Put

a = \frac{p _{0}}{1 - p _{0}}, b = \frac{p _{1}}{1 - p _{1}} .

θ

, the approximate power of the Wald test increases as its variance

V (n_{0}, n_{1}) = \frac{a}{n _{0}} + \frac{b}{n _{1}}

decreases. For fixed total sample size

n = n_{0} + n_{1}

, the allocation problem is therefore

min_{n_{0}, n_{1} > 0} {\frac{a}{n _{0}} + \frac{b}{n _{1}} : n_{0} + n_{1} = n},

with integer rounding applied after solving the continuous problem. The second-order condition is positivity of the second derivative at the stationary point.

ii

Solution

R = n_{0} / n_{1}

gives

n_{0} = n R / (1 + R)

and

n_{1} = n / (1 + R)

, so apart from the positive factor

1/ n

f (R) = (1 + R) (\frac{a}{R} + b) .

Hence

f^{'} (R) = - \frac{a}{R ^{2}} + b, f^{''} (R) = \frac{2 a}{R ^{3}} > 0.

The unique minimum is

R^{*} = \frac{a}{b} = \frac{p _{0} ( 1 - p _{1} )}{p _{1} ( 1 - p _{0} )} .

This is the Neyman allocation for the log relative risk of failure.

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