Let $F_{\lambda^0}$ denote the joint distribution of two independent random variables $Y_{\lambda^0}$ and $Z_{\lambda^0}$. The paper investigates properties of the joint distribution $F_\lambda$ of the linearly transformed random variables $Y_\lambda$ and $Z_\lambda$. Let $\Im_0$ be the Spearman rank correlation test, $\Im_1$ the difference sign correlation test, $\Im_2$ the unbiased grade correlation test (which is asymptotically equivalent to $\Im_0$), $\Im_3$ the medial correlation test, and $\mathcal{R}$ the ordinary (parametric) correlation test. (Whenever discussing $\mathcal{R}$ we assume existence of fourth moments.) Properties of the power of these tests are found for alternatives of the above-mentioned form, particularly for alternatives "close" to the hypothesis of independence and for large samples. Against these alternatives the efficiency of $\Im_3$ is found to depend strongly on local properties of the densities of $Y_{\lambda_0}$ and $Z_{\lambda^0}$, which should invite caution; and the efficiency of $\Im_1$ with respect to $\Im_0$ is often unity. Incidentally, Pitman's result on efficiency is extended in several directions.