Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 2 / d

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- Solution d

Solution

Let

v = p_{1} (1 - p_{1}) + p_{0} (1 - p_{0}), I_{j} = \frac{m _{j}}{v},

where

m_{j}

is the cumulative sample size per arm. The canonical joint distribution for group sequential test statistics is

(Z_{1} Z_{2}) \tilde{˙} N_{2} [(δ I_{1} δ I_{2}), (1 I_{1} / I_{2} I_{1} / I_{2} 1)] .

Thus

Corr (Z_{1}, Z_{2}) = m_{1} / m_{2}

. This correlation arises because the second statistic reuses all first-stage observations.

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