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If a regression analysis shows that 1 unit increase in independent variable A will increase the odds of a user conversion (dependent variable) by 2 times, is there a way to infer the change it can have on the average conversion rate too? Is it as simple as multiplying the average conversion rate with the predicted odds ratio so that if the avg. conversion rate is 20% then:

predicted conversion rate following 1 unit increase in variable A = 0.20 * 2 = 0.40 
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No. It's an odds ratio, so it changes the odds.

If your formula was correct, then you could easily get probabilities greater than 1 and less than 0.

To go from probability to odds, divide P by 1-P. To go back to probability, divide the odds by 1 + the odds.

The effect of a unit increase in the IV will be constant on the odds, but not on the probability.

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