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


{{DISPLAYTITLE:Hazard Ratio Calculation}}

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== gpt-5-2 ==
=== User: How do you calculate the hazard ratio? ===
How do you calculate the hazard ratio? For instance, in the SLS Phase 3 regal trial, if you look at the November 14, 2022 press release, it says a hazard ratio of 0.636 would determine statistical efficacy. How is that 0.636 calculated?

=== 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!

=== 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.

=== User: Explain how delayed immunotherapy effects can make median-based intuition misleading ===
Explain how delayed immunotherapy effects can make median-based intuition misleading

=== Assistant: This is an important point for cancer vaccines and immunotherapies like GPS, and it’s where a lot of investors and even… ===
This is an important point for cancer vaccines and immunotherapies like GPS, and it’s where a lot of investors (and even clinicians) get tripped up.

I’ll walk through it step-by-step and then tie it directly back to why medians can mislead you.

==== 1️⃣ What “median-based intuition” assumes ====

When people think:

: “Treatment median OS is 12.6 months vs control 8 months — that’s a big win”

they are implicitly assuming:
* The treatment effect starts immediately
* The hazard reduction is constant over time
* The survival curves separate early and stay separated
* The hazard ratio is roughly constant (proportional hazards)

That mental model is basically an exponential survival world.

This is often wrong for immunotherapy.

==== 2️⃣ What delayed immunotherapy effects actually look like ====

For cancer vaccines and immune-based therapies:

===== Early period (first months) =====
* Immune system is still being primed
* Patients may progress or die at similar or even higher rates than control
* Survival curves often overlap or cross

===== Later period =====
* A subset of patients mount a durable immune response
* Their risk of death drops substantially
* A long tail appears in the survival curve

This produces:
* Non-proportional hazards
* A hazard ratio that changes over time

==== 3️⃣ Why medians fail in this situation ====

===== 🔴 Median OS only reflects the 50th percentile =====

The median is the time when half the patients have died.

If:
* Many early events happen before the immune benefit kicks in
* The benefit is concentrated in a subset of responders

Then:
* The median may move very little
* Even if long-term survival is meaningfully improved

====== Example ======
* Control median OS = 8 months
* Treatment median OS = 9 months

That looks unimpressive.

But…
* 2-year OS: - Control = 5% - Treatment = 20%

That is a huge clinical benefit — completely invisible at the median.

==== 4️⃣ How this distorts “median → HR” intuition ====

Recall the shortcut:

HR≈mCmTHR \approx \frac{m_C}{m_T}HR≈mTmC
This assumes:
* Constant hazard ratio
* No delayed effect

But with immunotherapy:
* Early HR ≈ 1.0 or >1.0
* Late HR ≪ 1.0
* Overall HR is a weighted average across time

So:
* Median OS might barely change
* Yet the true HR (from Cox/log-rank) can still be favorable
* Or the opposite can happen: median improves, but HR misses significance

==== 5️⃣ Visual intuition (very important) ====

===== Chemotherapy-like curves =====

<syntaxhighlight>Survival
|
|\
| \
|  \      Control
|   \____
|        \____ Treatment
+---------------------- Time

</syntaxhighlight>
* Early separation
* Median shifts nicely
* HR ≈ median ratio

===== Immunotherapy-like curves =====

<syntaxhighlight>Survival
|
|\
| \_____ Control
|   \
|    \______
|           \_____ Treatment (long tail)
+------------------------- Time

</syntaxhighlight>
* Curves overlap early
* Separate late
* Median barely moves
* Tail drives benefit

==== 6️⃣ Why this matters for statistical testing ====

===== Cox model & log-rank test =====
* Assume proportional hazards
* Lose power when hazards are delayed
* Can underestimate benefit for immunotherapy

This is why immunotherapy trials increasingly look at:
* Landmark survival (e.g., 12-, 18-, 24-month OS)
* Restricted mean survival time (RMST)
* Piecewise hazard ratios
* Milestone survival percentages

==== 7️⃣ REGAL / GPS relevance (without speculating on results) ====

For a WT1-targeted cancer vaccine like GPS:
* Mechanistically, benefit would not be immediate
* Immune priming + memory formation take time
* A subset of patients may derive durable benefit

Therefore:
* Median OS may be a blunt instrument
* HR thresholds derived from medians (like 0.636) may misrepresent true benefit
* Long-term OS and tail separation are often more informative

This is exactly the pattern historically seen with:
* Sipuleucel-T
* Checkpoint inhibitors (early days)
* Several cancer vaccine programs

==== 8️⃣ One-sentence takeaway ====

: Delayed immunotherapy effects shift benefit into the survival tail, where medians and median-derived intuition break down — making long-term survival metrics far more informative than early median OS.

If you want, next we can:
* Quantify how much tail benefit is needed to “beat” a weak median
* Compare HR vs RMST numerically
* Walk through a toy dataset showing median ≈ unchanged but OS benefit is real

Just say the word.

== References ==
<references />


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