Run MNAR delta sensitivity¶

MAR is not testable (why), so doubt about it is addressed by delta adjustment: impute under MAR, shift the imputed values by a specified departure, and see how the pooled inference moves.

Pattern¶

import numpy as np
from longmi import DeltaAdjustment, pool_rubin

fit = imputer.fit(data)          # fit ONCE

scenarios = {
    "mar":        None,
    "means_x0.8": DeltaAdjustment(np.log(0.8), label="dropouts 20% lower"),
    "means_x1.25": DeltaAdjustment(np.log(1.25), label="dropouts 25% higher"),
}

results = {}
for name, delta in scenarios.items():
    completed = fit.impute(m=50, random_state=20260723, delta=delta)
    results[name] = pool_rubin(
        completed.analyze(analysis), validity=completed.declaration
    )

Three details that matter:

• Reuse the fit. Scenarios share one fitted imputation model; nothing is re-estimated, so runs are cheap and comparable.
• Reuse the seed. With equal random_state, scenario runs share the same underlying draws — differences between scenarios are the delta's effect, with Monte Carlo noise cancelled in the comparison.
• The scale is the link scale. For log-link count models, delta = log(r) multiplies the imputed means by r before the counts are drawn (the model-based pattern-mixture adjustment). Outcome-scale shifts are only available for continuous responses; for counts they are refused as a non-model-based transformation.

Choosing deltas¶

Anchor them substantively: "dropouts take 20% fewer medications than MAR predicts" is a defensible sentence; an arbitrary ±0.3 is not. Run a symmetric range (e.g. ×0.8, ×0.9, MAR, ×1.1, ×1.25) and report where — if anywhere — the conclusion changes sign or loses significance. The result's declaration records the scenario (Missingness assumption: MNAR(delta=...), MNAR sensitivity performed: Yes).

A worked example (epil seizure counts, deltas ×0.8 / ×1.25) lives in examples/epilepsy_counts.md.