Concept Architecture
Concept
Theoretically, Schoenfeld Residuals are residuals derived from the Cox proportional hazards model that measure the difference between the observed value of a covariate for an individual experiencing an event and the expected value of that covariate within the corresponding risk set. They were developed specifically to assess whether the effect of a covariate remains constant over time, thereby providing a diagnostic tool for evaluating the proportional hazards assumption.
Mathematically, a Schoenfeld residual is calculated at each observed event time and is defined for each covariate included in the Cox regression model. Under the proportional hazards assumption, the residuals should be randomly distributed with no systematic relationship to time. Patterns or trends in the residuals indicate that the covariate effect may vary over time, suggesting that the proportional hazards assumption has been violated.
In practice, Schoenfeld residuals are examined graphically by plotting them against event time and statistically through tests such as the Grambsch?Therneau test. They are routinely used during survival model validation in health technology assessment and health economic evaluation to determine whether Cox regression is an appropriate modelling framework or whether time-dependent effects should be incorporated.
Purpose
Used to diagnose whether covariate effects remain constant over time in Cox proportional hazards models, thereby assessing the validity of the proportional hazards assumption before interpreting hazard ratios or extrapolating survival outcomes.
Mathematical Formulae
Primary Formula
r? = x? ? E(X | t?)
where:
- r? = Schoenfeld residual
- x? = observed covariate value for the individual experiencing the event
- E(X | t?) = expected covariate value within the risk set at event time t?
Supporting Formulae
Expected covariate value:
E(X | t?) = ?x?w? / ?w?
where:
w? = exp(??x?)
Cox proportional hazards model:
h(t | X) = h?(t) ? exp(??X)
Related Mathematical Methods
- Cox Proportional Hazards Model
- Scaled Schoenfeld Residuals
- Grambsch?Therneau Test
- Survival Analysis
- Time-Dependent Covariate Models
- Maximum Partial Likelihood Estimation
Example
A Cox regression model evaluates overall survival following treatment with a new oncology therapy.
At one event time:
Observed treatment indicator:
x = 1
Expected treatment indicator from the risk set:
E(X | t) = 0.76
Schoenfeld residual:
r = 1 ? 0.76 = 0.24
Residuals calculated for all event times are plotted against follow-up time. The absence of a systematic trend supports the proportional hazards assumption.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Calculate Cox model risk weights |
| SUMPRODUCT | =SUMPRODUCT(C2:C100,D2:D100)/SUM(D2:D100) | Calculate expected covariate values within the risk set |
| IF | =A2-B2 | Calculate Schoenfeld residuals after expected values have been derived |
| CORREL | =CORREL(Time,Residuals) | Assess correlation between residuals and time |
| Scatter Chart | Residuals versus time | Visual assessment of the proportional hazards assumption |
VBA (Optional)
A VBA procedure can automate calculation of Schoenfeld residual summaries and generate diagnostic plots for multiple Cox regression models.
Sources
- Schoenfeld D. Partial Residuals for the Proportional Hazards Regression Model. Biometrika. 1982.
- Cox DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B. 1972.
- Grambsch PM, Therneau TM. Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika. 1994.
- Collett D. Modelling Survival Data in Medical Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 21: Flexible methods for survival analysis — Rutherford, Lambert, Sweeting, Pennington, Crowther, Abrams & Latimer, TSD 21 ed., 2020 (NICE Decision Support Unit (University of Sheffield))
Guidance extending standard survival analysis to flexible parametric methods — spline-based models, fractional polynomials, mixture and cure models, and relative-survival approaches — for capturing complex hazard functions in economic evaluation.
Frequently Asked Questions (7)
What is a schoenfeld residual?
A residual specific to the Cox model, calculated at each event time, used to test whether the proportional hazards assumption holds.
Source: Schoenfeld 1982
What is distinctive about how a Schoenfeld residual is defined?
Unlike residuals that give one value per patient, a Schoenfeld residual is defined for each covariate at each event time, based on the difference between a covariate's value for the patient who had the event and its average among all those still at risk then. This ties the residual to the event times rather than to individuals, which is what lets it reveal whether a covariate's effect drifts over follow-up. Its per-event structure suits testing the proportional hazards assumption. Collett (2015) describes it.
Source: Collett 2015
What is a Schoenfeld residual?
A Schoenfeld residual is a residual specific to the Cox proportional hazards model, calculated at each event time for each covariate, defined as the difference between the covariate value of the individual who had the event and the weighted average covariate value among those at risk. Introduced by Schoenfeld, these residuals are used to test whether the proportional hazards assumption holds, since a systematic pattern in them over time indicates that a covariate's effect changes, violating proportionality.
Source: Schoenfeld 1982
How are Schoenfeld residuals calculated?
Schoenfeld residuals are calculated at each event time, for each covariate, as the difference between the covariate value of the individual who experienced the event and the expected covariate value under the model, which is the average over those at risk weighted by their estimated hazards. There is one residual per covariate per event time. Because they compare the observed covariate of the event case with what the model expects, they capture information about whether covariate effects are stable over time.
Source: Schoenfeld 1982
How are Schoenfeld residuals used to test proportional hazards?
Schoenfeld residuals are used to test proportional hazards by examining whether they show a systematic trend against time for each covariate: under proportional hazards, they should have no trend, scattered around zero, whereas a trend indicates the covariate's effect changes over time, violating the assumption. Often the scaled Schoenfeld residuals, which enhance interpretability and power, are plotted and tested against time. A significant trend signals non-proportional hazards, prompting model extension.
Source: Grambsch & Therneau 1994
What do Schoenfeld residuals reveal?
Schoenfeld residuals reveal whether covariate effects in a Cox model are constant over time, as proportional hazards requires, or change, by showing whether the residuals for a covariate trend with time. A trend indicates the hazard ratio for that covariate is not constant, so its effect strengthens or weakens over follow-up. This identifies which covariates violate proportional hazards and how, guiding whether the model needs to allow time-varying effects for those covariates. A lack of trend supports the assumption.
Source: Schoenfeld 1982
Why are Schoenfeld residuals important for the Cox model?
Schoenfeld residuals are important because the Cox model's hazard ratios are valid summaries only if proportional hazards holds, and these residuals provide the standard means to check that assumption. By testing whether covariate effects are constant over time, they confirm whether the model's estimates can be trusted or whether the effect varies, requiring extension. Because checking proportional hazards is a routine and necessary step in Cox regression, Schoenfeld residuals, often in their scaled form, are a key diagnostic accompanying the model.
Source: Cox 1972
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Oct 2025
Content version: 1.0.0
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