The Data
Slope Estimates
What's Happening
◆ THE BLUEPRINT
Simpson's Paradox

A trend that appears in aggregated data can reverse when the data is split into groups. This is not rare. It happens whenever a confounding variable creates different subpopulations.

Why It Matters

If your data has group structure and you ignore it, your model can give you the wrong sign on a coefficient. The overall slope and the within-group slopes can point in opposite directions.

Overall: \(\hat{\beta}_1 < 0\), but within every group: \(\hat{\beta}_{1j} > 0\)
The Fix

Account for the group structure in your model. The next tabs explore three ways to do this: ignore groups entirely (complete pooling), treat each group independently (no pooling), or model groups as drawn from a distribution (partial pooling).

Explore More
Link Functions GLM foundations this app builds on
Model Comparison
Complete Pooling
No Pooling (Fixed Effects)
◆ THE BLUEPRINT
Complete Pooling

Fit one model to all data, ignoring groups entirely.

\(y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\)

One intercept, one slope. If groups differ, this misses that variation entirely. The model acts as if all observations come from the same population.

No Pooling (Fixed Effects for Group)

Fit a separate model per group, or include group as a factor with dummy variables.

\(y_{ij} = \beta_{0j} + \beta_1 x_{ij} + \varepsilon_{ij}\)

Each group gets its own intercept. But small groups have very few data points. Their estimates are noisy and unreliable.

The Tradeoff

Complete pooling has too much bias (ignores group differences). No pooling has too much variance (small-group estimates are wild). This is the bias-variance tradeoff applied to group estimation.

Bias-Variance Tradeoff the same tension, seen differently
Fit Lines
Shrinkage Map
◆ THE BLUEPRINT
The Mixed Model

A mixed model treats group intercepts as random draws from a shared distribution rather than as fixed parameters.

\(y_{ij} = (\beta_0 + b_{0j}) + \beta_1 x_{ij} + \varepsilon_{ij}\)
\(b_{0j} \sim N(0, \sigma^2_{b_0})\), \quad \varepsilon_{ij} \sim N(0, \sigma^2_e)\)
Shrinkage

The group-level estimate is a weighted average of the group's own data and the overall mean. The weight depends on group size and variance components.

\(\hat{b}_{0j} \approx \frac{n_j / \sigma^2_e} {n_j / \sigma^2_e + 1 / \sigma^2_{b_0}} \cdot (\bar{y}_j - \hat{\beta}_0)\)

When \(n_j\) is small, the group estimate shrinks heavily toward the grand mean. When \(n_j\) is large, the group keeps its own estimate. This is partial pooling.

Why It Works

Shrinkage reduces variance without adding much bias. The net effect: lower MSE, especially for small groups. The model borrows strength from the other groups.

Bias-Variance Tradeoff shrinkage as variance reduction
Spaghetti Plot
Random Effects Scatter
Variance Components
◆ THE BLUEPRINT
Random Intercepts + Random Slopes
\(y_{ij} = (\beta_0 + b_{0j}) + (\beta_1 + b_{1j}) x_{ij} + \varepsilon_{ij}\)
\(\begin{pmatrix} b_{0j} \\ b_{1j} \end{pmatrix} \sim N\left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \sigma^2_{b_0} & \rho \sigma_{b_0} \sigma_{b_1} \\ \rho \sigma_{b_0} \sigma_{b_1} & \sigma^2_{b_1} \end{pmatrix} \right)\)
Why Correlated Random Effects?

Groups that start high might change more slowly (negative correlation) or more quickly (positive correlation). The covariance structure captures this pattern.

Model Selection in R
Random intercepts: lmer(y ~ x + (1 | group))
Random slopes: lmer(y ~ x + (0 + x | group))
Both: lmer(y ~ x + (1 + x | group))
When Do You Need Random Slopes?

If the relationship between x and y truly varies across groups, a random-intercepts-only model underfits. Check whether the no-pooling slopes look different across groups. If they do, add random slopes.

The Sandbox
Coefficient Summary
Shrinkage Map
◆ THE BLUEPRINT
GLMMs: Mixed Effects + Link Functions

A Generalized Linear Mixed Model (GLMM) combines the GLM framework (link functions, exponential families) with random effects.

\(g(\mu_{ij}) = (\beta_0 + b_{0j}) + (\beta_1 + b_{1j}) x_{ij}\)
Gaussian
Identity link: \(\mu_{ij} = \eta_{ij}\)
Fitted with lmer()
Binomial
Logit link: \(\log\frac{p_{ij}}{1-p_{ij}} = \eta_{ij}\)
Fitted with glmer(..., family=binomial)
Poisson
Log link: \(\log(\mu_{ij}) = \eta_{ij}\)
Fitted with glmer(..., family=poisson)
Explore More
Link Functions deep dive on link functions and GLMs