Survival Analysis
Find the required sample size for a two-group exponential survival comparison when patient accrual is not uniform over the recruitment period. The calculation uses the Lachin and Foulkes (1986) truncated-exponential accrual model, which allows recruitment to be front-loaded, back-loaded, or uniform, together with the Schoenfeld-style requirement on the number of events.
To allow for a non-linear (front-loaded or back-loaded) rate of patient entry over an accrual period of length R, Lachin and Foulkes (1986) modeled the entry time using a truncated exponential distribution with shape parameter γ, density g(r) = γe−γr/(1 − e−γR) for 0 < r ≤ R. A negative γ produces a concave (front-loaded, fast-then-slow) accrual pattern, a positive γ produces a convex (slow-then-fast) pattern, and the uniform accrual case is recovered as γ → 0.
For a total study duration T = R + (follow-up), an exponential event hazard λ, and an exponential loss-to-follow-up hazard η, the probability that a subject is observed to have the event is
This event probability is computed separately for each group using its own hazard rate, then combined with the standard Schoenfeld requirement on the number of events for a two-sided (or one-sided) log-hazard-ratio test:
where p₁ and p₂ are the group allocation fractions. This is the same events-based search used by this site's uniform-accrual exponential survival calculator, with the uniform-accrual event probability replaced by the Lachin–Foulkes non-uniform expression above; setting γ to (numerically) zero reproduces the uniform-accrual result exactly.
The procedure assumes exponential (constant-hazard) event and loss-to-follow-up distributions in both groups, a truncated-exponential accrual pattern with a common shape parameter γ for both groups, and that the hazard ratio between groups is constant over the whole study period.
Lachin (2013, Statistics in Medicine, Section 9) illustrates the event-probability expression using a reference hazard rate of λ = 0.0875 per year, an R = 3 year accrual period with 40% of subjects recruited in the first half and 60% in the second (corresponding to a shape parameter γ = −0.27), a total duration of T = 7 years, and 4% per year losses to follow-up (η = 0.04). Lachin reports a resulting event probability of π = 0.335, which this calculator reproduces exactly.
As a further check, setting γ to a value very close to zero (uniform accrual) reproduces this site's separately validated uniform-accrual exponential survival calculator's event probabilities exactly for the same hazard rate, accrual period, and follow-up.
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.
Lachin, J. M. (2013). Sample Size and Power for a Logrank Test and Cox Proportional Hazards Model with Multiple Groups and Strata, or a Quantitative Covariate with Multiple Strata. Statistics in Medicine, 32(25), 4413–4425.
Schoenfeld, D. A. (1983). Sample-Size Formula for the Proportional-Hazards Regression Model. Biometrics, 39(2), 499–503.