🔍 Problem With Current Time-Varying Visualizations
Duration analyses commonly deal with nonproportional hazards by interacting covariates with analysis time and then visualizing time-varying coefficients or hazard ratios. Those postestimation plots are a useful starting point but can be misleading: when a coefficient changes sign over time, its instantaneous reversal does not necessarily mean the overall effect on outcomes is reversed. Relying on coefficient or hazard-ratio plots alone can therefore produce faulty inference.
📊 Show What Really Changes: Survivor Functions
Survivor functions provide a clearer picture of the net, cumulative consequences of a time-varying effect. By translating time-varying hazards into survival probabilities, survivor functions show whether and how the overall likelihood of the event changes across groups or covariate values over time.
Key points:
- A coefficient that crosses zero can coexist with an unchanged or only partially changed overall effect.
- Survivor functions summarize the accumulated impact of time-varying coefficients on event probabilities.
- Visual comparison of survivor curves (and their differences) avoids ambiguous interpretation from pointwise coefficient plots.
🧭 How Survivor Functions Are Calculated For Time-Varying Models
An outline is provided for computing survivor functions when covariate effects vary with time. The approach describes how to:
- Construct the instantaneous hazard including time-interaction terms for chosen covariate values,
- Integrate the hazard over time to obtain cumulative hazard,
- Convert cumulative hazard to survivor probabilities, and
- Plot survivor curves and their differences (with uncertainty bands) to assess net effects.
📌 Demonstration: Mediation and Interstate Conflict
The approach is applied to a prominent empirical finding: a reported time-varying effect of mediation on interstate conflict. Reanalysis using survivor-function visualizations shows that earlier conclusions about mediation are misleading when judged only by time-varying coefficients or hazard ratios. The survivor-based plots clarify the true, accumulated impact of mediation over time.
⚖️ Why This Matters
Survivor functions are essential for interpreting time-varying effects in event-history models. They resolve ambiguities that arise from sign changes in coefficients and provide researchers and consumers of research with a more accurate, policy-relevant understanding of how covariates influence event probabilities over time.
