A Numerical Algorithm for Calculating Group Sequential Boundaries

Calculating group sequential boundaries involves recursive evaluations of complex multivariate normal integrals. The Markov property of the standardized test statistics in a group sequential setting simplifies these calculations into successions of univariate integrals. Interestingly, Schoenfeld (2001) showed that the probability of crossing group sequential boundaries at each analysis can be calculated as a multiplication of a sequence of matrices, which can be used recursively to obtain the boundary values through numerical approximations. This post introduces this algorithm and provides an implementation using the SAS IML procedure.

Suppose a group sequential trial with up to 𝐾 analyses yields a sequence of standardized test statistics {𝑍₁, ..., 𝑍𝐾} following the canonical joint distribution with information levels {𝐼₁, ..., 𝐼𝐾} for the parameter 𝜃. Let ∆ₖ = 𝐼ₖ - 𝐼ₖ₋₁ for 𝑘 = 2, ..., 𝐾, then 𝑍₁ ∼ 𝑁(𝜃√𝐼₁, 1) and for each 𝑘 = 2, ..., 𝐾,

𝑍ₖ√𝐼ₖ - 𝑍ₖ₋₁√𝐼ₖ₋₁ ∼ 𝑁(𝜃∆ₖ, ∆ₖ) (1)

independently of 𝑍₁, ..., 𝑍ₖ₋₁.

Consider the two-sided test of the null hypothesis 𝐻₀:  𝜃 = 0 and assume that the overall Type I error (α) is partitioned into α₁, ..., α𝐾. The critical value for 𝑍₁ is simply 𝑐₁ = Φ⁻¹(1 - α₁/2), where Φ denotes standard normal cumulative distribution function. The critical values 𝑐ₖ > 0 for 𝑍ₖ, 𝑘 =2, ..., 𝐾, are calculated by solving

εₖ(𝑐₁, ..., 𝑐ₖ) = Ρ{𝑍ₖ > 𝑐ₖ, 𝑍ₖ₋₁ ∈ 𝙲ₖ₋₁, ..., 𝑍₁ ∈ 𝙲₁} = αₖ/2,

where Ρ{∙} denotes the probability under the null hypothesis 𝜃 = 0 and 𝙲ₛ denotes the interval [-𝑐ₛ, 𝑐ₛ] for 𝑠 = 1, ..., 𝑘-1. Divide each interval 𝙲ₛ into 𝑁 equally spaced subintervals and let 𝙲ₛᵢ be the 𝑖-th subinterval, 𝑖 = 1, ..., 𝑁. The Schoenfeld (2001) algorithm is developed based on the following matrices.

• 𝑀₁ - the 𝑁×1 matrix with (𝑀₁)ᵢ = Ρ{𝑍₁ ∈ 𝙲₁ᵢ}, 𝑖 = 1, ..., 𝑁.

• 𝑀ₛ - the 𝑁×𝑁 matrix with (𝑀ₛ)ᵢⱼ = Ρ{𝑍ₛ ∈ 𝙲ₛᵢ|𝑍ₛ₋₁ ∈ 𝙲₍ₛ₋₁₎ⱼ}, 𝑠 = 2, ..., 𝑘-1, and 𝑖, 𝑗 = 1, ..., 𝑁.

• 𝑈ₖ - the 1×𝑁 matrix with (𝑈ₖ)ᵢ = Ρ{𝑍ₖ > 𝑐ₖ|𝑍ₖ₋₁ ∈ 𝙲₍ₖ₋₁₎ᵢ}, 𝑖 = 1, ..., 𝑁.

Using the Markov property of {𝑍₁, ..., 𝑍𝐾}, it can be shown that

εₖ(𝑐₁, ..., 𝑐ₖ) = 𝑈ₖ𝑀ₖ₋₁∙∙∙𝑀₁ (2)

holds for all 𝑘 = 2, ..., 𝐾. When 𝑁 is large, the entries of these matrices can be approximated using the areas of rectangles corresponding to the individual subintervals.

It follows from (1) that, for 𝑘 = 2, ..., 𝐾, the conditional density of 𝑍ₖ given 𝑍₁ = 𝑧₁, ..., 𝑍ₖ₋₁ = 𝑧ₖ₋₁ depends only on 𝑧ₖ₋₁ and is equal to

where ϕ(𝑥) = exp(-𝑥²/2)/√(2π) is the standard normal density function. For each 𝑘 = 2, ..., 𝐾, define

Let 𝑑ₖ = 2𝑐ₖ/𝑁 be the width of the subintervals for 𝑘 = 1, ..., 𝐾-1, and let 𝑥ₖᵢ = -𝑐ₖ + (𝑖 - 0.5) 𝑑ₖ be the midpoint for the 𝑖-th (𝑖 = 1, ..., 𝑁) subinterval. Then we have the following approximations:

• (𝑀₁)ᵢ = Ρ{𝑍₁ ∈ 𝙲₁ᵢ} ≈ 𝑑₁ϕ(𝑥₁ᵢ), 𝑖 = 1, ..., 𝑁.

• (𝑀ₛ)ᵢⱼ = Ρ{𝑍ₛ ∈ 𝙲ₛᵢ|𝑍ₛ₋₁ ∈ 𝙲₍ₛ₋₁₎ⱼ} ≈ 𝑑ₛ𝑓ₛ(𝑥ₛᵢ, 𝑥₍ₛ₋₁₎ⱼ), 𝑠 = 2, ..., 𝑘-1, and 𝑖, 𝑗 = 1, ..., 𝑁.

• (𝑈ₖ)ᵢ = Ρ{𝑍ₖ > 𝑐ₖ|𝑍ₖ₋₁ ∈ 𝙲₍ₖ₋₁₎ᵢ} ≈ 𝐹ₖ(𝑐ₖ, 𝑥₍ₖ₋₁₎ᵢ), 𝑖 = 1, ..., 𝑁.

The right-hand side of equation (2) can be recursively evaluated using the above approximations to obtain the critical values 𝑐ₖ for 𝑍ₖ, 𝑘 =2, ..., 𝐾.

The following document (gsd_apprx.docx) provides an implementation of this algorithm using the SAS IML procedure. Boundaries for one-sided tests under 𝜃 = 0 and those under the alternative hypotheses can be obtained in the same manner.

gsd_apprx.docx

Download DOCX • 16KB

References

Schoenfeld, D. A. (2001). A simple Algorithm for Designing Group Sequential Clinical Trials. Biometrics 57, 972-974.