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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

• 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

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:

1. 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
2. Model the log hazard as a linear function of treatment The model estimates a coefficient (β\betaβ) for the treatment effect.
3. Exponentiate that coefficient The hazard ratio = eβe^{\beta}eβ. Easy Made Stat • 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

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

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

✅ It’s estimated via a regression model (Cox model), usually by exponentiating the treatment coefficient β\betaβ. Easy Made Stat ✅ HR < 1 indicates benefit for the treatment over control. Statistical Aid: A School of Statistics

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​=mT​ln(2)​
• Control hazard: λC=ln⁡(2)mC\lambda_C = \frac{\ln(2)}{m_C}λC​=mC​ln(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=λC​λT​​=ln(2)/mC​ln(2)/mT​​=mT​mC​​

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.

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• 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≈mT​mC​​ 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

Survival
|
|\
| \
|  \      Control
|   \____
|        \____ Treatment
+---------------------- Time

• Early separation
• Median shifts nicely
• HR ≈ median ratio

Immunotherapy-like curves

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

• 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.