An Exponential Effect Persistence Model for Intensive Longitudinal Data

Abstract

We develop an effect persistence model for intensive longitudinal data under a general assumption of an exponential loss of association between exposure and outcome over time. The working model proposed may be useful for understanding the complexity of phenomena where subjects can be repeatedly exposed to an intervention or a naturally occurring event while at the same time, the effect of any one exposure is expected to diminish over time. Under the main assumption, we specify a semi-linear model with extensions to generalized linear models. These methods are motivated by, and applied to, data from a study of adolescent exposure to pro-smoking advertisement where the impact of pro-smoking media exposure on young adults’ susceptibility to smoking is assessed along with the decay of the effect over time. We investigate the performance of the proposed method when the model assumptions are correctly specified or not. Supplementary materials for this article are available online.

1. Introduction

The effects of discrete interventions and naturally-occurring events on a person’s thoughts, emotions, behaviors, and physical functioning often vary over time (Vallacher, Read, & Nowak, 2002). For example, quitting smoking typically involves cycling between smoking at a regular rate; smoking at a reduced rate; achieving short and more extended periods of abstinence; lapsing; and relapsing (McCarthy, Ebssa, Witkiewitz, & Shiffman, 2015). The emotional effects of grief following the death of a loved one sometimes resolve quickly (Bonanno & Kaltman, 2001) and sometimes persist long after the initiating event (Lundorff, Holmgren, Zachariae, Farver-Vestergaard, & O’Connor, 2017). Even changes that appear to be irregular or random may be governed by identifiable causes. For example, weight loss interventions have been known to help overweight individuals attain their weight loss goals in the short term, but require continual maintenance to avoid a decay of the initial effect or the re-emergence of entrenched behaviors such as unhealthy diet and sedentary behavior (Jeffery et al., 2000). In the persuasion literature, multiple studies have demonstrated large and consequential initial effects of exposure to mass communication messages that are nevertheless short lived, decaying in a few days or weeks depending on the content of the communication (Gerber, D.P., & Shaw, 2011). In the context of health education interventions, it is common to see an immediate improvement followed by a deterioration (i.e., “decay”) of the interventions impact (Green, 1977). Even without a direct intervention or treatment, decay effects have been observed for naturally occurring phenomenon (see for example Setodji, Martino, Scharf, & Shadel, 2014). The notion of effect-decay is also important to understanding the persistence of memories. According to one theory, information will tend to “evaporate” with time unless it is actively rehearsed (Brown, 1958; Peterson & Peterson, 1959).

As such, understanding temporal persistence or decay of effects, especially in the case of repeated events is of central importance in behavioral sciences and it entails recognizing cognitive, behavioral, and physiological changes after a person has been subject to a treatment or a phenomenon of interest, establishing a cause-and-effect elucidation of the changes and evaluating the persistence or decay of the effect with the passage of time. In some cases, determining how long a detrimental effect lasts is necessary to establish the need for efforts to ameliorate the effect or the frequency of exposure that would put a person in a state of chronic risk or danger. Investigating the persistence of cognitive, behavioral, and physiological effects necessitates tracking the sequence of states as they unfold in real time and requiring intensive, longitudinal information on the needed constructs, a well-articulated theoretical model of change, and a statistical model that is a good approximation of the theoretical model (Collins, 2006). Studies must be designed to provide a detailed view of the change process as it unfolds, typically through repeated, intensive observations. Studies that employ longitudinal panel designs typically assess participants two or three times with long gaps (months or years) between measurement occasions. These types of designs are inadequate for the assessment of persistence or decay processes that may occur over a much shorter time frame. Recent methodological advances, such as “wearable” data collection devices, allow for repeated, intensive longitudinal assessment of various aspects of human behavior over time in real-world settings. Such methods generate rich longitudinal data sets that allow for a more nuanced examination of dynamic processes of behavior change over time. Modern intensive longitudinal data collection procedures (Hektner, Schmidt, & Csikszentmihalyi, 2007; Mehl & Conner, 2012; Walls & Schafer, 2006) are now being used to collect tens, hundreds, or even thousands of measurements on study participants over relatively short periods of time. When coupled with new innovative statistical methods such as modeling the time course of a variable or the within-person causal processes (e.g. Bolger & Laurenceau, 2013), data from these studies can be used to provide a detailed, never-before-seen look at cognitive, behavioral, and physiological changes and estimate the persistence of a wide range effects.

To our knowledge, very few statistical methods have been designed to model the complexity of persistence and decay processes with repeated events in a deterministic way. Persistence has been conceptualized as the extent to which events today have an effect on the whole future history of a stochastic process. For example, Nelson and Plosser (1982) and Cochrane (1991) argued that, because any time series with a unit root can be decomposed into a stationary series and a random walk, and the latter can have arbitrarily small variance, persistence should be measured as the ratio of the variance of the change in the random walk component to the variance of the actual change. Additional time series derivations have been proposed to quantify short-term, long-term and permanent effects, momentum, and repetition wear-out (Dekimpe & Hanssens, 1995; Pauwels et al., 2004) of different effects. One of the conditions of most of the time series approaches is that they assess only one shock or event or intervention and follow its changing effect over time using linear model assumptions. Beyond time series, other temporal behavior-change models have been used in the past to assess persisting effects. To estimate time effects, Zeger and Diggle (1994) proposed semi-parametric models where only the intercept in a linear part of the model depends on time. Hastie and Tibshirani (1993) and others proposed more general time-varying coefficient models where the models are linear in the predictors, but their coefficients are allowed to change smoothly with time. Recently, Tan, Shiyko, Li, Li, and Dierker (2012) proposed a semi-parametric time-varying effect model where previous parametric assumptions (e.g. Hastie & Tibshirani, 1993) about the nature of the change in the relationship between predictors and outcomes are relaxed, providing flexibility. Nevertheless, none of these methods attempt to model the complex process of effects persisting in the presence of repeated interventions or events. The purpose of this article is to introduce a new statistical model for the estimation of effect-persistence that takes into account various design complexities including repeated exposures. Additional background and an example of a study with intensive longitudinal data are presented in Section 2. Our proposed exponential persistence model is developed in Section 3 with a simulation study of its performance reported in Section 4. Results from an example of its use are shown in Section 5 and some model extensions in Section 6. Concluding comments are provided in Section 7.

2. Intensive longitudinal data example

2.1. General Setting

Longitudinal data collected using traditional methods (e.g. panel surveys or clinical trials) are useful for explaining or documenting phenomena, but only to a certain extent. Measuring individuals at infrequent intervals is valuable for studying changes that accrue over relatively macro-time scales (e.g. estimating a long-term treatment effect). However, as observations typically occur months or even years apart, such a design does not provide the opportunity to observe short-term changes that occur between observations. This can severely limit understanding of the change process if complex, short-term changes underlie and explain longer-term effects. This could lead to a mis-characterization of the intervention’s impact and a misunderstanding of the mechanisms underlying change. Rather than a snapshot of a specific phenomenon at two or three points in time, intensive longitudinal data (ILD) methods provide a means of gathering extensive information about a persons thoughts, feelings, behaviors, and physiology. These methods allow for the collection of information at frequent intervals over an extended period and thus provide the potential to inform complex mechanisms of influence and elucidate theories of change (Vallacher et al., 2002). Modern technological innovations provide easy modes of data collection through tablet computers, cell phones, global positioning systems (GPS), and portable devices (e.g. heart rate monitors, electronic pill dispensers for elderly, activity trackers and pedometers), making frequent repeated observation feasible. Data collected through these methods have been used to provide descriptive information about the frequency of occurrence of events and to estimate the instantaneous impact of events as they occur naturally in a person’s environment (e.g. Shiffman, 2009; Martino, Scharf, Setodji, & Shadel, 2012). They are not typically used to examine persistence as statistical methods for doing so are in their infancy. The exponential persistence model proposed in this study article fills this gap. The development of the method will be discussed in the context of a study that was designed to examine the impact of smoking-related media on young people’s risk for cigarette smoking.

2.2. Motivating study: Ad Spotter

The study that motivated the development of the exponential persistence model was designed to assess the impact of exposure to smoking-related media (i.e., all forms of advertising and promotion of tobacco products and portrayals of smoking in movies and on television) on college students’ risk for smoking (Martino et al., 2012). It is well known that exposure to pro-smoking media (e.g., cigarette advertisements and portrayals of smoking in movies) promotes smoking initiation and progression to regular smoking (DiFranza et al., 2006; Wellman, Sugarman, DiFranza, & Winickoff, 2006). Methods of assessing exposure to pro-smoking media have varied greatly across studies but typically rely on participants’ recall of past exposures and thus are limited because of the inaccuracy of memory. The goal of the motivating study was to develop a measurement approach to capture detailed and valid information on exposure to smoking-related media, model the immediate effects of that exposure on smoking-related cognitions, and investigate the persistence or decay of those effects.

The study enrolled 134 college students between the ages of 18 and 24 (Martino et al., 2012). The sample was 37% male and 66% Caucasian. The majority (61%) reported, at baseline, that they had smoked at some point in their lives, even if they had not smoked in the past month; these individuals were classified as “smokers” (Choi, Gilpin, Farkas, & Pierce, 2001). The remaining 39% of the sample were “non-smokers”, i.e., they had never smoked a cigarette, even a puff.

Participants were issued a handheld data collection device so that they could record all of their naturally-occurring exposures to pro-smoking media for 21 consecutive days. Participants were extensively trained in the use of the device, and were instructed to carry the device with them at all times so that they could initiate data entry each time they encountered an instance of pro-smoking media, and respond to random prompts (see below) when issued by the device.

At each exposure to smoking-related media, participants provided descriptive information about the exposure and answered questions about their intentions to smoke cigarette in the future. In particular, participants completed a 3-item scale that was adapted from Choi et al. (2001) and that has been shown to predict smoking initiation: “Do you think you will try a cigarette anytime soon?”, “Do you think you will smoke a cigarette anytime in the next year?”; and “If one of your best friends offered you a cigarette, would you smoke it?” Responses were made on a 1 (definitely not) to 10 (definitely yes) scale and averaged (reliability α = 0.94) to produce a smoking intention scale score (range: 1 − 10), with higher scores indicative of a stronger intentions to smoke. Participants similarly reported their smoking intentions in response to control prompts that occurred randomly three times per day in the absence of exposure. The data collection device automatically recorded the time of each data entry, whether in response to an exposure or random prompt. Overall, after dropping missing values, participants responded to 6,779 random prompts with an average of 51 prompts per participant and reported 1,112 exposures to pro-smoking media (average 8 and range 1 − 27 per participant) over the course of the study.

Using these data, Shadel, Martino, Setodji, and Scharf (2012) showed that exposure to pro-smoking media is associated with an increase in adolescents’ intention to smoke, when simply comparing assessments of intention at time of exposure with assessments of this outcome at random control prompts. This analysis used the control prompt assessments as equivalent controls ignoring the possibility of the persistence of exposure effects beyond the moment of exposure. That is, the assumption underlying this analysis was that intentions assessed at random prompts were free from the effect of exposure to smoking-related media. It is nearly impossible, however, to assess smoking-related intentions at a “pure” baseline (comparison) state that does not reflect at least some influence of past exposures to pro-smoking media. To the extent that exposure effects persist beyond exposure, the estimates reported in Shadel et al. (2012) likely understate the true effect of pro-smoking exposure on intentions. Later, Setodji et al. (2014) used these same data to assess the persistence of the exposure effect reported by Shadel et al. (2012), and showed that the impact of advertisement single exposure to pro-smoking media on young people’s intentions is detectable for up to 7 days. This persistence analysis used a non-parametric model that set aside some complexities having to do with the impact of repeated measurement (see below) and for each exposure, only modelled the observations up to the next exposure. Furthermore, the assessment of how long the exposure effect endured was achieved through visual inspection of a decay graph. The non-parametric estimation in Setodji et al. (2014) is a crude assessment of persistence when compared to the proposed exponential persistence model described in this article. A key advantage of the exponential persistence model is that it is designed to model, in a parametric way, the complex mechanism of the accumulation of effects over time as would be expected if an outcome of an event or exposure were still under the influence, to any extent, of a previous event or exposure.

3. An exponential persistence model for intensive longitudinal data

3.1. Notation

ILD has a form similar to traditional longitudinal data. For subject i’s observation j (with i = 1, 2,…, n and j = 1, 2,…, J_i), we will denote Y_ij, X_ij = (X_1ij, X_2ij,…,X_pij), and E_ij as the outcome, covariate and exposure respectively recorded at the j^th observation, where n represents the total number of subjects and J_i the total number of measurements for subject i, which can vary by subject. In addition, let t_ij be the measurement time of the j^th observation for the i^th subject with the stipulation that, for j₁ < j₂, we will have t_ij1 < t_ij2, i.e., the observations are ordered and each observation is recorded at a different time. The exposure variable E_ij is assumed to be an indicator variable that takes value 1 if an exposure occurred at the j^th observation and 0 otherwise.

3.2. Model formulation: semi-linear exponential persistence model

For ease of illustration, we first consider a linear model in which the outcome is a continuous variable. The classical hierarchical linear model for studying the conditional distribution of Y_ij given E_ij and X_ij considering the repeated measures is of the form

$$Y_{ij}={\nu }_i+{\mu }_{ij}+{\beta }^{\top }{\mathit{X}}_{ij}+{\epsilon }_{ij}$$ (1)

where ν_i is a subject level intercept and μ_ij is a subject level outcome that can change over time due to the impact of the exposures E_ij up to time t_ij after controlling for the p confounder variables in X_ij. Errors ε_ij are assumed normally distributed with zero mean and constant variance. In a linear decay process (e.g. Hennessy et al., 1999) with only one exposure at a time t₀, the assumption μ_ij = α(t_ij − t₀)1_(tij>t0) is often used with α the decay slope and 1_(.) an indicator function. Non-linear growth curve models have been proposed when a linear decay assumption is untenable (Burchinal & Appelbaum, 1991) but none of those growth curve models take into account the complexity of repeated exposures and the possibility that each additional exposure may result in greater impact (accumulation of effects). Our proposed exponential persistence model provides a way to model such complexity.

Proportional decay assumption

Our key assumption which will allow for inference about the persistence of the effect of an exposure or intervention, emulates Jost’s second law on the principles of memory (Wixted, 2004). This law stipulates that the older of two associations of the same strength will fall off less rapidly than the newer of the two associations in a given length of time. In other words, the law postulates that the magnitude of decay weakens as time goes by. Multiple retention functions have been proposed to assess this phenomenon (Rubin & Wenzel, 1996). A commonly used retention function assumes a retroactive decay where the loss is proportional to the amount of association that is left (Simon, 1966). Our proposed exponential model or forgetting curve will take this approach.

Assumption

(Exponential loss of association). After exposure, the outcome of interest decreases at a rate proportional to its current value over time.

This assumption stipulates that, for a general level μ of the outcome that is attributed to a particular exposure, an observed decay is proportional to the outcome level, no matter the time interval over which the decay measurement was taken. As the time interval varies, the decay is proportional to the size of that interval. That is to say, the size of the decay in a given infinitesimal time interval is proportional to μ. Mathematically, this process can be expressed as

$$\frac{d\mu }{dt}=-\mathrm{\lambda }\mu$$ (2)

where t is a general time and λ is the constant of proportionality that will be called the decay constant. This differential equation has a solution

$$\mu =\omega \exp (-\mathrm{\lambda }t)$$ (3)

for which ω being the level of the outcome at time t = 0 when the event occurred. In physics, this assumption is called the law of exponential decay or radioactive decay and commonly used for the description of the process by which a nucleus of an unstable atom loses energy by emitting ionizing radiation. In the study that motivated the development of this method, this assumption will postulate that after a college student’s smoking intention has been influenced by an exposure to smoking-related media, the impact of that exposure will decrease exponentially over time at a rate proportional to the initial impact of the exposure event. It will capture the main structure of the impact of an ad where a decline of the impact will occur but not be expected to be linear. In many situations (e.g. weight loss programs, daily diary of mediation taken), this assumption can be a good proxy for the process by which a time-varying effect evolves over time.

Applying the assumption of exponential loss of association to the model posited in equation 1 yields the following:

$$\begin{gathered} \text{Before any exposure,}{\mu }_{ij}={\omega }_i \\ \text{At the time}{\tau }_{i1}\text{of first exposure,}{\mu }_{ij}={\omega }_i{e}^{\theta } \\ \text{At time}{t}_{ij}\text{after the exposure,}{\mu }_{ij}={\omega }_i{e}^{\theta }{e}^{-\lambda \left(t_{ij}-{\tau }_{i1}\right)}={\omega }_i{e}^{\theta -\lambda \left(t_{ij}-{\tau }_{i1}\right)} \end{gathered}$$ (4)

There is a level ω_i of the outcome for subject i that is affected by an exposure. Note that in equation 4, the value ω_i will capture each subject’s starting point that can be influenced by an exposure, which can differ across individuals because some exposures of interest might have happened prior to the observation period (i.e. individuals can vary in history of exposures at baseline, prior to initiating the study). Technically, ω_i can also be the realization of a decaying effect if exposures occurred before the start of the study. The occurrence of an event will cause a change in ω_i to ω_ie^θ with a positive or negative exposure effect parameter θ signaling an increase or decrease of the outcome. After a length of time t_ij − T_i1, under the exponential loss of association assumption, the state ω_ie^θ of the outcome will decay to ω_ie^{θ−λ(t_ij−T_i1)} as a consequence of Equation 3, where λ is the decay constant. If a second exposure occurs at a particular time T_i2, the full decaying portion of the outcome will be influenced by that additional exposure and at a subsequent time t_ij > T_i2 the outcome process will be

$$\begin{gathered} \text{Second exposure at}{\tau }_{i2},{\mu }_{ij}={\omega }_i{e}^{\theta -\lambda \left({\tau }_{i2}-{\tau }_{i1}\right)}{e}^{\theta }={\omega }_i{e}^{2\theta -\lambda \left({\tau }_{i2}-{\tau }_{i1}\right)} \\ \text{At subsequent time}{t}_{ij},{\mu }_{ij}={\omega }_i{e}^{\theta -\lambda \left(t_{ij}-{\tau }_{i1}\right)}{e}^{\theta -\lambda \left(t_{ij}-{\tau }_{i2}\right)}={\omega }_i{e}^{2\theta -\lambda \left[\left(t_{ij}-{\tau }_{ij}\right)+\left(t_{ij}-{\tau }_{i2}\right)\right]} \end{gathered}$$ (5)

Before the second exposure occurs, the outcome will be at ω_ie^{θ−λ(τ_i2−τ_i1)}; when the second exposure occurs, the outcome will change by e^θ. From there, subsequent non-exposure observations will continue to be influenced by the decaying effect of the first exposure as well as the decaying effect of the second exposure. This allows for the accumulation of multiple events and decay in their corresponding effects over time, thus accommodating the complexity of multiple exposures at different time points.

From these setups, in the general case of the observation of the j^th outcome on participant i at time t_ij, assuming that up to that time, $K_{ij}={\sum }_{l=1}^j{E}_{il}$ different events occurred at times T_il, T_i2,…,T_iKij, the outcome process will take the following form:

$${\mu }_{ij}={\omega }_i{e}^{\theta K_{ij}-\lambda {\sum }_{k=1}^{K_{\mathrm{ij}}}\left(t_{ij}-{\tau }_{ik}\right)}={\omega }_i{e}^{\theta K_{ij}-\lambda Dtim{e}_{ij}}$$ (6)

where $Dtim{e}_{ij}={\sum }_{k=1}^{K_{ij}}\left(t_{ij}-{\tau }_{ik}\right)$ is the total decaying time by the j^th observation or the sum of the decaying times over all the observed exposures at time t_ij for subject i. Note that in this process, any time an exposure occurs, the part of the outcome that is influenced by the exposure will increase by a multiplicative factor e^θ; this includes the first time an exposure occurs, when the state will move from ω_i to ω_ie^θ.

The following proposition, posits a semi-linear exponential persistence model that can model complexities in ILD.

Proposition 1.

Under an exponential loss of association assumption, a semi-linear exponential persistence model can be formulated as

$$\begin{gathered} Y_{ij}={\nu }_i+{\omega }_i{e}^{\theta K_{ij}-\lambda Dtim{e}_{ij}}+{\beta }^T{\mathit{X}}_{ij}+{\epsilon }_{ij} \\ {\nu }_i={\nu }_0+{\xi }_{i}with{\xi }_i~\mathcal{N}\left(0,{\sigma }_{\nu }^2\right) \\ {\omega }_i={\omega }_0+{\zeta }_{i}with{\zeta }_i~\mathcal{N}\left(0,{\sigma }_{\omega }^2\right) \\ and{\epsilon }_{ij}~\mathcal{N}\left(0,{\sigma }_{\epsilon }^2\right),corr\left({\nu }_i,{\omega }_i\right)=\varrho \end{gathered}$$ (7)

where ω_i, the portion of the outcome that is influenced by the exposure, intervention or event of interest, is assumed to be normally distributed with an average ω₀ and a variance ${\sigma }_{\omega }^2,$ θ estimates the log of the rate of change in the outcome due to the exposure, and λ is the rate of the exponential decay of the exposure effect. The subject-level random intercept ν_i that does not change even with exposure will also be assume to be normally distributed with mean ν₀ and variance ${\sigma }_{\nu }^2.$ As in general settings, exposures may not be scheduled. K_ij and Dtime_ij are observed time-dependent characteristics of each observation, and can differ from one subject to another. There is no requirement for the exposures to be unscheduled; this model can also be applied in settings where intervention sessions are scheduled, spaced equally or differentially apart.

In the next section we present inferential implications of our model.

3.3. Model parameters and interpretation

With our proposed model (7), at any point in time, only part of each subject’s outcome of interest is influenced by the exposure. We will hereafter refer to this portion as the “covariate adjusted influenceable outcome”. At the beginning of the study, the average of the covariate adjusted influenceable outcome is captured by ω₀. If an individual has ω₀ ≈ 0, such group will never by affected by an exposure; in contrast ω₀ ≠ 0 suggests that an individual is likely to be influenced by an exposure. The absolute size of this influence will depend on the number of exposures a participant has experienced prior to observation, which is likely to be unknown or only roughly estimable in most cases. The average size of the effect of an exposure at the beginning of the study can be estimated by ω₀ (1 − e^θ), the average change between the level of the outcome before and after an exposure. The parameter θ will estimate the log of the rate of change in the covariate adjusted influenceable outcome at any exposure. Assuming exponential loss of association, λ will estimate the decay rate of the exposure effect over time.

From the model (7), inference can also be made about the duration of the effect of an exposure or the timing of the influenceable outcome to drop below a given level, relative to a specification of interest. The following corollary summaries the decay or influenceable drop time as followed:

Corollary 1.

In a exponential persistence model, the length of time it takes a participant to return to a fraction ρ of his or her initial state of the covariate adjusted influenceable outcome after only one time exposure can be estimated as

$$duration=\frac{\theta -ln(\rho )}{\lambda },$$

where for ρ = 1 it will estimate the amount of time it takes to return to the starting value. In general, after K time exposures occurred, the duration of time it will take to return to a fraction ρ of the covariate adjusted influenceable outcome that exists right before such K^th exposure can be estimated as

$${duration}_K=\frac{1}{K}\times \frac{\theta -log(\rho )}{\lambda },$$

a K fraction of the time it will take at the first exposure.

Similarly, for a design where one is interested in the time needed for the influenceable outcome to drop below ρ times its value right after the first exposure occurred, such drop time can be estimated as

$$DropTime\le -\frac{log(\rho )}{\lambda }.$$

For the general setting where up to K exposures have happened so far during the observation period, the following drop time will be smaller and estimated as

$${DropTime}_K\le \frac{1}{K}\times (-\frac{log(\rho )}{\lambda }).$$

The duration for a covariate adjusted influenceable outcome to drop back to its previous value does not depend on the time interval between previous exposures and occurs slower at a rate proportional to the total number of exposures that a subject has been subject to at any point during the observation period. Also, time to drop below a given level after an exposure only depends on the constant of proportionality or the decay constant λ in addition to the number of exposures that have occurred up to the point in time of interest. The justification of the corollary is given in a technical appendix.

3.4. Estimations

The parameters ν₀, ω₀, θ, λ and β for inference and estimation can be done with maximum likelihood methods. Let η = (ν₀, ω₀, θ, λ and β^T) be the vector of the model parameters, if we posit model (7) in terms of general non-linear model formulation as

$$\begin{gathered} {\mathit{y}}_i=g\left({\mathit{\theta }}_i,{\mathit{W}}_i\right)+{\mathit{e}}_i \\ {\mathit{\theta }}_i={\mathit{A}}_i\mathit{\eta }+{\mathit{B}}_i{\mathit{\delta }}_i \end{gathered}$$

where ${\mathit{y}}_i={\left(y_{i1},\dots ,y_{i{J}_i}\right)}^{\text{T}}$ is the J_i × 1 response vector for the J_i observations for participant i, ${\mathit{W}}_i={\left(W_{i1},\dots ,W_{i{J}_i}\right)}^{\top }$ with $W_{ij}=\left(K_{ij},Dtim{e}_{ij},{\mathit{X}}_{ij}\right)$ the covariate incorporating the total number of exposure and decay time, ϑ_i = g(θ_i, W_i) the participant specific mean function and e_i ~ $\mathcal{N}$(0, Λ_i) the J_i × 1 vector of errors for observations of participant i assumed to be normally distributed. The matrix Λ_i, which is J_i × J_i, can be parameterized using a small number of parameters and Λ_i is assumed to be the identity matrix ${\sigma }_{\epsilon }^2{\mathit{I}}_{J_i}$ if observations can be assumed to be conditionally independent within participants. The J_i × 1 coefficient vector θ_i for the participant i allows the fixed effects η and the random effects ${\mathit{\delta }}_i={\left({\nu }_i,{\omega }_i\right)}^{\top }~\mathcal{N}(0,\mathrm{\Psi })$ to be incorporated in the model with A_i and B_i being matrices of known constants (usually zeros and ones) combining the mixed effects.

With the normality assumptions made, the marginal density of y_i can be specified as

$$f\left({\mathit{y}}_i\right)|\mathit{\eta })=\int \varphi \left({\mathit{y}}_i|{\mathit{\vartheta }}_i,{\mathbf{\Lambda }}_i\right)\varphi \left({\mathit{\delta }}_i|0,\mathrm{\Psi }\right)d{\mathit{\delta }}_i,$$

where ϕ(.|μ, Ω) denotes a multidimensional normal density with mean μ and variance-covariance matrix Ω. As such, the log-likelihood of η for y = { y₁,…,y_n} is given by

$$\begin{gathered} l(\mathit{\eta }|\mathit{y})=\sum _{i=1}^{n}log\{\int 2{\pi }^{-\left(J_i/2+1\right)}{\left|{\mathbf{\Lambda }}_i\right|}^{-J_i/2}|\mathrm{\Psi }{|}^{-1/2} \\ \times \exp \left\{-\frac{1}{2}\left[{\left({\mathit{y}}_i-{\mathit{\vartheta }}_i\right)}^{\top }{\mathbf{\Lambda }}_i^{-1}\left({\mathit{y}}_i-{\mathit{\vartheta }}_i\right)+{\mathit{\delta }}_i^{\top }{\mathrm{\Psi }}^{-1}{\mathit{\delta }}_i\right]\right\}d{\mathit{\delta }}_i\} \end{gathered}$$

While the proposed model offers flexibility in capturing a complicated longitudinal process, its estimation procedure relies on more complicated mathematical derivations and heavier computational requirements than typical linear models. The numerical integration in the log-likelihood function is usually intractable. However, over the past few decades, several estimation algorithms have been proposed (e.g. Pinheiro & Bates, 1995) and implemented in different software packages in the literature to maximize nonlinear model likelihood functions. Such algorithms can be used for exponential persistence models as well.

The linearization methods using a first-order Taylor series expansion (Vonesh, 1992) or the first order conditional estimation (Wolfinger & Lin, 1997) to approximate the nonlinear function in terms of a linear pseudo-data model are the most widely implemented approach to estimate non-linear models due to their numerical simplicity. These methods are implemented in the R package nlme, in the SAS procedure nlmixed and in the stata module menl. The Lapalce approximation method (Pinheiro & Bates, 1995) which approximates the likelihood function by a second-order Taylor expansion of the integrand around the conditional mode of ϑ_i and the integral approximation methods using Monte Carlo integration (Walker, 1996) or importance sampling (Wang, 2007) are also adequate methods for parameter estimation in these exponential persistence models. The Laplace method has also been implemented in the SAS procedure nlmixed and different approaches based on the stochastic approximation expectation maximization (Kuhn & Lavielle, 2005) have been implemented in the R package saemix.

Bayesian approaches provide an alternative for estimating the parameters in the proposed model (see for example Gelman, Carlin, Stern, & Rubin, 2013). The method makes available a way to incorporate prior knowledge on the likely values of the parameters (such as θ or λ), thus providing a principled approach to accumulating and incorporating scientific knowledge about the persistence or decay process to the proposed model and improving the accuracy of the results or predictions. In the Bayesian framework, the inferences on the parameters of interest are made by summarizing their posterior distributions, a summary that provides a wider representation of the statistical model instead of a single point estimate. Bayesian software programs, such as JAGS (Plummer, 2003), can be used to implement a model such as the propose exponential persistence model.

3.5. Initialization

When implementing any of these nonlinear model estimation procedures, a common difficulty users encounter is that the algorithms can be slow, perform poorly or even be non-convergent, especially in cases where the starting values of the parameters used in the iterative process are unfavorable. Such computational problems may be even more acute when the data are too sparse or the dimensions of the random effects proposed in the model are over-specified. To overcome such potential problems, we proposed a default procedure for automatically creating a set of good initial values below.

1. Because the semi-linear exponential persistence model has a linear component, a direct way of obtaining the initial values for (ν₀, β^T) is to fit a linear mixed effect model of the form $Y_{ij}={\nu }_i^{(0)}+\delta E_{ij}+{\beta }^{(0)\top }{\mathit{X}}_{ij}+{\epsilon }_{ij},$ still with ${\nu }_i={\nu }_0^{(0)}+{\xi }_i$ and E_ij is the indicator of exposures in the data. The estimators $\left(\widehat{{\nu }_0^{(0)}},\widehat{{\beta }^{(0)}{}^{\text{T}}}\right)$ of $\left({\nu }_0^{(0)},{\beta }^{(0)\top }\right)$ will be used as the starting values of (ν₀, β^T).

2. Estimate the influenceable outcome from the model in (i) as $\widehat{Y_{ij}^{inf}}=Y_{ij}-\widehat{{\nu }_0^{(0)}}-\widehat{{\beta }^{(0)\top }}{\mathit{X}}_{ij}.$ Because ω_i can be thought of as the value of the influenceable outcome before an exposure occurred, estimate the starting values of ω₀ as the average of $\widehat{Y_{ij}^{inf}}$ before any exposure i.e. ${\omega }_0^{(0)}=\frac{1}{n_0}{\sum }_{i,j|t_{ij}<{\tau }_{i1}}\widehat{Y_{ij}^{inf}}$ where $n_0={\sum }_{i,j|t_{ij}<{\tau }_{i1}}1$.

3. For the initial value of the exposure effect θ, compute the average influenceable outcome in (i) the first time an exposure occurred $\kappa =\frac{1}{n}{\sum }_{i,j|t_{ij}={\tau }_{i1}}\widehat{Y_{ij}^{inf}}$ and estimate the initial value as ${\theta }^{(0)}=log\left(\kappa /{\omega }_0^{(0)}\right).$.

4. The value of the decay rate λ can be initialized based on expert elicitation but if none exists they should be set at a small value close to 0. In the smoking study application as well as the simulations reported in this manuscript, the model failed to converge at λ⁽⁰⁾ = 0. However, we expected a decay to occur, and setting the initial value to a small decimal removed from 0 (e.g. λ⁽⁰⁾ = 0.01) led to convergence. As such, theoretical expectation of the decay factor can be considered when setting its initial value.

5. The initial values for random errors covariance structure elements, depending on the structure, are simply chosen to give a condition of nearly uncorrelated errors. Depending on theoretical expectation, users can also specify other starting values for the covariance structure.

4. Simulation studies

We conducted simulation studies with emphases on sample size (and the length of data collection), error variance and model formulation to obtain information on the persistence model proposed especially when the true model deviates from our proposed model.

1. Because in most intensive longitudinal data studies, cost can limit the number of study participants, the simulations to assess sample size performance will provide insight on sample sizes needed for our proposed model to produce reliable estimates. We conducted our simulations for sample sizes of 25 to 500. In addition, we explored the effect of the length of data collection period (from 5 to 28 days) on sample size determination.

2. Similarly, intensive longitudinal data collection often produces very granular time series, which can yield large error standard deviations in some cases, especially if there are other factors impacting the process of change in outcome. One may expect the data generating process with extremely large error standard deviations to mask underlying trends and make it difficult for models such as the one proposed to produce good estimates. For this reason, generating outcomes with means similar to the one observed in the illustrative data in section 2.2, our simulation will also assess different error standard deviations σ_ε between 0.1 and 1. In our illustrative example, the participant level intraclass correlation (ICC) of the outcome was 0.89 and the average within participant outcome standard deviation was 0.64. For a typical simulation with the model error standard deviation σ_ε = 0.1, 0.5 and 1 scenarios proposed below, the outcome ICCs observed in such simulation were typically in the order of 0.91,0.61 and 0.47 respectively and the average within participant standard deviations were 0.38, 0.88 and 1.27 respectively. Our standard deviations scenarios for σ_ε = 0.1,0.5 and 1 were thus designed to provide a reasonable range of simulation to mimic data similar to the ones in our example.

3. Finally, because the assumption of exponential loss of association cannot be directly tested, the simulations using a data-generating process that deviate from our model assumption will shed some light on whether our proposed model can still be used even when the model’s main assumption is not fully justified. A step function for generating a simulation and assessing its implications will be provided.

We observed a strong tendency for the persistence model to recover the parameters of interest, whether the model is correctly specified or not. In addition, the models converge fairly often when the steps proposed in section 3.5 are used. Finally, even for a study with small number of participants, the models still perform well with 21 days of data collection and 3 random observations each day. For our specific simulation setup, the added value of collecting data on more days disappear after 10 days for the random error of σ_ε = 0.1.

4.1. Simulation with correctly specified functional form

We first started with a simulation where the process of exposure and effect followed the exponential persistence model posited in (7). For n study participants, we generated a covariate X_i and a random level of influenceable outcome ω_i with mean 1.5, both normally distributed. Then we assumed data was collected on each participant for 21 days with 3 random control observations each day in addition to possible exposures. Each day d was split into 3 time frames 12AM to 8AM, 8AM to 4PM and 4 PM to 12AM. Within each time frame, for a subject i, a time of the random observation was selected to be from a random uniform distribution within the interval: ${\tilde{t}}_{id}^1~U([d,d+0.33])$ in the morning, ${\tilde{t}}_{id}^2~U\left(\right[d+0.33,d+0.66\left]\right)$ in the afternoon and ${\tilde{t}}_{id}^3~U([d+0.66,d+1])$ in the evening where U([a, b]) describes a uniform distribution in the range [a, b]. Within each day, a number of exposure Nevent_id was also simulated to have a poison distribution with an average exposure rate of 1 per day. The remaining data were simulated as follows:

$$\begin{gathered} Neven{t}_{id}~Poisson(\lambda =1),d=1,2,\dots ,21 \\ {\tau }_{i{k}_d}~d+U([0,1])\text{the time of an exposure in day d},k_d=1,\dots ,Neven{t}_{id} \\ {t}_{ij}\in \left\{{\tilde{t}}_{i1}^1,{\tilde{t}}_{i1}^2,{\tilde{t}}_{i1}^3,\dots ,{\tilde{t}}_{i21}^1,{\tilde{t}}_{i21}^2,{\tilde{t}}_{i21}^3,{\tau }_{i1},\dots ,{\tau }_{i{J}_i}\right\},\text{all the observation times} \\ {X}_i~\mathcal{N}(0,1),\text{the covariate} \\ {\omega }_i~\mathcal{N}(1.5,0.1),\text{the influenceable outcome} \\ {Y}_{ij}=3+1.2X_i+Z_{ij}+{\epsilon }_{ij},{\epsilon }_{ij}~\mathcal{N}(0,0.1),\text{the full observed outcome} \\ \text{with}{Z}_{ij}=\{\begin{array}{ll}{\omega }_i\hfill & \text{before the first exposure}\hfill \\ {\omega }_i{e}^{\theta }\hfill & \text{at the first exposure}\hfill \\ {\omega }_i\hfill & {\prod }_{k=1}^{K_{ij}}{\psi }_{ik}\left(t_{ij}\right)\hfill & \text{for any subsequent time}\end{array} \end{gathered}$$

where Z_ij follows the exponential persistence model described in (7) with ${\psi }_{ik}\left(t_{ij}\right)=e^{\theta }{e}^{-\lambda \left(t_{ij}-{\tau }_{ik}\right)}$ the impact of an exposure that occurred at time τ_ik and its decay and the products over the functions ψ_ik(.) are only over the K_ij exposures that occurred before the time t_ij. The normally distributed random error ε_ij was generated with a standard deviation σ_ε. A Poisson distribution with a rate of 1 can still produce a large number of exposures on some days. As such, if more than 3 exposures were generated by the Poisson distribution for a day, those exposures were reduced to 3 in order to avoid days with too many unrealistically large numbers of exposures. These simulations were reported for sample sizes of 50, 100 and 200 and error standard deviations σ_εof0.1,0.5 and 1.

The visual representation of an example of data generated for a participant by this simulation scheme is reported in Figure 1a for the different error standard deviation scenarios. For visualization purposes, each of the three simulated trajectories were shifted by a constant number (intercept) to allow for separation in the lines. As theorized, the larger the error standard deviation becomes, the more oscillation is observed in the time series and at least visually, the greater the possibility of a trend being hidden. For comparison to the college student smoking study, an example of trajectory of the smoking intentions for two study participants is reported in Figure 1b. The simulated participant trajectory showed that the proposed exponential persistence model is potentially suitable for the smoking study data. In addition, the standard deviation σ_ε used in the simulations was in the range of the illustrative example data.

Figure 1:

Smoking intentions trajectories for participants of the smoking study and simulated trajectory with exponential persistence. In the simulated trajectory, exposures were randomly chosen and additive random normal errors with varying standard deviation σ_ε = 0.1, 0.05, 1 was added to an exponential persistence. The lines are the trajectories and the red dots are for exposure occurrences

Using the generated data, we fitted the semi-linear persistence model proposed in (7) in order to estimate the different model parameters. For this set up, the parameters η = (ν₀, ω₀, θ, λ, β^T) in the persistence model will have the true value of η = (3,1.5,0.3,0.1,1.2). In addition, the duration for the influenceable outcome to return to its starting value will be duration = $\frac{0.3}{0.1}=3.$ This procedure of data creation followed by parameter estimation was repeated 1,000 times to obtain a distribution of the different parameters being estimated. Four measures were used to evaluate the performance of the various estimators of each parameter. They include:

The model convergence rate that measures the percentage of time the models converged out of the 1,000 simulated datasets using the initial value selection proposed in section 3.5. As convergence of the semi-linear model proposed can depend on the initial values selected, this measure allows for an assessment of the initial value selection method proposed.

Bias is the difference between the expected value of the estimated model parameters in $\widehat{\mathit{\eta }}$ and the true value η = (3, 1.5, 0.3, 0.1, 1.2). We report the average over the 1,000 simulated datasets.

Root mean squared error (RMSE) is the square root of the expected value of the square of the difference between the estimated parameters in $\widehat{\mathit{\eta }}$ and the true value η. It captures both the bias and the precision of an estimator. It was estimated by taking the square root of the mean of the squared differences between estimated and true parameters from the simulated datasets.

Standard error (SE) is the standard deviation of the estimated parameters across the simulated datasets.

The resulting performance measures are reported Table 1. The proposed model parameter starting value estimation procedure produced very good convergence of the models with all convergence rates more than 99%. In terms of bias, for very small error standard deviations, the parameters were estimated to be very close to their true values (all absolute bias estimates were less than 0.005). In the same setting, as expected, the bias gets smaller with increase in the number of study participants. Even for larger error standard deviations σ_ε, the biases remain small, even though slightly larger than for the small standard deviation simulation. For σ_ε = 0.5, all absolute biases were smaller 0.03 and for σ_ε = 1, they were smaller than 0.06. For the duration parameter that estimates the number of days before an influenceable outcome returns to its starting value, the observed bias was in the scale of 0.04 for a true duration of 3 days. Even in the scenarios of larger error standard deviations, the bias is still small on the order of 0.1 or less.

Table 1:

Estimated performance measures under a correctly specified persistence model

Parameter		100 Bias	100 RMSE	100 SE	100 Bias	100 RMSE	100 SE	100 Bias	100 RMSE	100 SE
		σε=0.1, Conv=99.9%			σε=0.5, Conv=100%			σε=1, Conv=99.7%
	ν0	0.02	0.23	0.23	0.32	1.22	1.18	0.62	2.43	2.35
Sample	β	0.01	0.21	0.21	0.01	0.98	0.98	−0.14	1.80	1.80
size	ω0	−0.22	1.45	1.43	−2.39	3.73	2.87	−4.55	7.11	5.45
N=50	θ	0.11	0.36	0.34	1.83	2.53	1.74	3.53	5.20	3.82
	λ	0.02	0.11	0.11	0.40	0.67	0.54	0.77	1.36	1.12
Duration		0.46	1.54	1.47	6.14	9.34	7.05	11.32	18.60	14.75
		σε=0.1, Conv=99.8%			σε=0.5, Conv=99.9%			σε=1, Conv=100%
	ν0	0.02	0.17	0.16	0.33	0.89	0.83	0.84	1.83	1.62
Sample	β	0.00	0.14	0.14	−0.01	0.64	0.64	−0.02	1.21	1.21
size	ω0	−0.13	1.01	1.01	−2.46	3.21	2.05	−5.57	6.65	3.64
N=100	θ	0.11	0.26	0.24	1.85	2.24	1.25	4.00	4.74	2.55
	λ	0.02	0.08	0.07	0.39	0.54	0.38	0.86	1.14	0.75
Duration		0.38	1.07	1.00	6.49	8.09	4.83	13.20	16.49	9.89
		σε=0.1, Conv=100%			σε=0.5, Conv=100%			σε=1, Conv=100%
	ν0	0.02	0.11	0.11	0.34	0.67	0.57	0.82	1.43	1.17
Sample	β	0.00	0.10	0.10	−0.01	0.46	0.46	−0.02	0.84	0.84
size	ω0	−0.10	0.74	0.74	−2.64	2.97	1.37	−5.67	6.26	2.66
N=200	θ	0.10	0.19	0.17	1.92	2.09	0.82	4.06	4.45	1.82
	λ	0.02	0.06	0.05	0.41	0.48	0.25	0.85	1.01	0.53
Duration		0.38	0.79	0.69	6.73	7.56	3.45	13.77	15.56	7.24

σ_ε = standard deviation of the model error; Conv=Model convergence rate; Bias, RMSE and SE were all multiplied by 100

Because the performance measures were very small, they were all multiplied by 100. The simulations were conducted for 21 days of data collection and 3 random prompt per day.

Across the board, there is definitely a bias increase when the error standard deviation gets larger. Both RMSE and standard errors were also estimated to be very small in all simulation scenarios with the same pattern of larger sample sizes leading to improved model precision and a lower error variance leading to improved estimators.

Intensity of data collection

To provide information on the effect of study duration on our ability to capture the different model parameters, the simulations were extended to studies lasting 5 (less than one week) to 28 days (a month) in a setting where it takes 3 days for the influenceable outcome ω_i to return to its starting value. In addition, for each scenario, the number of random-prompt assessments was expanded to include the possibility of having 1, 2 or 4 random-prompt assessment per day. For each of these simulations, the RMSE for estimating θ the log of the rate of change in the outcome due to the exposure, λ the exponential decay rate and the duration parameter $\frac{\theta }{\lambda }$ were recorded and correlated with study duration. For all these scenarios, for simplicity, we set the random error to σ_ε = 0.1. Figure 2a and b provide a visual representation of the relationship for θ and the duration. The results for λ are similar but not reported here. Clearly, for a fixed sample size, the more random prompts per day, the better RMSE (i.e. better estimation) is to be expected. Similarly, the larger the number of participants the better the RMSE will be for all parameters being estimated. Additionally, if data is collected for 10 days or more, the increase in number of random prompt collected each day can compensate for smaller sample sizes. For example, having 100 participants and 4 random prompts a day can be equivalent to having 200 participants and 2 random prompts a day, but under this simulation setting, such equivalence is only realized with more than 10 days data collection. When data is only collected for fewer than 10 days for the parameters θ and λ, adding random prompts per day can potentially be more beneficial than adding participants if, in the end, a similar amount of data is collected. For example, for those parameters, 100 participants responding to 4 random prompts per day yields a smaller RMSE than 200 participants responding to 2 random prompts per day for under 10 days. Similarly, 50 participants responding to 4 random prompts per day resulted in a better RMSE than 100 participants responding to 2 random prompts daily, which in turn yielded a better RMSE than 200 participants responding to 1 random prompt per day when the duration of data collection was less than 10 days.

Figure 2:

Association between intensity of data collection of mean square error. The different lines are for sample sizes N=50, 100 and 200 and 1, 2 or 4 random prompts per day

In sum, for our proposed method, there are trade-offs between the number of participants, the required number of repeated assessments in a given interval and length of time to collect such information. In a setting such as the one used in our simulation where the influenceable outcome returns to its starting value after 3 days, there seemed to be no added value in collecting data for more than 10 days for parameter estimation and for the estimation of the duration the threshold for number of days needed seemed smaller. We hypothesized that for data processes where it takes the influenceable outcome more days to return to its starting value, there will be an optimal number of days needed for data collection but such threshold will more likely be greater than the 10 days observed in our simulation setting. All these situations can be taken into account when designing an intensive longitudinal data study.

4.2. Simulation with step functions in the persistence design

The results of the previous simulation revealed that when the persistence model is correctly specified, the proposed estimation approach behaves as most regression methods when correctly specified. However, the persistence model night not necessarily specify the data generating process correctly. In this section, then, we consider a scenario where the process of exposure effect persistence follows a stepwise function and we assess the characteristics of the exponential persistence model in (7) approximating such process. In this simulation, the steps for data generation are identical to those presented in Section 4.1 with the exception that ψ_ik(.) is now chosen to be a step function. After an exposure, the influenceable outcome is assumed to drop by 20% of its value every one day with

$${\psi }_{ik}\left(t_{ij}\right)=e^{\theta \left[1-Step\left(t_{ij}-{\tau }_{ik}\right)\right]}$$

and the step function Step(t) taking values 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, … for 0 < t ≤ 1, 1 < t ≤ 2, 2 < t ≤ 3, 3 < t ≤ 4, 4 < t ≤ 5, 5 < t ≤ 6, 6 < t ≤ 7,... respectively.

For the first day after an exposure, the influenceable outcome will be ω_ie^θ, then for the entire second day, it will drop to ω_ie^0.8θ, for the third day ω_ie^0.8θ and so on. Between the fifth and sixth day it will drop to ω_ie⁰ = ω_i.

An example of the simulated data is reported in Figure 3 where the step functions can be observed in the absence of the error term ε_ij. The trajectory with the individual error will be the one modeled in this simulation. The performance of the persistence model for this process, one that does not follow correctly the posited model, is reported in Table 2. Because the true value of λ is unknown in the data generating process, its performance is not reported. Also, even though the duration, parameter that reports the amount of time it will take for an influenceable outcome to return to its original value after an exposure is also not defined as in the persistence model, one can recognize that with a step function where a 20% daily drop is defined in the simulation, the influenceable outcome will return to its starting value between day 5 and 6 with a mid-point of 5.5. We posited that the persistence model would estimate the duration at the mid-point and 5.5 was used as the true value for bias and RMSE estimation.

Figure 3:

Outcome trajectory in a stepwise persistence process. The black line is an error free outcome trajectory with the red diamonds being exposures and the black circles random observations. The blue line is the trajectory with the random error with σ_ε = 0.5 and it is shifted down for visualization purpose.

Table 2:

Estimated performance measures under a stepwise persistence setting

Parameter		100 Bias	100 RMSE	100 SE	100 Bias	100 RMSE	100 SE	100 Bias	100 RMSE	100 SE
		σε=0.1, Conv=99.3%			σε=0.5, Conv=100%			σε=1, Conv=99.5%
	ν0	−0.56	0.99	0.82	0.15	4.27	4.27	0.98	8.82	8.77
Sample	β	0.00	0.38	0.38	−0.02	1.36	1.36	−0.03	2.06	2.06
size	ω0	0.30	1.72	1.70	−1.10	5.27	5.16	−2.74	10.80	10.45
N=50	θ	0.87	0.87	0.10	1.05	1.16	0.49	1.29	1.66	1.04
Duration		−1.07	1.70	1.33	1.04	6.30	6.22	2.95	12.82	12.48
		σε=0.1, Conv=98.1%			σε=0.5, Conv=99.9%			σε=1, Conv=99.9%
	ν0	−0.58	0.84	0.61	0.42	3.01	2.99	1.81	6.19	5.92
Sample	β	0.01	0.25	0.25	−0.01	0.94	0.94	0.05	1.42	1.42
size	ω0	0.29	1.33	1.30	−1.40	3.88	3.62	−3.46	7.84	7.04
N=100	θ	0.87	0.87	0.07	1.06	1.11	0.34	1.32	1.49	0.70
Duration		−1.04	1.43	0.98	1.06	4.53	4.40	2.80	9.36	8.93
		σε=0.1, Conv=97.9%			σε=0.5, Conv=100%			σε=1, Conv=100%
	ν0	−0.54	0.68	0.41	0.44	2.07	2.02	1.61	4.68	4.40
Sample	β	−0.01	0.17	0.17	−0.01	0.63	0.63	0.06	0.96	0.95
size	ω0	0.29	0.93	0.88	−1.39	2.82	2.46	−3.41	6.23	5.22
N=200	θ	0.87	0.87	0.05	1.06	1.09	0.24	1.29	1.39	0.52
Duration		−1.09	1.28	0.67	1.01	3.36	3.21	3.16	7.05	6.30

σ_ε = standard deviation of the model error; Conv=Model convergence rate; Bias, RMSE and SE were all multiplied by 100

Because the performance measures were very small, they were all multiplied by 100. The simulations were conducted for 21 days of data collection and 3 random prompt per day.

Once again, the persistence model performed well even when the model is not correctly specified. In terms of bias, for the small error variation of σ_ε = 0.1, all the biases are less than 0.001. The biases increase slightly with larger error variation as expected, but even for a large standard deviation of σ_ε = 1, all biases remain less than 0.04. The RMSE and SE also behave similarly across the simulation scenarios with the absolute value of these performance estimators being slightly larger than the bias estimates. The duration also did estimate the length of time for the influenceable outcome to return to its initial value at 5.5 with biases less than 0.03 in all cases. This provides evidence that even if the posited model is not necessarily equivalent to the data generating process, it can still capture parameters of interest such as the duration accurately.

5. Illustration with AdSpotter

Emerging evidence has supported the idea that exposure to pro-smoking media can engage cognitions about smoking in a way that places people at risk for future smoking (Shadel et al., 2016), and this analysis aims to extend this work by understanding the magnitude of such effects. Exposure to pro-smoking media affects individuals differently, depending on personal level of experience with smoking. Although exposure contributes to smoking initiation, enables the progression to more established smoking status, and serves as a “cue” to smoke for established smokers, the effects of pro-smoking media on the progression from never smoking to experimental smoking have been shown to be stronger than the effects of pro-smoking media on the progression from experimental smoking to regular smoking (Wellman et al., 2006). These differences are thought to be a consequence of increasing engagement of processes relating to nicotine dependence as level of smoking increases such that effects of media exposure are dampened as nicotine dependence becomes more entrenched. In addition, many socio-economic characteristics of the population in this study have been reported to be associated with responses to smoking-related media including gender, race, and whether the exposure occurred on a weekend or a weekday (Martino et al., 2012).

Because of these known factors, we included participants’ smoking status, gender, race and whether an observation was on the weekend (defined as Friday, Saturday or Sunday) or not as covariates in the exponential persistence model and we also included the possibility of the exposure effect being different between non-smokers and smokers. The posited model was as followed:

$$\begin{gathered} Intentio{n}_{ij}={\nu }_i+{\omega }_i{e}^{\left({\theta }_1+{\theta }_{2}NonSmoke{r}_i\right)K_{ij}-\left({\lambda }_1+{\lambda }_{2}NonSmoke{r}_i\right)Dtim{e}_{ij}}+ \\ {\beta }_{1}NonSmoke{r}_i+{\beta }_{2}Femal{e}_i+{\beta }_{3}Minority+{\beta }_{4}Weekend+{\epsilon }_{ij} \\ {\nu }_i={\nu }_0+{\xi }_i\text{with}{\xi }_i~\mathcal{N}\left(0,{\sigma }_{\nu }^2\right) \\ {\omega }_i={\omega }_0+{\zeta }_i\text{with}{\zeta }_i~\mathcal{N}\left(0,{\sigma }_{\omega }^2\right) \\ \text{and}{\epsilon }_{ij}~\mathcal{N}\left(0,{\sigma }_{\epsilon }^2\right),corr\left({\nu }_i,{\omega }_i\right)=\varrho \end{gathered}$$

All analyses were conducted in R and the nonlinear estimation was done using the nlme package. Using the procedure proposed in section 3.5(i), a linear mixed effect model estimated the starting values of (ν₀, β₁, β₂, β₃, β₄) to be (3.38, −2.27, 0.16, −0.01, 0.05). For the initial value of the influenceable outcome ω₀, the procedure returned a value of 0.18, but since the exposure effect is assumed different between non-smokers and smokers, the average for estimating κ in section 3.5(iii) was conducted separately in the two smoking status group. This returned $\left({\theta }_1^{(0)},{\theta }_2^{(0)}\right)=(0.89,0.54)$ as the starting values of (θ₁, θ₂). As mentioned earlier, setting the starting values for λ₁ and λ₂ to 0 led to non-convergence but because we expected a decay in the impact of the exposure, we reset those starting values to 0.01 and they led to convergence. The starting value of the covariance-structure was set to the identity matrix for the estimation.

Based on the coefficients of the posited model, Table 3 summarizes the estimated effects as well as their transformations with direct implications.

Table 3:

Estimated effects of exposure and parameter combinations

Name	Estimator	Estimate	95%CI
Model parameters
Intercept	ν0	3.41	(2.79, 4.03)
Non-Smoker	β1	−2.15	(−2.87, −1.43)
Gender	β2	0.25	(−0.5, 0.99)
Race	β3	−0.08	(−0.84, 0.68)
Weekend	β4	0.03	(−0.01, 0.06)
Influenceable outcome	ω0	0.13	(0.07, 0.19)
Exposure effect for smokers	θ1	0.25	(0.07, 0.42)
Exposure effect deviation for non-smokers	θ2	0.08	(−0.16, 0.32)
Decay rate smokers	λ1	0.09	(0.03, 0.16)
Decay rate deviation for non-smokers	λ2	0.07	(−0.07, 0.21)
Residual standard deviation	σε	0.81
Non- linear estimands
Influenceable outcome	ω0	0.13	(0.07, 0.19)
Exposure effect rate increase for smokers	$e^{{\theta }_1}$	1.28	(1.06, 1.50)
Exposure effect rate increase for non-smokers	$e^{{\theta }_1+{\theta }_2}$	1.39	(1.02, 1.76)
Exposure vs non-exposure difference for smokers	${\omega }_0{e}^{{\theta }_1}-{\omega }_0$	0.04	(0.02, 0.06)
Exposure vs non-exposure difference for non-smokers	${\omega }_0{e}^{{\theta }_1+{\theta }_2}-{\omega }_0$	0.05	(0.01, 0.09)
Decay duration to starting point in smokers	$\frac{{\theta }_1}{{\lambda }_1}$	2.73	(1.55, 3.91)
Decay duration to starting point in smokers	$\frac{{\theta }_1+{\theta }_2}{{\lambda }_1+{\lambda }_2}$	2.05	(0.83, 3.27)
Smoker duration to half influenceable after exposure	$-\frac{log(0.5)}{{\lambda }_1}$	7.62	(2.22, 13.02)
Non-smoker duration to half influenceable after exposure	$-\frac{log(0.5)}{{\lambda }_1+{\lambda }_2}$	4.33	(0.56, 8.10)

Note: “Non- linear estimands” were obtained using a delta-method.

We first inspected the univariate difference in the smoking outcome between the different types of smoking status groups. Even though the outcome of smoking intention has an average of 2.7 in the sample studied, a large difference in outcome exists between non-smokers (mean =1.2) and smokers (mean = 3.5) and in the persistence model, the covariate adjusted difference between smokers and non-smokers was estimated at −2.2. None of the other covariates were significant confounders. For the persistence parameters, the model reveled that only an average value of ω₀= 0.13 (95%CI = [0.07, 0.19]) of smoking intention is influenceable by the exposure to pro-smoking media. An instantaneous log of the rate of change in smoking intention was estimated to be θ₁ = 0.25 (95%CI = [0.07, 0.042]) among smokers and the log rate for non-smokers was similar with a deviation of θ₂ = 0.08 (95%CI = [−0.16, 0.032]). Similarly, a significant decay rate was observed among smokers λ₁ = 0.09 (95%CI = [0.03, 0.16]) and the rate in non-smokers is similar with a deviation of λ₂ = 0.07 (95%CI = [−0.07, 0.21]). Using the results in Corollary 1, we next constructed different estimates with practical implications using the delta- method (Fox & Weisberg, 2011). Among smokers, after an exposure occurred, the influenceable outcome increased at the rate of $e^{{\theta }_1}=1.28(95\%CI=[1.06,1.50])$ or by 28%. This means that, immediately after an advertising exposure, intentions increased from 0.13 to 0.17 or a differential change in outcome equaling ${\omega }_0{e}^{{\theta }_1}-{\omega }_0=0.04(95\%CI=[0.02,0.06])$ which is the classical exposure effect estimate being the difference in outcome with and without exposure. A similar effect was observed among non-smokers where an advertising exposure led to an increase in the influenceable outcome at the rate of $e^{{\theta }_1+{\theta }_2}=1.39(95\%CI=[1.02,1.76])$ or a 39% increase and this will be equivalent to a classical difference of ${\omega }_0{e}^{{\theta }_1+{\theta }_2}-{\omega }_0=0.05(95\%CI=[0.01,0.09]).$ Assessing how long it takes the influenceable outcome to return to its initial value observed in participants after the first exposure recorded in our participants, the results suggested an average of $\frac{{\theta }_1}{{\lambda }_1}=2.73\text{days}(95\%CI=[1.55,3.91])$ for smokers to have the influenceable outcome drop from 0.17 back to 0.13 and $\frac{{\theta }_1+{\theta }_2}{{\lambda }_1+{\lambda }_2}=2.05\text{days}(95\%CI=[0.83,3.27])$ for non-smokers to see a drop from 0.18 back to 0.13. Furthermore, if there is a desire to see the level of the influenceable outcome right after the first exposure drop to half of its value from 0.17 to 0.085 in smokers and from 0.18 to 0.09 in non-smokers, it will take those two groups approximately $-\frac{log(0.5)}{{\lambda }_1}=7.62\text{days}(95\%CI=[2.22,13.02])$ and $-\frac{log(0.5)}{{\lambda }_1+{\lambda }_2}=4.33\text{days}(95\%CI=[0.56,8.10])$ respectively. This duration of the influenceable outcome dropping to half of its value, which is below the influenceable outcome ω₀ observed at the beginning of the study, is reflective of the fact that college students are commonly exposed to a myriad of pro-smoking advertisements and the first exposure observed in this study is not the first time participants have encountered such ads in their lives. Allowing smokers for example to have their levels of intention dropped even further than the average level observed ω₀ in the study will require a participant to go more than 7 days without any exposure. Unfortunately, in this study, even though some participants reported no smoking media exposure for the 21 day study period, an average of 8.24 exposures to pro-smoking advertisements was reported (Martino et al., 2012). This reflects participants possibly being at elevated level of smoking intentions most of the time because of the frequency of exposures to which they are subject.

In summary, for a conceptual model of how exposures to pro-smoking media influences adolescents’ smoking intention behavior, using the semi-linear exponential persistence model, we were able to estimate the effect of exposures even in a complex setting where adolescents are exposed to media at random times. The results suggest that exposures occurring before prior exposures fully decaying could cause the risk of smoking to accumulate and at some point, lead to experimentation with smoking.

6. Beyond the semi-linear exponential persistence model

In the proposed semi-linear exponential persistence model, the influenceable outcome depends on time and exposure only. However, this model can be easily extended to include participant covariates. For example, in the adolescent smoking study example that motivated the current approach, one can hypothesize that the exposure effect and /or the decay can depend on a participant’s smoking status (smoker versus non-smoker). As such a straightforward extension of the model will be to posit that

$$Y_{ij}={\nu }_i+{\omega }_i{W}_{1i}{e}^{\theta W_{2i}{K}_{ij}-\lambda W_{3i}Dtim{e}_{ij}}+{\beta }^{\top }{\mathit{X}}_{ij}+{\epsilon }_{ij}$$

where (W_i1, W_i2, W_i3) are different participant characteristics that can confound the magnitude and the decay of the effect.

In addition, in the development of the exponential persistence model, we focused primarily on the classic setting where the outcome Y_ij is continuous with a linear relationship on the predictors except for the influenceable outcome and an additive error structure that is assumed to have an approximately normal distribution. However, in many studies, the data to be modeled can clearly be non-normal. For instance, a study may record whether a participant has decided to smoke (Yes/No) at the moment of an observation or collect information that can lead to the estimation of the number of hours or days until the next smoking or substance use. Dichotomous data on whether a participant smoked is clearly nonnormal and a count such as the number of days until the next substance use may not be suitable for a normal distribution analysis. In general settings, binomial or Poisson distributions are often used to model dichotomous and count outcomes and can be adapted for the exponential persistence model with the appropriate outcome distribution. As in generalized linear models - GLM- (Nelder & Wedderburn, 1972), the exponential persistence model can also be extended to outcomes generated from distributions in the exponential family, a large range of probability distributions that includes the normal, binomial, Poisson and gamma distributions, among others. The outcome variable Y_ij will still be modeled as a combination of the predictors with the exponential persistence assumption, except that the relationship will not necessarily be semi-linear as in our previous setup. Instead, a non-linear link function will link the outcome to the predictors including the exposure and its persistence effect. In this generalized setting, the mean ϑ_ij of the outcome will be specified as

$$h\left(E\left(Y_{ij}\right)\right)=h\left({\mathit{\vartheta }}_{ij}\right)={\nu }_i+{\omega }_i{W}_{1i}{e}^{\theta W_{2i}{K}_{ij}-\lambda W_{3i}Dtim{e}_{ij}}+{\beta }^{\top }{\mathit{X}}_{ij}$$

where h will be the link function. The smooth and invertible linearizing link function h() transforms the expectation of the outcome variable to have a linear relationship with the predictors. The additive normally distributed error will be replaced by a choice of the outcome distribution. Table 4 shows the outcome distribution, the link function and the variance of the three commonly used outcome types in human behavioral studies and beyond. For binomial data where the outcome Y_ij represents the likelihood that a participant will say “Yes” to a question (e.g. did you smoke), the logit link will transform of the probability of “Yes” to be a linear combination of decaying part of the outcome and the participant level associations with a binomial distribution. In another word, the log-odd of the outcome will be assumed to have an exponential decay. Similarly, for a count data, the log link will assume that the logarithm of the expected value of the outcome can be modeled by similar linear combination of parameters.

Table 4:

Example of generalized linear model distributions and link functions

Distribution	Link function h(.)	Variance	Outcome Range
Normal	identity : ϑ	${\sigma }_{\epsilon }^2$	(−∞, +∞)
Bernoulli	Logit: log(ϑ/(1 − ϑ))	ϑ (1 − ϑ)	0/1 (Yes/ No)
Poisson	Log : 1og(ϑ)	ϑ	0, 1, 2,…

Note: “other” link functions can also be considered.

Other more complicated extensions can even be considered. The proposed exponential persistence model assumption that the association of interest decreases at a rate proportional to its current value over the time since an exposure occurred can be extended to allow the decrease to be faster or slower over time through a function φ(t) of the time. So a more general framework will express equation (3) as

$$\mu =\omega \exp (-\phi (t))$$

and this will lead to the same model formulation except that in the model, the expression λDtime_ij will be replaced by ${\sum }_{k=1}^{K_{ij}}\phi \left(t_{ij}-{\tau }_{ik}\right)$ thus reframing the model as

$$\begin{gathered} Y_{ij}={\nu }_i+{\omega }_i{W}_{1i}{e}^{\theta W_{2i}{K}_{ij}-W_{3i}{\sum }_{k=1}^{K_{ij}}\phi \left(t_{ij}-{\tau }_{ik}\right)}+{\beta }^{\top }{\mathit{X}}_{ij}+{\epsilon }_{ij} \\ {\nu }_i={\nu }_0+{\xi }_i\text{with}{\xi }_i~\mathcal{N}\left(0,{\sigma }_{\nu }^2\right) \\ {\omega }_i={\omega }_0+{\zeta }_i\text{with}{\zeta }_i~\mathcal{N}\left(0,{\sigma }_{\omega }^2\right). \end{gathered}$$

As an example, assuming that the decay goes faster over time and defining $\mu =\omega \exp (-\phi (t))=\omega \exp \left(-{\lambda }_1\left(t_{ij}-{\tau }_{ik}\right)-{\lambda }_2{\left(t_{ij}-{\tau }_{ik}\right)}^2\right),$ the outcome model will then be specified as

$$Y_{ij}={\nu }_i+{\omega }_i{W}_{1i}{e}^{\theta W_{2i}{K}_{ij}-{\lambda }_1{W}_{3i}Dtim{e}_{ij}-{\lambda }_2{W}_{4i}Dtimes{q}_{ij}}+{\beta }^{\top }{\mathit{X}}_{ij}+{\epsilon }_{ij}$$

where $Dtimes{q}_{ij}={\sum }_{k=1}^{K_{ij}}{\left(t_{ij}-{\tau }_{ik}\right)}^2.$ As in any model specification, users will need to be mindful of the fact that the more parameters are specified in the model the more likely it is that model convergence issues will start arising. At the end, the general exponential persistence model provides a general framework under which complex behavior phenomena with time varying effects can be modeled under the exponential loss of association assumption or beyond.

7. Conclusion

Our goal in this article was to provide a new method for characterizing and estimating complex behavior models where a population is repeatedly exposed to events (e.g., interventions, treatments, environmental stimuli) over a specific period. Our proposed exponential persistence model provides a good candidate for estimating the decay, persistence, and increase that occur as a function of those events over time. The proposed method relates every observation to all previously observed exposures, and assesses different intensities in association to each of those events. This is a departure from the traditional time-varying mixed effect (parametric as well as non-parametric) models (e.g. Tan et al., 2012; Setodji et al., 2014), or the growth curve models or time series persistence models (e.g. Haaijer & Wedel, 2001; J.-K. Kim, 2000; J.-K. Kim, Silvapulle, & Hyndman, 2007), where the impact process of having multiple exposures is usually not disentangled. Even if the model posited in (7) does not correctly specify the phenomena, the results from the simulation studies showed that the proposed semi-linear persistence model can still approximate the phenomena accurately. The model also provides an estimate of additional parameters of interest such as the duration of the persistence of an effect. The model in (7) is a semi-linear model and requires the initialization of parameters and the choice of such initial values is critical for model convergence; our proposed initialization-procedure consistently leads to model convergence in the simulations conducted. We believe that the proposed method has notable advantages over methods of constant effect over time (Diggle, Heagerty, Liang, & Zeger, 2002) or varying-coefficients effects (Hoover, Rice, Wu, & Yang, 1998) that have used non-parametric functions in which some of the complexities of the added value of repeated exposures are bound to be ignored.

The exponential aspect of our model is a potential limitation in that it assumes that the effects of an exposure impact can never fully disappear although those effects can become very small. However, in situations where exposures keep repeatedly occurring, the prospect of an effect free outcome is not judged as probable. The proposed methodology might be useful for many interventions such as weight loss programs where participants self-decide on when exposures occur and modern technological innovations such as portable devices allow for the intensive collection of longitudinal information that can help assess the longitudinal path of a complex mechanism.