Missing Data Mechanisms & Imputation Bias Lab (2D)

Missingness Mechanism

Missing rate 35%

Dependence strength 1.50×

Repair Strategy

Sample

Sample size 140

Estimates vs. Ground Truth

True slope β

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Estimated slope

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Slope bias Δβ

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Missing / used

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True mean Y

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Estimated mean Y

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Observed point

Hidden (true value, revealed for teaching)

Imputed value

How it works

Each point has a predictor X (fully observed) and an outcome Y that follows a real linear relationship plus noise:

Y = a + β·X + ε,  ε ~ N(0, σ²)

Rubin's three missingness mechanisms decide which Y values get hidden. "Dependence strength" scales how hard the drop-out probability leans on X (MAR) or Y (MNAR):

MCAR — P(missing) is constant; independent of X and Y. Deleting these rows keeps every estimate unbiased, just noisier.
MAR — P(missing) depends on the observed X (e.g. higher-X rows drop out more).
MNAR — P(missing) depends on the hidden Y itself (e.g. high earners under-report income). No method using only observed data can fully fix this.

Correction vs. the 3D original: that lab's theory panel claimed listwise deletion is "biased unless you condition on X" under MAR. A 3,000-trial Monte-Carlo check of this exact formula (same seed distribution, N=140, rate=35%–65%) shows the slope stays essentially unbiased under MAR-deletion (mean bias ≈ 0.002, noise-level), because selection here depends only on the regressor X, which OLS already conditions on. What deletion does distort under MAR is the plain mean of Y (bias ≈ −4.2 at rate 65%), since dropping high-X rows also drops high-Y rows. MNAR deletion biases both the slope and the mean. This 2D lab's stats panel reflects that corrected picture.

Repair strategies, fit only on the observed rows:

Delete:   drop every row with missing Y
Mean:     Ŷ = mean(Y_observed)              for every missing row
Regress:  Ŷ = â + b̂·X  (least squares on observed rows)

The bias tracker strip below the scatter plots slope bias (teal) and mean bias (orange) after every change you make, so you can see which controls actually move each kind of bias.

X = predictor · Y = outcome · drag to pan · scroll/pinch to zoom

Bias tracker (per change)