Statistical & Clinical Analysis
Freedman Log-Rank Sample Size Formula: Required Events ((k*HR + 1)/(HR - 1))²
Calculate conservative log-rank survival event targets under non-null variance using Laurence S. Freedman’s 1982 Statistics in Medicine formula E = ROUNDUP((z_alpha + z_beta)² * (1 + k*HR)² / (k * (1 - HR)²), 0).
Freedman (1982) vs. Schoenfeld (1983): why regulatory protocols compare both log-rank event formulas!
Whereas Schoenfeld’s 1983 formula approximates log-rank test variance under contiguous local alternatives (producing slightly smaller event counts via 4 / (LN(HR))^2), Laurence S. Freedman (1982, Statistics in Medicine — used alongside Schoenfeld in PASS, EAST, and nQuery clinical-trial software) derived the exact variance of the log-rank statistic under the alternative Hazard Ratio HR with allocation ratio k = N_exp / N_ctrl: E = ((z_(1 - alpha/2) + z_(1 - beta))^2 * (1 + k * HR)^2) / (k * (1 - HR)^2)!
Under equal 1:1 randomization (k = 1), Freedman simplifies to E = (z_alpha + z_beta)^2 * (1 + HR)^2 / (1 - HR)^2! Because ((1 + HR)/(1 - HR))^2 > 4 / (LN(HR))^2 for all HR != 1, Freedman yields a slightly more conservative safeguard against underpowering: at HR = 0.70 (alpha = 0.05 two-sided, 80% power, k = 1), Freedman requires E = 253 events (vs. 247 by Schoenfeld; rising to 636 events at HR = 0.80)!
| Randomized Trial Design (alpha=0.05, 80% Power) | Freedman Events E (Conservative) | Schoenfeld Events Comparison | Allocation Ratio k = N_exp/N_ctrl |
|---|---|---|---|
| HR = 0.70, 1:1 Allocation (k = 1) | 253 Events | 247 Events (+6 safety margin) | Balanced 1:1 |
| HR = 0.80, 1:1 Allocation (k = 1) | 636 Events | 631 Events (+5 safety margin) | Balanced 1:1 |
| HR = 0.70, 2:1 Allocation (k = 2) | 272 Events | 278 Events | 2:1 Experimental:Control |
Freedman Required Log-Rank Events E = ROUNDUP((z_a + z_b)² * (1 + k*HR)² / (k * (1 - HR)²), 0)
Computes integer required events across both arms from z_(1-alpha/2) (B2), z_(1-beta) (B3), target Hazard Ratio HR (B4 != 1), and allocation ratio k = N_exp / N_ctrl (B5 > 0).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 = 1, B5 <= 0), "Invalid Freedman Log-Rank Inputs", ROUNDUP((((B2 + B3) ^ 2) * ((1 + B5 * B4) ^ 2)) / (B5 * ((1 - B4) ^ 2)), 0))Expected Experimental-Arm Event Share Under Alternative HR = (k * HR) / (1 + k * HR)
Evaluates the asymptotic proportion of observed events occurring in the experimental group under Hazard Ratio HR.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 = 1, B5 <= 0), "Invalid Freedman Log-Rank Inputs", ROUND((B5 * B4) / (1 + B5 * B4), 4))Try the example
Use the original values or change them to test your understanding.