Recurrent Event (Andersen-Gill) Sample Size Calculator

Methodology

This calculator implements the sample-size procedure proposed by Yongqiang Tang and Ronan Fitzpatrick for the robust Wald test from the Andersen-Gill model. The method accommodates recurrent events whose baseline rate may vary over time and accounts for within-subject heterogeneity through a dispersion parameter.

Andersen-Gill model

The treatment effect is represented by the recurrent-event rate ratio exp(β). For a Weibull control-arm rate function, the calculator uses:

Here ψ controls the scale of the baseline event-rate function and ν controls its shape. The treatment arm has rate λ₁(t) = exp(β)λ₀(t).

Robust variance

Under equal dropout distributions in the two treatment groups, Tang and Fitzpatrick express the large-sample variance component as:

where p₁ is the treatment allocation proportion, p₀ = 1 − p₁, and κ describes between-subject heterogeneity. With a common dispersion parameter κ, the second term uses the same κ in both groups.

The exposure-weighted quantities are:

For exponential loss to follow-up with rate δ and fixed treatment duration τ_c, the retention probability is π(t) = exp(−δt). The calculator evaluates the resulting integrals numerically, which is equivalent to the incomplete-gamma expressions given in the paper for the Weibull model.

Sample-size equation

For superiority or noninferiority testing, the required continuous sample size is:

For the one-sided option, the calculator uses z_1−α for the critical value. For the two-sided option it uses z_1−α/2. For a superiority calculation, M₀ = 1, so the denominator becomes β². The final sample size is rounded upward because fractional participants are not possible.

Staggered-entry design

Design 2 allows enrollment over an accrual period τ_a, followed by an additional treatment period τ_c. For uniform accrual (η = 0), the retention probability after the recruitment period is:

The 2022 correction to Tang and Fitzpatrick specifically affects the staggered-entry calculation. This implementation uses the corrected retention expression rather than the original erroneous Appendix A4 expression. The correction does not affect Design 1.

Validation example

The published CGD example uses ψ = 1.1, ν = 1.2, event-rate ratio = 0.6, κ = 0.8, δ = 0.25, τ_c = 1 year, 90% power, and a one-sided α = 0.025. With equal allocation, the calculated continuous sample size is approximately 364.61 and therefore rounds to 365 participants. With a 2:1 treatment:control allocation (p₁ = 2/3), the continuous result is approximately 389.26 and therefore rounds to 390 participants. These reproduce the published Design 1 values.

Important limitations

• The method assumes a proportional recurrent-event rate structure with a common treatment effect over time.
• The Weibull specification describes the control-arm event-rate function; other baseline rate functions require a different implementation.
• The robust variance is asymptotic. The authors note that simulation is recommended, particularly for relatively small trials.
• The Andersen-Gill formulation is not appropriate when an earlier event fundamentally changes the risk of subsequent events within a subject.

References

Tang, Y. & Fitzpatrick, R. (2019). Sample size calculation for the Andersen-Gill model comparing rates of recurrent events. Statistics in Medicine, 38(24), 4819–4827. DOI: 10.1002/sim.8335.

Tang, Y. & Fitzpatrick, R. (2022). Sample size calculation for the Andersen-Gill model comparing rates of recurrent events. Correction. Statistics in Medicine, 41(20), 4079. DOI: 10.1002/sim.9518.

Andersen, P.K. & Gill, R.D. (1982). Cox's regression model for counting processes: a large sample study. The Annals of Statistics, 10, 1100–1120.