Editing Openai/695f02d3-11f8-800e-993a-ad71db6d42aa (section)

=== Assistant: Here’s a clear explanation of how a hazard ratio HR like 0.636 is calculated and what it means — especially in the cont… ===
Here’s a clear explanation of how a hazard ratio (HR) like 0.636 is calculated and what it means — especially in the context of a clinical trial such as the SLS Phase 3 REGAL trial.

==== A hazard ratio compares the instantaneous risk of an event (e.g., death, progression, relapse) between two groups over time. It’s used in survival analysis and typically estimated using a statistical model such as the Cox proportional hazards model. Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Hazard_ratio|publisher=en.wikipedia.org|access-date=2026-01-10}}</ref> ====
* HR = 1.0 → no difference in hazard between groups
* HR < 1.0 → event occurs less often in the treatment group vs. control
* HR > 1.0 → event occurs more often in the treatment group vs. control Cancer.gov<ref>{{cite web|title=Cancer.gov|url=https://www.cancer.gov/publications/dictionaries/cancer-terms/def/hazard-ratio|publisher=cancer.gov|access-date=2026-01-10}}</ref>

==== In practice, hazard ratios are not computed by a simple ratio of two median survival times. Instead, they are estimated from time-to-event data across the entire follow-up period. ====

===== If a Cox model is used (as in most Phase 3 oncology trials), it works like this: =====
# Define the hazard function At time ttt, the hazard is the instantaneous risk of the event happening at that time, given that the person has survived up to ttt. Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Hazard_ratio|publisher=en.wikipedia.org|access-date=2026-01-10}}</ref>
# Model the log hazard as a linear function of treatment The model estimates a coefficient (β\betaβ) for the treatment effect.
# Exponentiate that coefficient The hazard ratio = eβe^{\beta}eβ. Easy Made Stat<ref>{{cite web|title=Easy Made Stat|url=https://help.easymedstat.com/support/solutions/articles/77000513111-how-to-calculate-hazard-ratios-hr-in-a-cox-model-|publisher=Easy Made Stat|access-date=2026-01-10}}</ref> • If β\betaβ is negative → HR < 1 (treatment reduces risk) • If β\betaβ is positive → HR > 1 (treatment increases risk) This is how the HR comes out mathematically. >

==== In the REGAL trial context (from the November 14, 2022 press release): ====
* An HR of 0.636 means that, at any point in time, patients in the treatment arm (e.g., GPS) had about 36% lower hazard (instantaneous risk) of death than patients in the control arm (e.g., BAT).
* This is a relative measure over the entire follow-up period under the proportional hazards assumption (i.e., the ratio is constant over time). Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Hazard_ratio|publisher=en.wikipedia.org|access-date=2026-01-10}}</ref>

So the trial planners prespecified that an HR ≤ 0.636 would be the threshold for demonstrating statistically significant improvement in overall survival (OS).

==== While median survival times might be reported (e.g., OS ~12.6 vs. 8 months), the hazard ratio does not come directly from those medians. Rather: ====
* The Cox model uses all survival times and censoring information across both groups.
* The hazard ratio is an overall measure of relative risk over time, not a simple ratio of medians. wiki.socr.umich.edu<ref>{{cite web|title=wiki.socr.umich.edu|url=https://wiki.socr.umich.edu/index.php/SMHS_SurvivalAnalysis|publisher=wiki.socr.umich.edu|access-date=2026-01-10}}</ref>

If you do convert medians into an approximate HR under simple exponential assumptions, there are formulas — but this is not how the final HR is usually derived in modern trials.

==== ✅ HR is a ratio of instantaneous event rates, not a ratio of survival probabilities at a fixed timepoint. Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Hazard_ratio|publisher=en.wikipedia.org|access-date=2026-01-10}}</ref> ====
✅ It’s estimated via a regression model (Cox model), usually by exponentiating the treatment coefficient β\betaβ. Easy Made Stat<ref>{{cite web|title=Easy Made Stat|url=https://help.easymedstat.com/support/solutions/articles/77000513111-how-to-calculate-hazard-ratios-hr-in-a-cox-model-|publisher=Easy Made Stat|access-date=2026-01-10}}</ref>
✅ HR < 1 indicates benefit for the treatment over control. Statistical Aid: A School of Statistics<ref>{{cite web|title=Statistical Aid: A School of Statistics|url=https://www.statisticalaid.com/hazard-ratio/|publisher=Statistical Aid: A School of Statistics|access-date=2026-01-10}}</ref>

If you want, I can walk you through a numerical example — e.g., how the Cox model estimates β from a small dataset and turns that into an HR. Just let me know!