# Measurement theory and sample size calculation for multivariate testing

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 $$P$$?