# Comparing magnitude of coefficients in a logistic regression

I have a logistic regression:

ln(p/(1-p)) = a + b * Age + c * Balance + d * Tenure

Let's say that:

b = 1.4

c = 2.5

d = -0.7

We can compare signs of the coefficients. I.e. right away we can say that an increase in Tenure will cause a decrease in probability of the dependent variable.

My question is: can we compare magnitudes of the coefficients to each other?

I know that we can take the exp() of any coefficient and get the odds ratio and make conclusions then. But what I am referring to is different - without taking the exp() can we simply say the following:

Since c > b then the impact from a 1 unit change in Balance on the DV's probability is greater than the impact from a 1 unit change in Age. (without saying how much greater. Just simply greater)

I did quite a lot of search and only found this thread: Comparing coefficients in logistic regression

Where this approach seems to be implied. However, I would like to confirm that you can interpret coefficients like this.

I look forward to your thoughts,

Thank you,

Kirill

I think you cannot. For a continuous variable such as Age, you can make the coefficient as big or small as you want, if you change your measurement unit(such as from second to 1000 years). For multiple regression you only can study the relation of one predictor variable with your outcome variable at one time and hold all other variable constant. I think you can compare the effect of variable on goodness of fit such as $R^2$, but I don't think you can compare coefficient directly.

• Thank you! That's a valid point Does this change if both variables are measured on the same scale? For instance Balance in dollars and Income in dollars. – Kirill Eremenko Aug 19 '15 at 8:34
• I thought about your response and actually now I disagree. I see how changing the unit of measurement can inflate/deflate coefficients, but isn't that why we say impact per unit change in underlying IV? Your logic could also be applied to multiple linear regression, but there we freely compare coefficients this way and make conclusions about how many units change the DV experiences per unit change in each IV. – Kirill Eremenko Aug 19 '15 at 8:44
• Such as for weight, Ton,Kg,Gram and Pound are all units,which unit will you use? – Deep North Aug 19 '15 at 9:05
• And your per unit change will be one Ton's change, one Kg's change or one Pound's change, they will correspond to different coefficients with the same data. – Deep North Aug 19 '15 at 10:12
• (+1) @Kirill: Does it make sense to equate each 1\$in your bank account to each 1\$ you earn in a year? If you already have some meaningful way - even a rough & ready one - to put two quantities on a common scale then you can compare their regression coefficients using that scale. Standardization is sometimes advocated for putting predictors on a common scale; but it's not always true that the distribution of predictors in a sample reflects that in a population, or that the latter's of much interest. You're right that you can compare numerical factors, stating the units, but there's ... – Scortchi - Reinstate Monica Aug 19 '15 at 11:29

The marginal effects from a logistic regression is the following:

The partial derivative essentially tells you the effect of a unit change in some variable x

The first part of the equation,, is always positive and would look like the curve below:

First thing to notice is that the marginal effect will depend on X. So normally we would evaluate the marginal effects at the mean. Having said that, regardless of where you evaluate the marginal effects

c > b implies that the balance has a higher effect on the probability of event than the age. (bcos the first part of that marginal effects equation is always positive)