bio-clinical-biostatistics-survival-analysis

What this skill does

## Version Compatibility

Reference examples tested with: lifelines 0.27+, scikit-survival 0.21+, statsmodels 0.14+, pandas 2.1+, numpy 1.26+. R packages cited (still the SOTA for survival): survival 3.8+, survRM2, cmprsk, riskRegression, mstate, flexsurv, icenReg, rpsftm.

Before using code patterns, verify installed versions match. If versions differ:
- Python: `pip show <package>` then `help(module.function)` to check signatures
- R: `packageVersion('<pkg>')` then `?function_name`

If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.

# Time-to-Event Analysis for Clinical Trials

**"Analyze time-to-event endpoint"** -> Estimate a hazard, survival probability, cumulative incidence, or restricted mean time using a method calibrated to (a) whether proportional hazards holds, (b) whether competing events exist, (c) whether censoring is informative, and (d) which estimand the trial targets under ICH E9(R1).

## The Single Most Important Modern Insight -- PH Almost Never Holds

In modern oncology with checkpoint inhibitors, targeted therapies, crossover, and depleted high-risk subjects over follow-up, **proportional hazards (PH) violations are the rule, not the exception**. The Cross-Pharma NPH Working Group (Lin et al 2020 *Stat Biopharm Res*; Magirr-Burman 2021 *Stat Biopharm Res* 15(2):293) documented systematic PH violations across phase III oncology trials, particularly delayed-effect patterns from checkpoint inhibitors.

The Cox HR is a *time-averaged log-hazard ratio* under PH violation (Xu-O'Quigley 2000), which may or may not be the estimand of interest. RMST (Royston-Parmar 2013) provides a clinically interpretable, hazard-free alternative.

## Algorithmic Taxonomy

| Method | Estimand | Inference | Strength | Fails when |
|--------|----------|-----------|----------|------------|
| Log-rank (unstratified) | Test of S_A(t) = S_B(t) all t | Permutation / asymptotic chi-square | Standard; preserves Type-I under PH | Underpowered under non-PH; treats all events equally |
| Stratified log-rank | Same null within strata, pooled | Asymptotic | Preserves stratification factor from randomisation | Stratification factor must be pre-specified |
| Weighted log-rank G(rho, gamma) | Direction-specific test under non-PH | Asymptotic | High power for delayed/early/middle effects | Weight choice must match true effect time profile; chasing weight = p-hacking |
| Cox PH | Conditional log-HR | Wald, LR, score | Standard; semi-parametric; covariate adjustment | PH violation makes HR a misleading summary; check via cox.zph |
| Stratified Cox | Same; baseline hazards differ by stratum | Wald | Handles non-PH by stratification | Loses inference on stratification variable; cannot interact treatment with strata |
| Time-varying Cox (`tt()`) | Time-dependent log-HR | Wald | Quantifies non-PH explicitly | Interpretability — no single "the HR"; choose g(t) carefully |
| Flexible parametric (Royston-Parmar) | Time-varying log-HR via splines | Wald | Smooth S(t), HR(t); supports extrapolation | Spline choice affects results; software in R `stpm2/stpm3` |
| RMST | Difference in mean survival truncated at tau | Wald with delta or pseudo-obs regression | Hazard-free; clinically interpretable in time units | tau choice; min follow-up across arms constrains tau |
| MaxCombo | Maximum over weighted log-rank family | Asymptotic multivariate normal | Robust to range of NPH patterns | Can reject in opposite directions on same data (Magirr 2022 critique) |
| Fine-Gray subdistribution HR | Conditional subdistribution HR | Wald | Direct CIF modeling | Andersen-Keiding 2012 critique: violates causal hazard semantics |
| Cause-specific Cox | Conditional cause-specific HR | Wald | Causally interpretable | Predicts hazards, not CIFs; need both for CIF prediction |
| Multi-state Cox (mstate) | Transition-specific HRs | Wald | Subsumes competing risks; handles relapse/remission | More complex; more parameters to estimate |
| Andersen-Gill (recurrent) | Rate ratio | Robust (cluster) Wald | Most efficient under exchangeability | Assumes exchangeable events |
| PWP (recurrent) | Conditional event-order HR | Stratified Wald | Handles event-order qualitative heterogeneity | More strata = more parameters; smaller per-stratum n |
| Interval-censored Cox (NPMLE) | Cumulative hazard | Likelihood ratio | Correct for periodic-assessment data | Slower; software in R `icenReg` |

**Postdoc reading list:**

- Royston-Parmar 2013 *BMC Med Res Methodol* 13:152 (RMST as primary)
- Uno et al 2014 *JCO* 32:2380 (RMST in oncology)
- Andersen-Keiding 2012 *Stat Med* 31:1074 (Fine-Gray semantic critique)
- Putter-Schumacher-van Houwelingen 2020 *Biom J* 62:790 (Fine-Gray revisited; reduction factor)
- Putter-Fiocco-Geskus 2007 *Stat Med* 26:2389 (competing risks tutorial)
- Magirr-Burman 2021 *Stat Biopharm Res* 15(2):293 (MaxCombo critique)
- Grambsch-Therneau 1994 *Biometrika* 81:515 (scaled Schoenfeld residuals)
- Buyse-Molenberghs 1998 *Biometrics* 54:1014 (PFS-OS surrogacy framework)
- Lewis et al 2023 (ICH E9(R1) censoring rules and estimand)
- Sun 2006 *The Statistical Analysis of Interval-Censored Failure Time Data* (Springer)

## Decision Tree by Scenario

| Scenario | Recommended approach | Why |
|----------|---------------------|-----|
| OS, drug expected to extend survival uniformly, PH plausible | Stratified log-rank + Cox HR with cox.zph diagnostic | Standard regulatory approach; pre-specify stratification factors from randomisation |
| PFS in immuno-oncology with expected delayed separation | MaxCombo with pre-specified G(0,0), G(0,1), G(1,0), G(1,1); RMST as sensitivity | Robust to NPH; explicit direction check (Magirr-Burman 2021) |
| OS with crossover to active arm | Treatment policy estimand (ITT) primary; RPSFT/IPCW as sensitivity for hypothetical | Lewis 2023; FDA/EMA accept both, ITT is primary |
| Time-to-event with assessment-schedule artifact (periodic scans) | Interval-censored Cox (R `icenReg::ic_par`) | Standard right-censoring at midpoint is biased |
| DOR (duration of response) | KM among responders; censor at next-therapy/death/dropout per pre-specified estimand | Weber 2023 *Pharm Stat*; responder-conditioned = doubly post-randomisation |
| Competing risk: drug efficacy on event A in context of high event-B mortality | Cause-specific Cox for A AND for B; report both; CIF via Aalen-Johansen | Andersen-Keiding 2012; Fine-Gray SHR is not causal |
| Prediction of CIF for clinical decision | Fine-Gray for CIF estimate; cause-specific Cox for etiology | Putter-Fiocco-Geskus 2007 split |
| Multi-state model (alive -> relapse -> death) | mstate framework with transition-specific Cox models | Subsumes competing risks |
| Recurrent events (exacerbations, hospitalisations) | Andersen-Gill with robust SE; cite event-order assumption | Most efficient; falls back to PWP if order matters |
| Single-arm OS extrapolation for HTA | Flexible parametric (Royston-Parmar) + external information | Smooth tail; supports extrapolation beyond trial follow-up |
| Non-PH with crossing hazards | RMST with pre-specified tau; OR multi-state model | HR loses meaning under crossing |

## Cox PH Diagnostics -- The Therneau-Grambsch Test (and Its Pitfalls)

**Goal:** Detect violations of the proportional hazards assumption that would invalidate the Cox HR as a meaningful summary statistic.

**Approach:** Compute scaled Schoenfeld residuals (Grambsch-Therneau 1994); regress against a time-transform g(t) under H0 of zero slope; supplement the asymptotic p-value with a graphical residual plot since the test is sensitive to g(t) choice and sample-size dependent.

```python
from lifelines import CoxPHFitter
from lifelines.statistics import proportional_hazard_test

cph = CoxPHFitter()
cph.fit(df, duration_col='time', event_col='event', formula='treatment + age + baseline_score')
cph.print_summary()

# Schoenfeld