Case studies / Healthcare
Cox Proportional Hazards Modelling to Identify Patients Most at Risk of Readmission
Modelling time-to-event rather than event-or-not, so a large public sector organisation could target risk assessment where it mattered.
The challenge
What the organisation came with
A large public sector healthcare organisation wanted to know which individuals were most likely to return after treatment, so that risk assessment and follow-up could be directed at the people most likely to need it.
The obvious framing — did the person return, yes or no — was the wrong one. The question was not whether an event occurred but how long until it occurred, and those are different statistical problems. Treating a time-to-event outcome as binary discards the timing entirely, so a return after three weeks and a return after three years count identically.
The harder problem was censoring. A substantial proportion of individuals had simply not experienced the event by the end of the five-year observation window. They are not non-events; they are people whose event time is unknown but known to be longer than their follow-up. Standard regression has no way to represent that. Dropping them biases the result, and coding them as non-events biases it further.
The approach
How it was analysed, and why that method
Why a Cox model rather than logistic regression
A Cox proportional hazards model was used. It estimates the hazard — the instantaneous risk of the event at any given time, conditional on having survived to that point — which is precisely the quantity the organisation needed in order to rank individuals by risk.
Logistic regression was explicitly rejected. It ignores follow-up time, and it cannot represent censored observations at all. With five years of data and a large censored group, it would have thrown away most of the information in the dataset and produced risk estimates that depended on an arbitrary cut-off date.
Choosing the model class, not tuning it. Once the outcome was correctly framed as time-to-event with right-censoring, the analysis followed. Framing it as a binary outcome would have produced a plausible-looking result that answered a different question.
Building and checking the model
Covariates were assessed individually and jointly, and the proportional hazards assumption — that the effect of a covariate on the hazard is constant over time — was tested rather than assumed. This is the assumption most often left unexamined in applied survival analysis, and where it fails the model's central claim does not hold.
The output was expressed as hazard ratios with confidence intervals, so each factor could be read as a multiplicative change in risk, and survival curves were produced for the groups the organisation cared about operationally.
Delivered
What the client received
The value
What changed as a result
The organisation used the findings to improve its risk assessment process and to inform future operational decisions.
Because the model ranks individuals by hazard rather than sorting them into a binary category, follow-up could be prioritised by degree of risk rather than by a threshold — and because the code was delivered alongside the report, the model can be rerun as new data accumulates.
Written from the assigned statistician's own project notes. The client is not named and no identifying detail, data or figures are published. Scale and timeframe are approximate.
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