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Sample Size for Survival Analysis

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Comprehensive power analysis for clinical trials with time-to-event endpoints (e.g., overall survival, progression-free survival, time to recurrence).

1. When to Use This Method

Survival analysis sample size calculations are appropriate when your primary endpoint is time-to-event, and the analysis will use either:

•Log-rank test — comparing two Kaplan-Meier survival curves
•Cox proportional hazards model — estimating hazard ratios with covariate adjustment

Appropriate For

✓Overall survival (OS), progression-free survival (PFS)
✓Time to recurrence, time to treatment failure
✓Event-driven trials with censoring expected
✓Phase III oncology trials, cardiovascular outcome studies

Not Appropriate For

✗Fixed time-point binary outcomes (use proportion methods)
✗Continuous endpoints without time component
✗Recurrent events (use different methods)

2. Mathematical Formulation

2.1 Schoenfeld Formula (Required Events)

The Schoenfeld (1981) formula calculates the total number of events (d) required:

d = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2}{p_1 \cdot p_2 \cdot (\log \text{HR})^2}

For two-sided test with allocation ratio p₁:p₂

For equal allocation ( $p_1 = p_2 = 0.5$ ), this simplifies to:

d = \frac{4(z_{1-\alpha/2} + z_{1-\beta})^2}{(\log \text{HR})^2}

Equal allocation formula

Where:

$d$ = total number of events required
$z_{1-\alpha/2}$ = critical value for Type I error (1.96 for α=0.05)
$z_{1-\beta}$ = critical value for power (0.842 for 80%, 1.282 for 90%)
$\text{HR}$ = hazard ratio (treatment/control)
$p_1, p_2$ = allocation proportions to each group

2.2 Freedman Formula (Alternative)

The Freedman (1982) formula uses a slightly different parameterization:

d = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2 \cdot (1 + \text{HR})^2}{(\text{HR} - 1)^2}

Freedman formula for equal allocation

The Freedman formula tends to give slightly more conservative (larger) event estimates compared to Schoenfeld, particularly for moderate effect sizes.

2.3 Converting Events to Sample Size

Once the required number of events is known, convert to total sample size:

N = \frac{d}{P_{\text{event}}}

Total sample size calculation

The overall event probability depends on the survival function in each arm and the follow-up duration. For exponential survival:

P_{\text{event}} = p_1 \cdot P_1 + p_2 \cdot P_2

Weighted average event probability

Where $P_i = 1 - S_i(t)$ is the probability of event by time t in group i.

2.4 Lachin-Foulkes Method (With Accrual)

For trials with a defined accrual period ( $T_a$ ) and additional follow-up ( $T_f$ ), the expected probability of event under exponential survival is:

P_i = 1 - \frac{e^{-\lambda_i T_f}(1 - e^{-\lambda_i T_a})}{\lambda_i T_a}

Event probability with uniform accrual

Where:

$T_a$ = accrual (enrollment) period
$T_f$ = additional follow-up after accrual closes
$\lambda_i = \frac{\log(2)}{\text{median}_i}$ = hazard rate in group i

2.5 Unequal Allocation

For allocation ratio k:1 (treatment:control):

d = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2 \cdot (k+1)^2}{k \cdot (\log \text{HR})^2}

Events for k:1 allocation

Allocation	Multiplier vs 1:1	Common Use Case
1:1	1.00×	Most efficient, standard design
2:1	1.13×	Improve treatment arm safety data
3:1	1.33×	Rare disease, maximize treatment exposure

2.6 Non-Inferiority Design

For non-inferiority trials testing $H_0: \text{HR} \geq \text{HR}_{NI}$ vs $H_1: \text{HR} < \text{HR}_{NI}$ :

d = \frac{4(z_{1-\alpha} + z_{1-\beta})^2}{(\log \text{HR}_{NI})^2}

One-sided test (assuming HR₀ = 1)

Common non-inferiority margins:

• $\text{HR}_{NI} = 1.25$ — Preserves 50% of log(HR) effect
• $\text{HR}_{NI} = 1.30$ — More lenient margin
• $\text{HR}_{NI} = 1.15$ — Stricter margin (large trials)

2.7 Dropout Adjustment

To account for loss to follow-up at rate $r$ :

N_{\text{adjusted}} = \frac{N}{1 - r}

Sample size inflated for dropout

Alternatively, incorporate dropout into the event probability calculation by treating dropout as an additional censoring mechanism.

3. Assumptions

Assumption	How to Check	If Violated
Proportional hazards	Schoenfeld residuals; log-log survival plots parallel	Use weighted log-rank (Fleming-Harrington), RMST, or piecewise methods
Non-informative censoring	Clinical review of dropout reasons; compare censored vs events	Sensitivity analyses; IPCW methods
Exponential survival (for conversion formulas)	Compare KM curve to exponential fit; historical data	Use simulation-based methods or Weibull assumptions
Uniform accrual	Review enrollment projections; historical accrual patterns	Piecewise accrual models; simulation
Stable hazard ratio estimate	Review prior data; Phase II confidence intervals	Power for range of HRs; adaptive designs

Understanding Hazard Ratios

The hazard ratio is the primary effect size measure. Key interpretation:

Hazard Ratio	Risk Reduction	Clinical Interpretation	Events Needed (80% power)
0.50	50%	Strong effect (rare in practice)	~65
0.60	40%	Large effect (breakthrough therapies)	~120
0.70	30%	Moderate-large effect	~246
0.75	25%	Moderate effect (typical target)	~380
0.80	20%	Small-moderate effect	~630
0.85	15%	Small effect (large trials needed)	~1,150

4. Regulatory Guidance

FDA Guidance

ICH E9 (Statistical Principles): "For survival analysis, the sample size is usually expressed in terms of the number of events (e.g., deaths) rather than the number of subjects to be randomized."

Oncology Guidance: FDA expects event-driven designs for OS and PFS endpoints. Sample size justification should include assumptions about median survival, accrual pattern, and expected hazard ratio with supporting evidence.

Non-Inferiority Trials: FDA guidance recommends preserving at least 50% of the historical treatment effect when defining the non-inferiority margin for survival endpoints.

EMA Guidance

CPMP/EWP/2158/99: "The number of events should be the primary determinant of sample size for time-to-event endpoints. The trial duration and sample size should be planned to achieve the target number of events."

Proportional Hazards: EMA expects assessment of proportional hazards assumption and pre-specified alternative analyses if violations are anticipated (e.g., immunotherapy trials with delayed effects).

5. Validation Against Industry Standards

Zetyra's survival power calculations have been validated against industry-standard software. All comparisons use Schoenfeld formula with equal allocation.

Scenario	α	Power	HR	Zetyra	PASS 2024	nQuery 9.5
Log-rank (Schoenfeld)	0.05	80%	0.70	246	246	246
Log-rank (Schoenfeld)	0.05	90%	0.70	329	329	329
Log-rank (Schoenfeld)	0.05	80%	0.75	380	380	380
Log-rank (Schoenfeld)	0.01	90%	0.70	436	436	436
Log-rank (Freedman)	0.05	80%	0.70	259	259	259
2:1 allocation	0.05	80%	0.70	277	277	277

6. Example SAP Language

Overall Survival Primary Endpoint (Oncology)

The primary endpoint is overall survival (OS), defined as the time from randomization to death from any cause. The trial is designed to detect a hazard ratio of 0.70 (30% reduction in risk of death) favoring the experimental arm compared to control. With a two-sided log-rank test at significance level α = 0.05 and 80% power, a total of 246 events are required (Schoenfeld formula). Assuming a median OS of 12 months in the control arm, an accrual period of 24 months with uniform enrollment, an additional follow-up period of 12 months, and a 10% dropout rate, approximately 440 subjects (220 per arm) are required to observe 246 events. The primary analysis will use the log-rank test stratified by [strata]. The hazard ratio and 95% confidence interval will be estimated using a Cox proportional hazards model stratified by the same factors.

PFS Co-Primary Endpoint

Progression-free survival (PFS) is a co-primary endpoint, defined as time from randomization to disease progression per RECIST 1.1 or death from any cause, whichever occurs first. For PFS, the trial is powered to detect HR = 0.65 with 90% power at α = 0.025 (one-sided, with multiplicity adjustment). This requires 182 PFS events. Based on assumed median PFS of 6 months in the control arm and the accrual pattern above, these events are expected to accrue within 18 months of first patient enrolled.

Non-Inferiority Survival Trial

This non-inferiority trial is designed to demonstrate that the novel agent is not inferior to the active comparator with respect to overall survival. The non-inferiority margin is HR = 1.25, which preserves 50% of the historical treatment effect (HR = 0.64 vs. placebo). The margin was selected based on [justification per regulatory guidance]. With a one-sided test at α = 0.025 and 80% power, assuming the true HR is 1.0 (no difference), 608 events are required. To observe these events within 36 months, approximately 850 subjects are needed, accounting for median survival of 24 months in both arms and 15% dropout.

7. R Code

Using gsDesign Package

library(gsDesign)

# Schoenfeld formula for required events
nEvents <- function(hr, alpha = 0.05, power = 0.80,
                    sided = 2, ratio = 1) {
  za <- qnorm(1 - alpha/sided)
  zb <- qnorm(power)
  p1 <- ratio / (1 + ratio)
  p2 <- 1 / (1 + ratio)
  d <- (za + zb)^2 / (p1 * p2 * log(hr)^2)
  ceiling(d)
}

# Example: HR = 0.70, 80% power
nEvents(hr = 0.70)  # Returns 246

# Using gsDesign for more comprehensive calculations
# with accrual and follow-up
x <- nSurv(
  lambdaC = log(2)/12,  # Control median = 12 months
  hr = 0.70,
  eta = 0.05/12,        # 5% annual dropout
  R = 24,               # 24 month accrual
  T = 36,               # 36 month total study duration
  alpha = 0.025,        # One-sided
  beta = 0.20           # 80% power
)
print(x)

# gsDesign with interim analyses
gsd <- gsDesign(
  k = 3,                # 2 interims + final
  test.type = 2,        # Two-sided symmetric
  alpha = 0.05,
  beta = 0.20,
  n.fix = 246,          # Fixed-design events
  sfu = sfOF            # O'Brien-Fleming spending
)
print(gsd)

Using survival Package for Simulation

library(survival)

# Simulate survival trial and estimate power
sim_survival_trial <- function(n_per_arm, hr, median_ctrl,
                                n_sims = 1000) {
  lambda_ctrl <- log(2) / median_ctrl
  lambda_trt <- lambda_ctrl * hr

  p_values <- replicate(n_sims, {
    # Generate survival times
    time_ctrl <- rexp(n_per_arm, lambda_ctrl)
    time_trt <- rexp(n_per_arm, lambda_trt)

    # Combine data
    df <- data.frame(
      time = c(time_ctrl, time_trt),
      event = 1,  # All events observed (no censoring for simplicity)
      group = rep(c("ctrl", "trt"), each = n_per_arm)
    )

    # Log-rank test
    survdiff(Surv(time, event) ~ group, data = df)$chisq
  })

  mean(p_values > qchisq(0.95, 1))  # Power estimate
}

# Validate: should give ~80% power for n=246 events
# (n per arm = 123 with 100% event rate)
sim_survival_trial(123, 0.70, 12, n_sims = 5000)

Freedman Formula Comparison

# Freedman (1982) formula
nEvents_freedman <- function(hr, alpha = 0.05, power = 0.80) {
  za <- qnorm(1 - alpha/2)
  zb <- qnorm(power)
  d <- (za + zb)^2 * (1 + hr)^2 / (hr - 1)^2
  ceiling(d)
}

# Compare formulas
hr_values <- c(0.50, 0.60, 0.70, 0.80)
comparison <- data.frame(
  HR = hr_values,
  Schoenfeld = sapply(hr_values, nEvents),
  Freedman = sapply(hr_values, nEvents_freedman)
)
print(comparison)

#     HR Schoenfeld Freedman
# 1 0.50         65       73
# 2 0.60        120      133
# 3 0.70        246      259
# 4 0.80        630      649

8. References

Schoenfeld D. (1981). The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika, 68(1), 316-319.

Freedman L.S. (1982). Tables of the number of patients required in clinical trials using the logrank test. Statistics in Medicine, 1(2), 121-129.

Lachin J.M., Foulkes M.A. (1986). Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics, 42(3), 507-519.

Collett D. (2015). Modelling Survival Data in Medical Research (3rd ed.). CRC Press.

ICH E9. (1998). Statistical Principles for Clinical Trials. International Council for Harmonisation.

FDA. (2007). Guidance for Industry: Clinical Trial Endpoints for the Approval of Cancer Drugs and Biologics.

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Sample Size for Survival Analysis

Contents

1. When to Use This Method

Appropriate For

Not Appropriate For

2. Mathematical Formulation

2.1 Schoenfeld Formula (Required Events)

Where:

2.2 Freedman Formula (Alternative)

2.3 Converting Events to Sample Size

2.4 Lachin-Foulkes Method (With Accrual)

Where:

2.5 Unequal Allocation

2.6 Non-Inferiority Design

2.7 Dropout Adjustment

3. Assumptions

Understanding Hazard Ratios

4. Regulatory Guidance

FDA Guidance

EMA Guidance

5. Validation Against Industry Standards

6. Example SAP Language

Overall Survival Primary Endpoint (Oncology)

PFS Co-Primary Endpoint

Non-Inferiority Survival Trial

7. R Code

Using gsDesign Package

Using survival Package for Simulation

Freedman Formula Comparison

8. References

Ready to Calculate?

Related Guides

Complete Guide to GSD

Non-Inferiority & Equivalence