Discrete-time proportional hazards¶

A grouped-duration model whose conditional event probability in each interval is linked to covariates so their effects correspond to proportional underlying hazards or a specified discrete analogue.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4211
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Survival Analysis, Discrete Duration Models → Mathematics

Core Idea¶

Discrete-time proportional hazards models represent interval-specific failure risk as a baseline time effect modified multiplicatively on an underlying hazard scale by covariates. Person-period records encode whether failure occurs in each at-risk interval; a link function separates baseline duration dependence from a constant covariate effect and yields a binary-response likelihood. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

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The Once-a-Year Checkup

Imagine checking a garden once a year to see if each plant has wilted yet. Every year you ask, "Did it wilt this year, if it hadn't before?" Some things, like being in a sunny spot, can push that chance up in the same steady way every year. This model helps figure out how much each thing pushes.

Chunk-by-Chunk Risk Model

Sometimes we want to know when something will happen, like when a student leaves a club, but we only check in chunks of time, like once a year. A discrete-time proportional hazards model looks at each time chunk and asks: for someone who hasn't left yet, did they leave during this chunk? It has one part for how the risk naturally changes from chunk to chunk, and another part for how things about the person, like their age, raise or lower that risk. The model assumes a person's traits push the risk by the same amount every chunk, measured on a special math scale rather than the plain chance. People who stop being tracked are counted only for the chunks they were watched.

Grouped-Time Hazard Model

A discrete-time proportional hazards model studies the time until an event when time comes in intervals, either because events are grouped into periods or because time is naturally counted in steps. The hazard is the chance that the event happens in a given interval, for those still at risk at its start. Data are arranged as person-period records: one row per subject per interval they were at risk, marking whether the event happened in that interval. A link function, a transformation of the probability, lets the model write the hazard as a baseline effect for each time interval plus covariate effects that stay constant over time on that transformed scale. This turns the problem into a series of yes-or-no outcomes that can be fit with binary regression. You need to fix the intervals, who is at risk, how censored cases are handled, and which link is used, because the covariate effects are only constant on that declared scale.

Discrete-time proportional hazards models represent the risk of failure in each interval, conditional on survival to its start, as a baseline time effect modified multiplicatively on an underlying hazard scale by covariates. Data are expanded into person-period records, with a binary indicator of whether failure occurred in each interval during which the subject was at risk; censored subjects contribute records only for intervals they were observed. A link function applied to the interval hazard separates baseline duration dependence, typically one parameter per interval, from covariate effects that are constant across intervals on the declared scale, so the likelihood becomes that of a binary-response regression. The approach adapts proportional-hazards reasoning to grouped or intrinsically discrete event times. Its validity depends on fixed risk sets, interval grouping, censoring rules, and link, and on the covariate effect actually retaining its declared time-constant scale.

Scope of Application¶

Discrete-time proportional hazards belongs to survival analysis and is useful where the analyst can specify subjects at risk, discrete time intervals, event indicator and censoring, baseline interval hazard, covariates, complementary-log-log or other link, proportional effect coefficients and likelihood, then evaluate risk sets, interval grouping, censoring and link are fixed and covariate effects retain their declared time-constant scale. The scope is broad within that domain but bounded by the need for risk sets, interval grouping, censoring and link are fixed and covariate effects retain their declared time-constant scale. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making risk sets, interval grouping, censoring and link are fixed and covariate effects retain their declared time-constant scale the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Discrete-time proportional hazards can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Discrete-time proportional hazards. Discrete-time proportional hazards compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: subjects at risk, discrete time intervals, event indicator and censoring, baseline interval hazard, covariates, complementary-log-log or other link, proportional effect coefficients and likelihood. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express risk sets, interval grouping, censoring and link are fixed and covariate effects retain their declared time-constant scale independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of survival analysis because they reuse subjects at risk, discrete time intervals, event indicator and censoring, baseline interval hazard, covariates, complementary-log-log or other link, proportional effect coefficients and likelihood, Person-period records encode whether failure occurs in each at-risk interval; a link function separates baseline duration dependence from a constant covariate effect and yields a binary-response likelihood., and type the carrier, state every parameter and convention in the definition, test that risk sets, interval grouping, censoring and link are fixed and covariate effects retain their declared time-constant scale, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Discrete-time proportional hazards Domain-specific

Parents (1) — outgoing typed relations

Discrete-time proportional hazards is a kind of Statistical Inference Prime

The model infers duration dependence and covariate effects from censored grouped events.

Hierarchy paths (3) — routes to 3 parentless roots

Discrete-time proportional hazards → is a kind of Statistical Inference → is a kind of Inductive Reasoning

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Discrete-time proportional hazards sits in a crowded region of the domain-specific corpus (31^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Survival Analysis & Demographic Rates (16 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25