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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J R Stat Soc Ser C Appl Stat. Author manuscript; available in PMC 2017 April 1.
Published in final edited form as:
PMCID: PMC4812831
NIHMSID: NIHMS714834

Two-stage model for time varying effects of zero-inflated count longitudinal covariates with applications in health behaviour research

Summary

This study proposes a two-stage approach to characterize individual developmental trajectories of health risk behaviors and delineate their time-varying effects on short-term or long-term health outcomes. Our model can accommodate longitudinal covariates with zero-inflated counts and discrete outcomes. The longitudinal data of a well-known study of youth at high risk for substance abuse are presented as a motivating example to demonstrate the effectiveness of the model in delineating critical developmental periods of prevention and intervention. Our simulation study shows that the performance of the proposed model improves as the sample size or number of time points increases. When there are excess zeros in the data, the regular Poisson model cannot estimate either the longitudinal covariate process or its time-varying effect well. This result, therefore, emphasizes the important role that the proposed model plays in handling zero-inflation in the data.

Keywords: Functional data, Health risk behavior, Hurdle model, Measurement error, Mixed model, Substance abuse

1. Introduction

Health behavior research often collects longitudinal data on the frequency or quantity of health risk behaviors such as the number of drinking days in a month or the number of drinks consumed on a drinking day (Buu et al., 2012a). Characterizing the developmental trajectory of health risk behaviors can facilitate the assessment of disease progression or evaluation of long-term effects of treatment or intervention. It can also provide important information about the timing for prevention and intervention. Furthermore, the developmental trajectory may be predictive of a future health outcome. If we can establish such predictive validity, the developmental trajectory can be useful for identifying high risk individuals for prevention and intervention purposes. Our methodological work was motivated by a well-known alcohol study, the Michigan Longitudinal Study (MLS), that has followed children at risk for alcoholism from childhood to adulthood. Our statistical analysis aims to characterize the developmental trajectory of alcohol use behaviors during adolescence and evaluate whether the adolescent alcohol use pattern is predictive of the alcohol dependence diagnosis in adulthood.

In order to achieve these research goals, we have to overcome some methodological challenges. First, the covariate is measured at many time points but the outcome is collected at one future time point so standard longitudinal methods that were designed for longitudinal outcomes are not applicable in this setting. Second, the longitudinal covariates are usually count measures with excess zeros beyond what would be expected from a typical distribution for count data such as the Poisson (Buu et al., 2012a). Third, the effect of longitudinal patterns of health risk behaviors on a short-term or long-term health outcome may be a complex function of time. For example, those periods with frequent occurrence of binge drinking (defined as consuming more than 5 standard alcohol drinks in one episode) tend to have higher negative effects on alcohol related problems. Fourth, health risk behaviors are most of the time self-reported and thus are subject to measurement errors due to recall bias or embarrassment (Tourangeau and Yan, 2007). Fifth, the between-subject and within-subject variability tends to be large in this kind of data, especially for studies on high risk youth who has not yet developed regular patterns of health risk behaviors (Collins et al., 2008).

Zhang et al. (2007) developed a two-stage functional mixed model that was motivated by the need to investigate the effect of the follicle stimulating hormone (FSH) time profile during a menstrual cycle on total hip bone mineral density (BMD) measured at a single time point. The model consists of two stages: the first stage estimates the periodic subject-specific FSH profiles using a nonparametric measurement error model; the second stage plugs the estimated subject-specific profiles in a functional linear model and estimates the functional covariate effect. Another relevant model was proposed by Bhadra et al. (2012) who studied the association between longitudinal prostate-specific antigen exposure history (continuous covariate) and prostate cancer (binary outcome). Bayesian inferential techniques were used to estimate the parameters of the exposure and disease risk models simultaneously by using a joint likelihood.

In the statistics literature, two approaches are commonly used to deal with measurement errors: regression calibration and joint likelihood. The two existing models reviewed above (Zhang et al., 2007; Bhadra et al., 2012) represent these approaches, respectively. Since the longitudinal covariate process in our setting is nonparametric, it requires parameterization with high-dimensional random effects. Thus, the procedure proposed by Bhadra et al. (2012) may encounter computational issues because the reversible jump MCMC algorithm is expected to converge slowly in the presence of high-dimensional random effects. Furthermore, the procedure of Bhadra et al. (2012) was developed for the setting with continuous longitudinal covariates and a binary response. Extending their Bayesian framework to the setting studied in this paper may be conceptually straightforward and yet challenging for implementation, as our setting involves a zero-inflated Poisson covariate and a response with a conditional distribution in the exponential family. For these reasons, in this study, we adopt the regression calibration approach and extend the model proposed by Zhang et al. (2007), which was originally designed for a continuous longitudinal covariate process and a continuous scalar outcome. The resulting computational complexity is also addressed in our work.

Two models have been frequently adopted in applied fields to handle longitudinal zero-inflated count data: the hurdle model and the zero-inflated Poisson model (see Buu et al. (2012a) for a comprehensive review). Both of them are finite mixture models – the hurdle mixes 0 and a truncated Poisson, whereas the zero-inflated Poisson mixes 0 and a regular Poisson. Conceptually, the hurdle model assumes that everybody in the study population is at risk for being involved in health risk behaviors and the level of involvement changes over time, whereas the zero-inflated Poisson model assumes that the population consists of a group of people who are at risk and another group who are not. Thus, the conceptual framework of the hurdle model better fits the common research interest of longitudinal studies that examine health behaviors of the at-risk population developmentally. Furthermore, the hurdle model has greater flexibility (it can handle not only zero-inflated data but also zero-deflated data) and computational advantage. For these reasons, it is adopted in this study to characterize the longitudinal covariate process.

This paper is organized as follows. In Section 2, we present a generalized functional linear model that employs the hurdle model to characterize the longitudinal covariate process and also accommodates any response variable from the exponential family. In Section 3, the longitudinal data of a well-known study of substance abuse are presented as a motivating example. Section 4 delineates the design and results of the simulation study. We present discussion and concluding remarks in Section 5.

2. Statistical Model and Parameter Estimation

Suppose that from the i-th subject (i = 1, …, n), we collected a response variable yi, a p-dimensional vector of time-invariant covariates zi, and an observed longitudinal covariate process wij measured at time points tij [set membership] [T1, T2], j = 1, …, Ji. Because wij is a zero-inflated count variable, we use the hurdle model (Mullahy, 1986) to specify its distribution:

P(wij=0)=1πi,P(wij=w)=πieμijμijww!(1eμij)forw>0,
(1)

where 0 < πi < 1 and μij > 0. Let ui be the latent risk level and xi(t) be the latent longitudinal covariate process for the i-th subject. The parameters can be modeled as logit(πi) = ui and log(μij) = xi(tij). The unique contribution of these latent quantities ui and xi(t) is, therefore, to deal with measurement errors in wij, which is a self-reported health risk behavior in our setting.

Assume that the response yi comes from an exponential family fy(y; ηi) and g(·) is the corresponding link function. yi is related to the time-invariant covariates zi, the latent risk level ui and the latent longitudinal covariate process xi(t) through the following generalized functional linear model, which is a generalized version of the model in Müller and Stadtmüller (2005):

g(ηi)=ziTδ+uiβ+T1T2xi(t)γ(t)dt,
(2)

where δ is a vector of coefficients for the time-invariant covariates zi; β is a coefficient of ui indicating the effect of individual risk on health outcome; and γ(t) is a coefficient function of xi(t) that delineates the time-varying effect of the longitudinal covariate process.

Since both xi(t) and γ(t) are infinitely dimensional, we propose to estimate them by the natural cubic smoothing splines. Let t0=(t10,,tq0)T be a q-dimensional vector of ordered distinct values of all time points tij, and assume that xi(t) and γ(t) belong to the function space Sncs spanned by natural cubic splines in [T1, T2] with knots t0. According to Section 2.2 of Green and Silverman (1994) and Zhang et al. (2007), for any function s(t) [set membership] Sncs, there exists a set of q piecewise cubic polynomial basis functions c(t) = (c1(t), …, cq(t))T such that s(t) = sT c(t) where s = s(t0) is a q-dimensional vector formed by evaluating s(t) at knots t0, and ss(t) is a 1-1 mapping from Rq to Sncs. Under this setting, we have xi(t)=xiTc(t) and γ(t) = γT c(t). Therefore, Equation (2) becomes

g(ηi)=ziTδ+uiβ+xiTCγ,
(3)

where C=T1T2c(t)cT(t)dt is a q × q matrix. Each ci(t) is also a natural cubic spline function such that ci(t0) = ei, where ei is a q-dimensional vector with 1 for the i-th element and 0 otherwise. Hence, each element in C can be calculated straightforwardly.

We then smooth a nonparametric function s(t) by penalizing the second derivative through T1T2(s(t))2dt. For the particular settings of natural cubic spline, Equation (2.3) in Green and Silverman (1994) specifies a q × q symmetric matrix K such that T1T2(s(t))2dt=sTKs.

We adopt the two-stage approach proposed in Zhang et al. (2007) because it takes into account the measurement error of the observed longitudinal covariate process wij. This is essential as wij tends to be self-reported data of health risk behaviors in our setting. In Stage 1, we estimate ui and xi; in Stage 2, we estimate δ, β, and γ given the estimates obtained in Stage 1. The technical details are described in the following sections.

2.1. The first stage: estimating ui and xi

In the estimation procedure, we assume ui = u0 + υi where u0 is the population risk level and υi~N(0,συ2) is the random deviation of an individual’s risk from the population risk. Similarly, the individual profile xi(t) can be decomposed as xi(t) = x0(t)+di(t) where x0(t) is the population profile and di(t) is the random deviation of an individual profile from the population profile. We assume that both x0(t) and di(t) are natural cubic smoothing spline functions and di(t) is a zero-mean Gaussian process, that is, E(di(t)) = 0 for any t. In addition, υi and di(t) are assumed to be mutually independent. Under this assumption, the likelihood from Model (1) can be factored into two components: one involves a Bernoulli distribution and the other a zero-truncated Poisson distribution. The estimation can be, therefore, conducted in two independent steps.

2.1.1. Step 1 estimating ui

Let wij=0 if wij = 0 and wij=1 if wij > 0. We can estimate ui using a generalized linear mixed model: wij~Bernoulli(πi), where logit(πi) = ui = u0 + υi. The penalized quasi-likelihood (PQL) method (Breslow and Clayton, 1993) is adopted to carry out the estimation of u0 and υi.

2.1.2. Step 2 estimating xi

We estimate xi(t) from all wij > 0 (denoted by wij+) through the following model

wij+~ZeroTruncatedPoisson(μij),log(μij)=xi(tij)=x0(tij)+di(tij).
(4)

By stacking over j, log(μi) = Nix0+Nidi, where log(μi) = (log(μi1), …, log(μiJi))T, x0 = x0(t0), di = di(t0), and Ni is a Ji × q incidence matrix mapping (ti1, …, tiJi)T to t0 such that the (j, l)-th element is 1 if tij=tl0 and 0 otherwise.

We next transform the random subject-specific deviation as di = B*bi. Here, B* = (1q, t0, B) is a q × q matrix with B = L(LT L)−1, where L is a q × (q − 2) full-rank matrix satisfying LT (1q, t0) = 0 and LLT = K. In addition, bi ~ N(0, D) with a covariance matrix D=diag(σ12,σ22,τdI(q2)×(q2)). An advantage of this transformation is that τd along with B directly characterize the smoothness of di. It can be shown that B* is a q × q full-rank matrix, and di = B*bi is a 1-1 transformation. Hence, we should focus on finding the estimation of bi. Using the following notations: log(μ) = (log(μ1)T, …, log(μn)T)T, N=(N1T,,NnT)T, and Z = diag(N1B*, …, NnB*), we may write the model as a generalized additive mixed model (Lin and Zhang, 1999):

log(μ)=NX0+Zb,

where b=(b1T,,bnT)T~N(0,D(θ)) with the variance component θ=(σ12,σ22,τd) and D = diag(D, …, D).

Based on the results in Lin and Zhang (1999), the estimates of x0 and b can be derived by maximizing the double penalized quasi-likelihood (DPQL)

12ijhij12bTD1b12λxx0TKx0
(5)

for given λx and θ, where

hij=2uw1(wij+)μijmij(wij+uw(s))sds.

In this DPQL, uw(·) is the mean function of the zero-truncated Poisson distribution; λx > 0 is a smoothing parameter controlling the goodness-of-fit of the model and the smoothness of x0(t); K has been defined in the previous paragraph; and mij are the prior weights which are set to be 1 (i.e., equal weights) in most applications, unless we have prior knowledge of the relative contribution of observations. In our study, all mij are set to be 1.

Both x0 and b can be estimated by the method in Section 3 of Lin and Zhang (1999); and the variance component θ can be estimated by the marginal quasi-likelihood method and then bias-corrected following similar procedures described in Sections 4 and 5 of that paper. The tuning parameter λx is selected by the generalized cross validation (GCV). It is important to note that log(·) in Equation (4) is actually a pseudo-link function because it models μij rather than the mean of wij+ from the zero-truncated Poisson distribution, of which the mean and variance functions are uw(μ)=μ1eμ and υw(μ)=μ(1eμμeμ)(1eμ)2, respectively. Thus, all the terms corresponding to these functions in the original estimation procedures of Lin and Zhang (1999) should be modified accordingly.

2.2. The second stage: estimating δ, β and γ

We plug the estimates ûi (Step 1) and x̂i = x̂0 + B*bi (Step 2) obtained from the first stage into Equation (3) and re-write the model as

g(ηi)=ziTδ+u^iβ+x^iTCγ,

For a given λγ controlling the smoothness of γ, we can estimate δ, β and γ by maximizing the penalized pseudo-quasi-likelihood

12ihi12λγγTKγ,

where

hi=2yiηimi(yiu)υy(u)du.

Here, υy(·) is a specified variance function determined by the distribution fy; and mi are the prior weights which are set to be 1 (i.e., equal weights) in most cases, unless we have prior knowledge of the relative contribution of observations. In our study, all mi are set to be 1.

This penalized pseudo-quasi-likelihood from the semiparametric model has essentially the same form as DPQL (Equation 5) except for a penalty term from the random effects, thus similar strategies as those in Lin and Zhang (1999) can be applied to estimate δ* and γ. The tuning parameter λγ is also selected by GCV.

It is possible in practical settings that wij = 0 is observed at all time points for some subject i. In this case, ui can still be estimated (in Step 1 of Stage 1), but the subject-specific deviation di = B*bi cannot be estimated (in Step 2 of Stage 1). For such i, we propose to replace x̂i with the population profile x̂0 as the best estimate.

3. The Michigan Longitudinal Study (MLS)

In this section, we describe a well-known alcohol study, the Michigan Longitudinal Study (MLS), as a motivating example and present the model fitted on the empirical data. The MLS is an ongoing prospective study of people at high risk for alcoholism (Zucker et al., 1996). It is the developmentally earliest study currently extant (from early childhood to adulthood) and is also one of the longest running (funded by the National Institutes of Health for 27 years). The study is highly influential in the alcohol and substance abuse field. We apply our method to data from the MLS and adopt the fitted model as the true model in the simulation, because the features of the data are typical in the field. Thus, our methodological work may have high applicability to the field. More importantly, these rich longitudinal data provide a rare opportunity for us to examine the effect of adolescent risk behaviors on adult health outcomes. Such investigation could shed some light on future prevention and intervention work.

The MLS recruited participant families using fathers’ drunk driving conviction records and door-to-door community canvassing in a four-county area in mid-Michigan. All participants received extensive in-home assessments of their substance use and related risk factors and consequences at baseline, and thereafter at 3-year intervals. The children of participant families were followed from early childhood to adulthood. During the critical developmental period of alcohol use: ages 13-20 (Wagner and Anthony, 2002), these children were assessed annually in order to measure drinking onset and patterns more accurately. In this study, we use longitudinal data from a sample of 508 children (71% males) for analysis. The maximum number of time points available is 8, although some participants may skip certain time points. Because the proposed model employing latent covariates does not require every participant to have the same number of time points or a fixed assessment schedule, the proposed procedure is quite flexible in a variety of longitudinal data settings as long as the observed time points across all subjects are dense.

Our investigation aims to (1) characterize alcohol use behavior developmentally during adolescence, and (2) delineate the time-varying effect of adolescent alcohol use patterns on alcohol dependence diagnosis in adulthood. In our analysis, the time-varying covariate at each annual assessment is the number of drinking days in past month, which is a commonly adopted time frame for retrospective reports of drinking behaviors. Because it is a count measure with excess zeros that can barely be handled by the classical Poisson regression model (67% across waves), statistical models for zero-inflated count data such as the hurdle model are needed. The outcome variable is the DSM IV alcohol dependence diagnosis (APA, 1994) during early adulthood, which is a binary variable with the value 1 for positive diagnosis and 0 otherwise. Because the alcohol research literature shows that positive family history of alcoholism and male gender both contribute to higher risk for meeting alcohol dependence diagnosis (see Buu et al. (2012b) for a comprehensive review), we include two binary time-invariant covariates in the model: parental lifetime alcohol use disorder diagnosis (positive diagnosis from either biological parent was coded as 1) and gender (male was coded as 1).

We fitted the proposed model on the MLS data. Figure 1 shows the overall developmental trajectory of alcohol use behavior during adolescence x0(t) and the time-varying effect of adolescent alcohol use on adult alcohol dependence diagnosis γ(t). As demonstrated by the left panel, in general, the frequency of alcohol use increased during adolescence, and the curve became flat when development approached early adulthood. The right panel shows that drinking behavior in early adolescence (before age 15) had the highest effect on adult alcohol dependence. In addition, around the time when most of the youth entered college (age 18), the time-varying effect had a small peak. Moreover, the confidence interval at ages 15 and 18 tended to be wider because of the drastic change in curvature.

Fig. 1
The fitted model on the empirical data: the overall developmental trajectory of alcohol use behavior during adolescence x0(t) and the time-varying effect of adolescent alcohol use on adult alcohol dependence diagnosis γ(t).

In terms of those time-invariant effects, our model indicates that males were at higher risk for meeting alcohol dependence criteria ([delta with circumflex]1 = 1.4353 with 95%CI=[0.8586, 2.0120]); parental alcohol use disorder did not have a significant effect ([delta with circumflex]2 = 0.3256 with 95%CI=[-0.2194, 0.8706]); individual risk for alcohol use contributed to greater likelihood of adult alcohol dependence ([beta] = 1.2449 with 95%CI=[0.8973, 1.5924]). Our analysis also estimated the population risk level u0 to be -0.7310 (with 95%CI=[-0.8637, -0.5983]) that corresponds to 65% zeros. Moreover, the other variance parameters in the model were estimated as σ^υ2=1.06302, σ^12=0.61982, σ^22=0.30572, [tau]d = 1.73532, and the smoothing parameters were selected as λx = 1.3, λγ = 0.003.

4. Simulation

We conducted two simulation experiments in this study. The first experiment was designed to evaluate the performance of the proposed model with different sample sizes, numbers of time points, and proportions of zeros. The fitted model in Section 3 was adopted as the true model to generate simulated data that closely represent the structure of real data so that the results can be more generalizable to the substance abuse field (Burton et al., 2006). The other experiment was motivated by our previous work finding that zero-inflation was frequently ignored in practice and yet could result in poor estimation, if the classical Poisson model was employed in the settings involving longitudinal outcomes (Buu et al., 2012a) or variable selection (Buu et al., 2011). Thus, our second simulation experiment evaluated the potential negative consequence of ignoring zero-inflation in the setting involving longitudinal covariates with zero-inflated counts and a discrete outcome. The detailed designs and results of the two experiments are described in the following sections.

4.1. Experiment 1: evaluating the performance of the proposed model

In this experiment, we manipulated three factors: (1) the sample size with the small size n = 250, medium size n = 500, or large size n = 750; (2) the number of time points with q = 8, 16, or 32; and (3) the population risk level with u0 = −0.75 or u0 = 0.75. When u0 = −0.75, the data contained approximately 65% zeros and this setting was similar to that of the real data. When u0 = 0.75, there were around 35% zeros in the data and we used this setting to simulate the situation to which a regular Poisson may apply.

We first set t0 to be q equally spaced time points in [−1.5, 1.5], where q is 8, 16, or 32. In order to simulate unbalanced observation time points tij among n subjects, we allowed the longitudinal covariate process wij for subject i to be observed at tij for j = 1, …, Ji, where Ji was an integer randomly chosen from {34q,,q}, and tij were Ji distinct points among t0.

The longitudinal covariate data wij were zero-inflated counts generated by Model (1). Here, we set u0 to be −0.75 or 0.75, συ2=12, x0(t) = 0.5 + 0.8 arctan(2t), and di(t) was determined by di = B*bi with bi being a random sample from N(0,diag(σ12,σ22,τdI(q2)×(q2))), where σ12=0.62, σ22=0.32, and τd = 12. The response data yi were then generated from

yi~Bernoulli(ηi),logit(ηi)=ziδ+uiβ+1.51.5xi(t)γ(t)dt,

where β = 1.25, δ = 1.5, and γ(t) = −1.5t3 +t2 +1.5t − 0.5. In addition, zi were randomly drawn from {0, 1} with P(zi=1)=23 for each subject i.

In summary, we manipulated three factors, the sample size (n), number of time points (q), and the population risk level (u0), and considered 3 × 3 × 2 = 18 situations in total. For each combination, we generated NS = 100 data sets and applied the proposed two-stage method to estimate the parameters. For each parameter, the mean with standard error (se) and the mean squared error (MSE) were calculated from NS replications. For the nonparametric coefficient functions x0(t) and γ(t), the mean integrated squared error (MISE) was used to summarize the results.

Table 1 presents the results of Experiment 1 with varied sample sizes n and fixed number of time points (q = 16) and population risk level (u0 = −0.75). When the sample size increased, the performance of the proposed model improved (i.e. smaller MSE or MISE). In Table 2, the number of time points q was varied while holding the other two factors constant (n = 500, u0 = −0.75). The performance of the proposed model improved with the number of time points. The effect of the population risk level u0 is summarized in Table 3 with the other two factors fixed at their medium values (n = 500, q = 16). By varying the value of u0 from -0.75 to 0.75, we manipulated the proportion of zeros in the data from a zero-inflated situation (≈ 65%) to a situation similar to a regular Poisson (≈ 35%). The result shows that the performance of the proposed model was only slightly better in the regular Poisson situation. This means that our model can handle the zero-inflation pretty well.

Table 1
Simulation results of Experiment 1 with varied sample sizes n (q = 16, u0 = −0.75).
Table 2
Simulation results of Experiment 1 with varied numbers of time points q (n = 500, u0 = −0.75).
Table 3
Simulation results of Experiment 1 with varied population risk levels u0 (n = 500, q = 16).

According to the values of mean(se) reported in Tables 1--3, some3, some biases exist in the estimation of parameters. This may result from the Laplace approximation employed in the high-dimensional integration for quasi-likelihood calculation, or the two-stage approach. Nevertheless, given that the true values are covered by the 95% confdence intervals, these biases may still be at an acceptable level.

Since the MISE of estimates for γ(·) is noticeably larger than that for x0(·), we further examine the estimates of γ(·). Figure 2 depicts the true curve of γ(t) used in this simulation (the solid curve), the average of estimated curves over 100 replications for n = 500, q = 16, and u0 = −0.75 (the dashed line), and its pointwise 95% confidence intervals calculated as the average of estimated curves ±1.96 multiplying the standard error of the 100 estimated curves (the dotted curves). It can be seen from Figure 2 that the large MISE of [gamma with circumflex](·) is mainly due to the large variance of [gamma with circumflex](·) because the average of the estimated curves is very close to the true one.

Fig. 2
The estimates of γ(t) with pointwise 95% confidence intervals for n = 500, q = 16, and u0 = −0.75 in Experiment 1.

4.2. Experiment 2: evaluating the consequence of ignoring the zero-inflation

In the second experiment, we fitted the regular Poisson model (instead of the hurdle model) on the simulated data generated according to the true model described in Experiment 1. Here, we only manipulated the proportion of zeros in the data (through the two different values of u0) while holding the sample size and number of time points at their medium values (n = 500, q = 16).

Table 4 summarizes the results of the simulation. As expected, when the data contained excess zeros (i.e. u0 = −0.75 corresponding to 65% zeros), the regular Poisson did not perform as well as in the situation without zero-inflation (i.e. u0 = 0.75 corresponding to 35% zeros). In Figure 3, we present the fitted longitudinal covariate process x0(t) and its time-varying effect γ(t) under the two situations. The estimated curves were derived from evaluating the fitted functions of the 100 replications at a set of grid points and connecting the means at all grid points. As demonstrated in the left panel, the regular Poisson model estimated x0(t) well when there was no zero-inflation (the bottom graph) whereas the model did very poorly in the situation of zero-inflation (the top graph). On the other hand, the right panel of Figure 3 shows that the performance of the regular Poisson did not vary much with the existence of zero-inflation in the data. Specifically, the regular Poisson model estimated the curve of γ(t) to be pretty flat in both situations, whereas the true effect changed developmentally across time.

Fig. 3
The fitted models without accounting for zero-inflation in Experiment 2: the longitudinal covariate process x0(t) and its time-varying effect γ(t) under different levels of population risk: u0 = −0.75 (65% zeros), u0 = 0.75 (35% zeros. ...
Table 4
Simulation results of Experiment 2 with varied population risk levels u0 (n = 500, q = 16).

5. Discussion

This study proposes a generalized time-varying effect model that employs the hurdle model to characterize the longitudinal covariate process with zero-inflated counts and also accommodates any response variable from the exponential family. Our model delineates both the longitudinal covariate process and its time-varying effect on a scalar outcome using natural cubic smoothing splines. Our two-stage approach also takes into account measurement errors in the observed longitudinal covariate process. Further, the proposed model involves random effects to deal with individual differences in risk levels and developmental trajectories.

The longitudinal data of the MLS, a well-known study of youth at high risk for substance abuse, are presented as a motivating example to demonstrate that the proposed model is very useful for informing critical developmental periods of prevention or intervention. For example, Figure 1 shows that alcohol use in early adolescence or the freshman year of college could have profound impact on adult alcohol dependence. Further, our empirical analysis indicates that family history of alcoholism has a distal and nonsignificant effect on adult alcohol dependence in comparison to adolescent alcohol use. This indicates that prevention and intervention efforts during adolescence can still counteract genetic vulnerability. Moreover, the proposed model can be applied to not only multi-wave longitudinal studies like the MLS, but also to short-term studies that involve intensive data collection such as the ecological momentary assessment or the daily process data, which are collected through modern health communication technology (Cranford et al., 2010; Liu et al., 2013; Selya et al., 2013; Vasilenko et al., 2014).

The design of our simulation studies is unique because it simulates the features of the empirical data so that the results can be generalizable to the substance abuse field. The performance of the proposed model improves as the sample size or the number of time points increases. Our simulation study also shows that when there are excess zeros in the data, the two-stage model involving the regular Poisson cannot estimate either the longitudinal covariate process or its time-varying effect well. This emphasizes the important role that the hurdle model plays in handling zero-inflation in the data. Furthermore, for most settings in health behavior research, it is practical to assume that each individual carries a different level of risk ui for engaging in health risk behaviors. Our simulation experiment shows that when data are generated under this conceptual framework, the regular Poisson model which constraints the value of ui to be 0 for each individual does not estimate the time-varying effect γ(t) well, even when the degree of zero-inflation is minimal.

In this study, we only consider the Poisson link function for the hurdle model because it fits well with the real data example. In some applications where overdispersion is common, the negative binomial link should be considered. Furthermore, the two-stage approach ignores the uncertainty inherited from the first stage in the estimation of the second stage. There is, therefore, an issue related to uncertainty propagation in the practical implementation of the two-stage method. The Bayesian frame-work for joint models proposed by Bhadra et al. (2012) has been shown to be effective for dealing with this kind of issue in the setting with continuous longitudinal covariates and a binary response. Thus, future research may extend the Bayesian framework to the setting studied in this paper that involves zero-inflated count longitudinal covariates and a discrete response.

Our work in this paper focuses on the setting that involves a longitudinal covariate process with zero-inflated counts and a discrete outcome. Future studies may consider modeling multiple longitudinal covariate processes that tend to co-occur such as substance use behaviors, violence behaviors, and HIV sexual risk behaviors. Furthermore, although a scalar outcome such as the adult alcohol dependence diagnosis in our motivating example is quite common in practice, there are some settings that may involve multiple outcomes. For example, a health outcome may be recorded at several time points. In this kind of setting, we would add some terms with respect to time on the right hand side of Equation (2) to delineate the longitudinal change in the outcome. We may also add a random term to account for the dependence among repeated measures within each individual (i.e., a random effect model), or take a marginal (GEE) approach with robust standard errors (see Diggle et al. (2002) for a comprehensive account of these alternative approaches). This is, however, beyond the scope of this paper.

6. Software

Software in the form of R code, together with a sample input data set and complete documentation is available on request from the corresponding author (ude.hcimu@uub).

Acknowledgments

Funding

This work was supported by the National Institutes of Health (NIH) [P50 DA010075, P50 DA036107 & R01 CA168676 to H.Y. and R.L.; R37 AA007065 to R.A.Z.; and K01 AA016591 & R01 DA035183 to A.B.] and National Science Foundation DMS 1512422 to R.L. The content is solely the responsibility of the authors and does not necessarily represent the official view of the NIH.

Footnotes

Conflict of Interest: None declared.

Contributor Information

Hanyu Yang, Pennsylvania State University, University Park, USA.

Runze Li, Pennsylvania State University, University Park, USA.

Robert A. Zucker, University of Michigan, Ann Arbor, USA.

Anne Buu, University of Michigan, Ann Arbor, USA.

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