Modeling Temporal Dependencies in Crossover Designs

Introduction

In our previous lessons, we have constructed a solid foundation for representing causal relationships using static Directed Acyclic Graphs (DAGs). We've explored how treatments flow through mediators, how confounders create spurious associations, and how conditioning on colliders can paradoxically introduce bias. These static models provide a powerful "snapshot" of a system. However, your proposed N-of-1 crossover experiment is inherently dynamic; it unfolds over time, with treatments and observations occurring in sequence. A static model is insufficient to capture this evolution.

This lesson will bridge that gap by incorporating the dimension of time directly into our causal models. We will explore how to represent the temporal sequencing inherent in your crossover design, allowing us to model phenomena like adaptation, psychological carryover, and other lagged effects. This directly addresses the learning outcome: Encode temporal structure (lagged effects, adaptation) in the causal model to align with the crossover design. By the end of this lesson, you will be able to transform a static DAG into a dynamic one that more faithfully represents the evolving nature of the triadic relationship you are studying.

1. Why Time is Not Optional in Causal Graphs

The most fundamental principle of causality is that causes must precede their effects. While this seems obvious, standard DAGs often leave the temporal relationship between variables implicit. This ambiguity can be problematic, especially when dealing with systems involving feedback or measurements taken over time. To build a rigorous causal model for a longitudinal experiment, we must make time explicit.

The paper "The Case for Time in Causal DAGs" provides a compelling argument and a formal framework for this. It proposes we think of each variable in our DAG not as a timeless concept, but as an aggregation of measurements taken at specific points in time.

[PDF] The Case for Time in Causal DAGs - Alexander Reisach

This paper formalizes the intuition that causes must precede effects and provides the tools to explicitly encode time into our graphical models.

First, read the introduction, paying attention to the dialogue and the section The case for time. This will establish why an explicit treatment of time is necessary for interpretation and for validating the acyclicity assumption. Next, read the sections "Causal Relationships are Relative" and "A Formal Notion of Time". Focus on the concept of a causal variable being an aggregate of temporal components, denoted as X_{T_x}. This is the key formal step that allows us to time-index our variables.

The core idea is to replace a variable like Satisfaction with a series of time-indexed variables: Satisfaction(t=1), Satisfaction(t=2), and so on. This simple change has profound implications for how we draw and interpret our DAGs.

2. Modeling the Crossover Design and Carryover Effects

Your crossover design involves at least two periods. A simplified DAG for such a design might look like the one below, which already incorporates a time index for the period.

A basic causal model for a two-period crossover design. $R$ represents the randomization of the treatment sequence. $Z_1$ and $Z_2$ are the treatments administered in periods 1 and 2, respectively. $Y_1$ and $Y_2$ are the corresponding outcomes. The arrow from $Y_1$ to $Y_2$ represents a potential carryover effect.

In this graph:

Variables are indexed by period (1 or 2).
$R$ being randomized means it has no parents, and it determines the treatment in each period ( $Z_{1}$ and $Z_{2}$ ).
There's an arrow from the outcome in period 1 ( $Y_{1}$ ) to the outcome in period 2 ( $Y_{2}$ ). This represents a carryover effect: the experience and outcome from the first period might directly influence the outcome in the second period, independent of the second period's treatment.

The assumption of "no carryover effects" is fundamental to the analysis of many simple crossover trials. In graphical terms, this assumption is equivalent to stating that there is no arrow from any variable in period 1 to any variable in period 2, except through the stable traits of the participants (which are common causes of both $Y_{1}$ and $Y_{2}$ ).

The paper "Experimental designs for identifying causal mechanisms" discusses this assumption in the context of the potential outcomes framework.

Experimental designs for identifying causal mechanisms

This paper discusses experimental designs for mediation analysis, including a crossover design where the "no carry-over effect" assumption is critical.

Read section "3.2 Crossover design" and the formal statement in "Assumption 7". This will connect the statistical assumption of "no carryover" from the experimental design literature to the graphical concept of a lagged effect that we are modeling.

By making time explicit in our DAG, we are forced to be explicit about our assumption regarding carryover. If we believe that, for instance, a particularly good or bad experience with a living arrangement in period 1 could color perceptions in period 2, we must include an arrow like $Y_{1} \to Y_{2}$ . Deleting that arrow is a strong causal claim.

3. Unrolling Cycles: Modeling Adaptation and Feedback

One of the most powerful applications of temporal DAGs is in modeling systems with feedback loops. In your triadic scenario, it's highly plausible that satisfaction and behavior are mutually reinforcing. For example, high satisfaction in the triad might lead to more positive interactions, which in turn boosts satisfaction. A static DAG would represent this as a cycle: Satisfaction $\leftrightarrow$ Behavior. Since DAGs must be acyclic, this is an invalid representation.

The solution is to "unroll" the cycle over time. What appears as a cycle in a static, aggregated view becomes a perfectly valid acyclic chain when time is considered.

These three DAGs model the relationships between socio-economic factors, child neglect, and psychological outcomes over three time points (W7, W8, W9). They are excellent examples of "unrolled" temporal models.

Look closely at any of the models in the image above. You can see several types of temporal relationships:

Autoregressive effects: An arrow from a variable to itself at the next time step (e.g., Depressive Symptoms W7 $\to$ Depressive Symptoms W8). This models inertia or stability; a person's mood last week is a good predictor of their mood this week.
Lagged effects: An arrow from one variable at time $t$ to a different variable at time $t + 1$ (e.g., School Adaptation W7 $\to$ Depressive Symptoms W8).
Unrolled Feedback: The combination of these paths creates what might look like a feedback loop in aggregate. For example, School Adaptation affects Depressive Symptoms, and Depressive Symptoms affects School Adaptation at the next time point.

This "unrolling" is not just a graphical trick; it's a more accurate causal statement about how the system evolves.

[PDF] The Case for Time in Causal DAGs - Alexander Reisach

This section explicitly describes how to resolve causal cycles by incorporating time.

Please read the section "Unrolling Causal Cycles". Example 5 is a concise demonstration of how a cyclic graph Y <-> X is unrolled into an acyclic structure Y(t1) -> X(t2) -> Y(t3). This is precisely the technique you will use to model adaptation in the triad.

4. Constructing the Temporal DAG for the Vee-Triad

Let's now apply these principles to your experiment. We need to construct a DAG that represents the causal relationships within each period of your crossover design and, crucially, the temporal links between periods.

Let's denote period by the subscript $p$ . For each period, you have variables like:

Treatment (Living Arrangement): $T_{p}$
Satisfaction for Pivot (A), Partner B, Partner C: $S_{A, p}, S_{B, p}, S_{C, p}$
Mediators (e.g., Autonomy, Security): $M_{a u t o n o m y, A, p}, M_{sec u r i t y, B, p}$ , etc.

Within-period structure (for a single period $p$ ):
This part is similar to the static DAGs we've considered. Arrows would go from the treatment $T_{p}$ to mediators for each person, and from mediators to satisfaction outcomes. You would also include the interference pathways we've discussed, such as partner B's sense of security affecting partner A's satisfaction ( $M_{sec u r i t y, B, p} \to S_{A, p}$ ).

Between-period structure (links from period $p$ to $p + 1$ ):
This is where we encode the temporal dynamics. Here are some plausible and important paths to consider:

Treatment Carryover: Does the experience of a living arrangement in period $p$ have a direct effect on satisfaction in period $p + 1$ , even after the new treatment $T_{p + 1}$ is applied? This would be an arrow: $T_{p} \to S_{A, p + 1}$ . This is a strong form of carryover.
Outcome-to-Outcome Lag (Autoregression): A person's satisfaction level is likely to be stable over time. This is represented by autoregressive paths: $S_{A, p} \to S_{A, p + 1}$ , $S_{B, p} \to S_{B, p + 1}$ , etc. This models psychological adaptation and mood inertia.
Cross-Lagged Effects (Adaptation): This is where the rich dynamics of the triad appear. The state of the system in one period can alter how it functions in the next.
- For example, if partner B felt a low sense of security in period $p$ , they might change their behavior in period $p + 1$ , which in turn affects the pivot's satisfaction in that period. This could be modeled as: $M_{sec u r i t y, B, p} \to S_{A, p + 1}$ .
- Another example: The pivot's satisfaction in period $p$ might directly impact their own sense of autonomy in the next period, representing a process of self-reflection and adjustment: $S_{A, p} \to M_{a u t o n o m y, A, p + 1}$ .

By combining the within-period and between-period structures, you create a large but acyclic graph that represents the full, dynamic causal story of your experiment. The key is to think of your experiment not as two separate studies, but as one continuous process where the past causally influences the future.

Conclusion

In this lesson, we have moved beyond static causal models to embrace the dimension of time. This is an essential step for aligning your causal model with the dynamic nature of your N-of-1 crossover experiment.

Key Takeaways:

Static DAGs are insufficient for longitudinal designs; we must make time explicit by time-indexing our variables (e.g., $Y_{1}, Y_{2}, ...$ ).
The principle "causes precede effects" allows us to "unroll" apparent feedback cycles into valid acyclic chains over time. This is how we model adaptation and mutual influence.
Temporal DAGs enable us to explicitly model crucial assumptions of crossover designs, such as carryover effects, by either including or excluding arrows that cross time periods.
The resulting temporal DAG can represent complex dynamics like autoregressive effects (a variable's influence on itself over time) and cross-lagged effects (one person's state at time $t$ influencing another's at time $t + 1$ ).

You now have the tools to draw a causal graph that fully represents the hypothesized dynamics of the vee triad over the course of your experiment. In our next lesson, we will take this temporal DAG and address the challenge of identification. We will discuss how to adapt the backdoor criterion and other identification strategies to this more complex setting to determine if we can indeed isolate the causal effects of interest.

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