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Discrete-time proportional hazards
Definition
Discrete-time proportional hazards models describe event occurrence when follow-up is divided into intervals and each interval's event probability is conditional on surviving to its start. A widely used grouped-time formulation applies a complementary log-log link to those conditional probabilities. It corresponds to grouping an underlying continuous-time proportional hazards model under suitable covariate assumptions. Terminology varies, so the link and interpretation should be stated rather than treating all discrete-time event models as equivalent.
What it means
The central question is whether an event happens during the current interval among units still at risk. This differs from predicting whether an event ever occurs. A customer who already canceled no longer contributes ordinary at-risk intervals for a first-cancellation analysis. A unit whose follow-up ends without an event is censored, not known to remain event-free forever. Modeling intervals allows researchers to use data collected monthly or annually while describing how conditional event probabilities change with time and covariates.
How it works
Create one record for each observed at-risk unit-interval, with an event indicator and covariates appropriate to that interval. In the complementary log-log formulation, log[-log(1 - h_j(x))] = alpha_j + x beta, where h_j is the interval event probability and alpha_j represents baseline time dependence. Equivalently, h_j(x) = 1 - [1 - h_j(0)]^exp(x beta). Under an underlying continuous-time proportional hazards interpretation, exp(beta) is a hazard ratio for a one-unit covariate change. It is not generally the ratio of the finite-interval probabilities. Fit an appropriate likelihood, handle censoring and time structure explicitly, and examine whether covariate effects remain proportional over time.
Three practical examples
- Subscription analysis: model first cancellation by month among accounts that have remained subscribed until that month.
- Equipment research: analyze failure intervals when exact failure times are not observed but inspection intervals are known.
- Workforce research: model time to a defined transition while accounting for employees whose follow-up ends before that transition.
Think of it this way
Think of a sequence of gates. At each gate, only those who have passed all earlier gates without the event remain in the risk set. The model estimates the chance of an event at the current gate, conditional on reaching it. Multiplying the chances of passing each gate yields survival across the sequence.
Worked example
Suppose hypothetical baseline event probabilities are 0.10 in interval one and 0.20 in interval two. Consider a covariate contrast with exp(beta) = 2 in the complementary log-log model. Its interval probabilities are 1 - 0.90 squared = 0.19 and 1 - 0.80 squared = 0.36. The probability ratios are therefore 1.9 and 1.8, not a constant two. Baseline survival through both intervals is 0.90 x 0.80 = 0.72. For the covariate group, it is 0.81 x 0.64 = 0.5184, equal to 0.72 squared. Cumulative event probabilities are consequently 0.28 and 0.4816. This example shows how proportional underlying hazards translate into interval probabilities without simply doubling each probability. These are specified model values rather than estimates from observed customers.
Illustrative case study
Fictional case: Cedar Membership studies monthly cancellation histories. Analysts create at-risk account-month records and distinguish cancellation from the end of available follow-up. They allow baseline risk to vary with membership duration and check whether a promotional-plan association changes over time. Reports translate the model into interval and cumulative probabilities rather than describing every exponentiated coefficient as a probability multiplier. This invented case illustrates grouped event-time modeling, not a causal effect of a promotion or a validated forecasting improvement.
Common misconceptions and mistakes
Do not confuse a hazard conditional on survival with an unconditional probability of an event at any time. A logistic discrete-time model describes proportional odds of interval hazards under its usual specification; it is not identical to the complementary log-log model. Repeated unit-interval rows are parts of a survival likelihood, not arbitrary independent new people. Censoring assumptions, omitted heterogeneity and time-dependent covariates need attention. A proportional effect can fail, and unequal interval definitions must be reflected in baseline modeling. Exact event times, interval censoring and right censoring are different observation patterns.
Business use
Businesses can use these models when event timing matters but data are naturally grouped into periods. The results can inform follow-up schedules and probability forecasts, provided the event definition, eligibility and censoring rules are sound. Evaluate calibration over relevant horizons and examine changes across cohorts or operating conditions. Coefficients summarize modeled associations unless a separate design and assumptions support causality. A model that describes first events also cannot automatically handle repeated events, competing outcomes or customers returning after cancellation without additional structure.
People also ask
Must baseline risk be constant?
No. Interval-specific terms or suitable time functions can allow it to vary.
Does a hazard ratio of two mean twice the monthly event probability?
Generally not for finite intervals, as the worked example demonstrates.
Can a person with no observed event contribute data?
Yes. Their observed event-free follow-up contributes information under the appropriate censoring assumptions.
Check your understanding
Why is the second interval's event probability evaluated only for survivors of the first?
Answer: A first-event hazard is conditional on remaining event-free. The cumulative two-interval event probability therefore uses one minus the product of the two survival probabilities, rather than simply adding the hazards.
Related terms
- Survival analysis
- Proportional hazards
- Risk set
- Time-varying covariate
- discrete hazard
- complementary log-log link
- right censoring
- interval censoring
- baseline hazard
- person-period data
Learn next
construct at-risk interval records for a small set of event histories. Compare logit and complementary log-log links and calculate survival from predicted interval probabilities. Then investigate censoring assumptions, proportionality checks and the distinction between an underlying hazard ratio and an observed-period risk ratio.
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Sources
German Rodriguez, Discrete Time Models: https://grodri.github.io/glms/notes/c7s6 Stata, Discrete-time survival analysis: https://www.stata.com/manuals/stdiscrete.pdf SAS, Complementary Log-Log Model for Interval-Censored Survival Times: https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/statug_logistic_sect065.htm
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