The Hazard Ratio (HR) and the Area Under the Receiver Operating Characteristic (AUROC) curve serve different purposes in the context of survival analysis and predictive modeling. While both are valuable metrics, they measure different aspects of model performance, and their interpretation should be considered in specific contexts.

Hazard Ratio (HR):The Hazard Ratio is commonly used in survival analysis to quantify the instantaneous risk of an event occurring in one group compared to another. A higher HR indicates a higher risk of the event happening in the group being compared. However, the HR does not directly measure the discriminatory ability or accuracy of a predictive model.

Area Under the Receiver Operating Characteristic (AUROC) Curve:The AUROC curve, on the other hand, is commonly used to assess the discriminatory power of a binary classification model. It evaluates the trade-off between sensitivity and specificity across different decision thresholds. A higher AUROC indicates better discriminatory performance.

Considerations:

In summary, while a higher HR indicates a higher risk in survival analysis, it should not be directly compared to the AUROC, which is more appropriate for assessing discriminatory accuracy. If you're looking for an AUROC-like metric for survival models, the C-index could be a suitable alternative.

Puthachad Namwaing has given a very comprehensive answer above. An alternative perspective is to consider the following. Imagine we have a HR for two predictors that come from a multivariable model.

i) In the simple model these two predictors are independent of each other they both predict the outcome. The hazard of death in the model is the product of the two factors. One predictor may have a larger effect than the other but they are different sources of variation and both may have an important influence on the outcome.

ii) The size of the HR for each is entirely dependent on the nature / scale of the predictor. Imagine (say) 'weight' as a predictor - the HR associated with a unit change will vary depending on whether weight is measured in g of Kg.

iii) where predictors are categorical a predictor that occurs infrequently (say a rare diagnosis) could have a very large HR associated with it but have very little impact on the overall ability of the model to correctly classify/predict death - for the simple reason that so few people are affected and so omission of this variable does not lead to incorrect prediction for many people.

iv) The accuracy of the estimated effect of each factor will depend on the error associated with it. A 'strong' effect estimated with low precision will be associated with a large prediction error.

Peter Donald Griffiths Thank you very much as well for the ideas and patterns that have added very detailed ideas this time from you.