Variables with time-varying effects and the Cox model: Some statistical concepts illustrated with a prognostic factor study in breast cancer

The Cox model relies on the proportional hazards (PH) assumption, implying that the factors investigated have a constant impact on the hazard - or risk - over time. We emphasize the importance of this assumption and the misleading conclusions that can be inferred if it is violated; this is particularly essential in the presence of long follow-ups.

We illustrate our discussion by analyzing prognostic factors of metastases in 979 women treated for breast cancer with surgery. Age, tumour size and grade, lymph node involvement, peritumoral vascular invasion (PVI), status of hormone receptors (HRec), Her2, and Mib1 were considered.

Median follow-up was 14 years; 264 women developed metastases. The conventional Cox model suggested that all factors but HRec, Her2, and Mib1 status were strong prognostic factors of metastases. Additional tests indicated that the PH assumption was not satisfied for some variables of the model. Tumour grade had a significant time-varying effect, but although its effect diminished over time, it remained strong. Interestingly, while the conventional Cox model did not show any significant effect of the HRec status, tests provided strong evidence that this variable had a non-constant effect over time. Negative HRec status increased the risk of metastases early but became protective thereafter. This reversal of effect may explain non-significant hazard ratios provided by previous conventional Cox analyses in studies with long follow-ups.

Investigating time-varying effects should be an integral part of Cox survival analyses. Detecting and accounting for time-varying effects provide insights on some specific time patterns, and on valuable biological information that could be missed otherwise.