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What is power in logistic regression? Is it the ability of the test to reject the null hypothesis when it is actually false?

Second, if you're trying to maximize your statistical power when doing a logistic regression, is it better to use predictor values that are only high or low or a range of predictor values?

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Power by definition is what you wrote. The ability to reject a false null hypothesis. That is how assertive a model is to say that a predictor x has something to do with the dependent variable y. Power is a probability so closer is it to 1, better it is.

For the second question, there is no fixed answer to this question. It depends on the data. Lets say you are predicting defaulter out of people in which one variable is delay in payment. Now if you are doing a high low categorization you are essentially saying that everybody ( lets say) who is above 90 days has beta = .4 and everybody less than 90 has beta = .15 . In essence you are trying to segregate a 89 day delay from a 90 day delay while a different view could be that both of 'em are same guys. So in my view low/high or range of values should come from curve fitting analysis where you should be watchful of inflections in the curve. Where is it peaking ?? where does it show almost equal behavior in defaulters/non defaulters. The final aim is to multiply the person's delay with the right beta and not an imposed beta.

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