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Suppose I have $n=100$ observations of ordinal data and get threshold coefficients $b_1, \dots, b_3$ and probit slope $b_4$. I want to test the hypothesis $H_{0}: \frac{b_{3}}{b_{4}} = \frac{1}{2}$ vs. $H_a: \frac{b_{3}}{b_{4}} \neq \frac{1}{2}$. So I want to count the proportion of times we fail to reject the null hypothesis. To do this, would we just keep sampling 100 observations similar to the previous ones and return the proportion of p-values greater than $0.05$?

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To do power analysis you need to specify the effect size - if $\frac{b_3}{b_4} = \frac{1}{2} + 10^{-20}$ you will have much lower power than if $\frac{b_3}{b_4} = 5000$. So, you need to decide what is a "small" and "large" effect size in the context of the application, simulate data under those circumstances, then empirically look at how often you reject $H_0$. (btw, I think you meant proportion of $p$-values less than $.05$:)) – Macro Jan 17 at 16:27

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