Truncated Regression Model¶
Infer a population response–covariate relation from records admitted only when the response falls inside a known region, by conditioning on that inclusion.
Core Idea¶
A truncated regression model studies a population relation between outcome \(Y\) and covariates \(X\) when records are observed only if \(Y\) enters a declared range \(A\). The observed outcome density is \(f(y\mid x;\theta)/P_\theta(Y\in A\mid X=x)\) for \(y\in A\). This inclusion probability corrects the likelihood for the selected sample under the stated population model; ordinary regression on included cases alone generally does not recover the intended population coefficients.[^ref-83c15884285d]
Scope of Application¶
A constructed normal earnings record with conditional mean \(1\), standard deviation \(1\), observed \(y=1\) and upper cutoff \(2\) has admission probability \(\Phi(1)\approx0.84134\) and conditional density \(\phi(0)/\Phi(1)\approx0.47417\). Separately, Stata's worked lower-truncation demonstration fits positive work hours on 150 records and marks 100 other records as truncated from that fit; its Tobit contrast retains all 250 under a censoring model. Normal linear truncated regression is one parametric form, but the essential pattern is outcome-dependent record inclusion plus a justified conditional population model.[ref-83c15884285d][ref-9af2a09f479a]
Clarity¶
Truncation removes excluded records from the standard analysis file. Censoring retains those units with boundary or interval information about their outcomes. An auxiliary frame might still reveal counts or covariates for excluded units, so “nothing whatsoever is known” is too strong. A marginal truncated distribution also lacks the covariate relation that makes this a regression model.[^ref-dca5f542e051]
Manages Complexity¶
The model separates the response law from the observation rule, then joins them through one inclusion-probability normalizer. This makes the selection adjustment inspectable and exposes whether the threshold, density and target population have been specified correctly.[^ref-83c15884285d]
Abstract Reasoning¶
Name the target population, response, covariates and inclusion region. Check whether outside units disappear or are merely censored. Specify \(f(y\mid x;\theta)\), compute \(P_\theta(Y\in A\mid X=x)\), and use the conditional observed-data likelihood only if those assumptions reflect the actual sampling process.[ref-83c15884285d][ref-dca5f542e051]
Knowledge Transfer¶
Upper- and lower-threshold applications share the same model structure; only the probability used in the denominator changes. The strict genus is Statistical Model: outcome truncation and inclusion-normalized likelihood are its differentia. Regression is a closer subject-matter neighbor, while Truncated Normal Distribution is a component only when the response law is normal.
[^ref-83c15884285d]: Stata, official truncated-regression manual. [^ref-dca5f542e051]: Stata, official censored/truncated-outcome guidance. [^ref-9af2a09f479a]: Amemiya, original truncated-normal regression article (1973), bibliographic record.
Relationships to Other Abstractions¶
Current abstraction Truncated Regression Model Domain-specific
Parents (1) — more general patterns this builds on
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Truncated Regression Model is a kind of Statistical Model Domain-specific
An outcome-truncated regression law is a specialized statistical model.
Hierarchy paths (6) — routes to 4 parentless roots
- Truncated Regression Model → Statistical Model → Representation → Abstraction
- Truncated Regression Model → Statistical Model → Probability Distribution → Random Variable → Function (Mapping)
- Truncated Regression Model → Statistical Model → Probability Distribution → Probability → Measure → Set and Membership
- Truncated Regression Model → Statistical Model → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Truncated Regression Model → Statistical Model → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Truncated Regression Model → Statistical Model → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Truncated Regression Model sits in a sparse region of the domain-specific corpus (96th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- M-Estimator — 0.78
- Learnable Function Class — 0.77
- Ecological Correlation — 0.77
- Base Rate — 0.77
- Log-Linear Analysis — 0.76
Computed from structural-signature embeddings · 2026-10-08