A Bayesian zero-inflated beta-binomial model for longitudinal data with group-specific changepoints

Abstract

Timeline followback (TLFB) is often used in addiction research to monitor recent substance use, such as the number of abstinent days in the past week. TLFB data usually take the form of binomial counts that exhibit overdispersion and zero inflation. Motivated by a 12-week randomized trial evaluating the efficacy of varenicline tartrate for smoking cessation among adolescents, we propose a Bayesian zero-inflated beta-binomial model for the analysis of longitudinal, bounded TLFB data. The model comprises a mixture of a point mass that accounts for zero inflation and a beta-binomial distribution for the number of days abstinent in the past week. Because treatment effects appear to level off during the study, we introduce random changepoints for each study group to reflect group-specific changes in treatment efficacy over time. The model also includes fixed and random effects that capture group- and subject-level slopes before and after the changepoints. Using the model, we can accurately estimate the mean trend for each study group, test whether the groups experience changepoints simultaneously, and identify critical windows of treatment efficacy. For posterior computation, we propose an efficient Markov chain Monte Carlo algorithm that relies on easily sampled Gibbs and Metropolis–Hastings steps. Our application shows that the varenicline group has a short-term positive effect on abstinence that tapers off after week 9.

1 |. INTRODUCTION

Bounded discrete data with an abundance of zeros arise commonly in many public health and medical applications. In smoking cessation studies, for example, patients will often report the number of abstinent days during the prior week using a method known as timeline followback (TLFB).¹ Due to their bounded and discrete nature, TLFB data often resemble binomial counts ranging, for example, from 0 to 7 in the case of weekly recall data. However, because smoking behavior is typically correlated from one day to the next, bounded TLFB data are generally overdispersed relative to the binomial distribution, which assumes independence among the underlying Bernoulli trials. Moreover, in smoking cessation studies, some patients will report no abstinence during the study, leading to zero inflation relative to the standard binomial distribution.^2–4 Despite these defining features, TLFB data are often analyzed using linear regression,^5–7 which ignores the inherent discreteness of the data. Alternatively, some investigators model TLFB data using a Poisson distribution,^8,9 which is appropriate for unbounded counts, such as the number of cigarettes smoked in the past week, but not for bounded counts, such as abstinent days. In the latter case, a more suitable approach is to adopt a model specifically designed for overdispersed, zero-augmented binomial data, such as the zero-inflated beta-binomial (ZIBB) model.

To date, ZIBB models have been applied in a broad range of cross-sectional studies. Luo and Paul¹⁰ investigated the ZIBB model in the presence of missing data and compared its performance under three different missingness mechanisms. Hu et al¹¹ introduced a ZIBB model for overdispersed microbiome data to estimate the association between multiple phenotypes and microbiome abundance among patients with pediatric Crohn’s disease. Motivated by an application to dental caries, Albert et al¹² developed a frequentist approach to estimating marginal mean effects from the ZIBB model. Zhang et al¹³ presented an analogous Bayesian approach to estimating marginal mean effects in a cross-sectional case-control study. Despite this growing interest in ZIBB models, relatively little attention has been devoted to ZIBB mixed models in the context of longitudinal studies. Recently, Azimi et al¹⁴ proposed a joint model to analyze correlated ZIBB and normal responses, where random intercepts were used to link the ZIBB and linear mixed models. However, they did not apply their model to longitudinal data, nor did they allow the random effects to be correlated within the ZIBB model, which has been shown to yield biased inferences in applications involving zero-inflated and related two-part mixture models.^15,16

Our goal, therefore, is to extend this body of work by developing a Bayesian ZIBB mixed model for the analysis of bounded longitudinal TLFB data. Our approach is motivated by a 12-week placebo-controlled trial evaluating the efficacy of varenicline,¹⁷ a smoking cessation medication designed to curb the severity of withdrawal symptoms and the desire to smoke. One of the study’s aims was to determine whether varenicline increased the number of abstinence days reported at each weekly visit over the course of the study. To address this aim, we propose a ZIBB model that comprises a mixture of a point mass at zero that accounts for zero inflation and a beta-binomial distribution that models the number of abstinent days in the past week while allowing for overdispersion. In our motivating application, we observed that the effect of varenicline leveled off rapidly toward the end of the study. To accommodate this abrupt change in efficacy, we introduce random changepoints for each study group to capture group-specific changes in treatment efficacy over time. We adopt a piecewise linear model with group-specific changepoints rather than a higher-order spline model because identifying the precise timing when treatment efficacy levels off is often of clinical interest. The model additionally includes fixed and random effects corresponding to the group- and subject-level slopes before and after the changepoints. The random effects are modeled jointly across the binary and count components of the ZIBB model, improving inferences by permitting the zero-inflated process to be correlated with the beta-binomial process. Using the proposed model, we can accurately estimate mean abstinent trends for each study group, test whether groups experience changepoints at the same time, and detect critical windows during which treatment is most impactful. For posterior computation, we develop an efficient hybrid Markov chain Monte Carlo (MCMC) algorithm that relies on easily sampled Gibbs and Metropolis–Hastings (MH) steps. By averaging over MCMC iterations, we can obtain smooth estimates of the temporal trends for each study group. We apply the model to a sample of 145 patients enrolled in the varenicline trial, which was conducted at the Medical University of South Carolina from August 15, 2012, to October 20, 2017.

2 |. ADOLESCENT SMOKING CESSATION STUDY

For our motivating application, we analyzed data from a randomized, placebo-controlled clinical trial for treatment-seeking adolescent and young adult cigarette smokers.¹⁷ The study aimed to evaluate the efficacy and safety of varenicline tartrate for smoking cessation among adolescents aged 14 to 21 years at an outpatient clinic in Charleston, South Carolina. Participants under the age of 18 required parental consent to enter the study. Varenicline tartrate is designed to promote early abstinence by reducing the feeling of craving and withdrawal caused by smoking cessation. The trial enrolled participants who smoked daily for at least 6months, had at least 1 failed prior quit attempt and were seeking to quit. In our study, subjects in the varenicline group received 0.5-mg or 1.0-mg tablets in addition to smoking cessation counseling, whereas the placebo group received matched placebo tablets in addition to counseling. The institutional review board of the Medical University of South Carolina, Charleston, approved the study protocol, which was overseen by an independent National Institute on Drug Abuse–appointed data and safety monitoring board.

The study followed smoking participants for 12 weeks to collect data on a variety of outcomes, including weekly TLFB assessments of abstinence days during the prior week. Specifically, at each weekly visit, participants reported the number of abstinent days during the past week. Thus, each patient had up to 12 weekly assessments, with each assessment representing a binomial count ranging from 0 to 7 abstinent days. Of the 157 participants enrolled in the study, 12 subjects had no TLFB data, resulting in a final sample of 145 subjects (67 in varenicline and 78 in placebo) with a total of 1282 weekly visits. Further details on study design and participant demographics can be found in Gray (2019).¹⁷

Figure 1 presents the mean abstinence days at each visit for the two study groups. Local polynomial regression (LOESS) smoothing lines are used to summarize temporal trends for the placebo (solid line) and varenicline (dashed line) groups. Throughout the study, the varenicline group reported more abstinent days on average compared to the placebo group, particularly early in the study. These trends support previous research suggesting that varenicline promotes early abstinence among adolescent smokers.¹⁷ The placebo group, meanwhile, displayed a slow but steady increase in abstinence over the course of the study, which is expected given that these patients received usual care in the form of counseling. We also observed a plateau effect toward the end of the study whereby the mean abstinence days stabilized around week 8, particularly for the varenicline group. This potential changepoint is highlighted in Figure 1 by the vertical dotted line. By the end of the study, the two groups were nearly identical, with the varenicline group reporting an average of 3.6 abstinent days at week 12 compared to 3.5 for the placebo group.

FIGURE 1.

Mean abstinence days for two treatment groups in the smoking cessation study. LOESS-smoothed lines are used to show the abstinence trends for each group. The vertical line represents a potential changepoint at week 8 for the varenicline group.

Next, we investigated the potential for zero inflation, as the dataset contained 51% zeros overall. Figure 2 presents the histogram of abstinent days in the past week at weeks 1, 6, and 12. We observed a high percentage of zeros throughout the study, particularly early in the study when participants were more inclined to smoke. To further check for the presence of zero inflation, we fit several fixed effects regression models with time, treatment, and time × treatment interaction as covariates to help determine the most appropriate modeling strategy for these data. Specifically, we fit Poisson, zero-inflated Poisson (ZIP), negative binomial (NB), zero-inflated negative binomial (ZINB), binomial, zero-inflated binomial (ZIB), ordinary beta-binomial, and ZIBB models via maximum likelihood and used Akaike’s information criterion (AIC) to select the best-fitting model. The results are presented in Table 1. The ZIBB model had the lowest AIC, followed by the ordinary beta-binomial and ZIP models, suggesting that a ZIBB model accounting for both overdispersion and zero inflation is ideally suited for these data.

FIGURE 2.

Histograms of abstinent days at weeks 1, 6, and 12.

TABLE 1.

Model comparison statistics for the smoking cessation study (bold indicates preferred model).

Candidate model	AIC
Poisson	6050.7
ZIP	4342.2
NB	4764.0
ZINB	4344.0
Binomial	7563.0
ZIB	4697.0
BB	3959.3
ZIBB	3923.4

3 |. ZERO-INFLATED BETA-BINOMIAL MODEL

Given that the ZIBB showed the best fit in preliminary analyses, we sought to develop an appropriate ZIBB model for our study. Generally speaking, zero-inflated models are mixtures of a point mass at zero representing excess zeros and a discrete distribution for the remaining values. The term “excess” denotes the fact that the data contain more zeros than expected under a standard discrete distribution such as the binomial. By design, zero-inflated models partition zeros into two types. The first type is referred to as a “structural” zero; these zeros correspond to individuals who are not at risk for an event, and who therefore have no opportunity for a nonzero value. The second type, termed the “chance” or “at-risk” zero, applies to a latent class of individuals who are at risk for an event but nevertheless have an observed response of zero. For example, in our smoking cessation application, the structural zeros might represent patients who are unwilling to abstain from smoking, and thus report no abstinence during the prior week. The chance zeros, on the other hand, might correspond to patients who have a desire to abstain, but for various reasons are unable to do so. In short, then, zero-inflated models can be viewed as latent class models in which the classes are formed by the two types of zeros.

3.1 |. Zero-inflated beta-binomial mixed effects model

We propose a flexible ZIBB mixed effects model to address both zero inflation and overdispersion. For patient $i(i=1,\dots ,n)$ at week $j(j=1,\dots ,J)$ of the study, let $Y_{ij}$ denote the number of abstinent days in the past $m=7$ days. We assume that $Y_{ij}$ follows a ZIBB distribution, which can be expressed as a mixture of a point mass at zero that accounts for zero inflation and a beta-binomial (BB) distribution; that is,

$$Y_{ij}\sim \left(1-{\pi }_{ij}\right){\mathbb{1}}_{\left(R_{ij}=0\wedge Y_{ij}=0\right)}+{\pi }_{ij}BB\left(m,a_{ij},b_{ij}\right){\mathbb{1}}_{\left(R_{ij}=1\right)},$$ (1)

where ${\mathbb{1}}_{(\cdot )}$ is the indicator function and $R_{ij}$ is a latent abstinence indicator such that with probability $1-{\pi }_{ij}$, $R_{ij}=Y_{ij}=0$ implying a structural zero, and with probability ${\pi }_{ij}$, $R_{ij}=1$ and $Y_{ij}$ is in turn drawn from a BB distribution of the form

$$p\left(y_{ij}\mid R_{ij}=1\right)=\left(\genfrac{}{}{0pt}{}{m}{y_{ij}}\right)\frac{B\left(a_{ij}+y_{ij},m+b_{ij}-y_{ij}\right)}{B\left(a_{ij},b_{ij}\right)}$$ (2)

with observation-level shape parameters $a_{ij}$ and $b_{ij}$. The variable $R_{ij}$ can be viewed as a “potential” abstinence indicator that distinguishes between the two types of zeros: structural zeros and chance zeros. When $R_{ij}=0$, then $Y_{ij}=0$ with probability 1, and hence no abstinent days are recorded (i.e. a “structural zero” is observed). Conversely, when $R_{ij}=1$, $Y_{ij}$ is drawn from a $\mathrm{B}\mathrm{B}\left(m,a_{ij},b_{ij}\right)$ distribution, and there is the potential to observe abstinent days. Because the BB distribution allows for zeros, it is possible to observe no abstinence (ie, a “chance” zero) even if $R_{ij}=1$.

Following Arostegui et al¹⁸ and Najera-Zuloaga et al,¹⁹ we model the shape parameters as $a_{ij}={\varphi }_{ij}(1-\rho )/\rho$ and $b_{ij}=\left(1-{\varphi }_{ij}\right)(1-\rho )/\rho$, where ${\varphi }_{ij}$ denotes the proportion of abstinent days given $R_{ij}=1$ and $\rho$ is a correlation parameter controlling the degree of overdispersion $(0<\rho <1)$. Under this set-up, given $R_{ij}=1$, the mean and variance of $Y_{ij}$ are $\mathrm{E}\left(Y_{ij}\mid R_{ij}=1\right)=m{\varphi }_{ij}\equiv {\mu }_{ij}$ and $\mathrm{V}\mathrm{a}\mathrm{r}\left(Y_{ij}\mid R_{ij}=1\right)=m{\varphi }_{ij}\left(1-{\varphi }_{ij}\right)[1+\rho (m-1\left)\right]$, resulting in overdispersion relative to the conventional binomial model. The marginal mean of $Y_{ij}$, averaged over the latent indicator $R_{ij}$, is $\mathrm{E}\left(Y_{ij}\right)={\pi }_{ij}{\mu }_{ij}\equiv v_{ij}$. Thus, ${\nu }_{ij}$ represents the marginal mean abstinent days at week $j$ across the structural and abstinent classes, while ${\mu }_{ij}$ represents the conditional mean given $R_{ij}=1$. As $\rho$ approaches 1, the conditional distribution of $Y_{ij}$ given $R_{ij}=1$ becomes increasingly dispersed relative to the binomial. In our application, $m$ is fixed at 7 for all $i$ and $j$ by study design; more generally, however, $m$ could vary as a function of $i$ and $j$. Likewise, we assume constant $\rho$, but this could be relaxed if needed by allowing $\rho$ to vary across observations if the data warrant such an approach.

We can extend model (1) to the longitudinal setting by modeling ${\pi }_{ij}=\mathrm{P}\mathrm{r}\left(R_{ij}=1\right)$ and ${\varphi }_{ij}$ in terms of fixed and random effects using logit links:

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ij}\right)=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left[\mathrm{P}\mathrm{r}\left(R_{ij}=1\mid \mathit{\alpha },{\mathit{b}}_{1i}\right)\right]={\mathit{x}}_{1ij}^T\mathit{\alpha }+{\mathit{u}}_{1ij}^T{\mathit{b}}_{1i}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t},$$ (3)

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\varphi }_{ij}\right)={\mathit{x}}_{2ij}^T\mathit{\beta }+{\mathit{u}}_{2ij}^T{\mathit{b}}_{2i,}\text{BB}\text{component,}$$ (4)

where ${\mathit{x}}_{1ij}$ and ${\mathit{x}}_{2ij}$ are $p_1\times 1$ and $p_2\times 1$ vector of design covariates for the ith individual at week $j$; $\mathit{\alpha }$ and $\mathit{\beta }$ are $p_1\times 1$ and $p_2\times 1$ fixed effect of coefficients corresponding to binary and BB components, respectively; ${\mathit{u}}_{1ij}$ and ${\mathit{u}}_{2ij}$ are $q_1\times 1$ and $q_2\times 1$ random effect design vectors; and ${\mathit{b}}_{1i}$ and ${\mathit{b}}_{2i}$ are $q_1\times 1$ and $q_2\times 1$ random effects for binary and BB components. Additionally, it may be reasonable to assume that there is correlation between the binary and BB components, thereby allowing for dependence between the probability of abstinence and the BB distribution among those in the abstinence class. For example, in our application, patients who are more likely to be in the abstinence class (higher ${\pi }_{ij}$) may exhibit more abstinent days per week (higher ${\varphi }_{ij}$). To accommodate this dependence, we assume that the $(q_1+q_2)\times 1$ vector of random effects ${\mathit{b}}_i={\left({\mathit{b}}_{1i}^T,{\mathit{b}}_{2i}^T\right)}^T$ follows a multivariate normal distribution:

$${\mathit{b}}_i\mid {\mathbf{\Sigma }}_b\stackrel{ind}{\sim }{\mathrm{M}\mathrm{V}\mathrm{N}}_{q_1+q_2}\left(0,{\mathbf{\Sigma }}_b\right),$$ (5)

where ${\mathbf{\Sigma }}_b$ is a $\left(q_1+q_2\right)\times \left(q_1+q_2\right)$ positive-definite covariance matrix. The off-diagonal elements of ${\mathbf{\Sigma }}_b$ control the covariances between the random effects, thus accommodating dependence-between the binary and BB components. As a special case, we can set the covariances to be $0$ to permit independent model components. However, recent work has demonstrated that misspecifying the random effect covariance as independent when they are in fact correlated can lead to biased inferences.^15,16 We, therefore, recommend starting with a general unstructured covariance matrix and reducing the dimension if the data suggest that a simpler covariance structure is warranted.

3.2 |. Extension to random changepoint model

Motivated by the smoking cessation study, we now extend the model to the setting where the group temporal trends exhibit random changepoints over the course of the study. We adopt a linear changepoint model rather than a higher-order spline or polynomial model because identifying the precise timing when treatment effects taper off is often of clinical interest. As noted in Figure 1, we observed a plateau effect toward the end of the study whereby the probability of abstinence leveled off around week 8, particularly for the varenicline group. As further support for this point, Figure S1 in the Online Supplement also presents an analogous figure on the empirical logit scale, since this is the linear predictor scale in which the splines operate. Upon observing the logit scale plot, we again noticed two segmented lines in the treatment group and a straight line in the placebo group. To accommodate these temporal changes, we consider a piecewise generalized linear mixed model that includes group-specific changepoints for each component:

$$\begin{gathered} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ij}\right)={\mathit{x}}_{1ij}^T\mathit{\alpha }+{\mathit{u}}_{1ij}^T{\mathit{b}}_{1i} \\ ={\alpha }_0+{\alpha }_1{t}_{ij}+{\alpha }_2{\mathrm{t}\mathrm{r}\mathrm{t}}_i+{\alpha }_3\left(t_{ij}\times {\mathrm{t}\mathrm{r}\mathrm{t}}_i\right)+{\alpha }_{41}{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\alpha }_{42}{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)} \\ +b_{1i1}+b_{1i2}{t}_{ij}+b_{1i3}\left[{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)}\right] \end{gathered}$$ (6)

$$\begin{gathered} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\varphi }_{ij}\right)={\mathit{x}}_{2ij}^T\mathit{\beta }+{\mathit{u}}_{2ij}^T{\mathit{b}}_{2i} \\ ={\beta }_0+{\beta }_1{t}_{ij}+{\beta }_2{\mathrm{t}\mathrm{r}\mathrm{t}}_i+{\beta }_3\left(t_{ij}\times {\mathrm{t}\mathrm{r}\mathrm{t}}_i\right)+{\beta }_{41}{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}}_i=0\right)}+{\beta }_{42}{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}}_i=1\right)} \\ +b_{2i1}+b_{2i2}{t}_{ij}+b_{2i3}\left[{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}}_i=0\right)}+{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}}_i=1\right)}\right], \end{gathered}$$ (7)

where $t_{ij}$ denotes study week $\left(t_{ij}=1,2,\dots ,12\right);{\mathrm{t}\mathrm{r}\mathrm{t}}_i=1$ if the ith subject is randomized to treatment group, and 0 if randomized to placebo group; the linear spline function is defined as ${\left(t_{ij}-{\kappa }_g\right)}_{+}=\left(t_{ij}-{\kappa }_g\right){\mathbb{1}}_{\left(t_{ij}>{\kappa }_g\right)}$, and thus takes the value 0 for $t_{ij}\le {\kappa }_g$ and $\left(t_{ij}-{\kappa }_g\right)$ for $t_{ij}>{\kappa }_g;{\kappa }_g$ denotes the random changepoint in the slope on the logit scale for group $g(g=1,2)$ for both the binary and BB components; ${\alpha }_0$ and ${\beta }_0$ are the intercept for the binary and BB component, respectively; ${\alpha }_1$ and ${\beta }_1$ represent the slopes before the changepoint for the placebo group; ${\alpha }_2$ and ${\beta }_2$ represent the baseline treatment effect for the binary and BB component; ${\alpha }_3$ and ${\beta }_3$ are interaction effects between time and treatment prior to the changepoints for the two model components; ${\alpha }_{41}$ and ${\alpha }_{42}$ are group-specific changes in slopes after the changepoints in the binary component, and ${\beta }_{41}$ and ${\beta }_{42}$ are the changes in slopes after the group-specific changepoints in the BB component. Furthermore, for each subject, random intercepts ($b_{1i1},b_{2i1}$) and random slopes ($b_{1i2},b_{2i2}$) before changepoints are included in both components. Finally, we allow for subject-specific changes in slopes following the changepoints; these are captured by $b_{1i3}$ for the binary component and $b_{2i3}$ for the BB component.

Figure 3 presents a schematic diagram for BB component of the proposed model (an analogous scheme can be developed for binary component). We assume two changepoints exist at week ${\kappa }_1$ and ${\kappa }_2$ for the placebo and treatment groups (vertical dot-dashed lines), respectively (${\kappa }_1=5$ and ${\kappa }_2=8$ in this hypothetical example). The y-axis denotes the logit scale of ${\varphi }_{ij}$, which is the linear predictor scale from Equation (7). The two solid lines represent hypothetical treatment trends over 12-weeks study. For the placebo group (represented using triangles), ${\beta }_1$ denotes the group-level slope before ${\kappa }_1$ and (${\beta }_1+{\beta }_{41}$) is the slope after ${\kappa }_1$; similarly, for the treatment group (highlighted in diamonds), (${\beta }_1+{\beta }_3$) is the group-level slope before ${\kappa }_2$ and (${\beta }_1+{\beta }_3+{\beta }_{42}$) is the slope after ${\kappa }_2$. Further, ${\beta }_{41}$ and ${\beta }_{42}$ denote the change in slopes following ${\kappa }_1$ and ${\kappa }_2$ for the placebo and treatment groups, respectively. The two dotted lines represent the subject-specific trajectories, one for each group. For subject $i$ in placebo group, $b_{2i2}$ is the random slope before the changepoint ${\kappa }_1$, and $b_{2i3}$ represents the change in random slope after ${\kappa }_1$; similarly, for subject $i^{\prime }$ in treatment group, $b_{2i^{\prime }2}$ is the random slope before the changepoint ${\kappa }_2$, and $b_{2i^{\prime }3}$ represents the change in random slope following ${\kappa }_2$. All corresponding slopes are labeled on the regression lines.

FIGURE 3.

A schematic diagram for the BB component of the proposed model. The two solid lines represent hypothetical group-level trends for the placebo and treatment groups. The two dotted lines represent the subject-specific trajectories. Corresponding slopes are labeled on the regression lines.

3.3 |. Bayesian inference

Statistical inference can be handled using maximum likelihood or Bayesian methods. However, maximum likelihood becomes challenging as the complexity of the model increases, particularly for models that include multiple changepoints. As an alternative, we develop a Bayesian approach for the changepoint model proposed in Section 3.2.

To ensure that the changepoint model is well identified, we assign weakly informative $\mathrm{M}\mathrm{V}\mathrm{N}\left({\mathit{\mu }}_0,{\mathbf{\Sigma }}_0\right)$ priors to the fixed effects $\mathit{\alpha }$ and $\mathit{\beta }$, with defaults ${\mathit{\mu }}_0=0$ and ${\mathbf{\Sigma }}_0=100{\mathit{I}}_6$. For the changepoints, ${\kappa }_1$ and ${\kappa }_2$, we assign independent truncated normal priors, denoted $\mathrm{T}\mathrm{N}\left({\mu }_0,{\sigma }_0^2,L_0,U_0\right)$ with defaults ${\mu }_0=5$, ${\sigma }_0^2=100$, lower bound $L_0=2$, and upper bound $U_0=11$. The lower and upper bounds are chosen to ensure a reasonable range for the potential changepoints in our application. Given that our study spans from week 1 to week 12, we anticipated that potential changepoints would likely occur between week 2 and week 11. In general, users have the flexibility to define lower and upper bounds for priors in random changepoints. Furthermore, a popular non-informative proper prior for $d\times d$ matrix of a random effect variance ${\mathbf{\Sigma }}_b$ is to select the inverse Wishart $\left(v_0,{\mathit{G}}_0\right)$ with $v_0=d+1$ and ${\mathit{G}}_0={\mathit{I}}_d$. This has the appealing property that each of the correlations in ${\mathbf{\Sigma }}_b$ are marginally a uniform prior distribution.²⁰ We also assign a proper but non-informative uniform $\mathrm{U}(0,1)$ prior to the BB correlation parameter $\rho$, thus allowing the data to fully inform the posterior in this case. Finally, we assign a MVN prior to the 6 × 1 vector of random effects, ${\mathit{b}}_i={\left({\mathit{b}}_{1i}^T,{\mathit{b}}_{i2}^T\right)}^T$, as outlined in Equation (5).

For posterior computation, we adopt a hybrid Gibbs-MH algorithm, whereby we update the parameters for the binary component using Gibbs sampling and the parameters for the BB component using MH steps, as there are no closed-form full conditionals for the BB component. Specifically, for the binary component in Equation (6), we adopt the convenient Pólya-Gamma data-augmentation algorithm.²¹ The sampler introduces latent Pólya-Gamma weights, ${\omega }_{ij}$, which can in effect enable us to represent the logistic regression model as a latent mixture of normal distributions with Pólya-Gamma precision (inverse variance) terms. For $b>0$ and $c\in \mathbb{R}$, ${\omega }_{ij}$ is said to have a Pólya-Gamma (PG) distribution if

$${\omega }_{ij}\sim \mathrm{P}\mathrm{G}(b,c)\stackrel{d}{=}\frac{1}{2{\pi }^2}\sum _{k=1}^{\mathrm{\infty }}\frac{d_k}{(k-1/2{)}^2+c^2/\left(4{\pi }^2\right)},$$

where $d_k$ ‘s are independently distributed to $\mathrm{G}\mathrm{a}(b,1)$. This leads to a convenient data-augmented Gibb-samplers for parameters in binary component, such as $\mathit{\alpha }$ and ${\mathit{b}}_{1i}={(b_{1i1},b_{1i2},b_{1i3})}^T$. If we draw ${\omega }_{ij}$ independently from $\mathrm{P}\mathrm{G}(1,{\mathit{x}}_{1ij}^T\mathit{\alpha }+{\mathit{u}}_{1ij}^T{\mathit{b}}_{1i})$, it can be shown that, conditional on ${\omega }_{ij}$, then the latent response $z_{ij}=\left(R_{ij}-1/2\right)/{\omega }_{ij}$ is distributed according to $\mathrm{N}({\mathit{x}}_{1ij}^T\mathit{\alpha }+{\mathit{u}}_{1ij}^T{\mathit{b}}_{1i},{\omega }_{ij}^{-1})$, where $R_{ij}$ is the latent abstinence indicator. Therefore, for posterior inference, we can update $\mathit{\alpha }$ from a MVN full conditional and $b_{1il}$ ($l=1,2,3$ and $i=1,\dots ,n$) from a univariate normal full conditional using an efficient data-augmented Gibbs sampling routine. Further details on Pólya-Gamma data augmentation for zero-inflated models can be found in Neelon.²²

For the BB component in Equation (7), we use a mixture of Gibbs and MH steps for posterior computation. First, we update the latent abstinence indicators $R_{ij}$ from their discrete full conditional distributions as outlined in Neelon.²² Next, we update the fixed effects, $\mathit{\beta }$, using an adaptive MH algorithm²³ with a MVN proposal density, where the proposal variance is tuned during the burn-in period to ensure the optimal acceptance rate. We then update the correlation parameter, $\rho$, using a random-walk MH step based on TN proposal density restricted to the interval (0,1). We next update the random effects ${\left(b_{2i1},b_{2i2},b_{2i3}\right)}^T$ using random-walk MH steps independent normal proposals. We use a Gibbs step to update ${\mathbf{\Sigma }}_b$ from its IW full conditional. Finally, we update the two random changepoints, ${\kappa }_1$ and ${\kappa }_2$, using independent random-walk MH steps with TN proposal densities restricted to plausible ranges based on the application. Of note, at each MCMC iteration, we assume a piecewise linear model for each study group. However, after averaging across MCMC iterations, we obtain smooth posterior mean estimates of the temporal trends for each group, since the locations of the changepoints vary across MCMC iterations, thus building an automated smoothing mechanism into the MCMC algorithm. Further details of the posterior algorithm, including derivations of full conditional distributions, can be found in Appendix A of the Online Supplement.

Convergence of the algorithm is monitored using standard Bayesian diagnostics such as trace plots and Geweke statistics,²⁴ which we implement in R software²⁵ using the coda package.²⁶ To select the optimal model, we adopt the Bayesian Watanabe Information Criteria (WAIC)²⁷ model-comparison measure, which combines a measure of predictive accuracy of the fitted model with a penalty for model complexity. Models with smaller WAIC values are preferred. Using this model selection procedure, we can assess the presence of various model features, such as zero inflation and overdispersion. To further evaluate the model fit, we generate data replicated from the posterior predictive distribution²⁸ and produce a histogram comparing the observed and posterior-predicted abstinent days at weeks 1, 6, 12. If the model fits well, the posterior predictive histogram should mirror the observed histogram from Figure 1.

4 |. SIMULATION STUDY

We conducted a simulation study to evaluate the performance of the changepoint model proposed in Section 3.2. The goal of the simulation was to determine whether we could accurately recover the group-specific trajectories and correctly estimate the changepoints for each group. To mimic the longitudinal design of the varenicline trial, we generated data from the following ZIBB changepoint model:

$$\begin{gathered} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\pi }_{ij}\right)={\alpha }_0+{\alpha }_1{t}_{ij}+{\alpha }_2{\mathrm{t}\mathrm{r}\mathrm{t}}_i+{\alpha }_3\left(t_{ij}\times {\mathrm{t}\mathrm{r}\mathrm{t}}_i\right)+{\alpha }_{41}{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\alpha }_{42}{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)} \\ +b_{1i1}+b_{1i2}{t}_{ij}+b_{1i3}\left[{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)}\right] \end{gathered}$$

$$\begin{gathered} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\varphi }_{ij}\right)={\beta }_0+{\beta }_1{t}_{ij}+{\beta }_2{\mathrm{t}\mathrm{r}\mathrm{t}}_i+{\beta }_3\left(t_{ij}\times {\mathrm{t}\mathrm{r}\mathrm{t}}_i\right)+{\beta }_{41}{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\beta }_{42}{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)} \\ +b_{2i1}+b_{2i2}{t}_{ij}+b_{2i3}\left[{\left(t_{ij}-{\kappa }_1\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=0\right)}+{\left(t_{ij}-{\kappa }_2\right)}_{+}{1}_{\left({\mathrm{t}\mathrm{r}\mathrm{t}}_i=1\right)}\right]. \end{gathered}$$

In particular, we generated 500 subjects from the above model and allowed the number of observations per subject, $n_i$, to vary from 1 to 12, resulting in 3224 total observations. For each subject, we simulated the $n_i$ weekly visits, $t_{ij}$, from a discrete uniform distribution taking values {1, 2,…,12}. We generated the treatment indicator, ${\mathrm{t}\mathrm{r}\mathrm{t}}_i$, from $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}(0.5)$ and set the fixed effect coefficients to be $\mathit{\alpha }=(0.59,0.86,0.45,-0.35,-1.15,-0.54{)}^T$ and $\mathit{\beta }=(-0.94,0.59,0.89,0.43,0.68,-0.66{)}^T$. We set the BB correlation parameter, $\rho$, to be 0.25 and the two changepoints to be ${\kappa }_1=8$ for the placebo group and ${\kappa }_2=7$ for the treatment group. For each subject, we generated the 6 × 1 random effect vector, ${\mathit{b}}_i={\left(b_{1i1},b_{1i2},b_{1i3},b_{2i1},b_{2i2},b_{2i3}\right)}^T$, independently from a $\mathrm{M}\mathrm{V}\mathrm{N}\left(0,{\mathbf{\Sigma }}_b\right)$ density; the values of ${\mathbf{\Sigma }}_b$ are presented in Table 2. Figure S2 in Appendix C of the Online Supplement presents a histogram of the simulated data, which contained 53% zeros.

TABLE 2.

Parameter estimates and 95% CrIs for random changepoint ZIBB model used in the simulation study.

Component	Parameter	Simulated value	Posterior mean	95% CrI
Binary	${\alpha }_0$	0.59	0.36	[−0.12, 0.98]
	${\alpha }_1$	0.86	0.92	[0.67, 1.21]
	${\alpha }_2$	0.45	0.67	[0.04, 1.26]
	${\alpha }_3$	−0.35	−0.33	[−0.69, −0.02]
	${\alpha }_{41}$	−1.15	−1.13	[−1.61, −0.65]
	${\alpha }_{42}$	−0.54	−0.68	[−1.07, −0.27]
BB	${\beta }_0$	−0.94	−0.85	[−1.24, −0.48]
	${\beta }_1$	0.59	0.55	[0.38, 0.73]
	${\beta }_2$	0.89	0.81	[0.42, 1.20]
	${\beta }_3$	0.43	0.47	[0.25, 0.68]
	${\beta }_{41}$	0.68	0.60	[0.18, 0.99]
	${\beta }_{42}$	−0.66	−0.39	[−0.67, −0.05]
	$\rho$	0.25	0.28	[0.23, 0.32]
Changepoints	${\kappa }_1$	8.00	8.08	[7.46, 8.72]
	${\kappa }_2$	7.00	6.84	[6.12, 7.60]
	${\kappa }_1-{\kappa }_2$	1.00	1.24	[0.25, 2.20]
Random	${\mathbf{\Sigma }}_{11}$	0.50	1.19	[0.46, 2.28]
Effect	${\mathbf{\Sigma }}_{22}$	0.35	0.57	[0.34, 0.89]
	${\mathbf{\Sigma }}_{33}$	0.35	0.45	[0.12, 1.23]
	${\mathbf{\Sigma }}_{44}$	0.50	0.26	[0.08, 0.55]
	${\mathbf{\Sigma }}_{55}$	0.50	0.47	[0.31, 0.61]
	${\mathbf{\Sigma }}_{66}$	0.35	0.31	[0.10, 0.88]
	${\mathbf{\Sigma }}_{12}$	0.05	0.02	[−0.17, 0.18]
	${\mathbf{\Sigma }}_{13}$	0.05	0.01	[−0.12, 0.15]
	${\mathbf{\Sigma }}_{23}$	0.05	−0.01	[−0.15, 0.10]
	${\mathbf{\Sigma }}_{14}$	0.05	−0.01	[−0.21, 0.13]
	${\mathbf{\Sigma }}_{24}$	0.05	0.03	[−0.08, 0.16]
	${\mathbf{\Sigma }}_{34}$	0.05	0.01	[−0.09, 0.12]
	${\mathbf{\Sigma }}_{15}$	0.05	−0.00	[−0.21, 0.20]
	${\mathbf{\Sigma }}_{25}$	0.05	0.06	[−0.05, 0.16]
	${\mathbf{\Sigma }}_{35}$	0.05	−0.03	[−0.18, 0.12]
	${\mathbf{\Sigma }}_{45}$	0.15	0.02	[−0.09, 0.12]
	${\mathbf{\Sigma }}_{16}$	0.05	−0.05	[−0.12, 0.26]
	${\mathbf{\Sigma }}_{26}$	0.05	0.03	[−0.12, 0.25]
	${\mathbf{\Sigma }}_{36}$	0.05	0.01	[−0.13, 0.14]
	${\mathbf{\Sigma }}_{46}$	0.05	0.03	[−0.05, 0.21]
	${\mathbf{\Sigma }}_{56}$	0.05	−0.02	[−0.19, 0.14]

For Bayesian inference, we implemented the data-augmented Gibbs sampler for the binary component and the hybrid Gibbs and MH sampler for the BB component, as outlined in the previous section. We assigned a $\mathrm{M}\mathrm{V}\mathrm{N}(0,100{\mathit{I}}_6)$ prior to the fixed effect regression coefficients for both components, a Unif(0,1) prior to the BB correlation parameter, and an $\mathrm{I}\mathrm{W}\left(7,{\mathit{I}}_6\right)$ prior to the random effect covariance matrix. Finally, we assigned two independent $\mathrm{T}\mathrm{N}\left(\mathrm{5,100},L_0=2,U_0=11\right)$ priors to the random changepoints, ${\kappa }_1$ and ${\kappa }_2$.We ran the MCMC algorithm for 30,000 iterations, discarding the first 10 000 iterations as burn-in and retained every 5th iteration to reduce autocorrelation. MCMC diagnostics, including trace plots and Geweke statistics, indicated excellent posterior mixing and convergence of the MCMC chain. Figure S3 in Appendix C of the Online Supplement presents the traceplots and Geweke diagnostic p-values for selected model parameters.

Table 2 presents the simulated parameter values along with the posterior mean estimates and 95% credible interval (CrIs). Overall, the estimates were close to the true values with 95% CrIs that contained the simulated values. The true changepoints for both groups were also accurately estimated. One appealing feature of our proposed model is that we can easily assess the equivalence of the group-specific changepoints. If the 95% CrI for the difference in the two changepoints includes 0, this suggests that the two changepoints are coincident. In such cases, one can adopt a more parsimonious model by assuming a single changepoint common to both study groups. In our simulation, the true difference between the changepoints is ${\kappa }_1-{\kappa }_2=1$; the posterior mean of this difference was 1.28 (95% CrI: [0.27, 2.22]), providing evidence that the two changepoints occur separately. To further investigate the behavior of the changepoints, we replicated the simulation with both changepoints occurring at week 7. In this case, the posterior mean difference between the two changepoints was −0.29 (95% CrI: [−1.13, 0.57]), indicating that the proposed model can detect simultaneous changepoints when supported by the data.

Figure 4 presents true and estimated population average marginal mean trends for each study group, where the marginal mean for the ($i,j$)-th observation is given by $v_{ij}={\pi }_{ij}{\mu }_{ij}=m{\pi }_{ij}{\varphi }_{ij}$ and ${\pi }_{ij}$ and ${\varphi }_{ij}$ are defined in Equations (6) and (7). The y-axis represents weekly mean abstinent days for the placebo (panel a) and treatment (panel b) groups. The vertical lines indicate the estimated changepoints and their 95% CrIs. The true changepoint values are shown in the bottom right corner of each figure. To estimate the population average curves, at the sth iteration of the MCMC sampler, we computed the marginal mean, ${\nu }_{il}^{(s)}={\pi }_{il}^{(s)}{\mu }_{il}^{(s)}=m{\pi }_{il}^{(s)}{\varphi }_{il}^{(s)}$, for subject $i$ over a grid of $L$ points ($l=1,\dots ,L$). Next, at each grid point $l$, we averaged across subjects in each group to obtain the group population average estimates at the $s$th iteration. Finally, we averaged across the MCMC iterations to obtain final posterior mean estimates and 95% CrIs.

FIGURE 4.

Marginal mean abstinent days for simulation study: (A) placebo group and (B) treatment group.

As expected, the posterior mean trends in Figure 4 mirror the observed data and are able to capture the occurrence of the changepoints. Specifically, at week 6, the estimated abstinent days (2.20 days for placebo and 3.82 days for treatment) were close to the true values (2.38 days for placebo and 3.82 days for treatment). Let ${\delta }_j$ be the true difference in abstinent days between the groups at week $j$, where $j=1,\dots ,12$. The true difference at week 6 between the groups, ${\delta }_6$, was 1.45 days, whereas the estimated difference, ${\hat{\delta }}_6$, was 1.61 (95% CrI: [1.24, 1.99]), indicating a discernible difference between groups at this time point. Likewise, at week 9, the estimated mean abstinent days were 4.03 and 4.05 for placebo and treatment groups, while the mean true abstinent days were 4.20 and 4.05 days, respectively. The true difference between the groups at week 9, ${\delta }_9$, was −0.16 (${\hat{\delta }}_9=-0.02$; 95% CrI: [−0.25, 0.31]) days, suggesting no substantial difference between groups at week 9. This highlights the flexibility of the model—in particular, its ability to accurately detect the group-specific changes in trends and to identify the occasions when the treatment effect is most prominent. To better highlight the slopes before and after the changepoints, we also provide the trend figures on the logit scale in Figure S4 in Appendix C of the Online Supplement. To further assess the importance of accounting for zero inflation, we compared the proposed ZIBB model to an analogous ordinary BB model, which is identical to the ZIBB model with the binary component removed. The WAIC value was lower in the proposed ZIBB model (6359) compared to the BB model (6770), suggesting that the ZIBB provides a better fit when the data are truly zero-inflated. Additionally, Figure S5 in Appendix C of the Online Supplement presents the marginal mean trends between the two study groups, indicating the poor alignment between simulated data and the fitted BB model. This further supports the use of the ZIBB model in this setting.

As a final evaluation, we re-ran the simulation with a sample of n=1000. The results, presented in Table S1 and Figure S6 in Appendix C of the Online Supplement, suggest that the predicted trends are quite accurate with relatively narrow CrIs, indicating that as the sample size increases, the accuracy of the model improves substantially. Of note, for n=500, the mixture of Gibbs sampling and MH algorithm took 22 min and 5 s to run 30 000 iterations on a Dell XPS 8940 ×64-based PC; for n=1000, the algorithm took 32min and 30 s to run. Lastly, to assess the potential efficiency gains of the proposed Pólya-Gamma sampler for the binary component, we compared hybrid Gibbs-MH algorithm to a full MH algorithm for both components in terms of computation time and model performance. Overall, our findings indicate that the hybrid algorithm outperforms full MH updates. The proposed hybrid algorithm appeared to stabilize after about 25 000 iterations, whereas under the full MH algorithm, most parameters had not converged by 30 000 iterations, especially in the binary component. Figure S7 in Appendix C of Online Supplement presents the selected traceplots and MCMC diagnostics for selected parameters in the binary component.

5 |. APPLICATION TO THE SMOKING CESSATION STUDY

Next, we fit the proposed ZIBB model from Section 3.2 to the smoking cessation study. The model specification was similar to Equations (6) and (7). Prior specification was identical to that outlined in Section 4, although we restricted the range of TN prior for the changepoints to weeks 5–11 to better reflect the pattern observed in Figure 1. We ran the Gibbs sampler conservatively for 80 000 iterations, discarding 30 000 iterations as burn-in and retained every 20th iteration. Figure S8 in Appendix D of the Online Supplement presents the traceplots and Geweke diagnostic p-values.

For model development, we started with the two-changepoint ZIBB mixed model from Section 3.2, assuming changepoints existed for both groups. We found that the 95% CrIs of slopes after the changepoint in the placebo group included 0 for both components, suggesting that there was no discernible changepoint for this group. Therefore, we discarded the changepoint in the placebo group and re-fit the model with a changepoint in the varenicline group only. For model comparison, we additionally fit an ordinary BB model with single changepoint in the varenicline group to evaluate the zero-inflated assumption. We computed the WAIC values for the candidate models. The resulting WAIC values were 2564 for the ZIBB model and 2580 for the ordinary BB model, suggesting better fit for the one-changepoint ZIBB model. We also attempted to fit a quadratic version of the ZIBB model with a quadratic trend in addition to linear terms, but the model demonstrated poor convergence based on traceplots, Geweke’s diagnostics, and effective sample size (ESS) evaluation (see Figure S9 in Appendix D of the Online Supplement for details). This was observed even after running the MCMC for 120 000 iterations, suggesting better fit in linear time trend compared to the quadratic model.

Table 3 presents the posterior means and 95% CrIs for time, treatment, and interaction covariates for the one-changepoint ZIBB model. The random effect covariance estimates are provided in Table S2 in Appendix D of the Online Supplement. The reference group for treatment group was the placebo group and the time variable was centered at week 5. The changepoint for varenicline group was estimated to occur at week 9.09 (95% CrI: [7.63, 10.57]). For the binary component, the large estimates of ${\alpha }_0$ and ${\alpha }_1$ suggest that placebo patients had a high probability of belonging to the abstinent class throughout the study (or, conversely, a low probability of observing a structural zero). For example, the percentage of structural zeros in the placebo group was 5.71% at week 1 and this dropped to 0.31% by week 5. The positive estimates for the treatment indicator (${\hat{\alpha }}_2$ and ${\hat{\beta }}_2$) indicated that the varenicline group had a higher abstinence rate at week 5 (the “baseline” week after centering time). In the BB component, the slope before the changepoint (${\hat{\beta }}_1+{\hat{\beta }}_3=0.55$; 95% CrI: [0.38, 0.74]) indicated a positive trend on the logit scale for the varenicline group prior to week 9 for subjects in the abstinence class. Moreover, the BB slope after the changepoint for the varenicline group was 0.08 ($={\hat{\beta }}_1+{\hat{\beta }}_3+{\hat{\beta }}_{42}$; 95% CrI: [−0.36, 0.44]), indicating that the treatment effect stabilized after week 9 for subjects in the abstinence class. Lastly, the BB correlation parameter was 0.13 (95% CrI: [0.10, 0.16]), suggesting that overdispersion remained present in the data even after accounting for between-subject variation through the inclusion of random effects.

TABLE 3.

Posterior means and 95% CrIs from the analysis of the smoking cessation study.

Component	Variable	Interpretation	Posterior mean	95% CrI
Binary	${\alpha }_0$	Intercept	7.12	[4.01, 11.92]
	${\alpha }_1$	Time a	0.56	[−0.36, 1.60]
	${\alpha }_2$	Treatment b	0.67	[−3.90, 5.55]
	${\alpha }_3$	Interaction (Time × Treatment)	0.06	[−1.57, 1.82]
	${\alpha }_{42}$	Change in slope after CP for varenicline group	−0.55	[−5.28, 3.57]
BB	${\beta }_0$	Intercept	−2.82	[−3.97, −1.94]
	${\beta }_1$	Time a	0.45	[0.32, 0.61]
	${\beta }_2$	Treatment b	1.24	[0.15, 2.53]
	${\beta }_3$	Interaction (Time × Treatment)	0.09	[−0.15, 0.35]
	${\beta }_{42}$	Change in slope after CP for varenicline group	−0.48	[−0.96, −0.04]
	$\rho$	BB correlation parameter	0.13	[0.10, 0.16]
Changepoints	${\kappa }_2$	CP (varenicline group)	9.09	[7.63, 10.57]

CP = changepoint.

^aTime was centered at week 5.

^bReference: placebo group.

Figure 5 presents the observed and estimated population average trends for the marginal mean number of abstinent days for the two groups: (a) placebo group and (b) varenicline group. The y-axis represents the marginal mean abstinent days at each weekly visit. The vertical lines on panel (b) represent the changepoint on varenicline group and its 95% CrI. Dotted and solid lines represent the observed and estimated mean abstinent days trends, respectively. Overall, we observed an increasing trend for the placebo group over the course of study, which is anticipated since the placebo group received usual care and is therefore expected to display a steady improvement in abstinence. Similarly, for the varenicline group, the estimated trend increased from week 1 to the estimated changepoint at ${\hat{\kappa }}_2=9.09$ weeks. After week 9, the varenicline effect leveled off and remained at about 3.5 abstinent days/week for the remainder of the study. At week 12, the observed mean abstinent days for the placebo and varenicline groups were 3.46 and 3.62 days, while the posterior mean estimates from the changepoint model were 3.70 (95% CrI: [3.29, 4.08]) and 3.62 (95% CrI: [3.16, 4.05]) for the two groups.

FIGURE 5.

Marginal mean abstinent days for smoking cessation study: (A) placebo group and (B) varenicline group.

Figure 6 presents the observed and estimated difference in mean number of abstinent days between groups. Let $v_{1j}$ and $v_{2j}$ denote the group-average marginal mean abstinent days at week $j$ for the placebo and varenicline groups, respectively. The y-axis denotes the difference in mean number of abstinent days between groups (eg, ${\delta }_j=v_{2j}-v_{1j}$) at week $j$. The horizontal line at 0 represents no difference in mean number of abstinent days between two groups. The estimated difference trend increased from week 1 through week 5 (${\hat{\delta }}_5=1.03$; 95% CrI: [0.74, 1.34]) before leveling off in week 7 (${\hat{\delta }}_7=0.99$; 95% CrI: [0.60, 1.39]). After week 7, the varenicline effect declined toward the end of the study. At week 12, the difference in abstinent days between groups was −0.08 (95% CrI: [−0.68, 0.51]), suggesting that the abstinence rate did not differ between groups. Moreover, the 95% CrIs did not include 0 from week 1 through 9, with the peak difference occurring between weeks 5 and 7, suggesting that this period might constitute a critical window in which varenicline is most efficacious. As a final check of model fit, Figure 7 compares histograms of the observed and posterior-predictive abstinent days generated from the proposed model at weeks 1, 6, and 12. The proposed model showed excellent fit, yielding a predictive histogram that closely mirrored the observe histogram at each week.

FIGURE 6.

Difference in mean abstinent days between the groups. Values above zero represent greater abstinence for varenicline group.

FIGURE 7.

Histograms of observed and posterior-predictive (“fitted”) abstinent days from the proposed model.

6 |. DISCUSSION

We have introduced a ZIBB mixed model with group-specific changepoints for the analysis of bounded TLFB data. To our knowledge, this is the first application of the ZIBB model to the longitudinal setting; it is also the first to introduce group-specific changepoints to flexibly capture the temporal trends for each study arm. For posterior inference, we proposed an efficient MCMC algorithm that relied on a combination of full-conditional Gibbs and MH steps. This estimation approach offers advantages over maximum likelihood, which can become challenging when the complexity of the model increases, particularly for models including high-dimensional random effects and multiple changepoints.

Our simulation results suggest that the model can accurately recover group-specific trajectories and precisely estimate the changepoints for each group. We can also detect whether group changepoints occur at the same time. In the smoking cessation trial, our analysis revealed that the varenicline group experienced higher abstinence rates early in the study, particularly between weeks 5 and 7, but these rates ebbed after week 9. This result conforms to a previous study that found no difference between the varenicline and placebo group at week 12.¹⁷ These findings may have important implications for the clinical use of varenicline. For example, clinicians may wish to prescribe the medication early in a patient’s course of treatment to help promote rapid, short-term abstinence until other therapies can take effect.

In some settings, quantifying the difference between groups is the primary (or perhaps sole) aim, and in such cases, one may be able to express a changepoint in terms of the difference between groups—for example, by introducing changepoint into the interaction terms that capture group differences. A potential limitation of this approach is that it assumes a common changepoint for both groups, which may not accurately reflect the underlying dynamics if treatment responses vary widely. This approach also fails to provide information regarding which specific groups experience changepoints. Of note, if we assume a common changepoint in our proposed model [ie, ${\kappa }_1={\kappa }_2$ in Equations (6) and (7)], it can be shown that our model is equivalent to a model that introduces a changepoint for the group × time interaction. Thus, our proposed model offers a flexible analytic strategy in which simpler models arise special cases, including models in which the changepoint is expressed in terms of the difference between groups.

There are a number of future directions for this work. In this article, we assumed independence among the group-specific changepoints. A more general approach might allow the group changepoints to be correlated through a joint prior structure. This might be appealing in settings where we expect the group changepoints to occur at similar but not identical times. The model also assumed identical group changepoints for the binary and BB components of the model, and hence we have two changepoints, one for each group; this could be extended to allow for different changepoints across model components. The model could further be extended to allow for subject-specific changepoints in addition to group-specific changepoints. Additionally, based on the preliminary analysis, we considered a relatively simple model with at most two changepoints—one for each group. Because we did not observe a change in slope for the placebo group, we ultimately settled on a single changepoint in the treatment group only. In more complex settings, one could allow for an unknown number of changepoints in each group.²⁹ However, such an approach should adopt an appropriate dimension-switching computational algorithm, such as reversible jump MCMC,³⁰ which can be challenging to fit, particularly for mixture models such as the ZIBB. It is possible to further extend the model to a marginalized zero-inflated model in order to effectively model the marginal mean count of the population, providing straightforward inference for overall exposure effects.^31,32 Moreover, in our application, we chose a piecewise linear spline for the varenicline group, as this choice provided an adequate fit to our data. More generally, higher-order splines may be preferable, and in such cases, our model can be easily extended to incorporate higher-order piecewise polynomials if the data support such an extension. Finally, rather than assuming the BB correlation parameter is constant across observations, one could model the correlation as a function of covariates to accommodate heterogeneity in the level of overdispersion across subjects.

In general, the proposed model should prove useful in a broad range of longitudinal studies involving overdispersed and zero-augmented binomial data. A relevant instance is provided by the Adverse Childhood Experiences (ACEs) questionnaire.³³ In this survey, participants document the cumulative count of adverse childhood events, ranging from 0 to 10 occurrences, leading to count data that are both bounded and overdispersed. The overdispersion arises due to the interrelation between adverse outcomes (if a participant reports one event, then the likelihood of reporting others increases). In the context of participants being enrolled in a longitudinal study with interventions randomly assigned, the potential exists for a decline in adverse events as time progresses. In such cases, changepoint analysis can play a crucial role in understanding the effectiveness of the interventions and identifying the point at which a significant change or reduction in adverse events occurs. More broadly, the method proposed here could be expanded to settings where the focus is on investigating temporal changes within the realm of overdispersed binomial data.

Supplementary Material

SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.

DATA AVAILABILITY STATEMENT

Data and R code used in the simulation studies are available upon request from the first author.