Carryover Effects: Modeling & Adjustment

1. The Nature of Carryover Effects in a Behavioral Experiment

In a crossover design, a carryover effect occurs when the influence of a treatment administered in one period persists and affects the outcomes measured in a subsequent period. While a washout period is designed to mitigate this, its effectiveness is not guaranteed, especially for non-pharmacological interventions.

For our experiment with the vee triad, we are primarily concerned with what is termed behavioral carryover, as opposed to the biological carryover seen in drug trials. Let's delve into this distinction.

As the paper highlights, a behavioral carryover is the impact a treatment has on subsequent outcomes by altering a participant's behavior, habits, or psychological state. In our context, this could manifest in several ways:

• Contrast Effect: After a period of Triadic Cohabitation, the subsequent Dispersed Living arrangement might feel disproportionately lonely or, conversely, liberating. The perception of the second arrangement is not absolute but is contrasted with the immediately preceding one.
• Adaptation/Habituation: A period of high logistical complexity, like the Anchor-Satellite model, might make the participants more adept at communication and scheduling. This learned skill could then carry over into the next period, artificially boosting satisfaction in a way that isn't attributable to the new arrangement itself.
• Expectation Hangover: Positive or negative feelings from one arrangement could create an optimistic or pessimistic bias when entering the next, a phenomenon related to the demand characteristics we discussed previously but specifically tied to the treatment sequence.

Ignoring these effects is not an option. Doing so violates the core assumption that the outcome in a given period depends only on the treatment in that period, potentially leading to biased estimates of our causal effects.

2. The Statistical Impact of Unmodeled Carryover

Your background in experimental design has already made you familiar with the ways violations of assumptions can corrupt inferences. Carryover effects are a classic example. When present but unmodeled, they can systematically distort our findings.

The paper on behavioral carryover provides a concise analysis of this problem. In essence, if we denote the carryover effect from the control condition as ${\lambda }_0$ and from the treatment condition as ${\lambda }_1$, the bias in the standard treatment effect estimator is proportional to the sum ${\lambda }_0+{\lambda }_1$.

The key takeaway is that a non-zero carryover effect introduces a systematic error. Depending on its direction, we might falsely conclude an arrangement is effective when it isn't (inflated Type I error), or we might fail to detect a truly effective arrangement because the carryover effect is masking its impact (reduced power). The solution is not to simply hope for the best, but to explicitly incorporate carryover into our analytical model.

3. Analytical Strategy: Mixed-Effects Models

Our N-of-1 design involves repeated measurements on a single system (the triad). This structure means the observations are inherently correlated; the satisfaction reported in week 5 is not independent of the satisfaction reported in week 4. This violation of the independence assumption, which is fundamental to basic regression models, requires a more sophisticated tool. The ideal framework for this is the mixed-effects model.

Before diving into the equations, let's build a strong conceptual understanding of what these models do.

The video makes it clear that mixed models allow us to estimate an average, or "fixed," effect for our predictors (like the living arrangement) while simultaneously modeling the variability that comes from the data's clustered nature.

Now, let's see how to apply this to our specific problem of carryover. A mixed model can be specified to include terms that explicitly represent the treatment from the previous period.

Based on this paper, our analytical plan will be to fit a mixed-effects model to the time-series of satisfaction data from the triad. For a single N-of-1 trial, the model simplifies but the principle remains. A general form for the satisfaction $Y$ in period $j$ would be:

Where:

• ${\beta }_0$ is the intercept.
• ${\text{Treatment}}_j$ is a categorical variable for the living arrangement in period $j$. ${\beta }_1$ is the fixed effect we want to estimate.
• ${\text{Period}}_j$ is a continuous or factor variable for the experimental period, to control for any underlying time trends (e.g., the triad getting more comfortable with the study over time). ${\beta }_2$ is the period effect.
• ${\text{Carryover}}_{j-1}$ is a categorical variable indicating which treatment was in place during period $j-1$. The coefficient ${\beta }_3$ directly estimates the carryover effect.
• ${ϵ}_j$ is the error term. The mixed model allows us to specify a structure for these errors to account for their correlation over time (e.g., an autoregressive structure).

4. A Complementary View: Cross-Lagged Models

Another powerful perspective, originating from econometrics and time-series analysis, is the cross-lagged panel model. This framework is explicitly designed to investigate reciprocal effects over time and provides a strong theoretical complement to the mixed-model approach.

The convergence of these two analytical traditions is powerful. Both the mixed-effects model approach and the Random Intercept Cross-Lagged Panel Model (RI-CLPM) point to the same solution: we must model the stable, underlying characteristics of our unit of analysis (the triad) to properly isolate the dynamic, within-subject changes caused by our interventions and their carryover.

5. Final Analytical Strategy

Synthesizing these concepts, we can now finalize our analytical strategy for the pre-registration document. We will employ a linear mixed-effects model to analyze the satisfaction outcomes.

The model will be specified as follows:

• Outcome: The triadic satisfaction score (which we will design in the next module).
• Fixed Effects:
  1. Treatment: A factor with three levels for our living arrangements. This will give us the main effect estimates.
  2. Period: A factor to account for systematic changes across the six experimental periods.
  3. Carryover: A factor representing the treatment from the immediately preceding period. This term will absorb the variance attributable to carryover, providing a less biased estimate of the Treatment effect.
• Random Effects:
  1. Random Intercept: Although we have only one "subject" (the triad), the repeated nature of the measurements across many days or weeks creates a nested data structure. Modeling a random intercept for the triad (or potentially for each experimental block) properly accounts for the non-independence of these observations.

This approach is robust, theoretically sound, and directly addresses the challenge of carryover, allowing us to parse the complex temporal dynamics of our experiment.

Conclusion

In this final lesson of the module, we have built the last crucial piece of our experimental design: a plan to manage carryover effects. We recognized that while procedural controls like washout periods are helpful, they are not foolproof for behavioral interventions. An analytical strategy is therefore essential.

Our key takeaways are:

• Behavioral carryover is a credible threat in our experiment, where psychological states and learned habits from one period can influence the next.
• If ignored, carryover effects can bias treatment estimates and either inflate Type I error or reduce statistical power.
• Mixed-effects models are the ideal tool for analyzing our data, as they are designed for the correlated, repeated-measures structure of an N-of-1 trial.
• Our specific analytical plan involves fitting a mixed model with fixed effects for treatment, period, and a carryover term (the previous period's treatment), along with a random intercept to account for the nested data structure.

With this, we conclude our deep dive into the design of the N-of-1 crossover trial. We have a complete blueprint covering randomization, blocking, blinding, control of demand characteristics, and now, the management of carryover.

In the next module, we pivot from design to measurement. Our first task will be to construct a valid and reliable multi-dimensional satisfaction instrument—the very outcome variable this entire sophisticated experiment is designed to measure.