13.2 Example: Admission decisions and gender

The UC Berkeley admission data illustrate how varying slopes can reveal or remove apparent associations — a classic case of Simpson's paradox.

13.2.1 Simpson's paradox

In the aggregate data, women appear less likely to be admitted. Within departments, the pattern often reverses or disappears. A varying slopes model by department reveals how the gender-admission association varies:

$\text{logit}(p_{ij}) = \alpha_{\text{dept}[j]} + \beta_{\text{dept}[j]} \cdot \text{male}_i$

# Simulate Simpson's paradox: admission data
set.seed(42)
n_depts <- 6

# Departments vary in selectivity and gender bias

# Some departments favor males, some favor females, some are neutral

dept_intercept <- rnorm(n_depts, 0, 1.5)
dept_slope <- rnorm(n_depts, 0, 0.5) # gender effect varies by dept

# More women apply to selective (low intercept) departments

# This creates the paradox: aggregate shows women disadvantaged

n_per <- 100
d <- do.call(rbind, lapply(1:n_depts, function(j) {
male <- sample(c(0, 1), n_per, replace = TRUE, prob = c(0.7, 0.3)) # more women apply
logit_p <- dept_intercept[j] + dept_slope[j] * male
p <- 1 / (1 + exp(-logit_p))
admit <- rbinom(n_per, 1, p)
data.frame(dept = factor(j), male = male, admit = admit)
}))

# Aggregate model (ignores department)

m_agg <- glm(admit ~ male, family = binomial, data = d)
cat("Aggregate model (Simpson's paradox):\n")
cat(" male coefficient:", round(coef(m_agg)[2], 3), "\n")

# Varying slopes by department

library(lme4)
m_vary <- glmer(admit ~ male + (1 + male | dept), family = binomial, data = d)

cat("\nVarying slopes model:\n")
cat(" Fixed male effect:", round(fixef(m_vary)[2], 3), "\n")
cat(" Dept intercept SD:", round(attr(VarCorr(m_vary)$dept, "stddev")[1], 3), "\n")
cat("  Dept slope SD:    ", round(attr(VarCorr(m_vary)$dept, "stddev")[2], 3), "\n")
cat(" Correlation: ", round(attr(VarCorr(m_vary)$dept, "correlation")[1, 2], 3), "\n")

# Department-specific effects

cat("\n Department-specific male effects:\n")
dept_slopes <- fixef(m_vary)[2] + ranef(m_vary)$dept[, 2]
for (j in 1:n_depts) {
cat(sprintf(" Dept %d: %.3f\n", j, dept_slopes[j]))
}

cat("\n The aggregate effect may differ from department effects!\n")

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13.2.2 Interpreting the varying slopes

# Visualize department-specific effects
set.seed(42)
n_depts <- 6
dept_intercept <- rnorm(n_depts, 0, 1.5)
dept_slope <- rnorm(n_depts, 0, 0.5)
n_per <- 100

d <- do.call(rbind, lapply(1:n_depts, function(j) {
male <- sample(c(0, 1), n_per, replace = TRUE, prob = c(0.7, 0.3))
logit_p <- dept_intercept[j] + dept_slope[j] * male
p <- 1 / (1 + exp(-logit_p))
admit <- rbinom(n_per, 1, p)
data.frame(dept = factor(j), male = male, admit = admit)
}))

library(lme4)
m <- glmer(admit ~ male + (1 + male | dept), family = binomial, data = d)

# Extract department effects

dept_int <- fixef(m)[1] + ranef(m)$dept[, 1]
dept_slp <- fixef(m)[2] + ranef(m)$dept[, 2]

# Plot: department-specific admission probabilities

plot(0, 0, type = "n", xlim = c(-0.5, 1.5), ylim = c(0, 1),
xlab = "Male (0 = female, 1 = male)", ylab = "P(admit)",
main = "Department-specific admission curves")

for (j in 1:n_depts) {
x_seq <- c(0, 1)
logit_p <- dept_int[j] + dept_slp[j] * x_seq
p <- 1 / (1 + exp(-logit_p))
lines(x_seq, p, col = j, lwd = 2)
}
legend("topright", legend = paste("Dept", 1:n_depts), col = 1:n_depts, lwd = 2,
cex = 0.8)

cat("Each department has its own admission curve.\n")
cat("Some favor males, some favor females, some are neutral.\n")
cat("The aggregate effect hides this variation.\n")

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Rethinking

Simpson's paradox is a pooling problem. Simpson's paradox is not really a paradox — it is a consequence of ignoring grouping structure. When groups differ in both the outcome and the predictor, the aggregate relationship can reverse the within-group relationship. The varying slopes model solves this by estimating department-specific effects. But it also tells us something deeper: the variation across departments is itself informative. If the gender effect varies a lot across departments, the "average" gender effect is not very meaningful. We need to understand why departments differ, not just what the average is.

Summary of Section 13.2

Simpson's paradox: aggregate effect can reverse within-group effects.
Varying slopes by department reveal how the gender-admission association varies.
The covariance between intercepts and slopes captures how selectivity and bias relate.
The variation across departments is as important as the average effect.
Varying slopes solve the paradox by not pooling when they shouldn't.

Varying slopes by construction

Example: Cross-classified chimpanzees with varying slopes