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Load ei and prepare the 2x2 margins
Data preparation — shapes the raw inputs into what the estimator expects.
Load the package and the matproii data, then specify the formula relating the row margin x to the column margin t.
# Install: install.packages("ei")
library(ei)
data(matproii)
form <- t ~ x
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Fit the ecological inference model
The core estimate — where the causal quantity itself is computed.
Estimate the 2x2 EI model, supplying the precinct totals so counts can be recovered.
dbuf <- ei(formula = form, total = "n", data = matproii)
Before using this step, make its assumptions and its output explicit. A useful peer note explains what you would check, what could go wrong, or what would change your interpretation.
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Summarize aggregate estimates
Uncertainty quantification — standard errors, intervals, and aggregation.
Print the district-level aggregate Betab and Betaw estimates with their standard deviations and bounds.
summary(dbuf)
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Read out precinct-level betas
Reporting — turn the numbers into a figure or table a reader can act on.
Use eiread() to extract the precinct-specific point estimates of betab and betaw for mapping or further analysis.
betab <- eiread(dbuf, "betab")
betaw <- eiread(dbuf, "betaw")
head(cbind(betab, betaw))
Before using this step, make its assumptions and its output explicit. A useful peer note explains what you would check, what could go wrong, or what would change your interpretation.
Questions to explore- Which assumption matters most here, and what evidence could challenge it?
- What would you check before passing this output to the next step?
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Inputs & data
Aggregate unit-level data: for each district, two observed margins (e.g. fraction black and turnout) and the population size.
Show data format & exampleHide example
| precinct | x_black | t_turnout | n |
|---|---|---|---|
| 1 | 0.20 | 0.55 | 800 |
| 2 | 0.65 | 0.48 | 1200 |
| 3 | 0.10 | 0.61 | 950 |
| 4 | 0.40 | 0.52 | 700 |
Result figure rendered by StatsOtter from the package's documented example — unofficial community showcase; all credit to the original authors.
The estimate
T_i=\beta_i^{b}\,X_i+\beta_i^{w}\,(1-X_i),\qquad 0\le\beta_i^{b},\beta_i^{w}\le 1
⚠️ Unofficial community showcase of ei (docs). Not affiliated with the authors — all credit to Gary King & coauthors; this summarizes public documentation.
What it does: The ei package implements Gary King's solution to the ecological inference problem—estimating unobserved individual-level relationships (e.g. how subgroups voted) using only aggregate data such as precinct or district marginals. How it works: For 2x2 tables it combines the deterministic accounting identity (which bounds each unit's unknowns given its margins) with a statistical model: a truncated bivariate normal distribution over the unit-level parameters, estimated by maximum likelihood and simulation. This yields district-level and aggregate point estimates with uncertainty, while respecting the logical bounds each precinct imposes. Assumptions: The model's distributional form (truncated bivariate normal), no spatial autocorrelation beyond what's modeled, and absence of certain aggregation bias; diagnostics and extended models address some violations. Output: estimated internal cell quantities (e.g. fraction of each group voting), with bounds and credible intervals, at both unit and aggregate levels.
What you get — Estimated internal cell quantities (e.g. group-specific turnout) with bounds, at unit and aggregate levels.
Example output
Maximum likelihood results in scale of estimation (and se's)
[,1] [,2]
Zb 0.314 0.184
Zw 0.733 0.114
Aggregate Bounds
betab betaw
lower 0.025 0.601
upper 0.444 0.762
Estimates of Aggregate Quantities of Interest
mean sd
Bb 0.612 0.028
Bw 0.358 0.014
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