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I am testing for zero Pearson correlation and my null hypothesis is $\rho=0$. It is easy to find Type 1 error or rejection of Null hypothesis and hence the false positive rate. However, my alternate hypothesis is $\rho \neq 0$. How to find Type 2 error or the false negative rate, when I have multiple alternate hypotheses? Am I clear in my question? Please throw me some in sights. Thanks

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1 - Probability of type 2 error is power. So talking about type 2 error is same as talking about power. The power is calculated at fixed alternate hypothesis. For the fixed sample size, you can select several fixed alternative hypotheses, and calculate the power. Then draw a plot with hypotheses on x-axis and power on y-axis, you will get a power curve. In your case, you can take $\phi = 0.0 \text { to 1 by } 0.05$ 21 alternative hypotheses, and calculate 21 powers, then get the power curve. Here is example of power curve. http://www.stat.wisc.edu/~st571-1/HW10.pdf

For your question: How to find Type 2 error or the false negative rate, when I have multiple alternate hypotheses? The answer is you need take the alternative hypothesis one by one, and calculate the power one by one. Given there are infinite number of alternative hypotheses, the power curve is needed.

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