there is little similarity across patterns. In scenario (C), the mvn-mar assumption produces highly biased mean estimates as expected.

Simulation (D) has distinct covariance structures across the patterns, and we find that shrinkage on both the mean and dependence parameters produces the minimum risk. Estimation using a pattern-specific dependence structure also leads to low risk for Σ but poorer performance for the mean parameters. The risk in estimation of the covariance matrices and the conditional intercepts is much higher when common dependence is imposed, although estimation of mean structure (and hence, treatment effects) is impacted less.

Typically, one would fit the data to multiple models, and choose the best model using a selection method such as deviance information criterion (dic; Spiegelhalter et al., 2002). However, evaluation of the dic statistic demonstrated mixed performance in data such as ours. dic tends to systematically favor simpler models: shrink mean to mvn-mar and pattern mean structures and equal dependence to the shrink and pattern choices. Further simulation experiments also indicate poor performance with other model selection criteria: log psuedo-marginal likelihood (Geisser and Eddy, 1979) and posterior predictive loss based criteria (Ibrahim and Laud, 1994; Daniels et al., 2012). Details about dic simulation studies can be found in Section A.3 of the Web Appendix.

Based on the results from risk simulations and the unsatisfactory performance of several model selection criteria, we recommend using the shrinkage framework for the mean structure and either the equal or shrink model for the dependence, with the sparse GARP prior. The choice between equal and shrink dependence will be guided by the level of balance between dropout patterns. When patterns are unbalanced or sample sizes are small, the equal model should be favored to stabilize estimation of the covariance matrix; shrink can be used with large sample sizes and more balanced dropout times. Consequently, we base our analysis of the CTQ2 data in Sections 5 and 6 on the model formed by using the shrink framework for the conditional intercepts, equal for the dependence, and sparse for the GARPs.

5. CTQ2 data analysis under non-ignorable MAR

We now turn to the analysis of the CTQ2 data using the shrink/equal/sparse model under the MAR assumption. As in the simulation, our MCMC chain runs for 90,000 iterations, and we throw out the first 15,000 and retain every 50th sample.

The main targets of inference are the probability of abstaining at the final time point T = 8 for each treatment (marginally over patterns) and the expected weight change over the course of the study, as well as significance tests for a treatment effect due to exercise. As a key concern of this study is the interaction between smoking and weight change, we also consider the correlation between Q_iT and W_iT. To obtain estimates of these quantities, we draw 5000 fully-observed responses ${\{{\mathbf{Y}}_i^{\text{new}}\}}_{i=1}^{5000}$ from p(d_i, y_i) at each parameter value in the posterior sample and compute sample means. Details of the algorithm are found in the Web Appendix, Section A.4.

The estimated probability that a patient abstains in the final week, P(Q_iT = 1), is 0.47 for the exercise treatment and 0.53 for the control with a posterior probability of 0.25 that the exercise treatment is superior. For the weight measurements, we find an expected weight change of 3.0% from baseline for both the wellness and exercise treatments, and the posterior probability that patients gain less under exercise is 0.51. For the exercise treatment the correlation between Q_iT and W_iT is 0.13, and it is 0.18 for the control group. These positive values support the study motivation that women who successfully abstain tend to gain weight and that the exercise treatment may reduce this interaction as seen by the smaller correlation, although the 95% credible interval covers zero (see Table 4). Overall, under MAR we fail to find evidence that the exercise treatment produces better results than wellness in terms of quit rates, weight changes, or their relationship. Credible intervals for these quantities can be found in Table 4.

Table 4.

Posterior mean and 95% credible interval for the quantities of interest under each missingness assumption. The posterior probability row gives the probability that the exercise treatment is superior: higher cessation rate, lower weight change, lower correlation.

		Missing data assumption
Quantity of interest	Treatment	Zit MAR Wit MAR	Zit MNAR Wit MNAR	Zit MNAR Wit MAR	Qit = 0 if missing Wit MAR
	Wellness	0.53 (0.40, 0.65)	0.50 (0.37, 0.63)	0.50 (0.37, 0.63)	0.30 (0.21, 0.39)
P (QiT = 1)	Exercise	0.47 (0.35, 0.59)	0.44 (0.32, 0.56)	0.43 (0.32, 0.56)	0.28 (0.20, 0.37)
	Post. prob.	0.25	0.24	0.23	0.42
	Wellness	3.0% (2.3, 3.8)	2.9% (2.2, 3.6)	3.0% (2.2, 3.7)	2.6% (1.8, 3.3)
E(WiT)	Exercise	3.0% (2.3, 3.7)	2.8% (2.0, 3.4)	3.0% (2.3, 3.6)	2.7% (2.1, 3.4)
	Post. prob.	0.51	0.61	0.51	0.38
	Wellness	0.18 (− 0.08, 0.42)	0.18 (− 0.08, 0.42)	0.18 (− 0.07, 0.42)	0.21 (0.02, 0.39)
corr(QiT, WiT)	Exercise	0.13 (− 0.14, 0.36)	0.13 (− 0.13, 0.36)	0.13 (− 0.14, 0.36)	0.17 (− 0.04, 0.35)
	Post. prob.	0.61	0.59	0.61	0.62

Comparing our results under non-ignorable MAR to the previous analysis under ignorable MAR in Liu et al. (2009), we note that our estimates of the quit probabilities are 7 to 9 percentage points higher, although their estimates are contained within our credible intervals. By allowing the response distribution to vary across dropout times, our model is more flexible whereas their model implicitly assumes a common distribution across D_i. However, this increased flexibility does come at a cost of wider credible intervals.

We additionally run MCMC chains using the CTQ2 data with changes to hyperparameter values in the priors to test the sensitivity of our prior choices. The models with pattern dependence are somewhat sensitive to the priors, but this is expected as inference for the sparsely observed patterns will be more influenced by the prior. The parameter estimates and conclusions are relatively unchanged for the equal and shrink dependence structures, including the selected model.

6. Missing not at random PMM

6.1. Specifying the extrapolation distribution

To this point we have considered missingness to be MAR by using (2) to define the extrapolation distribution. As stated earlier, this is a questionable assumption, if not wholly unreasonable, as we expect patients who leave the study are more likely to smoke than those that continue on, even after conditioning on their history. Hence, we need to extend the model of Section 3 to allow informative missingness by considering alternative specifications of the extrapolation distributions f_d;t(y_it|ȳ_it) (d < t) through sensitivity parameters.

To do this, we consider non-future dependent missing data mechanisms. An MDM is said to satisfy non-future dependence (NFD) if P(D = d|y) = P(D = d|y₁, …, y_d+1), that is, the probability a patient’s last observation occurs at time d depends on her observed measurements Y₁, …, Y_d and the first missed observation Y_d+1 but is independent of all future missed measurements (Kenward et al., 2003). Clearly, if the extrapolation distributions are chosen through MAR, it will satisfy NFD (MDM is also independent of Y_d+1), but other choices of the extrapolation distributions generally will not. NFD only impacts the form of p(y_mis|y_obs, θ_E) and thus, cannot be tested from the data. However, it provides a realistic and intuitive starting point for defining the extrapolation distribution under MNAR. Kenward et al. (2003) show that for a PMM with NFD the distributions for patterns d < t − 1 have the form

$$f_{d;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})=p({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D\ge t-1)=\sum _{s=t-1}^T\frac{{\pi }_s{f}_{s;1}({\mathbf{y}}_{i1})\cdots f_{s;t-1}({\mathbf{y}}_{i,t-1}|{\overline{\mathbf{y}}}_{i,t-1})}{{\sum }_{j=t-1}^T{\pi }_j{f}_{j;1}({\mathbf{y}}_{i1})\cdots f_{j;t-1}({\mathbf{y}}_{i,t-1}|{\overline{\mathbf{y}}}_{i,t-1})}{f}_{s;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}}).$$ (14)

In comparison with MAR (2), we are now conditioning on the observed patterns as well as the d = t − 1 first missed observation pattern. In particular, (14) depends on the unidentified f_t−1;t(y_it|ȳ_it), the model for the first missed observation. A further benefit to the NFD assumption is that the number of extrapolation distributions to define decreases from (T − 1)(T − 2)/2 to T − 1.

To specify these first post-dropout distributions f_t−1;t(y_it|ȳ_it), we apply a location shift to the MAR mixture distribution which will allow a sensitivity parameter specification. This idea was previously considered in the univariate context by Wang and Daniels (2011, with correction). To that end we rewrite the MAR distribution (2) as ${\sum }_{s=t}^T\alpha (\mathsf{S},{\overline{\mathbf{y}}}_{\mathit{\text{it}}})f_{s;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})$, letting α(s, ȳ_it) be the mixing weight P(D = s|ȳ_it, D ≥ t). The location shift on the MAR distribution implies

$$\mathrm{E}\{{\mathbf{Y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D=t-1\}=\mathrm{E}\{{\mathbf{Y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D\ge t\}+{\mathbf{\Delta }}_t,$$

that is, the mean of the mixture is shifted by Δ_t = (Δ_t1, Δ_t2)^⊤. This is accomplished by choosing $f_{t-1;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})={\sum }_{s=t}^T\alpha (\mathsf{S},{\overline{\mathbf{y}}}_{\mathit{\text{it}}}){\stackrel{\sim }{f}}_{s;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})$, where f̃_d;t(y_it|ȳ_it) is multivariate normal with mean ζ_d;t + Δ_t + (Φ_d;1t, …, Φ_d;t−1,t)^⊤ȳ_it and covariance matrix Ω_d;t. The choice Δ_t = 0₂ produces f_t−1;t(y_it|ȳ_it) as the MAR distribution (2), and plugging that into (14) gives the MAR distribution (2) for all terms in the extrapolation (d < t − 1). It is also possible to introduce sensitivity parameters for the GARP coefficients Φ and/or the variance Ω, but in the interest of model parsimony, we do not pursue such models here. Also, note incorporating a location shift for missing data is closely related to the exponential tilting model (e.g., Birmingham et al., 2003; Kim and Yu, 2011).

As we are only making adjustments to the extrapolation term, the observed data distribution remain the same as under MAR. In fact, running a new MCMC chain for the MNAR analysis is not even needed. Letting π(Δ|θ_O) be a prior distribution for the sensitivity parameters potentially depending on the observed data distribution parameter θ_O, we can draw the MNAR posterior sample by first drawing θ_O from the observed data posterior (as sampled using non-ignorable MAR) and drawing Δ from π(Δ|θ_O); we provide details in the Web Appendix, Section A.4. In contrast, selection and shared parameter models typically do not exhibit a sensitivity parametrization; this implies that the observed data likelihood will depend on the assumptions made about the missing data mechanism. In such cases it will be necessary to refit the data and repeat any model selection procedure for each new MNAR assumption.

Using NFD and sensitivity parameters, we have reduced the problem of specifying (T − 1)(T − 2)/2 extrapolation distributions to that of choosing a distribution for T − 1 Δ_ts. We further assume the distribution of the sensitivity parameters is independent of t, leaving specification of a single Δ for each treatment. Next, we introduce a strategy to elicit expert opinion about their distribution.

6.2. Elicitation of distribution for sensitivity parameters

Elicitation of prior distributions from (non-statistician) subject-matter experts can be a challenging task. The best strategy is typically to ask for expected values or quantiles of observable measurements, and for the statistician to translate this into a distribution for the parameter (Bedrick et al., 1996; Chaloner, 1996). In the context of this smoking cessation trial, we do not explicitly ask about a distribution for Δ, but instead we inquire about the anticipated behavior of a dropout patient relative to a non-dropout patient (similar to Daniels and Hogan (2008), Section 10.2). To that end, our collaborator Dr. Marcus filled in the form in Table 3 with her beliefs about the status of the unobserved patients. Her answers are depicted in bold, and we use them to form our distribution for Δ as follows.

Table 3.

Elicitation of distributions for the sensitivity parameter ∆. Our subject-matter expert was asked to fill out this form to elicit the values for the sensitivity parameters. Her responses are shown in bold.

Consider two women who have the same history of smoking and weight change and are receiving the same intervention (exercise or wellness). Patient A is observed at time t − 1 but drops out of the study and is not seen at time t, while patient B remains in the study and is observed at time t.
If the probability that patient B smoked during week t is p, what is your best guess (median) for the probability that patient A (who dropped out) smoked during week t? Also provide a lower bound and upper bound on reasonable values.
Treatment	Prob. observed patient B smokes (p)	Best guess	Lower bound	Upper bound
		for the probability that the unobserved Patient A smokes
Wellness	25 %	50 %	40 %	60 %
Wellness	50 %	70 %	60 %	80 %
Wellness	75 %	95 %	90 %	100 %
Exercise	25 %	50 %	40 %	60 %
Exercise	50 %	70 %	60 %	80 %
Exercise	75 %	95 %	90 %	100 %
If the observed patient B has an expected percentage weight change from baseline of w at week t, what is your best guess (median) for the expected percentage weight change from baseline at week t for patient A who dropped out? Also provide a lower bound and upper bound on reasonable values.
For reference, the average weight change was 2.4% and the standard deviation was 2.6%. Also, negative values are allowed if it is believed that the patient will have lost weight since baseline.
Treatment	Weight change for observed patient B (w)	Best guess	Lower bound	Upper bound
		for the expected weight change of the unobserved Patient A
Wellness	2.5 %	2 %	0 %	5 %
Exercise	2.5 %	1.5 %	0 %	4 %

Translating the information in Table 3 into a prior for the sensitivity parameter Δ₂ = E [W_it|ȳ_it, D = t − 1] − E [W_it|ȳ_it, D ≥ t] is relatively straightforward as Δ₂ represents the difference in the expected weight change between patient A in pattern D = t − 1 and patient B with pattern D ≥ t. For each treatment we let δ_k (k = med, LB, UB) be the difference between the elicited value for patient A and the provided value for patient B at percentile k (median, lower bound/minimum, upper bound/maximum). The assumed prior for Δ₂ is a 50-50 mixture of Unif(δ_LB, δ_med) and Unif(δ_med, δ_UB) which will match the elicited percentiles (Wang et al., 2010). For example, ${\mathrm{\Delta }}_2~\frac{1}{2}\text{Unif}(-2.5,-1)+\frac{1}{2}\text{Unif}(-1,1.5)$ for the exercise treatment.

Dealing with Δ₁ = E [Z_it|ȳ_it, D = t − 1] − E [Z_it|ȳ_it, D ≥ t] is more challenging because we elicit the expert opinion not in terms of the (fictional) latent variables Z on which the sensitivity parameter is defined, but in terms of the smoking status Q. This choice is consistent with our previous observation that we obtain higher quality information when we elicit in terms of potential measurements. From Table 3, let ψ_LB(p), ψ_med(p), ψ_UB(p) denote the lower bound, median, and upper bound for p̃ = P (Q_it = 0|ȳ_it, D = t − 1) (patient A’s smoking probability) given p = P (Q_it = 0|ȳ_it, D ≥ t − 1) (patient B’s smoking probability) at the elicited values p = 0.25, 0.50, 0.75. We obtain the functions ψ_LB(p), ψ_med(p), ψ_UB(p) that cover the full range of p ∈ [0, 1] by linear interpolation as in Figure 2. Similarly to Δ₂ the implied distribution for p̃, the probability of smoking for a dropout patient with history ȳ_it, is the mixture $\frac{1}{2}\text{Unif}({\psi }_{\text{LB}}(\hat{p}),{\psi }_{\text{med}}(\hat{p}))+\frac{1}{2}\text{Unif}({\psi }_{\text{med}}(\hat{p}),{\psi }_{\text{UB}}(\hat{p}))$, where p̂ = P (Q_it = 0|ȳ_it, D ≥ t − 1) is the smoking probability for the observed counterpart. From our choice $f_{t-1;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})={\sum }_{s=t}^T\alpha (\mathsf{S},{\overline{\mathbf{y}}}_{\mathit{\text{it}}}){\stackrel{\sim }{f}}_{s;t}({\mathbf{y}}_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}})$, this distribution on p̃ implies a distribution on Δ₁ through

$$\stackrel{\sim }{p}=P(Q_{\mathit{\text{it}}}=0|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D=t-1)=\sum _{s=t}^T\alpha (s,{\overline{\mathbf{y}}}_{\mathit{\text{it}}})F\{{\mathrm{\Delta }}_1+\mathbf{E}[Z_{\mathit{\text{it}}}|{\overline{\mathbf{y}}}_{\mathit{\text{it}}},D=s]\},$$ (15)

where F{·} is the standard normal cumulative distribution function. This model can be viewed in a similar spirit to the marginalized models of Heagerty (1999). In practice to obtain a sample value of Δ₁, we calculate p̂, draw p̃ from the mixture distribution, and numerically solve (15) for Δ₁. In the special case where the MAR distribution is a single component (not a mixture), as in the MCAR and mvn-mar models, a simpler log-odds approximation exists that avoids to the need to numerically solve (15); see Section A.5 of the Web Appendix for details.

Figure 2.

Prior distribution of P (Q_it = 0|Ȳ_it, D = t − 1) given P (Q_it = 0|Ȳ_it, D ≥ t) for each treatment. The bold line represents the median value, the solid lines the lower and upper bounds, and the dotted line is P (Q_it = 0|Ȳ_it, D ≥ t). The points marked with the dot are elicited from Table 3, and the remainder of the prior is linearly interpolated.

Also note that we have assumed that the distribution of Δ₂ = E [W_it|ȳ_it, D = t − 1] − E [W_it|ȳ_it, D ≥ t] is constant in the expected weight change of the observed patient w = E [W_it|ȳ_it, D ≥ t]. This assumption could be easily relaxed by eliciting δ_k for a few values of w and interpolating functions δ_k(w). Given ŵ = E [W_it|ȳ_it, D ≥ t], Δ₂ would be drawn from the mixture defined by δ_k(ŵ).

6.3. Estimation of treatment effects under MNAR

Having specified a prior distribution for the sensitivity parameters, we may now estimate the main quantities of interest under MNAR. Recall that our inferential focus is the probability of smoking, the percentage weight change from baseline, and the correlation between smoking and weight change at the final week for each treatment. With only about 60% of patients completing the study, we also want to consider how the MAR/MNAR assumptions and the choice of priors on Δ affect our conclusions about these quantities.

To that end, we consider three MNAR missing data assumptions in addition to the MAR analysis of Section 5. First, we apply the elicited prior on the sensitivity parameters. Next, we consider a prior for Δ that uses the elicited prior for Δ₁ but sets Δ₂ = 0. This choice is motivated by the fact that the expert has strong intuition about the probability of smoking for the dropouts but may be less clear about how their weight will behave due to the competing factors that the patient is no longer attending exercise sessions (if a_i = 1) and has likely relapsed to smoking. Hence, we use the MNAR assumption on Z but treat the weight measurements as partially ignorable. Finally, we make the assumption that all unobserved smoking statuses are 0 (smoke) as has been previously used in the smoking cessation literature (Marcus et al., 2005). This extreme assumption falls outside of the framework of PMMs, partial ignorability, and sensitivity parameters, and it requires a new MCMC chain to fit a single pattern model to this augmented data with the sparse prior on the GARPs, assuming the missing weight changes are ignorable.

As in the MAR case, we estimate treatment effects by drawing full data for 5000 patients at each parameter value in the posterior sample. For the MNAR-PMMs this is the same MCMC sample used in Section 5 for the MAR analysis. Pseudo-code for the algorithm may be found in the Web Appendix, Section A.4. Table 4 contains the estimated quantities and 95% credible intervals.

When accounting for the informativeness of the study dropout on cessation in the PMM, we observe slightly lower quit probabilities whether we assume the post-dropout weight changes to be informative or not. The posterior probability of improved cessation rates under exercise is relatively unchanged, which is not surprising as our elicited prior assumes an equal change in smoking rates for both arms relative to the patients that remain under observation (Table 3 and Figure 2). When we consider the strong assumption that all missing Q_its are smoking, we see a more dramatic change in the estimated cessation rates. However, this assumption sets all missing cessation values to zero and will necessarily be biased low for the true cessation probability. For the expected weight change during the study period, estimates are stable across our three PMMs, and somewhat lower under the Q_it = 0 assumption due to the positive correlation between Q and W and the additional zeros in Q. The substantive conclusions for the relationship between smoking status and weight change do not differ from the MAR results. This is not surprising since our sensitivity prior assumes dropouts are less likely to abstain and have lower weight changes, agreeing with our MAR results. Overall, all conclusions are unchanged as there continues to be no evidence of a treatment difference. Further, our cessation rate and weight change estimates differ from the MAR results relatively little across the PMMs, indicating that the results are insensitive to small-to-moderate departures from MAR.

In Section A.6 of the Web Appendix, we consider the estimation error and efficiency of using our framework in a simulation study designed to mimic the CTQ2 data. When the missingness is MNAR in the true model, we find reduced bias and mean squared error in the quit rate and mean weight change estimates when using our modeling framework versus the MAR assumption or complete case analysis. In Section A.7, we also explore additional analyses of the CTQ2 data to better understand the sensitivity of our conclusions to the assumptions of non-future dependence and partial ignorability of the intermittent missingness. Additionally, we consider alternative choices of the distribution on the sensitivity parameters Δ including a dispersed version of our expert-elicited choice and a more extreme prior. We conclude that partial ignorability has the least impact on inference, followed by the choice of distribution on Δ, and the NFD assumption has the largest impact of the three.

7. Discussion

In this work we have proposed methodology to analyze mixed longitudinal data with informative dropout, which was motivated by a smoking cessation study. We consider pattern mixture models to allow the MNAR distributions to be defined through sensitivity parameters. As many patterns contain few observations, Bayesian shrinkage on the mean and dependence parameters is incorporated to share information across potentially similar patterns. Distributions for the sensitivity parameters are elicited from a subject-matter expert. Based on careful analysis of the CTQ2 data, we conclude that the exercise intervention has no effect on cessation rates or weight changes and that the conclusions are robust to post-dropout departures from missing at random.

One issue deserving additional consideration is that the sensitivity parameters for the MNAR model is defined in terms of the history Ȳ_it, which include not the smoking statuses Q_obs but the latent variables Z_obs. While it may be reasonable to compare the unobserved and observed patients with common weight changes and inclinations to smoke each week (richer information than just whether or not she smoked), this is a distinction likely to be lost on the clinician from whom the sensitivity parameter distribution is elicited. It is unclear how to avoid this issue when using a model with latent variables for the discrete process nor is it apparent what modeling scheme without latent variables would be appropriate for a longitudinal, mixed binary-continuous process with dropout. Defining the assumptions through Z leads to more accessible models and can be viewed as reasonably close to the assumptions in terms of Q, but further exploration of this issue is warranted. Similar issues arise with the definition of partial ignorability through Z instead of Q.

As noted in Section 4, model selection in this context proved particularly challenging. None of the usual methods (deviance information criteria, log psuedo-marginal likelihood, posterior predictive loss criteria) were able to discriminate between models in our simulation study. More research in this area is needed but beyond the scope of this paper.

While our model has considered the case of a single binary and continuous response at each time point, this methodology can easily be extended beyond the bivariate case and to allow alternative data types such as ordinal responses. Important considerations will include the necessary identifiability constraints in Ω_d;t for the latent variables corresponding to binary and ordinal responses, elicitation of sensitivity parameters or their distribution, and the potential need for sparsity in Ω_d;t or its inverse if many responses are observed at each time point.

Although exploratory analysis of covariates indicated little predictive value for the CTQ2 data, it is also possible to extend our model to adjust for covariates. The model for dropout time can easily be adapted by specifying π as a function of covariates. To allow the response model to depend on predictors, we add a β′x_it term to the mean of f_d;t(y_it|ȳ_it), although the interpretation of these regression parameters may be challenging due to the sequential nature of the distribution.