Concept Architecture
Concept
Theoretically, Scaled Schoenfeld Residuals are transformed Schoenfeld residuals used to assess the proportional hazards assumption in Cox proportional hazards regression. They are derived by scaling the original Schoenfeld residuals using an estimate of the covariance matrix of the regression coefficients, producing residuals whose expected value is constant over time when the proportional hazards assumption is satisfied. They exist to provide both graphical and formal statistical diagnostics for detecting time-varying covariate effects.
Mathematically, scaled Schoenfeld residuals are obtained by multiplying the Schoenfeld residuals by an estimate of the covariance matrix of the regression coefficients. Under the proportional hazards assumption, the scaled residuals should exhibit no systematic trend when plotted against event time. A non-zero trend indicates that the effect of one or more covariates changes over time, suggesting violation of the proportional hazards assumption.
In practice, scaled Schoenfeld residuals are routinely examined using residual plots with smoothed trend lines and formal hypothesis tests, most notably the Grambsch?Therneau test. They are widely used in survival analyses supporting health technology assessment and health economic evaluation to verify whether Cox regression provides an appropriate representation of treatment effects over time.
Purpose
Used to evaluate 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?* = V(??) ? r?
where:
- r?* = scaled Schoenfeld residual
- r? = Schoenfeld residual
- V(??) = estimated covariance matrix of the regression coefficients
Supporting Formulae
Schoenfeld residual:
r? = x? ? E(X | t?)
Cox proportional hazards model:
h(t | X) = h?(t) ? exp(??X)
Grambsch?Therneau test statistic:
?� = (??(t))?V????(t)
Related Mathematical Methods
- Schoenfeld Residuals
- Cox Proportional Hazards Model
- Grambsch?Therneau Test
- Survival Analysis
- Proportional Hazards Assumption
- Time-Dependent Covariate Models
Example
A Cox regression model estimates the treatment effect for a new oncology intervention. Scaled Schoenfeld residuals are plotted against follow-up time for the treatment covariate.
The smoothed residual curve remains approximately horizontal, and the Grambsch?Therneau test produces ?� = 1.42 with p = 0.23.
Because no significant time trend is detected, there is no evidence that the proportional hazards assumption has been violated.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SLOPE | =SLOPE(B2:B51,A2:A51) | Estimate the trend in scaled residuals over time |
| CORREL | =CORREL(A2:A51,B2:B51) | Assess association between residuals and follow-up time |
| LINEST | =LINEST(B2:B51,A2:A51,TRUE,TRUE) | Estimate regression coefficients for diagnostic assessment |
| IF | =IF(P2>0.05,"No Evidence of Violation","Possible Violation") | Summarise proportional hazards diagnostic results |
| Scatter Chart | Residuals versus time | Visual assessment of proportional hazards assumption |
VBA (Optional)
A VBA procedure can automate the production of scaled Schoenfeld residual diagnostic plots and summarise proportional hazards test results across multiple Cox regression models.
Sources
- Schoenfeld D. Partial Residuals for the Proportional Hazards Regression Model. Biometrika. 1982.
- Grambsch PM, Therneau TM. Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika. 1994.
- Collett D. Modelling Survival Data in Medical Research.
- Kleinbaum DG, Klein M. Survival Analysis: A Self-Learning Text.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What are scaled Schoenfeld residuals?
A diagnostic technique testing the proportional hazards assumption by scaling Schoenfeld residuals by their variance to improve test power.
Source: Grambsch & Therneau 1994
How do scaled Schoenfeld residuals test proportional hazards?
Scaled Schoenfeld residuals are plotted against time to test whether a covariate's effect stays constant, as the proportional hazards assumption requires. If the assumption holds, the residuals scatter around a flat line with no trend, whereas a systematic slope indicates the effect is changing over time and the assumption is violated. Scaling the residuals by their variance sharpens this test compared with the raw version. The plot turns an abstract assumption into a visible pattern. Grambsch and Therneau (1994) developed this method.
Source: Grambsch & Therneau 1994
How are scaled Schoenfeld residuals used?
Scaled Schoenfeld residuals are used to test the proportional hazards assumption by plotting them against time, or a function of time, for each covariate and looking for a systematic trend: a non-zero slope indicates that the covariate's effect changes over time, violating proportional hazards, while a flat pattern supports the assumption. A formal test of the slope accompanies the plot. Because their expectation relates to the time-varying coefficient, scaled Schoenfeld residuals give a direct and interpretable check of proportionality.
Source: Grambsch & Therneau 1994
How do scaled Schoenfeld residuals differ from Schoenfeld residuals?
Scaled Schoenfeld residuals are the raw Schoenfeld residuals scaled by their estimated variance, a transformation introduced by Grambsch and Therneau that gives them the property that their expected value plus the estimated coefficient approximates the time-varying coefficient. This makes them more directly interpretable and improves the power of tests for non-proportional hazards compared with the unscaled residuals. So Schoenfeld residuals are the basic per-event residuals, and scaling them enhances their usefulness for detecting and characterising departures from proportional hazards.
Source: Schoenfeld 1982
What does a trend in scaled Schoenfeld residuals indicate?
A trend in scaled Schoenfeld residuals against time indicates that a covariate's effect on the hazard changes over follow-up, meaning the hazard ratio is not constant and the proportional hazards assumption is violated for that covariate. An upward or downward slope shows the effect strengthening or weakening over time. A roughly flat pattern with no trend supports proportional hazards. Detecting such a trend prompts extending the model to allow the effect to vary with time or otherwise accommodating the non-proportionality.
Source: Grambsch & Therneau 1994
How do scaled Schoenfeld residuals support the Cox model?
Scaled Schoenfeld residuals support the Cox model by providing a means to check its key proportional hazards assumption, on which the validity of its hazard ratios depends. By plotting and testing the residuals against time for each covariate, analysts can detect whether effects are constant or change over follow-up. Confirming proportional hazards justifies interpreting the hazard ratios, while detecting violations signals the need to extend the model. Scaled Schoenfeld residuals are thus a standard diagnostic accompanying the fitting of a Cox 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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- Term code
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