The three independent sample means satisfy

\overline{Y}_{i} \sim N (μ_{i}, v_{i})

. A linear transformation of their joint multivariate normal distribution is again normal. The two treatment contrasts have covariance

Cov (\overline{Y}_{1} - \overline{Y}_{0}, \overline{Y}_{2} - \overline{Y}_{0}) = Var (\overline{Y}_{0}) = v_{0} .

m_{i} = δ_{i} / s_{i}

and

ρ = v_{0} / (s_{1} s_{2})

, their joint distribution is

(W_{1} W_{2}) \sim N_{2} ((m_{1} m_{2}), (1 ρ ρ 1)) .

The positive correlation coefficient arises from the shared control arm. Under positive variances,

0 < ρ < 1

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