Modeling Dynamic Causal Effects in Repeated Episodes

In our previous lessons, we have built a sophisticated toolkit for analyzing emotional influence over time, culminating in the Random Intercept Cross-Lagged Panel Model (RI-CLPM). This model allowed us to estimate the strength and direction of causal effects, such as how the wife's guilt influences the husband's jealousy. However, a core assumption of the RI-CLPM is that these causal links are stable—that the effect of one person's emotion on another's is constant across every measurement occasion. Today, we challenge that assumption.

The goal of this lesson is to specify a model to estimate how the causal effect of an activity on an emotion changes across repeated episodes. Human responses are rarely static. We adapt, learn, and tire. An activity that initially provokes strong jealousy might have a diminished effect over time (habituation), or conversely, repeated exposure might amplify the emotional response (sensitization). Your expertise in Design of Experiments involves understanding how effects change under different conditions; here, the "condition" that is changing is time or experience itself. We will explore statistical models that move beyond a single effect coefficient, allowing the effect to be a dynamic function of time.

Conceptualizing Changing Effects: Habituation and Sensitization

At the heart of our inquiry are two fundamental psychological processes: habituation and sensitization.

Habituation is a decrease in response to a stimulus after repeated presentations. For example, the husband's jealousy in response to the wife recounting details of an encounter might be high initially but lessen over subsequent encounters.
Sensitization is an increase in response to a stimulus after repeated presentations. For example, feelings of guilt in the wife might be minimal at first but accumulate and intensify with each successive encounter.

These processes are not mutually exclusive and can be influenced by other factors. The images below illustrate these concepts.

These graphs show the theoretical change in response over repeated trials for habituation (a decrease) and sensitization (an increase). The observed behavior is a net result of underlying processes.

This figure illustrates how the nature of the stimulus ('Picture Content') can alter the repetition effect. The effect of an 'unpleasant' stimulus might decrease (habituate), while a 'pleasant' one could increase (sensitize). In your research, this could be analogous to how the perceived emotional tone of a wife-bull encounter (e.g., purely physical vs. emotionally intimate) might change its effect on the husband's jealousy over time.

Our task is to find a modeling framework that can capture these dynamic patterns, moving from a static causal parameter, $β$ , to a time-varying one, $β (t)$ .

Modeling Time-Varying Effects

We will explore several ways to model an effect that changes over time, starting with simpler parametric methods and building to a highly flexible non-parametric approach.

1. Parametric Approach: Interaction with Time in Growth Models

A straightforward method, building on the logic of multilevel models, is to specify an interaction between your predictor and a polynomial of time. Imagine you are modeling the husband's jealousy as a function of a specific activity (e.g., WifeActivity, coded 0/1). The time variable, Episode, is the sequential number of the encounter (1, 2, 3, ...).

A simple linear interaction model would be:

Jealousy_{i t} = (γ_{00} + u_{0 i}) + (γ_{10} + u_{1 i}) Episode_{t} + γ_{20} WifeActivity_{i t} + γ_{30} (WifeActivity_{i t} \times Episode_{t}) + ϵ_{i t}

In this model:

$γ_{20}$ is the effect of the activity at Episode = 0.
$γ_{30}$ is the key coefficient: it represents the linear change in the activity's effect for each one-unit increase in Episode.
- A significant negative $γ_{30}$ would suggest habituation.
- A significant positive $γ_{30}$ would suggest sensitization.

This approach can be extended with quadratic ( $A c t i v i t y \times E p i so d e^{2}$ ) or cubic terms to capture more complex, non-linear changes in the effect.

The following video discusses the related concept of time-varying covariates (TVCs) in growth models. While it focuses on the predictor changing over time, the logic of modeling its changing impact is directly applicable.

Growth Curve Episode 7: Time-Varying Covariates

This video from CenterStat introduces time-varying covariates and how their effects are modeled.

Watch the section explaining how TVCs account for the "pokes and pulls" around a growth trajectory. This is conceptually similar to how a specific activity might cause a deviation in an emotional trajectory. Then, pay close attention to the discussion on testing for changing effects over time by either constraining coefficients to be equal or allowing them to differ, and the concept of lagged effects.

2. Discrete Approach: Phase Models in DSEM

If your study involves distinct phases—for example, a baseline period followed by an intervention period where a new communication rule is introduced—you can use Dynamic Structural Equation Modeling (DSEM) to estimate different effect parameters for each phase.

You can specify a model where the coefficients themselves change from the baseline phase to the intervention phase. The resource below outlines this logic for a single-arm trial, which can be adapted to your context by defining "phases" based on time or experimental manipulations.

Intensive Longitudinal Data Analyses and Sample Size ...

This dissertation chapter provides detailed specifications for DSEM in intervention studies.

First, read the section Model 1 specification (pp. 37-39). Notice how separate equations are defined for the baseline and intervention phases, allowing for a shift in the mean (\mu). Next, jump to the section on Potential Extensions (pp. 49-50). This section explicitly shows how you can model an intervention's effect not just on the mean of an outcome, but on its dynamic properties, like the autoregressive coefficient (\phi) or residual variance (\sigma^2). You could use the same logic to model a change in the coefficient for an activity's effect between phases.

This phase-based approach is powerful for analyzing planned interventions, but it's less suited for exploring gradual, continuous change over a long period. For that, we turn to a non-parametric method.

3. Non-Parametric Approach: The Time-Varying Effect Model (TVEM)

The most flexible way to answer our question is with the Time-Varying Effect Model (TVEM). Instead of assuming a parametric form (like linear or quadratic) for the change in an effect, TVEM uses non-parametric smoothing techniques (specifically, P-splines) to let the data define the shape of the change.

The model equation looks simple, but holds a profound difference:

Emotion_{i t} = β_{0} (t) + β_{1} (t) \cdot Activity_{i t} + ϵ_{i t}

Here, both the intercept $β_{0} (t)$ and the slope $β_{1} (t)$ are functions of time $t$ .

$β_{0} (t)$ represents the average trajectory of the emotion over time, controlling for the activity.
$β_{1} (t)$ is the time-varying effect of the activity on the emotion. It is a curve, not a single number, showing how the strength and direction of the causal effect evolve with each episode.

The following paper provides an excellent, accessible introduction to TVEM.

Using the Time-Varying Effect Model (TVEM) to Examine ...

This article introduces TVEM and applies it to smoking cessation data, which provides a clear analogy for your work on relationship dynamics.

First, read the rationale for TVEM, which explains the limitations of standard MLM for capturing complex, non-parametric changes. Next, study the conceptual overview, paying close attention to Equation (1) and the explanation of the intercept and slope functions. Review the section on model selection, which describes the practical process of using knots, AIC, and BIC to determine the complexity of the functions. Your background in statistics will make this process intuitive. Finally, examine the results for the time-varying effect of negative affect and the time-varying effect of self-efficacy. Focus on how the authors interpret the graphs of the slope functions (Figures 3 and 4) to tell a story about how these effects change over time.

Applying TVEM to Your Triadic Research

Let's synthesize this for your study. Suppose you want to test how the effect of the bull's communication style (e.g., BullCommunicatesEmpathy, a score from 1-10) during an encounter affects the wife's subsequent guilt. You have data from 50 such encounters.

Define the Model: You would specify a TVEM where Episode (from 1 to 50) is your time variable.
$WifeGuilt_{i t} = β_{0} (Episode_{t}) + β_{1} (Episode_{t}) \cdot BullCommunicatesEmpathy_{i t} + ϵ_{i t}$
Fit and Estimate: Using a package that supports TVEM (e.g., the %TVEM macro in SAS mentioned in the paper, or packages in R), you would fit the model. The procedure would involve selecting the optimal number of knots for the $β_{0} (t)$ and $β_{1} (t)$ functions based on information criteria like AIC.
Interpret the Output: The key output is the plot of the slope function, $β_{1} (t)$ , against Episode.
- If $β_{1} (t)$ is consistently negative and becomes more negative over time, it would suggest that the guilt-reducing effect of the bull's empathy becomes stronger with repeated encounters (a form of positive reinforcement or learning).
- If $β_{1} (t)$ starts negative but trends towards zero, it suggests habituation: the bull's empathy matters a lot initially, but its impact diminishes as the wife becomes more accustomed to the situation.
- If the function is flat and non-significant, it implies the effect is negligible and does not change over time.

This method allows you to generate nuanced, data-driven hypotheses about the longitudinal evolution of your triad's emotional ecosystem.

Conclusion

Today, we have moved beyond the restrictive assumption of static causality. You've learned that phenomena like habituation and sensitization are not just theoretical concepts but can be explicitly modeled to understand how the impact of activities and behaviors evolves over time.

Key Takeaways:

Causal effects in longitudinal processes are often time-varying, not fixed.
Parametric interactions with time (e.g., in growth models) offer a simple way to test for linear or polynomial changes in an effect.
DSEM phase models are ideal for comparing effects across discrete, pre-defined periods, such as before and after an intervention.
The Time-Varying Effect Model (TVEM) is a powerful non-parametric approach that estimates an effect as a smooth function of time, allowing for flexible, data-driven discovery of how causal relationships evolve.

You are now equipped with a set of advanced tools to model the dynamic nature of causal effects. This knowledge forms a critical bridge to our next module, where we will focus on Designing Experiments for Triadic Interventions. Knowing that effects can change over time is crucial when designing an experiment. For example, you might design a study not just to see if an intervention works, but how its effectiveness changes across repeated applications, a question for which TVEM is the perfect analytical tool.

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