Let $\mathbf{Y}$ be a vector of independent, normally-distributed random variables. Let $S_1$, $S_2$ and $S_3$ be three non-overlapping samples of sizes $N_1$, $N_2$ and $N_3$, respectively. Let $M_A$ and $M_B$ be two measurement methods which allow us to sample $\mathbf{Y}$ independently; $M_A$ allows $\mathbf{Y}$ to be sampled without error, whereas this is not the case for $M_B$. We measure $\mathbf{Y}_1$ from $S_1$ using both $M_A$ and $M_B$, $\mathbf{Y}_2$ from $S_2$ using only $M_A$, and $\mathbf{Y}_3$ from $S_3$ using only $M_B$. For $i=2$ and $i=3$, we solve the linear equation $\mathbf{Y}_i=\mathbf{X}_i \mathbf{B}_i+\mathbf{E}_i$, where $\mathbf{X}_i$ is a multivariate vector whose values were measured errorlessly. We wish to test the null hypothesis that the multivariate correlation between $\mathbf{Y}_2$ and $\mathbf{X}_2$ is not significantly different from that between $\mathbf{Y}_3$ and $\mathbf{X}_3$. What value of $N_1$ is required to quantify the reliability of $M_B$ compared to $M_A$ so that the null hypothesis can be tested at significance level $\alpha$ and with statistical power $P$?


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