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I want to obtain a standardised effect size for a regression with a binary outcome. If I standardise the (continuous) predictor and run a probit regression, the resulting coefficient can be interpreted as in the difference in Z-score outcome per one unit increase in the change in the Z-score predictor. Therefore, is this the same as a standardised regression coefficient?

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I don't think you can interpret the coefficient of a probit model like that. For instance, I think that your independent variable going from 1 to 2 is different than going from 2 to 3. You may want to read up on interpreting probit parameters. There's a good explanation here: https://stats.idre.ucla.edu/stata/output/probit-regression/

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