time-to-event-methods

# Time-to-Event Methods ## When to Use This Skill - Selecting appropriate analysis methods for survival endpoints - Handling non-proportional hazards scenarios - Implementing weighted logrank tests - Designing MaxCombo tests - Using RMST or milestone endpoints ## Analysis Methods Overview ### Standard Logrank Test **When Optimal:** - Proportional hazards assumption holds - Treatment effect constant over time **Formula:** ``` Z = Σ(O_trt - E_trt) / √(Var) ``` **simtrial Implementation:** ```r data |> wlr(weight = fh(rho = 0, gamma = 0)) ``` ### Fleming-Harrington Weighted Logrank **Weight Function:** ``` w(t) = S(t)^ρ × (1 - S(t))^γ ``` **Parameter Effects:** | ρ | γ | Emphasis | Best For | |---|---|----------|----------| | 0 | 0 | Uniform (standard LR) | Proportional hazards | | 0 | 0.5 | Moderate late | Moderate delayed effect | | 0 | 1 | Strong late | Strong delayed effect | | 1 | 0 | Early | Early divergence | | 0.5 | 0.5 | Balanced | Crossing hazards | **simtrial Implementation:** ```r # Late emphasis data |> wlr(weight = fh(rho = 0, gamma = 0.5)) # Early emphasis data |> wlr(weight = fh(rho = 1, gamma = 0)) ``` ### Magirr-Burman (MB) Weights **Design:** Zero weight before delay, then increasing weight. **Parameters:** - `delay`: Time before weights increase - `w_max`: Maximum weight cap **Formula:** ``` w(t) = min(w_max, S(min(t, τ*))^(-1)) ``` **When to Use:** - Known delay in treatment effect - Clear scientific rationale for delay period **simtrial Implementation:** ```r # 4-month delay, max weight 2 data |> wlr(weight = mb(delay = 4, w_max = 2)) # Unlimited weight growth data |> wlr(weight = mb(delay = 6, w_max = Inf)) ``` ### Early Zero Weights (Xu et al., 2017) **Design:** Exactly zero weight for early period, then standard logrank. **When to Use:** - Want to completely ignore early period - Regulatory acceptance of early exclusion **simtrial Implementation:** ```r # Zero weight for first 6 months data |> wlr(weight = early_zero(early_period = 6)) ``` ### MaxCombo Test **Concept:** Combine multiple weighted logrank tests, take maximum Z-score. **Advantages:** - Robust across NPH patterns - Maintains power under uncertainty - Single pre-specified p-value **Common Combinations:** | Combo | Tests | Use Case | |-------|-------|----------| | 2-test | FH(0,0) + FH(0,1) | Unknown late effect | | 3-test | FH(0,0) + FH(0,0.5) + FH(0.5,0.5) | Comprehensive | | Custom | FH(0,0) + FH(0,1) + FH(1,1) | Maximum robustness | **simtrial Implementation:** ```r # Two-test MaxCombo data |> maxcombo(rho = c(0, 0), gamma = c(0, 1)) # Three-test MaxCombo data |> maxcombo(rho = c(0, 0, 0.5), gamma = c(0, 0.5, 0.5)) ``` **Correlation Handling:** MaxCombo accounts for correlation between tests using multivariate normal distribution. ### Restricted Mean Survival Time (RMST) **Definition:** Area under survival curve up to time τ. **Formula:** ``` RMST(τ) = ∫₀^τ S(t) dt ``` **Advantages:** - Interpretable (expected survival time) - Valid under non-PH - No proportionality assumption **Considerations:** - Choice of τ is critical - τ must be within follow-up - Less powerful than logrank under PH **simtrial Implementation:** ```r data |> rmst(tau = 24) # RMST at 24 months ``` ### Milestone Analysis **Definition:** Compare survival probability at fixed time point. **Test Statistic:** ``` Z = (S_trt(t*) - S_ctrl(t*)) / SE ``` **Advantages:** - Easy to interpret - Clinically meaningful time point - Valid under non-PH **simtrial Implementation:** ```r data |> milestone(ms_time = 12, test_type = "naive") ``` ## Non-Proportional Hazards Patterns ### Delayed Treatment Effect **Pattern:** HR = 1 initially, then HR < 1 **Analysis Recommendations:** 1. Primary: FH(0, γ) with γ > 0 or MaxCombo 2. Sensitivity: Standard logrank 3. Alternative: RMST with appropriate τ **Simulation Setup:** ```r fail_rate <- data.frame( stratum = rep("All", 4), period = rep(1:2, 2), treatment = c(rep("control", 2), rep("experimental", 2)), duration = c(4, 100, 4, 100), # 4-month delay rate = log(2) / c(12, 12, 12, 18) # HR=1 then HR=0.67 ) ``` ### Crossing Hazards **Pattern:** Early benefit reverses over time **Analysis Recommendations:** 1. Consider if crossing is clinically meaningful 2. FH(0.5, 0.5) may be appropriate 3. MaxCombo provides robustness 4. RMST with carefully chosen τ ### Diminishing Effect **Pattern:** Strong early effect that wanes **Analysis Recommendations:** 1. FH(ρ, 0) with ρ > 0 2. Early milestone analysis 3. Consider if effect is clinically durable ### Cure Model **Pattern:** Proportion of patients cured (never event) **Analysis Recommendations:** 1. Standard logrank often adequate 2. Long-term milestone helpful 3. Consider cure fraction estimation ## Method Selection Algorithm ``` START │ ├─ Is proportional hazards expected? │ ├─ Yes → Standard logrank FH(0,0) │ └─ No → Continue │ ├─ Is delayed effect expected? │ ├─ Yes, delay known → MB weights │ ├─ Yes, delay uncertain → FH(0, 0.5) or MaxCombo │ └─ No → Continue │ ├─ Is crossing possible? │ ├─ Yes → RMST or FH(0.5, 0.5) │ └─ No → Continue │ ├─ Maximum robustness needed? │ ├─ Yes → MaxCombo │ └─ No → FH(0, γ) based on expected pattern │ END ``` ## Power Comparison Under Different Scenarios ### Proportional Hazards (HR = 0.7) | Method | Relative Power | |--------|---------------| | Logrank FH(0,0) | 100% (optimal) | | FH(0, 0.5) | ~95% | | MaxCombo | ~98% | | RMST | ~90% | ### Delayed Effect (3-month delay, HR = 0.6 after) | Method | Relative Power | |--------|---------------| | Logrank FH(0,0) | 70% | | FH(0, 0.5) | 90% | | MB(delay=3) | 95% | | MaxCombo | 92% | ### Crossing Hazards | Method | Relative Power | |--------|---------------| | Logrank FH(0,0) | Variable | | FH(0.5, 0.5) | Better | | RMST | Depends on τ | | MaxCombo | Robust | ## Practical Considerations ### Regulatory Acceptance - FDA generally accepts weighted logrank with justification - Pre-specification is critical - MaxCombo gaining acceptance - RMST as sensitivity analysis ### Pre-specification Requirements 1. Analysis method must be specified before unblinding 2. Weight parameters (ρ, γ) must be fixed 3. MaxCombo test components must be defined 4. τ for RMST must be justified ### Sample Size Implications - Weighted tests may require larger sample under PH - MaxCombo has slight efficiency loss - Consider this in planning ## Best Practices 1. **Primary Analysis**: Choose method aligned with expected NPH pattern 2. **Sensitivity Analyses**: Include standard logrank and alternatives 3. **Justification**: Document scientific rationale for method choice 4. **Simulation**: Validate power across plausible scenarios 5. **Pre-specification**: Lock method before any data review