The Day You Turn 65, You Become Healthier

Policy Creates Cliff Edges Everywhere

Medicare at 65 is just one example. Policy creates cliff edges everywhere:

• Age 21: Legally purchase alcohol
• Income below $X: Qualify for Medicaid
• GPA ≥ 3.0: Make the Dean’s List
• Birth weight < 1500g: Admission to NICU
• Vote share > 50%: Win the election

Pictures Reveal What Statistics Summarize

Each of the following slides presents a different RDD scenario.

• What do you observe in each case?
• What questions arise about the identification strategy?
• How do these patterns inform whether an RDD assumption is valid?

RDD in Pictures: The Alcohol Example

Question: What is age doing here? What’s the story being told?

RDD in Pictures: Binned Means

Question: What do you think each dot represents? What do you think each line represents? What’s the story being told?

RDD in Pictures: The Treatment Effect

Question: Where is the treatment effect on the graph? Treatment effect of what? Give a rough approximation of the treatment effect.

The Jump Tells the Causal Story

# Generate RDD example data
n <- 500
rdd_example <- tibble(
  x = runif(n, -1, 1),
  treated = ifelse(x >= 0, 1, 0),
  # True effect of 3
  y = 2 + 0.8 * x + 3 * treated + rnorm(n, 0, 0.5)
)

ggplot(rdd_example, aes(x = x, y = y, color = factor(treated))) +
  geom_point(alpha = 0.5, size = 2) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray, linewidth = 1) +
  geom_smooth(method = "lm", se = FALSE, linewidth = 1.5) +
  scale_color_manual(
    values = c("0" = accent_coral, "1" = secondary_teal),
    labels = c("Below cutoff (Control)", "Above cutoff (Treated)")
  ) +
  annotate("segment", x = 0, xend = 0, y = 2, yend = 5,
           arrow = arrow(ends = "both", length = unit(0.1, "inches")),
           color = primary_blue, linewidth = 1.5) +
  annotate("text", x = 0.12, y = 3.5, label = "Treatment\nEffect",
           color = primary_blue, fontface = "bold", size = 5, hjust = 0) +
  labs(
    title = "Regression Discontinuity in Action",
    subtitle = "The jump at the cutoff identifies the causal effect",
    x = "Running Variable (centered at cutoff)",
    y = "Outcome",
    color = NULL
  ) +
  theme_health_econ() +
  theme(legend.position = "bottom") +
  coord_cartesian(ylim = c(0, 7))

Figure 1: The core intuition: a jump at the cutoff reveals the treatment effect

RDD Was Invented, Forgotten, and Reborn

Every Design Needs Its Language

The Twins Test

If the answer is yes, you have a good RDD. The cutoff acts like a coin flip for people near it.

From Concept to Classification

Now that we understand why cliff edges create natural experiments, we need to distinguish between two kinds. Sometimes the edge is absolute — cross the line and treatment is guaranteed. Other times, the edge is softer — crossing makes treatment more likely but not certain.

This distinction shapes everything about how we estimate the effect.

Two Flavors of Discontinuity

# Sharp RDD
sharp_data <- tibble(
  x = seq(-1, 1, length.out = 200),
  treatment_prob = ifelse(x >= 0, 1, 0)
)

p_sharp <- ggplot(sharp_data, aes(x = x, y = treatment_prob)) +
  geom_line(linewidth = 2, color = secondary_teal) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray) +
  labs(
    title = "Sharp RDD",
    subtitle = "Treatment is deterministic at cutoff",
    x = "Running Variable",
    y = "P(Treatment)"
  ) +
  theme_health_econ() +
  scale_y_continuous(limits = c(0, 1), labels = scales::percent) +
  annotate("text", x = -0.5, y = 0.1, label = "No one treated",
           color = accent_coral, fontface = "bold", size = 4) +
  annotate("text", x = 0.5, y = 0.9, label = "Everyone treated",
           color = secondary_teal, fontface = "bold", size = 4)

# Fuzzy RDD
fuzzy_prob <- function(x) {
  plogis(3 * x) * 0.6 + ifelse(x >= 0, 0.3, 0)
}

fuzzy_data <- tibble(
  x = seq(-1, 1, length.out = 200),
  treatment_prob = fuzzy_prob(x)
)

fuzzy_jump_left <- fuzzy_prob(-1e-6)
fuzzy_jump_right <- fuzzy_prob(1e-6)

p_fuzzy <- ggplot(fuzzy_data, aes(x = x, y = treatment_prob)) +
  geom_line(linewidth = 2, color = secondary_teal) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray) +
  labs(
    title = "Fuzzy RDD",
    subtitle = "Treatment probability jumps but isn't deterministic",
    x = "Running Variable",
    y = "P(Treatment)"
  ) +
  theme_health_econ() +
  scale_y_continuous(limits = c(0, 1), labels = scales::percent) +
  annotate("segment", x = 0, xend = 0, y = fuzzy_jump_left, yend = fuzzy_jump_right,
           arrow = arrow(ends = "both", length = unit(0.1, "inches")),
           color = primary_blue, linewidth = 1.5) +
  annotate("text", x = 0.2, y = 0.5, label = "Jump",
           color = primary_blue, fontface = "bold", size = 4)

p_sharp + p_fuzzy

Figure 3: Sharp RDD: treatment jumps from 0 to 1. Fuzzy RDD: treatment probability jumps but not completely.

Sharp RDD Means the Rule Is Absolute

Definition: Treatment is a deterministic function of the running variable

\[ D_i = \begin{cases} 1 & \text{if } X_i \ge c_0 \\ 0 & \text{if } X_i < c_0 \end{cases} \]

Examples in Health Economics:

• Medicare eligibility at age 65
• Medicaid income thresholds
• Hospital quality ratings with discrete cutoffs

Fuzzy RDD Means the Rule Leaks

Definition: Treatment probability jumps at the cutoff, but not from 0 to 1

\[ \lim_{x \downarrow c_0} P(D_i = 1 | X_i = x) \neq \lim_{x \uparrow c_0} P(D_i = 1 | X_i = x) \]

Examples:

• SAT cutoff for college admission (some below get in, some above don’t)
• Union certification elections (not all certified workplaces unionize)
• Recommendation for surgery based on BMI threshold

From Description to Identification

We can now recognize an RDD when we see one, and we know whether it’s sharp or fuzzy. We can apply the Twins Test to check if the design is credible.

But why does comparing twins at the edge give us a causal effect? The answer lies in a single, elegant assumption.

Everything Except Treatment Must Be Smooth

What we assume:

The potential outcomes \(E[Y^0 | X]\) and \(E[Y^1 | X]\) are continuous (smooth) at the cutoff.

What this means:

• Without treatment, outcomes would change gradually across the threshold
• Any jump in observed outcomes must be due to treatment
• Confounders also vary smoothly (no sudden changes at the cutoff)

Graduate Students: Formal Smoothness Condition

\(\lim_{x \uparrow c_0} E[Y^j | X = x] = \lim_{x \downarrow c_0} E[Y^j | X = x]\) for \(j \in \{0, 1\}\). Both conditional expectation functions must be continuous at \(c_0\).

Show R Code

smooth_data <- tibble(
  x = seq(-1, 1, length.out = 200),
  y0 = 2 + 0.8 * x,
  y1 = 2 + 0.8 * x + 2.5
)

ggplot(smooth_data) +
  geom_line(aes(x = x, y = y0), color = accent_coral, linewidth = 1.5, linetype = "dashed") +
  geom_line(aes(x = x, y = y1), color = secondary_teal, linewidth = 1.5, linetype = "dashed") +
  geom_vline(xintercept = 0, linetype = "dotted", color = slate_gray) +
  annotate("text", x = -0.7, y = 1.5, label = "E[Y⁰|X]",
           color = accent_coral, size = 5) +
  annotate("text", x = 0.7, y = 5, label = "E[Y¹|X]",
           color = secondary_teal, size = 5) +
  labs(
    title = "Smoothness Assumption",
    subtitle = "Both potential outcomes are continuous at the cutoff",
    x = "Running Variable", y = "Outcome"
  ) +
  theme_health_econ()

Smoothness vs. Treatment: A Visual Comparison

set.seed(6320)
n_sv <- 300
sv_data <- tibble(
  x = runif(n_sv, -1, 1),
  y0 = 2 + 0.8 * x + rnorm(n_sv, 0, 0.4),
  y1 = 2 + 0.8 * x + 2.5 + rnorm(n_sv, 0, 0.4),
  treated = ifelse(x >= 0, 1, 0),
  y_obs = ifelse(x >= 0, y1, y0)
)

# Left panel: Potential outcomes (both visible, with scatter)
p_left <- ggplot(sv_data) +
  geom_point(aes(x = x, y = y0), color = accent_coral, alpha = 0.3, size = 1.5) +
  geom_point(aes(x = x, y = y1), color = secondary_teal, alpha = 0.3, size = 1.5) +
  geom_smooth(aes(x = x, y = y0), method = "lm", se = FALSE,
              color = accent_coral, linewidth = 1.5) +
  geom_smooth(aes(x = x, y = y1), method = "lm", se = FALSE,
              color = secondary_teal, linewidth = 1.5) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray, linewidth = 1) +
  annotate("text", x = -0.7, y = 1, label = "Y\u2070 (untreated)",
           color = accent_coral, fontface = "bold", size = 4.5) +
  annotate("text", x = 0.7, y = 5.5, label = "Y\u00b9 (treated)",
           color = secondary_teal, fontface = "bold", size = 4.5) +
  annotate("text", x = 0, y = 6.8, label = "Cutoff",
           color = slate_gray, size = 3.5) +
  labs(title = "Potential Outcomes",
       subtitle = "Both are smooth everywhere",
       x = "Running Variable", y = "Potential Outcomes") +
  theme_health_econ() +
  coord_cartesian(ylim = c(0, 7))

# Right panel: Realized outcomes (only observed arm, with LATE annotation)
p_right <- ggplot(sv_data, aes(x = x, y = y_obs, color = factor(treated))) +
  geom_point(alpha = 0.4, size = 1.5) +
  geom_smooth(method = "lm", se = FALSE, linewidth = 1.5) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray, linewidth = 1) +
  scale_color_manual(values = c("0" = accent_coral, "1" = secondary_teal),
                     guide = "none") +
  annotate("segment", x = 0, xend = 0, y = 2, yend = 4.5,
           arrow = arrow(ends = "both", length = unit(0.1, "inches")),
           color = primary_blue, linewidth = 2) +
  annotate("label", x = 0.18, y = 3.25, label = "LATE",
           color = "white", fill = primary_blue, fontface = "bold",
           size = 5, label.size = 0) +
  annotate("text", x = 0, y = 6.8, label = "Cutoff",
           color = slate_gray, size = 3.5) +
  labs(title = "Actual Outcomes",
       subtitle = "Treatment switches which outcome we observe",
       x = "Running Variable", y = "Observed Outcome") +
  theme_health_econ() +
  coord_cartesian(ylim = c(0, 7))

p_left + p_right

Figure 6: Left: Both potential outcomes are smooth. Right: Treatment assignment creates the observable jump.

Question: Why is the left panel different from the right? In the right panel, where did the second line go?

RDD Identifies a Local Treatment Effect

The treatment effect at the cutoff:

\[ \delta_{\text{SRD}} = \lim_{x \downarrow c_0} E[Y_i | X_i = x] - \lim_{x \uparrow c_0} E[Y_i | X_i = x] \]

Estimating the Treatment Effect with Regression

In practice, we estimate this with regression:

\[ Y_i = \alpha + \tau D_i + \beta_1 (X_i - c_0) + \beta_2 D_i \cdot (X_i - c_0) + \varepsilon_i \]

where \(D_i = \mathbf{1}(X_i \ge c_0)\) and \(\tau\) is the treatment effect.

From Theory to Practice

The smoothness assumption tells us why the jump is causal. Now we need to decide how to measure it. This turns out to be harder than it looks — and the answer involves a fundamental trade-off.

The Bandwidth Trade-off

Narrow bandwidth:

• Units closer to cutoff → more comparable
• Less bias from functional form
• But: fewer observations → more variance

Wide bandwidth:

• More observations → more precision
• But: units farther from cutoff may differ
• More sensitive to functional form assumptions

Show R Code

# Show different bandwidths
bw_data <- rdd_example %>%
  mutate(
    in_narrow = abs(x) <= 0.3,
    in_wide = abs(x) <= 0.7
  )

ggplot(bw_data, aes(x = x, y = y)) +
  # Wide bandwidth region
  annotate("rect", xmin = -0.7, xmax = 0.7, ymin = -Inf, ymax = Inf,
           fill = primary_blue, alpha = 0.1) +
  # Narrow bandwidth region
  annotate("rect", xmin = -0.3, xmax = 0.3, ymin = -Inf, ymax = Inf,
           fill = secondary_teal, alpha = 0.2) +
  geom_point(aes(color = factor(treated)), alpha = 0.5, size = 2) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray) +
  scale_color_manual(values = c("0" = accent_coral, "1" = secondary_teal), guide = "none") +
  annotate("text", x = 0, y = 6.5, label = "Narrow BW",
           color = secondary_teal, fontface = "bold", size = 4) +
  annotate("text", x = 0.5, y = 6.5, label = "Wide BW",
           color = primary_blue, fontface = "bold", size = 4) +
  labs(
    title = "Bandwidth Selection",
    subtitle = "The eternal bias-variance trade-off",
    x = "Running Variable", y = "Outcome"
  ) +
  theme_health_econ() +
  coord_cartesian(ylim = c(0, 7))

Modern Bandwidth Selection Methods

Flexible Regression Specifications

The saturated model allows different slopes on each side of the cutoff:

\[ Y_i = \alpha + \tau D_i + \beta_1 (X_i - c_0) + \beta_2 D_i \cdot (X_i - c_0) + \varepsilon_i \]

For more flexibility, add polynomial terms (where \(\tilde{X}_i = X_i - c_0\)):

\[ \begin{aligned} Y_i = \alpha + \tau D_i &+ \beta_1 \tilde{X}_i + \beta_2 \tilde{X}_i^2 \\ &+ \gamma_1 D_i \cdot \tilde{X}_i + \gamma_2 D_i \cdot \tilde{X}_i^2 + \varepsilon_i \end{aligned} \]

From Estimation to Skepticism

We now have the tools to estimate the jump. But a good researcher asks: What could go wrong? The most dangerous threat is the one that undermines the entire design.

Manipulation Destroys the Design

If people can manipulate the running variable…

• Units sort themselves above/below the cutoff
• Those just above differ systematically from those just below
• The “as-if-random” assumption fails

Examples of manipulation:

• Students retaking exams to cross a scholarship threshold
• Hospitals recoding diagnoses to avoid quality penalties
• Birth weight recorded at exactly 1500g to qualify for NICU

Show R Code

# Simulated manipulation - bunching above cutoff
set.seed(123)
manip_data <- tibble(
  x = c(rnorm(300, -0.3, 0.25), rnorm(200, 0.1, 0.15))
)

ggplot(manip_data, aes(x = x)) +
  geom_histogram(binwidth = 0.05, fill = primary_blue, color = "white", alpha = 0.8) +
  geom_vline(xintercept = 0, linetype = "dashed", color = accent_coral, linewidth = 1.5) +
  annotate("text", x = 0.25, y = 50, label = "Suspicious\nbunching!",
           color = accent_coral, fontface = "bold", size = 5) +
  labs(
    title = "Evidence of Manipulation",
    subtitle = "Excess mass just above the cutoff",
    x = "Running Variable", y = "Count"
  ) +
  theme_health_econ()

Detecting Manipulation: The McCrary Test

Idea: If there’s no manipulation, the density of the running variable should be continuous at the cutoff.

# Valid density
valid_density <- tibble(
  x = seq(-2, 2, length.out = 200),
  density = dnorm(x, 0, 0.8)
)

p_valid <- ggplot(valid_density, aes(x = x, y = density)) +
  geom_area(fill = secondary_teal, alpha = 0.3) +
  geom_line(color = secondary_teal, linewidth = 1.5) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray) +
  labs(title = "No Manipulation", subtitle = "Density is smooth at cutoff",
       x = "Running Variable", y = "Density") +
  theme_health_econ() +
  annotate("text", x = 0, y = 0.55, label = "Continuous",
           color = secondary_teal, fontface = "bold", size = 5)

# Manipulated density
manip_density <- tibble(
  x = seq(-2, 2, length.out = 200),
  density = ifelse(x < 0, dnorm(x, 0, 0.8) * 0.7, dnorm(x, 0, 0.8) * 1.3)
)

p_manip <- ggplot(manip_density, aes(x = x, y = density)) +
  geom_area(fill = accent_coral, alpha = 0.3) +
  geom_line(color = accent_coral, linewidth = 1.5) +
  geom_vline(xintercept = 0, linetype = "dashed", color = slate_gray) +
  labs(title = "Manipulation Detected", subtitle = "Jump in density at cutoff",
       x = "Running Variable", y = "Density") +
  theme_health_econ() +
  annotate("segment", x = 0, xend = 0, y = 0.35, yend = 0.45,
           arrow = arrow(ends = "both", length = unit(0.08, "inches")),
           color = primary_blue, linewidth = 1.5) +
  annotate("text", x = 0.3, y = 0.4, label = "Gap!",
           color = accent_coral, fontface = "bold", size = 5)

p_valid + p_manip

Figure 11: Left: Valid RDD with continuous density. Right: Manipulation creates a discontinuity in the density.

Density Test: Visual Evidence from McCrary (2008)

A picture with and without a discontinuity in the density from McCrary, 2008.

NICU Saves Lives — But Can RDD Prove It?

Heaping: Visual Evidence

Illustration of heaping on the running variable (reproduced from Almond, 2010)

Why So Many Babies Weigh Exactly 1500 Grams?

Think about it: birth weight is a continuous biological measurement. You would not expect a spike at any particular number.

When Rounding Breaks the Design

• Standard density tests (McCrary) can miss heaping because they lack statistical power right at the cutoff
• The heaped babies at exactly 1500g tend to be sicker than babies on either side — they are outliers that distort the estimated treatment effect
• The bottom line: Always plot the raw data. Statistical tests are necessary but not sufficient. Your eyes can catch patterns that formal tests miss.

Robustness Checks for RDD

RDD Effects Are Local — Generalize with Caution

From Methods to Medicine

We’ve built the toolkit — now let’s use it. The next application puts every concept we’ve learned to work on a real health policy question.

Sojourner et al. (2015): Discussion Questions

Skim the paper and think about the following:

1. What is the main objective or research question?
2. What is the running variable and why is it a good choice?
3. What methods are they using?
4. What are the primary threats to validity and how do they address them?
5. What are the main results and policy implications?

The Research Question

What is the causal effect of unionization on nursing home performance?

The challenge:

• Unionized workplaces may differ systematically
• Workers who vote to unionize may be different
• Simple comparisons are confounded

The RDD solution:

• Compare nursing homes where the union barely won vs. barely lost
• At the 50% vote threshold, outcome is essentially random

Show R Code

# Conceptual union RDD
union_concept <- tibble(
  vote_share = seq(0.3, 0.7, length.out = 100),
  certified = ifelse(vote_share >= 0.5, 1, 0)
)

ggplot(union_concept, aes(x = vote_share, y = certified)) +
  geom_step(color = secondary_teal, linewidth = 2) +
  geom_vline(xintercept = 0.5, linetype = "dashed", color = accent_coral, linewidth = 1.5) +
  scale_x_continuous(labels = scales::percent) +
  scale_y_continuous(breaks = c(0, 1), labels = c("No", "Yes")) +
  labs(
    title = "Sharp RDD at 50%",
    subtitle = "Union certification determined by majority vote",
    x = "Vote Share for Union",
    y = "Union Certified"
  ) +
  theme_health_econ() +
  annotate("text", x = 0.35, y = 0.5, label = "Barely lost",
           color = accent_coral, fontface = "bold", size = 4) +
  annotate("text", x = 0.65, y = 0.5, label = "Barely won",
           color = secondary_teal, fontface = "bold", size = 4)

The First Stage: Union Certification

The probability of union certification jumps discretely at the 50% vote threshold

Results: Staffing Levels

Results: Quality of Care

Summary: RDD in Health Economics

RDD Checklist for Applied Work

Best Practices for RDD

• Always plot the data
• Run McCrary density tests
• Check covariate balance at the cutoff
• Test sensitivity to bandwidth and functional form
• Be humble about external validity

The Cliff Edge Principle

RDD works because:

• Policy creates discontinuities that nature would not
• People near the edge are comparable — they just happened to land on different sides
• The jump at the edge is causal — as long as nothing else jumps there too

This is the most visually transparent causal method we have. If you can see the jump, you can believe the effect.

Looking Ahead