Survival Analysis

Punchline

For time-to-event data:

1. Kaplan-Meier (KM) curves — visualize survival probability over time for each group
2. Log-rank test — test whether the curves are statistically significantly different
3. Cox proportional hazards model — estimate the hazard ratio (HR)
   • Check PH assumption: Schoenfeld residuals plot + log-log survival plot
   • If violated: use Restricted Mean Survival Time (RMST) instead
4. HR < 1 = treatment group has better survival

Definition

Survival analysis refers to a class of statistical methods for analyzing time-to-event data, where the outcome is the time from a defined origin (e.g., randomization) to the occurrence of a specified event (e.g., death, disease progression).

Key characteristics of survival data:

• The outcome is time until an event, not just whether the event occurs
• Data are often censored, meaning the exact event time is not observed for all individuals

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Kaplan–Meier Estimation and Log-Rank Test

Kaplan–Meier Estimator

The Kaplan–Meier (KM) method is a nonparametric estimator of the survival function: [ S(t) = P(T > t) ]

• It produces a step function that decreases at observed event times
• It accounts for right-censored data, allowing inclusion of patients with incomplete follow-up
• The KM curve provides an estimate of the probability of remaining event-free over time

This is the standard method for visualizing survival outcomes in clinical trials.

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Censoring

Censoring occurs when the exact event time is unknown but is known to exceed a certain time.

Common types:

• Administrative censoring: study ends before the event occurs
• Loss to follow-up: patient exits the study early

A key assumption:

Censoring is non-informative, meaning censored individuals have the same future risk as those remaining under observation.

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Log-Rank Test

The log-rank test is a nonparametric hypothesis test used to compare survival distributions between groups.

• Null hypothesis: survival functions are identical across groups
• It compares observed vs expected events over time
• Most powerful when the proportional hazards assumption holds

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Cox Proportional Hazards Model

The Cox model is a semi-parametric regression model for time-to-event data:

[ h(t \mid X) = h_0(t)\exp(\beta X) ]

• ( h(t \mid X) ): hazard function (instantaneous event rate at time (t))
• ( h_0(t) ): baseline hazard (unspecified)
• ( \beta ): regression coefficients

Key concepts

• Hazard: instantaneous risk of experiencing the event at time (t), given survival up to (t)
• Hazard ratio (HR): relative hazard between groups
• Risk set: individuals still at risk of the event just prior to time (t)

Assumption

The Cox model assumes proportional hazards, meaning the hazard ratio between groups is constant over time.

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Clinical Context (Example: Oncology)

In oncology, survival outcomes are often analyzed using endpoints such as:

• Overall Survival (OS): time to death from any cause
• Progression-Free Survival (PFS): time to disease progression or death

Clinical covariates (e.g., TNM staging system) are often included in models:

• T: tumor size
• N: lymph node involvement
• M: metastasis

These factors may be used as prognostic variables in survival models.

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Learn from the papers

Reading Guide (Survival Analysis Papers)

Study & Endpoint

1. What is the primary endpoint? How is it defined (event, time origin)?
2. What are the secondary endpoints?

Population 3. What analysis population is used (ITT, per-protocol, subgroup)?

Censoring & Follow-up 4. What are the censoring rules? 5. What is the follow-up duration (median or range)?

Methods 6. What statistical methods are used (KM, log-rank, Cox)? 7. Is the analysis stratified? If yes, by what factors?

Results Interpretation 8. What is the hazard ratio (HR) and 95% CI? 9. How do the Kaplan–Meier curves behave (separation, crossing, convergence)? 10. Does the proportional hazards assumption appear reasonable?

Subgroup & Robustness 11. Are subgroup effects consistent? 12. Are interaction tests performed?

Clinical Interpretation 13. Is the effect clinically meaningful? 14. Any key limitations or biases?

Paper Summary 1

Reference: Pembrolizumab versus Chemotherapy for PD-L1–Positive Non–Small-Cell Lung Cancer ^[1]

Category	Item	Summary
Study	Disease	Advanced NSCLC
	Population	PD-L1 ≥ 50%
	Design	Randomized controlled trial
	Comparison	Pembrolizumab vs Chemotherapy
Endpoints	Primary	Progression-Free Survival (PFS)
	Secondary	Overall Survival (OS), Objective Response Rate (ORR), Safety
Censoring	Rules	Censored if alive without progression at last follow-up or lost to follow-up
	Assumption	Assume non-informative censoring
Methods	Survival estimation	Kaplan–Meier
	Group comparison	Stratified log-rank test
	Effect estimation	Cox proportional hazards model
Results	HR (PFS)	0.50 (95% CI: 0.37–0.68)
	P-value	< 0.001
	Interpretation	The Pembrolizumab group has ~50% lower risk of progression or death
KM Curve	Pattern	Early and sustained separation
	PH assumption	Reasonable (no crossing)
	Conclusion	Pembrolizumab consistently superior
Subgroup	Consistency	Generally consistent across subgroups
	Limitation	Some wide CIs; no clear effect modification
Clinical	Interpretation	Strong, clinically meaningful benefit
	Signal	Clean survival signal (early + sustained separation)
Notes	Limitations	Subgroups exploratory; PH not formally tested; OS may be immature

Paper Summary 2

Reference: Trastuzumab Deruxtecan in Previously Treated HER2-Low Advanced Breast Cancer ^[2]

Category	Item	Summary
Study	Disease	Unresectable or metastatic HER2-low breast cancer
	Population	Patients with HER2-low disease, defined as IHC 1+ or IHC 2+/ISH-negative, previously treated with 1 or 2 lines of chemotherapy in the metastatic setting; 494 patients had hormone receptor (HR)-positive disease and 63 had HR-negative disease
	Design	Phase 3, randomized, open-label trial
	Comparison	Trastuzumab deruxtecan vs physician’s choice of chemotherapy
Endpoints	Primary	Progression-free survival (PFS) by blinded independent central review in the HR-positive cohort
	Secondary	Overall survival (OS) in the HR-positive cohort; PFS in all patients; OS in all patients; objective response; safety
Censoring	Rules	For PFS, patients without documented progression or death were censored at the last adequate tumor assessment; sensitivity analyses also examined choices such as not censoring at new anticancer therapy, handling progression after missed assessments, and alternative censoring for randomized-but-untreated patients
	Assumption	Assume non-informative censoring
Methods	Survival estimation	Kaplan–Meier
	Group comparison	Stratified log-rank test
	Effect estimation	Stratified Cox proportional-hazards model
	Stratification factors	HER2 IHC status (1+ vs 2+/ISH-negative), number of prior chemotherapy lines in metastatic disease (1 vs 2), and HR/CDK4/6 status
Results	HR (PFS, primary endpoint)	0.51 (95% CI: 0.40–0.64) in the HR-positive cohort
	P-value	< 0.0001
	Interpretation	The trastuzumab deruxtecan group had about a 49% lower hazard of progression or death than the chemotherapy group in the HR-positive cohort
	Median PFS	10.1 months vs 5.4 months in the HR-positive cohort
	Key OS result	HR for OS in the HR-positive cohort: 0.64 (95% CI: 0.48–0.86; P = 0.0028)
KM Curve	Pattern	Early and sustained separation favoring trastuzumab deruxtecan
	PH assumption	Appears reasonable from the reported KM curves; no major crossing emphasized in the main report
	Conclusion	Trastuzumab deruxtecan consistently outperformed chemotherapy on PFS and OS
Subgroup	Consistency	Benefit was generally consistent across prespecified subgroups
	Limitation	The HR-negative subgroup was small, so those results are exploratory and less precise
Clinical	Interpretation	Strong, clinically meaningful improvement in both PFS and OS
	Signal	Clear efficacy signal with improvement in both the primary endpoint and key secondary survival endpoints
Notes	Limitations	Open-label design; HR-negative subgroup underpowered for firm conclusions; PH assumption was not formally highlighted in the main paper; HRQoL was not powered for definitive conclusions

Paper Summary 3

Reference: Osimertinib in Untreated EGFR-Mutated Advanced Non–Small-Cell Lung Cancer ^[3]

Category	Item	Summary
Study	Disease	Advanced (locally advanced or metastatic) NSCLC
	Population	Treatment-naïve patients with EGFR-mutated NSCLC (exon 19 deletion or L858R)
	Design	Phase 3, randomized, double-blind controlled trial
	Comparison	Osimertinib vs standard EGFR-TKI (gefitinib or erlotinib)
Endpoints	Primary	Progression-Free Survival (PFS) (investigator-assessed)
	Secondary	Overall Survival (OS), Objective Response Rate (ORR), Duration of Response, Safety
Censoring	Rules	Patients without progression or death were censored at the date of last tumor assessment; censoring also applied for patients starting new anticancer therapy before progression
	Assumption	Assumes non-informative censoring (implicit)
Methods	Survival estimation	Kaplan–Meier
	Group comparison	Stratified log-rank test (by mutation type [exon 19 vs L858R] and race [Asian vs non-Asian])
	Effect estimation	Stratified Cox proportional hazards model
Results	HR (PFS)	0.46 (95% CI: 0.37–0.57)
	P-value	< 0.001
	Interpretation	The osimertinib group had about a 54% lower hazard of progression or death compared to standard EGFR-TKI in full analysis set
	Median PFS	18.9 months vs 10.2 months
KM Curve	Pattern	Early and sustained separation favoring osimertinib
	PH assumption	Reasonable (no major crossing observed)
	Conclusion	Osimertinib consistently outperformed standard EGFR-TKIs in delaying progression
Subgroup	Consistency	Treatment benefit consistent across major subgroups (mutation type, race, CNS metastases)
	Limitation	Some subgroups have wider CIs; subgroup analyses are exploratory
Clinical	Interpretation	Strong and clinically meaningful improvement in PFS
	Signal	Large magnitude benefit with durable separation of survival curves
Notes	Limitations	OS immature at initial publication; crossover and subsequent therapies may confound OS; PH assumption not formally tested

Paper Summary 4

Non-Proportional Hazards Reference: Nivolumab versus Docetaxel in Advanced Nonsquamous Non–Small-Cell Lung Cancer^[4]

Category	Item	Summary
Study	Disease	Advanced nonsquamous NSCLC
	Population	Patients with advanced NSCLC who had disease progression during or after platinum-based chemotherapy
	Design	Phase 3, randomized, open-label trial
	Comparison	Nivolumab vs Docetaxel
Endpoints	Primary	Overall Survival (OS)
	Secondary	Objective Response Rate (ORR), Progression-Free Survival (PFS), Safety
Censoring	Rules	Patients alive at last follow-up were censored at last known alive date; for PFS, patients without progression or death were censored at last tumor assessment
	Assumption	Assumes non-informative censoring (implicit)
Methods	Survival estimation	Kaplan–Meier
	Group comparison	Stratified log-rank test
	Effect estimation	Stratified Cox proportional hazards model
	Stratification factors	PD-L1 expression level, prior maintenance therapy
Results	HR (OS, primary endpoint)	0.73 (95% CI: 0.59–0.89)
	P-value	0.002
	Interpretation	Nivolumab reduced the hazard of death by ~27% compared to docetaxel
	Median OS	12.2 months vs 9.4 months
KM Curve	Pattern	Delayed separation: curves overlap early, then diverge
	PH assumption	Likely violated (non-proportional hazards suggested by delayed effect)
	Conclusion	Nivolumab shows survival benefit despite delayed treatment effect
Subgroup	Consistency	Greater benefit observed in patients with higher PD-L1 expression
	Limitation	Some subgroups have wide CIs; exploratory interpretation
Clinical	Interpretation	Clinically meaningful OS benefit with improved tolerability vs chemotherapy
	Signal	Delayed but durable survival benefit characteristic of immunotherapy
Notes	Limitations	Evidence of non-proportional hazards; HR represents an average effect over time; alternative methods (e.g., RMST) not used; open-label design

Paper Summary 5

Reference: Pembrolizumab plus Chemotherapy in Metastatic Non–Small-Cell Lung Cancer^[5]

Category	Item	Summary
Study	Disease	Metastatic nonsquamous NSCLC
	Population	Previously untreated patients with metastatic nonsquamous NSCLC, without EGFR or ALK alterations
	Design	Phase 3, randomized, double-blind, placebo-controlled trial
	Comparison	Pembrolizumab + chemotherapy vs placebo + chemotherapy
Endpoints	Primary	Overall Survival (OS) and Progression-Free Survival (PFS)
	Secondary	Objective Response Rate (ORR), Duration of Response, Safety
Censoring	Rules	Patients without event were censored at last known alive date (OS) or last tumor assessment (PFS); censoring applied for patients without progression or death at cutoff
	Assumption	Assumes non-informative censoring (implicit)
Methods	Survival estimation	Kaplan–Meier
	Group comparison	Stratified log-rank test
	Effect estimation	Stratified Cox proportional hazards model
	Stratification factors	PD-L1 tumor proportion score (<1% vs 1–49% vs ≥50%), choice of chemotherapy (cisplatin vs carboplatin)
Results	HR (OS, primary endpoint)	0.49 (95% CI: 0.38–0.64)
	P-value	< 0.001
	Interpretation	Pembrolizumab plus chemotherapy reduced the hazard of death by ~51% compared to chemotherapy alone in the overall population
	Median OS	Not reached vs 11.3 months (at initial analysis)
	HR (PFS)	0.52 (95% CI: 0.43–0.64)
KM Curve	Pattern	Early separation with sustained benefit; slight early overlap possible but no crossing
	PH assumption	Generally reasonable; no strong evidence of violation
	Conclusion	Combination therapy consistently improved survival outcomes
Subgroup	Consistency	Benefit observed across PD-L1 subgroups (including <1%)
	Limitation	Magnitude of effect varies by PD-L1 expression; subgroup analyses exploratory
Clinical	Interpretation	Strong and clinically meaningful improvement in both OS and PFS
	Signal	Robust benefit across populations, including those with low PD-L1 expression
Notes	Limitations	Early OS data immature (median not reached); subgroup analyses exploratory; PH assumption not formally tested

5-Paper Survival Analysis Comparison Sheet

Paper #	Trial	Disease / Setting	Comparison	Primary Endpoint(s)	Main Survival Result	KM Pattern	PH Assumption	Key Survival Lesson
1	reck2016pembrolizumab [1:1]	Advanced NSCLC, PD-L1 ≥50%, first-line	Pembrolizumab vs Chemotherapy	PFS	HR (PFS) = 0.50 (95% CI: 0.37–0.68), p < 0.001	Early and sustained separation	Reasonable	Textbook Kaplan–Meier + Cox example; clean PH case
2	modi2022trastuzumab [2:1]	Unresectable/metastatic HER2-low breast cancer	Trastuzumab deruxtecan vs physician’s choice chemotherapy	PFS in HR-positive cohort	HR (PFS, HR-positive cohort) = 0.51 (95% CI: 0.40–0.64), p < 0.0001	Early and sustained separation	Reasonable	Strong example of defining endpoint and analysis population precisely
3	soria2017osimertinib [3:1]	EGFR-mutated advanced NSCLC, first-line	Osimertinib vs gefitinib/erlotinib	PFS	HR (PFS, FAS) = 0.46 (95% CI: 0.37–0.57), p < 0.001	Early and durable separation	Reasonable	Very clean targeted-therapy survival result; strong Cox interpretation
4	borghaei2015nivolumab [4:1]	Advanced nonsquamous NSCLC after platinum chemotherapy	Nivolumab vs docetaxel	OS	HR (OS) = 0.73 (95% CI: 0.59–0.89), p = 0.002	Delayed separation; early overlap	Likely violated	Example where HR is an average over time and may hide delayed immunotherapy effect
5	gandhi2018pembrolizumab [5:1]	Metastatic nonsquamous NSCLC, untreated, no EGFR/ALK alteration	Pembrolizumab + chemotherapy vs placebo + chemotherapy	OS and PFS	HR (OS) = 0.49 (95% CI: 0.38–0.64), p < 0.001; HR (PFS) = 0.52 (95% CI: 0.43–0.64)	Early separation with sustained benefit; no major crossing	Generally reasonable	Example of subgroup heterogeneity without obvious PH violation

Main Takeaways Across the 5 Papers

Theme	What I Learned
Canonical workflow	Most papers follow: Kaplan–Meier curves + log-rank test + Cox model + subgroup forest plot
Need to specify endpoint and population	Hazard ratios must be tied to a specific endpoint and analysis set (e.g., PFS in FAS, or PFS in HR-positive cohort)
PH can hold cleanly	reck2016pembrolizumab [1:2], modi2022trastuzumab[2:2], and soria2017osimertinib[3:2] are examples where Cox HR is straightforward to interpret
HR can be imperfect	borghaei2015nivolumab[4:2] shows delayed effect and likely non-proportional hazards, so HR is only an average summary
Subgroup heterogeneity vs non-PH	gandhi2018pembrolizumab[5:2] shows subgroup differences (PD-L1), which do not necessarily imply PH violation
Regulatory-style reporting	Survival results are typically reported using HR, 95% CI, p-value, median survival, and KM plots—even when assumptions are imperfect

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1. https://www.nejm.org/doi/full/10.1056/NEJMoa1606774 ↩︎ ↩︎ ↩︎

2. https://www.nejm.org/doi/full/10.1056/NEJMoa2203690 ↩︎ ↩︎ ↩︎

3. https://www.nejm.org/doi/full/10.1056/NEJMoa1713137 ↩︎ ↩︎ ↩︎

4. https://www.nejm.org/doi/full/10.1056/NEJMoa1507643 ↩︎ ↩︎ ↩︎

5. https://www.nejm.org/doi/full/10.1056/NEJMoa1801005 ↩︎ ↩︎ ↩︎