# Intervention Analysis: RDD and Stepped-Wedge Designs

In many research settings, we are not interested in estimating where a change occurred, but rather in testing whether a change occurred at a known time or threshold. Common examples include:

1. Regression Discontinuity Designs (RDD): Testing for a jump or slope change at a specific policy threshold.
2. Intervention Studies: Testing for an effect after a specific event (e.g., a medical intervention or marketing campaign).
3. Stepped-Wedge Trials: Testing for effects across multiple clusters that receive an intervention at different, pre-determined time points.

`smoothbp` provides the `fixed()` helper to handle these cases efficiently within the spike-and-slab framework.

    library(smoothbp)
    library(ggplot2)
    library(dplyr)

## 1. Regression Discontinuity Design (RDD)

In an RDD, we assume that the treatment is assigned based on a “running variable” crossing a threshold. We want to know if there is a discontinuity at that threshold.

Traditionally, RDD models use simple piecewise linear regression. `smoothbp` allows for a smooth transition if the effect is not instantaneous, or a sharp transition if it is.

### Simulated RDD Data

Let’s simulate a scenario where a policy change at \(x = 0\) leads to a change in the slope of the outcome.

    set.seed(123)
    n <- 200
    x <- runif(n, -5, 5)
    # True effect: slope increases by 2 after x=0
    y <- 5 + 1 * x + 2 * (x - 0) * (x > 0) + rnorm(n, 0, 1)

    dat_rdd <- data.frame(x = x, y = y)

    ggplot(dat_rdd, aes(x, y)) +
      geom_point(alpha = 0.6) +
      geom_vline(xintercept = 0, linetype = "dashed", color = "red") +
      theme_minimal() +
      labs(title = "Simulated RDD Data", subtitle = "Red line marks the known threshold at x = 0")

### Testing the Discontinuity

We use `smoothbp_ss()` to estimate the probability that a slope change exists at \(x = 0\). By using `fixed(0)` for `omega`, we tell the model exactly where the potential break is.

    fit_rdd <- smoothbp_ss(
      formula = y ~ x,
      omega   = list(fixed(0)),
      rho     = list(fixed(100)), # Sharp transition
      data    = dat_rdd,
      iter    = 2000,
      warmup  = 1000
    )

    # Posterior Inclusion Probability (PIP)
    pip(fit_rdd)

The PIP close to 1.0 indicates very strong evidence for a structural break at the threshold.

## 2. Stepped-Wedge Design

In a stepped-wedge design, different clusters (e.g., hospitals, schools) cross over from control to intervention at different points in time. The timing for each cluster is known.

### Simulated Stepped-Wedge Data

We’ll simulate 5 clusters. Each cluster receives the intervention at a different month.

    n_clusters <- 5
    n_time     <- 24
    dat_sw <- expand.grid(
      time    = 1:n_time,
      cluster = paste0("Cluster_", 1:n_clusters)
    )

    # Pre-determined intervention times
    switch_times <- setNames(c(6, 10, 14, 18, 22), paste0("Cluster_", 1:n_clusters))
    dat_sw$interv_t <- switch_times[dat_sw$cluster]

    # Simulate data: intervention adds +1.5 to the slope
    dat_sw$y <- unlist(lapply(1:nrow(dat_sw), function(i) {
      t      <- dat_sw$time[i]
      switch <- dat_sw$interv_t[i]
      # Base slope = 0.5, Intervention effect = +1.5
      5 + 0.5 * t + 1.5 * (t - switch) * (t > switch) + rnorm(1, 0, 1)
    }))

    ggplot(dat_sw, aes(x = time, y = y, color = cluster)) +
      geom_line() +
      geom_point(aes(shape = time >= interv_t), size = 2) +
      theme_minimal() +
      labs(title = "Simulated Stepped-Wedge Trial", shape = "Intervention Active")

### Modeling the Shared Effect with Varying Breakpoints

We want to estimate a single “treatment effect” (\(\delta\)) that applies to all clusters, but occurs at different times (\(\omega\)) for each cluster.

Using `fixed(dat_sw$interv_t)` allows us to specify observation-specific breakpoints.

    fit_sw <- smoothbp_ss(
      formula = y ~ time,
      b0      = ~ cluster, # Cluster-specific intercepts
      omega   = list(fixed(dat_sw$interv_t)),
      rho     = list(fixed(100)),
      data    = dat_sw
    )

    # Probability of intervention effect
    pip(fit_sw)

The model correctly identifies the shared intervention effect across all clusters, even though they switched at different times.

## Summary

By using the `fixed()` helper, you can: - Simplify the model: If the location of a break is known, the sampler is faster and more stable. - Perform Hypothesis Testing: The PIP provides a direct Bayesian measure of evidence for an intervention effect. - Handle Complex Designs: Observation-specific fixed points enable the analysis of stepped-wedge and other group-sequential designs.