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

Solution

For equal arm size

n

, write

v^{*} = p_{1} (1 - p_{1}) + p_{0} (1 - p_{0})

at the clinically relevant alternative. A one-sided level-

α

Wald test rejects when

Z > z_{1 - α}

, and its approximate statistical power is

1 - Φ (z_{1 - α} - \frac{δ ^{*} n}{v ^{*}}) .

Equating this to

1 - β

gives the per-arm sample size

n = \frac{v ^{*} ( z _{1 - α} + z _{1 - β} ) ^{2}}{( δ ^{*} ) ^{2}},

rounded up. If the design uses a null-based critical standard error

v_{0}

but an alternative standard error

v^{*}

, the corresponding more general formula is

n = \frac{( z _{1 - α} v _{0} + z _{1 - β} v ^{*} ) ^{2}}{( δ ^{*} ) ^{2}} .

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