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:

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:

# 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:

# 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:

# 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:

# 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:

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:

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:

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