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:

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 0 or MaxCombo

1. Sensitivity: Standard logrank
2. Alternative: RMST with appropriate τ

Simulation Setup:

fail_rate  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

## Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

- **Author:** [choxos](https://github.com/choxos)
- **Source:** [choxos/BiostatAgent](https://github.com/choxos/BiostatAgent)
- **License:** MIT

Install and usage instructions live in the source repository linked above.