Testing and Modeling Time-Varying Effects

Why Let a Covariate Effect Change With Time?

A Cox proportional hazards model represents a covariate’s effect as constant over follow-up: for two individuals who differ by one unit in that covariate but otherwise have the same modeled covariates, their hazard ratio does not depend on time. This can be too restrictive. For example, a treatment might be associated with a larger hazard difference soon after enrollment and a smaller difference later. A single hazard ratio would conceal that pattern.

The key modeling distinction is between a time-varying covariate, whose value changes during follow-up, and a time-varying effect, in which the coefficient multiplying a covariate changes with time. A covariate can be fixed while its effect changes. Conversely, a covariate can change over time while its coefficient remains constant. The earlier material on time-varying covariates treats the first issue; here the focus is on the second.

A time-by-covariate interaction is one way to model a changing effect. It makes the hazard ratio an explicit function of time, so the effect can be reported at meaningful follow-up times rather than summarized by one constant number.

Model and Interpretation

Let \(X_i\) be a covariate for individual \(i\), and let \(Z_i\) denote other covariates whose effects are modeled as constant over time. A Cox model with a time-varying effect of \(X_i\) is

Here \(h_0(t)\) is an unspecified baseline hazard, \(g(t)\) is a specified function of follow-up time, and \(\gamma\) is the interaction coefficient. For two individuals with the same \(Z_i\), the hazard ratio comparing \(X=x+1\) with \(X=x\) at time \(t\) is

This formula follows by subtracting the two log hazards. The baseline term and the contribution of \(Z_i\) cancel, leaving \(\beta+\gamma g(t)\) for a one-unit difference in \(X\). For a difference of \(d\) units, the hazard ratio is \(\exp\{d[\beta+\gamma g(t)]\}\).

If \(g(t)=0\) at a chosen reference time \(t_0\), then \(\beta\) is the log hazard ratio at \(t_0\), and \(\gamma\) describes how that log hazard ratio changes with \(g(t)\). A convenient choice is \(g(t)=\log(t/t_0)\), which is zero at \(t=t_0\) and compares time on a multiplicative scale. Time must be positive for this choice. Centering time this way also makes the main-effect coefficient interpretable at a useful reference time.

This model relies on the usual Cox-model conditions: the event and censoring information are recorded appropriately; censoring is independent of the event time conditional on the modeled information; individuals contribute independent event histories; and the specified covariates and functional form are adequate for the intended comparison. It also assumes that effects represented by \(\boldsymbol{\theta}\) are proportional over time. The interaction only relaxes proportionality for \(X\); it does not automatically fix other model misspecifications.

Testing for a Time-Varying Effect

For a single interaction coefficient, the null hypothesis of proportional hazards for \(X\) is \(H_0:\gamma=0\), against \(H_a:\gamma\ne0\). A Wald test uses the estimated coefficient and its standard error:

Under \(H_0\), and with the usual large-sample approximation, \(Z\) is approximately standard normal. A two-sided p-value is \(2P(N\geq |Z|)\), where \(N\sim N(0,1)\). For multiple time-interaction terms, test their coefficients jointly; for example, a Wald test uses the estimated coefficient vector and its covariance matrix. A likelihood-ratio test instead compares the maximized partial likelihoods of nested models, with and without the interaction terms. The test should match the intended restriction, and both models must be fitted to the same analysis data.

Uncertainty for a Hazard Ratio at a Selected Time

Testing \(\gamma=0\) answers whether the specified interaction is supported by the data. It does not by itself give the uncertainty for the hazard ratio at a particular time. At time \(t\), the estimated log hazard ratio is \(\hat{\beta}+\hat{\gamma}g(t)\). Its variance must include the covariance between the estimates:

An approximate confidence interval is formed on the log hazard-ratio scale, then exponentiated at both endpoints. This procedure uses the estimated covariance matrix from the fitted model. Simply combining separate confidence intervals for \(\beta\) and \(\gamma\) does not generally give the correct interval.

Choosing the Time Function and Checking the Model

The interaction is only as flexible as the chosen \(g(t)\). A log-time interaction is useful when a gradual change with follow-up time is scientifically plausible. A linear-time interaction gives a different pattern: it makes the log hazard ratio linear in elapsed time rather than in log time. The choice should be made with the time scale and scientific question in mind, and should not be selected solely because it produces a small p-value.

Diagnostics help assess whether the selected form is credible. The earlier material on Schoenfeld residuals provides a way to examine evidence of nonproportionality. In this setting, inspect whether the residual pattern for \(X\) is consistent with the proposed time trend, and consider whether the trend is approximately represented by \(g(t)\). Also check that the time variable is defined consistently, that the reference time is meaningful, and that the fitted interaction is not driven by very few individuals at late follow-up. Sparse risk sets can make late-time estimates unstable even when the model converges.

A useful comparison is with stratification. Stratifying on a categorical variable allows its baseline hazard to differ across strata but does not estimate a hazard ratio for that variable. It is suitable when adjustment is needed but its effect is not the target. By contrast, an interaction model estimates a time-dependent hazard ratio for the covariate of interest. A time-varying covariate model addresses changing covariate values, not necessarily a changing coefficient. These approaches answer different questions and are not interchangeable.

A single interaction also has limitations. If the actual effect changes sharply or nonlinearly, a linear function of time or log time may misrepresent it. Adding many time functions can overfit, especially with few events, and makes interpretation less direct. In such cases, carefully chosen piecewise effects or more flexible functions may be considered, with the extra modeling complexity justified by the data and scientific aims. Whatever form is used, report the hazard ratio at selected times with uncertainty, and specify the covariate contrast and time units.

Common Pitfalls

• Confusing changing values with changing effects. A covariate measured repeatedly is not automatically a time-varying effect. The interaction term specifically allows the coefficient to depend on time.
• Interpreting \(\exp(\beta)\) as the hazard ratio at all times. With an interaction, \(\exp(\beta)\) is the hazard ratio only where \(g(t)=0\); elsewhere use \(\exp\{\beta+\gamma g(t)\}\).
• Ignoring the time scale or reference. With log time, changing units without adjusting the reference can change coefficient interpretation. State the units and reference time explicitly.
• Claiming that a significant interaction proves a particular biological mechanism. The test concerns the specified statistical form and does not establish causation or explain why the effect changes.
• Omitting covariance when constructing an interval. Use the full estimated variance of \(\hat{\beta}+\hat{\gamma}g(t)\), including its covariance term.