Non-Inferiority and Equivalence
Sample-size calculation for a two-group non-inferiority survival trial using a Cox proportional hazards model. Specify the non-inferiority hazard-ratio margin, anticipated hazard ratio, event probabilities, alpha, and power. Runs entirely in your browser.
This calculator uses the proportional-hazards sample-size formulation for a two-group non-inferiority test based on the Cox proportional hazards model. The treatment hazard ratio is defined as HR = h2/h1, where Group 1 is the control and Group 2 is the treatment. When higher hazards are worse, the non-inferiority hypotheses are H0: HR ≥ HR0 versus Ha: HR < HR0.
This method uses this procedure as a non-inferiority test for two survival curves using Cox's proportional hazards model. It notes that the Cox regression coefficient is log(HR), that the hazard ratio is assumed constant over time, and that the Cox test and corresponding logrank formulation have the same asymptotic power and sample-size basis.
Let P1 and P2 denote the proportions assigned to the control and treatment groups, respectively, and let Pev1 and Pev2 denote the probabilities of observing the event during the study in the two groups. The overall event probability is
For an actual hazard ratio HR1 below the non-inferiority boundary HR0, the required total sample size is calculated as
Here z1−α is the standard-normal quantile associated with the one-sided Type I error rate and zPower is the standard normal quantile associated with the desired power. Because this calculator uses equal allocation, P1 = P2 = 0.50. The total sample size is rounded upward to the next even integer so that both groups contain the same whole number of participants.
The event count is a consequence of the anticipated event probabilities. The Cox proportional-hazards calculation is fundamentally driven by the amount of event information; therefore, specifying lower event probabilities generally requires more participants to obtain the same information.
The calculation assumes proportional hazards, meaning that the hazard ratio remains constant over the study. It also uses the large-sample normal approximation underlying the Cox/logrank power calculation. The event probabilities are assumed to represent the probabilities of observing the event over the course of the study.
Chow, S.C., Shao, J., & Wang, H. (2008). Sample Size Calculations in Clinical Research, 2nd Edition. Chapman & Hall/CRC.
Schoenfeld, D.A. (1983). “Sample-size formula for the proportional-hazards regression model.” Biometrics, 39(2), 499–503.
the software. this method: Non-Inferiority Tests for Two Survival Curves using Cox's Proportional Hazards Model, Chapter 731.
Advanced documentation includes survival-analysis procedures based on Cox regression and non-inferiority testing, including “Non-Inferiority Test for Two Survival Curves using Cox Regression.”