Selection-Model Sensitivity Analysis¶
Method — instantiates Missingness-Aware Estimator Selection
Models the response process jointly with the outcome to examine how non-ignorable missingness would affect estimates.
Selection-Model Sensitivity Analysis writes an explicit equation for the response process — the probability that a value is observed as a function of the outcome itself, including the part of the outcome that went missing — and estimates it jointly with the outcome model. The link between the two equations is a dependence parameter: when it is zero, missingness is ignorable; as it moves away from zero, the model encodes that people with worse (or better) unobserved outcomes were systematically less likely to be observed. Its defining move is to name the mechanism as a model — the non-ignorable missingness lives in a fitted selection equation — and then vary its strength to see how the outcome estimate responds. This is the opposite factorization from stratifying by pattern: here the outcome drives observation through a written-down response equation, not a per-stratum shift.
Example¶
A labor economist wants the mean offered wage in a regional workforce, but wages are observed only for people who are employed, and the whole concern is that those who took jobs are not a random slice — the workers who stayed out may be precisely those facing lower offers. A selection model writes two coupled equations: one for the wage, and one for the probability of being employed (observed) that is allowed to depend on the same latent factors as the wage. A single correlation parameter, rho, ties the unobservables of the two equations — rho of zero means employment is unrelated to the unseen wage (ignorable), while a strongly negative rho means low-wage individuals were the ones who dropped out of observation. The analyst estimates the wage mean across a spectrum of rho values and reports how the estimate slides: near-ignorable assumptions give one figure, and progressively stronger selection pulls the corrected mean down. The deliverable is the trajectory of the estimate against the assumed selection strength, so a reader can see how much the headline depends on believing employment was benign.
How it works¶
- Write the response equation. Model the probability of observation as a function of the outcome (and covariates), so missingness can depend on the missing value itself.
- Couple it to the outcome model. Join the two equations through a dependence parameter that governs how strongly the outcome drives observation.
- Sweep the dependence. Fix or vary that parameter across a range, since it is at best weakly identified, and re-estimate the outcome at each setting.
- Read the trajectory. Report how the estimate moves from the ignorable case to strong non-ignorability.
Tuning parameters¶
- Dependence-parameter range — the span of selection strength (e.g. rho) explored; wider is more honest about non-ignorability, narrower more decisive.
- Fixed versus estimated dependence — whether to sweep the parameter as a sensitivity dial or attempt to estimate it, trading transparency against fragile identification.
- Exclusion restriction — whether a variable affecting observation but not the outcome is used to aid identification; a strong one stabilizes estimates but is a heavy, often disputable, assumption.
- Functional form of the selection equation — the assumed shape of the response model, which the results can be quite sensitive to.
When it helps, and when it misleads¶
Its strength is that it takes the mechanism seriously as an object of analysis: when the plausible story is "the outcome itself drove who we saw," a selection model states that story mathematically and quantifies its consequences, rather than assuming it away.[n1]
Its failure mode is fragile identification: the dependence parameter is barely informed by the data, so the answer is really driven by the assumed functional form and any exclusion restriction — get those wrong and the "correction" is an artifact. The classic misuse is to report a single fitted selection model as though it had recovered the truth, when its point estimate rests on untestable shape assumptions. The guarding discipline is to treat the dependence parameter as a sensitivity dial to be swept and reported as a range, and to be explicit that any exclusion restriction is an assumption, not a finding.
How it implements the components¶
missingness_mechanism_assumption_frame— it makes the mechanism assumption a fitted equation: the response process is modeled explicitly as a function of the outcome, so the non-ignorability claim is written down rather than assumed away.sensitivity_and_tipping_point_plan— sweeping the dependence parameter across its range is the sensitivity plan, tracing how the estimate responds as assumed selection strengthens.
It does not stratify the sample by observed pattern or issue a pooled estimand-preservation statement across strata — missingness_pattern_inventory and estimand_preservation_statement are Pattern-Mixture Sensitivity Model's; the selection model instead writes a single response equation for how the outcome drives observation, and varies its strength.
Related¶
- Instantiates: Missingness-Aware Estimator Selection — the joint response-and-outcome route to MNAR sensitivity.
- Consumes: Process-Based Missingness Audit supplies the process story that justifies modeling the response as outcome-driven.
- Sibling mechanisms: Doubly Robust Missingness Adjustment · Full-Information Maximum Likelihood Path · Inverse-Probability Weighting Model · MCAR Diagnostic Test and Balance Review · Missingness Indicator Matrix · Multiple Imputation Workflow · Pattern-Mixture Sensitivity Model · Process-Based Missingness Audit · Tipping-Point Analysis
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Selection-Model Sensitivity Analysis operates as an analytical, modeling, inference, comparison, or optimization procedure that derives insight or a solution because it models the response process jointly with the outcome to examine how non-ignorable missingness would affect estimates.
Independent corroboration: The frozen evidence defines Selection-Model Sensitivity Analysis as 'Models the response process jointly with the outcome to examine how non-ignorable missingness would affect estimates', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Statistics & Experimental Design
Origin pattern: Single lineage
Present-day reach: Multi-domain
Rationale: Joint outcome-response models and sensitivity to non-ignorable missingness are established statistical missing-data methodology.
Related originating lineages:
- Data Science & Analytics — Modern predictive workflows use analogous selection models to test robustness under observation bias.
- Mathematics — Mathematical modeling, proof, and abstract-structure practice supplies a parallel or contributing lineage for the mechanism's defining operation: models the response process jointly with the outcome to examine how non-ignorable missingness would affect estimates.
- Medicine & Healthcare — Longitudinal clinical and epidemiological studies made attrition and informative response central applied problems.
Review resolution: The blind reviewers agree that statistics_experimental_design is the primary origin and differ only on alternate origin disagreement, domain reach disagreement, encyclopedia synthesis disagreement. I preserve every independently explained alternate from both records rather than imposing a numeric cap. I retain single_lineage because the combined record shows one traceable formative lineage. The broader reach of multi_domain records portability separately from historical provenance, and encyclopedia_synthesis=true preserves the affirmative synthesis judgment where either reviewer identified one.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The Heckman selection model — the archetypal joint model of an outcome equation and a selection (observation) equation linked by correlated errors, developed for wages observed only among the employed. Its identification leans heavily on functional form and any exclusion restriction, which is why sensitivity sweeping is the honest way to use it. ↩