Statistical & Clinical Analysis
George-Desu & Rubinstein-Lachin-Foulkes Survival Trial Sample Size: Accrual & Follow-Up
Calculate required Phase III survival events via George & Desu (1974) d = CEILING(4*(z_alpha/2 + z_beta)^2 / (LN(HR))^2, 1) and convert events into enrolled patients N under uniform accrual A and additional follow-up F.
Why survival trials are powered by EVENTS d—and how uniform accrual A + follow-up F sets total patients N!
Derived by Stephen L. George & M. M. Desu (1974, Journal of Chronic Diseases), a 1:1 randomized exponential time-to-event trial testing Hazard Ratio HR = lambda_E / lambda_C != 1 at two-sided alpha (z_alpha/2 = 1.96) and power 1 - beta (z_beta = 1.2816 for 90% power) requires d = CEILING(4 * (z_alpha/2 + z_beta)^2 / (LN(HR))^2, 1) total events across both arms! For HR = 0.70 (30% hazard reduction) at 90% power, d = CEILING(4 * (1.96 + 1.2816)^2 / (LN(0.70))^2, 1) = 331 events!
To translate 331 events into enrolled patients N when subjects accrue uniformly over A years and are followed for an additional F years after accrual closes (total trial duration T = A + F; Rubinstein, Gail & Santner 1981 / Lachin & Foulkes 1986), an arm with exponential hazard lambda = LN(2) / Median_Survival has exact administrative event probability P_event(lambda) = 1 - (EXP(-lambda * F) - EXP(-lambda * (A + F))) / (lambda * A)! Averaging 0.5 * (P_C + P_E) gives required total sample size N = CEILING(d / (0.5 * (P_C + P_E)), 2)!
| Oncology Survival Design (Control Median = 18 mo, HR = 0.70, 90% Power) | Exact Formula Result | Clinical Trial Impact |
|---|---|---|
| George-Desu Required Total Events d | CEILING(4*(1.96+1.2816)^2 / LN(0.7)^2, 1) | 331 Total Progression/Death Events |
| Pooled Event Probability (Accrual A=2 yr, Follow-up F=1.5 yr) | 0.5 * (P_event(lambda_C) + P_event(HR*lambda_C)) | 0.6088 (60.88% Experience Event by Study End) |
| Total Enrolled Patients N (1:1 Even Integer) | CEILING(331 / 0.6088, 2) | 544 Randomized Patients (272 per Arm) |
George-Desu (1974) Required Total Events d Across Both Arms (1:1 Allocation)
Computes CEILING(4 * (z_alpha/2 + z_beta)^2 / (LN(HR))^2, 1) from HR (B2), z_alpha/2 (B5), and z_beta (B6).
=IF(OR(B2 <= 0, B2 = 1, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 < 0), "Invalid Survival Trial Design Inputs", CEILING(4 * ((B5 + B6) ^ 2) / (LN(B2) ^ 2), 1))Total Enrolled Sample Size N Under Uniform Accrual A (yr) + Additional Follow-Up F (yr)
Divides required events by exact uniform-accrual exponential event probability across Control median B3 (years) and Experimental hazard HR * lambda_C, rounded up to an even integer for 1:1 randomization.
=IF(OR(B2 <= 0, B2 = 1, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 < 0), "Invalid Survival Trial Design Inputs", CEILING(CEILING(4 * ((B5 + B6) ^ 2) / (LN(B2) ^ 2), 1) / (0.5 * ((1 - (EXP(-(LN(2) / B3) * B7) - EXP(-(LN(2) / B3) * (B4 + B7))) / ((LN(2) / B3) * B4)) + (1 - (EXP(-(B2 * LN(2) / B3) * B7) - EXP(-(B2 * LN(2) / B3) * (B4 + B7))) / ((B2 * LN(2) / B3) * B4)))), 2))Try the example
Use the original values or change them to test your understanding.