Dividing predictor by a constant increases the coefficient in logistic regression?

[spidersgeorg]

I ran a logistic regression and then decided to change the scale of one of my predictors by dividing it by 100. The resulting beta coefficient for this new variable in the new logistic regression ended up increasing. Is this mathematically sound? Would anybody be able to explain why this happens?

[PaigeMiller · Accepted solution]

Sure. If you run a logistic regression and get a regression coefficient of 0.3 for a variable X, this would indicate that the predicted Y value of the logistic regression (in this case the log odds ratio) change by 0.3 units for every one unit that X changes. So now, you divide X by 100, and previuosly there was a 0.3 unit change for every one unit the X changes before you divided by 100, then you need to change the rescaled X to have a coefficient of 30 — now a 1 unit change in X is rescaled to a 0.01 change in X, and this will cause the predicted Y to change by 30*0.01 = 0.3.

Or algebraically:

Pred Y = b0 + b1x1

but now you divide x1 by 100 the formula becomes

Pred Y = b0 + (100*b1)*(x1/100) which is algebraically the same as above

[spidersgeorg]

Okay, so is this true:

• multiply predictor by constant --> results in smaller beta coefficient
• divide predictor by constant --> results in larger beta coefficient

Thanks for your insight, just want to make sure I understand.

[PaigeMiller]

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@spidersgeorg wrote:

Okay, so is this true:

• multiply predictor by constant --> results in smaller beta coefficient
• divide predictor by constant --> results in larger beta coefficient

Thanks for your insight, just want to make sure I understand.

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It should say:

• multiply predictor by constant --> the coefficient is divided by the constant
• divide predictor by constant --> the coefficient is multiplied by the constant