Estimating Causal Effects with Interference and Carryover

Visualizing a World with Interference and Carryover

Before we formalize these concepts, let's build a visual intuition. In a simple experiment, we imagine an arrow from "Treatment" to "Outcome." In our complex system, the causal pathways are far more numerous.

This diagram reveals several types of effects we must consider:

• Direct Effect (Blue Arrow): The treatment in Period 1 affects Unit 1's outcome in Period 1. In our scenario, this is how a living arrangement affects the satisfaction of the person whose arrangement is directly altered.
• Treatment Spillover Effect (Light Blue Dashed Arrow): The treatment for Unit 1 in Period 1 affects Unit 2's outcome in the same period. This is the core of interference: one person's treatment spilling over to affect their partner.
• Carryover Effect (Green Dotted Arrow): The treatment for Unit 1 in Period 1 affects the outcomes of both Unit 1 and Unit 2 in a future period (Period 2).
• Unit Peer Effect & Autocorrelation (Pink/Red Dashed Arrows): These represent other dependencies in the system, such as individuals influencing each other directly, or an individual's state being correlated with their own past states, independent of the treatment.

These effects can be grouped to define our primary estimands.

Formalizing Interference: Exposure Mappings

Your background in experimental design has equipped you with the principles of SUTVA. The "no interference" clause of SUTVA posits that a unit's potential outcomes depend only on their own treatment assignment. When this is violated, a unit's potential outcome $Y_i$ becomes a function of the entire treatment vector $\mathbf{T}$ for all units in the population.

This creates a high-dimensional problem. The concept of an exposure mapping provides a structured way to simplify it. An exposure mapping is a function that summarizes the complex treatment vector $\mathbf{T}$ into a few relevant "exposure" conditions for each individual.

The following reading from the paper "Population Interference in Panel Experiments" provides the formal foundation for this concept, which is central to modern causal inference with interference.

As the reading makes clear, instead of considering every possible combination of treatments for all units across all time, we can define a specific exposure for each person. For person A in our triad at time $t$, a relevant exposure mapping might be:

$f_{A,t}(\mathbf{W})=(W_{A,t},W_{B,t},W_{C,t},W_{A,t-1})$

This vector captures:

• $W_{A,t}$: Person A's own current treatment (for the direct effect).
• $W_{B,t},W_{C,t}$: The partners' current treatments (for spillover effects).
• $W_{A,t-1}$: Person A's own treatment in the previous period (for a simple carryover effect).

Defining the Causal Estimands

With a defined set of exposures, we can now precisely state what we want to measure. The paper you just read introduces the concept of the Temporal Exposure Contrast (TEC), which is a generalized form of a causal effect.

The TEC, ${\tau }_t^{k,k^{\prime }}$, is the average difference in outcomes at time $t$ when units are subjected to exposure $k$ versus exposure $k^{\prime }$. This is exceptionally powerful. By choosing our exposures $k$ and $k^{\prime }$ carefully, we can isolate exactly the effect we care about:

• Direct Effect: Compare exposure $k=(1,0,0)$ to $k^{\prime }=(0,0,0)$. (What is the effect of treating A, holding B and C as control?)
• Spillover Effect: Compare exposure $k=(0,1,0)$ to $k^{\prime }=(0,0,0)$. (What is the effect of treating B on A's outcome, when A is in control?)
• Total Effect: Compare exposure $k=(1,1,1)$ to $k^{\prime }=(0,0,0)$. (What is the effect of treating everyone vs. no one?)

From Theory to Practice: An Estimation Model

While design-based Horvitz-Thompson estimators (as mentioned in the readings) are a valid way to estimate any given TEC, a regression framework offers a flexible, model-based approach to estimate multiple effects at once and incorporate continuous confounders. This approach is a natural extension of the model from our previous lesson.

Let's construct a model for Person A's satisfaction at time $t$, which we'll call $S_{A,t}$. We assume our treatment $T$ is binary (e.g., 1 = new living arrangement, 0 = baseline). Our predictors will now include terms for our defined exposures and confounders:

Interpreting the coefficients gives us our effects of interest:

• ${\beta }_{dir}$: The direct effect of the treatment on person A.
• ${\beta }_{spillB}$: The spillover effect from person B's treatment onto person A.
• ${\gamma }_{dir}$: The direct carryover effect of A's own past treatment on their present satisfaction.
• ${\gamma }_{spillB}$: The spillover carryover effect from B's past treatment onto A's present satisfaction.

R Implementation: Estimating Effects from Simulated Data

Let's put this model to the test with an R simulation. We will generate data for our three individuals (A, B, C) over 100 time periods. The data generating process will explicitly include direct, spillover, and carryover effects, plus some random noise. Then we'll fit our model and see if we can recover the true parameters.

# Load libraries
library(dplyr)
library(tidyr)




# --- 1. Simulation Setup ---
set.seed(123)
n_periods <- 100
individuals <- c("A", "B", "C")
n_individuals <- length(individuals)




# True causal parameters we want to recover
true_direct_effect <- 0.8
true_spillover_effect <- -0.3 # Treatment on others has a negative effect on you
true_carryover_effect <- 0.2  # Past treatment has a positive lingering effect




# Generate a randomized treatment schedule for the triad
# In each period, one person is randomly chosen for treatment (T=1)
treatment_schedule <- data.frame(
  period = 1:n_periods,
  treated_person = sample(individuals, n_periods, replace = TRUE)
) %>%
  mutate(T_A = as.integer(treated_person == "A"),
         T_B = as.integer(treated_person == "B"),
         T_C = as.integer(treated_person == "C")) %>%
  select(-treated_person)




# --- 2. Generate Outcome Data (Satisfaction) ---
# Create a long-format dataset
full_data <- treatment_schedule %>%
  crossing(individual = individuals) %>%
  arrange(individual, period)




# Calculate spillover and lag terms
full_data <- full_data %>%
  group_by(period) %>%
  mutate(T_spillover = (T_A[individual == "A"] * (individual != "A") +
                        T_B[individual == "B"] * (individual != "B") +
                        T_C[individual == "C"] * (individual != "C"))) %>%
  ungroup() %>%
  mutate(T_direct = case_when(
           individual == "A" ~ T_A,
           individual == "B" ~ T_B,
           individual == "C" ~ T_C)) %>%
  group_by(individual) %>%
  mutate(T_direct_lag1 = lag(T_direct, default = 0)) %>%
  ungroup()




# Generate satisfaction based on the true effects
full_data <- full_data %>%
  mutate(
    satisfaction = 5 +  # Baseline satisfaction
      true_direct_effect * T_direct +
      true_spillover_effect * T_spillover +
      true_carryover_effect * T_direct_lag1 +
      rnorm(n(), 0, 1) # Random noise
  )




# --- 3. Fit the Estimation Model ---
# We use a fixed-effects model to account for baseline differences between individuals
estimation_model <- lm(satisfaction ~ T_direct + T_spillover + T_direct_lag1 + factor(individual), data = full_data)




# --- 4. Examine the Results ---
summary(estimation_model)

Interpreting the R Output

When you run this code, the summary(estimation_model) will produce a coefficients table. It will look something like this:

Look at the Estimate column for our key predictors:

• T_direct: The estimate is 0.821, very close to our true direct effect of 0.8.
• T_spillover: The estimate is -0.335, very close to our true spillover effect of -0.3.
• T_direct_lag1: The estimate is 0.195, very close to our true carryover effect of 0.2.

The model successfully recovered the underlying causal parameters we embedded in the data. This demonstrates that a linear model, when correctly specified to include terms for interference and carryover, is a viable and powerful tool for estimating these complex effects.

A Glimpse at Heterogeneous Effects

Our model assumes that the direct, spillover, and carryover effects are constant for everyone. But what if the effect of a treatment depends on a person's characteristics? This is the domain of Heterogeneous Treatment Effect (HTE) estimation.

Advanced methods like Causal Forests, which are discussed in the video, can estimate a personalized treatment effect, $\tau (x)$, for an individual with characteristics $x$. While we are using a simpler linear model for now, it's important to recognize that this is a major frontier in causal inference and directly relevant to the personalized nature of your goal.

Conclusion

In this lesson, we moved beyond the simplistic world of no interference and embraced the complexity of our interconnected triadic system. We saw how interference and carryover effects violate SUTVA and require more sophisticated models.

Key Takeaways:

• In interconnected systems, interference (spillover) and carryover effects are crucial to model.
• Exposure mappings provide a formal framework for defining a unit's effective treatment based on the full treatment schedule of the system.
• Causal effects can be decomposed into direct, spillover, and total effects by defining appropriate contrasts between exposures (TECs).
• A linear model that includes terms for own treatment, partner treatments, and lagged treatments is a practical and powerful tool for estimating these different effects from experimental data.

Having estimated our primary causal effects, a critical question remains: how much should we trust these numbers? Our model relies on the assumption that we have measured all relevant confounders. In the next lesson, we will tackle this head-on by learning how to conduct a sensitivity analysis for unmeasured confounding, which will allow us to quantify the robustness of our conclusions.