Concept Architecture
Concept
Theoretically, a Time-Fixed Covariate is an explanatory variable whose value remains constant throughout the observation period for each individual. Such covariates represent baseline characteristics measured at study entry and are assumed not to change during follow-up. Time-fixed covariates are fundamental to regression and survival analysis because they permit estimation of constant associations between baseline factors and clinical outcomes without modelling changes over time.
Mathematically, a time-fixed covariate enters a regression or survival model as a constant predictor whose value does not vary with time. In the Cox proportional hazards model, its effect is represented by a regression coefficient that produces a constant hazard ratio throughout follow-up, provided the proportional hazards assumption holds. The covariate contributes a single parameter estimate regardless of follow-up duration.
In practice, time-fixed covariates commonly include age at baseline, sex, treatment allocation, disease stage at enrolment and genetic characteristics. They are routinely incorporated into survival models, generalised linear models and decision-analytic models used in health economic evaluations to adjust for baseline differences and estimate treatment effects.
Purpose
Used to represent baseline characteristics that remain unchanged throughout follow-up, enabling adjustment for fixed prognostic factors in survival analyses and health economic models.
Mathematical Formulae
Primary Formula
h(t | X) = h?(t) ? exp(??X? + ??X? + ? + ??X?)
where X represents time-fixed covariates.
Supporting Formulae
Hazard ratio for a time-fixed covariate:
HR = exp(?)
General linear predictor:
? = ?? + ??X? + ??X? + ? + ??X?
Related Mathematical Methods
- Cox Proportional Hazards Model
- Generalised Linear Model
- Logistic Regression
- Multiple Regression
- Survival Analysis
- Baseline Covariate Adjustment
Example
A survival analysis evaluates mortality following treatment for heart failure.
Baseline treatment group:
Treatment = 1
Control = 0
Estimated regression coefficient:
?? = ?0.405
Hazard ratio:
HR = exp(?0.405) = 0.67
Because treatment allocation is fixed at randomisation, it remains constant throughout follow-up and is modelled as a time-fixed covariate.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(A2="Treatment",1,0) | Create a binary baseline treatment indicator |
| XLOOKUP | =XLOOKUP(ID,BaselineID,Age) | Retrieve fixed baseline covariates |
| INDEX | =INDEX(BaselineData,MATCH(ID,PatientID,0),3) | Import baseline characteristics into the analysis dataset |
| EXP | =EXP(B2) | Convert regression coefficients into hazard ratios |
| LN | =LN(C2) | Calculate log hazard ratios for interpretation |
VBA (Optional)
A VBA procedure can automatically populate baseline covariates from patient records and prepare fixed predictor datasets for regression and survival analyses.
Sources
- Cox DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B. 1972.
- Therneau TM, Grambsch PM. Modeling Survival Data: Extending the Cox Model.
- 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 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 time fixed covariate?
A patient characteristic in a survival model measured once and assumed constant throughout follow-up, such as baseline age or sex.
Source: Cox 1972
What constant predictor variable is a time-fixed covariate?
A time-fixed covariate is an explanatory variable in a survival analysis whose value does not change over the follow-up period. Characteristics recorded at baseline, such as sex or the treatment group assigned at randomisation, are typical examples. Because the value stays constant, its effect on the hazard is straightforward to include and interpret. In a Cox model such covariates are usually assumed to act proportionally on the hazard throughout follow-up. Where a characteristic can change over time, a time-varying covariate is needed instead.
Source: Collett 2015
What is a time-fixed covariate?
A time-fixed covariate is a patient characteristic in a survival model measured once, usually at baseline, and assumed constant throughout follow-up, such as sex, baseline age, treatment assignment, or a baseline measurement. Its value does not change over time in the model, so its effect on the hazard is based on the single measured value. Time-fixed covariates are the usual predictors in survival models like the Cox model, entering as constant characteristics whose effect is estimated over the follow-up.
Source: Cox 1972
How are time-fixed covariates used in survival models?
Time-fixed covariates are used in survival models, such as the Cox model, as predictors whose constant baseline values determine their effect on the hazard, estimated as hazard ratios. Because their values do not change, each individual's covariate contributes the same throughout follow-up, and the model estimates how the characteristic affects the event rate. Most covariates in survival analysis are time-fixed, representing baseline characteristics, and they are handled straightforwardly, with the proportional hazards assumption concerning whether their effect is constant over time.
Source: Cox 1972
What are examples of time-fixed covariates?
Examples of time-fixed covariates include sex; age at baseline; treatment group assigned at randomisation; baseline disease stage or severity; genetic factors; and any characteristic measured once at the start and treated as unchanging. These are contrasted with time-varying covariates, such as a repeatedly measured biomarker, whose values change during follow-up. Time-fixed covariates capture baseline features, and they are the standard form of predictor in survival regression, representing characteristics fixed at the outset of observation.
Source: Cox 1972
How does a time-fixed covariate differ from a time-varying covariate?
A time-fixed covariate is measured once and assumed constant throughout follow-up, so its value does not change, whereas a time-varying covariate can change value over follow-up, such as a repeatedly measured biomarker or a treatment that starts partway through. Time-fixed covariates contribute a constant value to the model, while time-varying covariates require methods that update their value over time. The distinction matters because handling time-varying covariates needs special techniques, whereas time-fixed covariates are handled straightforwardly.
Source: Kalbfleisch & Prentice 2002
What assumption applies to a time-fixed covariate in the Cox model?
In the Cox model, a time-fixed covariate is assumed to have a proportional hazards effect, meaning its hazard ratio is constant over time even though its value is fixed. The covariate's value does not change, but the assumption concerns whether its effect on the hazard stays the same throughout follow-up. If the effect of a time-fixed covariate changes over time, proportional hazards is violated for it, which is checked with diagnostics such as Schoenfeld residuals and addressed by allowing a time-varying effect.
Source: Cox 1972
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Oct 2025
Content version: 1.0.0
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