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

=== Assistant: Great question — that highlighted sentence is getting at an approximation method that’s often used for trial design and… ===
Great question — that highlighted sentence is getting at an approximation method that’s often used for trial design and intuition, but not for the final regulatory analysis. Let’s unpack it cleanly.

==== If you assume survival times follow an exponential distribution, then: ====
* The hazard rate (λ) is constant over time
* Survival function: S(t)=e−λtS(t) = e^{-\lambda t}S(t)=e−λt

Under this assumption, the median survival time (m) satisfies:

S(m)=0.5=e−λmS(m) = 0.5 = e^{-\lambda m}S(m)=0.5=e−λm
Solving:

λ=ln⁡(2)m\lambda = \frac{\ln(2)}{m}λ=mln(2)

==== If both arms are exponential: ====
* Treatment hazard: λT=ln⁡(2)mT\lambda_T = \frac{\ln(2)}{m_T}λT=mTln(2)
* Control hazard: λC=ln⁡(2)mC\lambda_C = \frac{\ln(2)}{m_C}λC=mCln(2)

The hazard ratio is:

HR=λTλC=ln⁡(2)/mTln⁡(2)/mC=mCmTHR = \frac{\lambda_T}{\lambda_C}
     = \frac{\ln(2)/m_T}{\ln(2)/m_C}
     = \frac{m_C}{m_T}HR=λCλT=ln(2)/mCln(2)/mT=mTmC
===== : =====

==== Suppose trial designers expect: ====
* Control median OS ≈ 8 months
* Treatment median OS ≈ 12.6 months

Then:

HR≈812.6=0.635HR \approx \frac{8}{12.6} = 0.635HR≈12.68=0.635
➡️ That’s essentially 0.636

So the 0.636 was not “measured” — it was reverse-engineered from an assumed survival improvement under exponential survival.

==== ### ====
* Power calculations
* Event-number planning
* Sample size justification
* Prespecifying a clinically meaningful effect size

Regulators and statisticians need one number that represents:

: 

HR is convenient because:
* It’s scale-free
* It works naturally with log-rank tests
* It maps directly to Cox models

===== - Final hazard ratio reporting =====
* Efficacy determination
* Labeling claims

Why? Because:
* Real survival curves are not exponential
* Hazards often change over time
* Crossing curves and delayed effects break the assumption

The actual HR is estimated from:
* All patient-level time-to-event data
* Censoring
* The Cox proportional hazards model

==== When the Nov 14, 2022 press release says: ====

: 

It means:
* Before the trial started, SELLAS assumed something like: - Control OS ≈ X - Treatment OS ≈ Y
* Those assumptions implied: HR=XY≈0.636HR = \frac{X}{Y} \approx 0.636HR=YX≈0.636
* The trial was powered so that if the true HR ≤ 0.636, the log-rank test would reach statistical significance at the prespecified alpha.

It does not mean:
* 0.636 came from interim data
* 0.636 was computed from medians
* 0.636 is a hard biological threshold

==== : ====

If you want, I can:
* Reconstruct the exact OS assumptions likely used in REGAL
* Show how many events were needed to detect HR = 0.636
* Explain how delayed immunotherapy effects can make median-based intuition misleading

Just tell me where you want to go next.