Missing Data Mechanisms & Imputation Bias Lab

Missingness Mechanism

Missing rate 35%

Repair Strategy

Estimates vs. Ground Truth

True slope β

—

Estimated slope

—

Slope bias Δβ

—

Missing / used

—

True mean Y

—

Estimated mean Y

—

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:

MCAR — P(missing) is constant; independent of X and Y. Deleting these rows keeps the estimate unbiased, just noisier.
MAR — P(missing) depends on the observed X (e.g. higher-X rows drop out more). Deletion is biased unless you condition on X; regression imputation, which uses X, can largely recover β.
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 — it's the hardest case in real pipelines.

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 + β̂·X  (least squares on observed rows)

The slope β̂ and mean shown on the left are computed from whichever rows survive your chosen strategy, then compared against the true β and true mean of the full, uncorrupted dataset — the gap is the bias a real ML pipeline would silently inherit.

X = predictor · Y = outcome (height) · Z = spread for readability

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.