# How can I interpret the coefficient of a logit transformed explanatory variable in linear regression?

I fit a linear regression model with continuous response. One of my predictor variables is given in percentage. So I transformed the predictor with logit transformation. My question is how can I interpret the coefficient of this transformed predictor? You can consider the following model as example:

$$Y = b_0 + b_1 logit(X) + e$$

How to interpret $$b_1$$?

• Cases one unit apart on the logit of X have an expected difference of b1 on Y. May 19 '19 at 18:55
• @HeteroskedasticJim (great name) is correct. My question is, why take the logit of x? May 20 '19 at 11:05

Welcome to the site, user248399.

As @Heteroskedastic Jim states, b1 allows you to calculate the expected change in Y for a given change in logit(X).

If the coefficient b1 is significantly different from 0 (check the standard error on the estimate of b1 and calculate confidence intervals, or check the p-value for b1), then it appears that logit(X) is a useful predictor for Y.

A couple of things to consider:

1. Have you tried the regression with the untransformed values of X, and checked to see if the residuals are normally distributed and without obvious patterns in the residual vs. fitted values plot? i.e. is the transformation necessary?

2. In the dataset you have used to fit the relationship, do you have any values very close to (or at) 0% or 100%? If you don't have these, be extra careful - you cannot be confident that the relationship will hold outside of the values of logit(X) for which you had data. It could be particularly dangerous to extrapolate to values close to 0% or 100%, because the tails of the logit transformation are quite severe, so at small or large percentages, you will get quite large changes in logit(X) (and therefore the predicted value of Y) for only a small change in X.

Less severe alternatives to the logit transformation are available - e.g. folded roots, see here, here and here.

• In my data there are some values exactly 0 and 100. That's why I tried different transformations i.e., log. But in the theory I found some authors suggested to use logit transformation. I was wondering how to interpret the coefficients in case of logit transformation. May 22 '19 at 7:16
• The interpretation of the coefficient is exactly the same as if you had just used X, except that you have the relationship of Y with logit(X) instead. If that is not clear to you, try predicting some values of Y from different values of logit(X), and then if it is still not clear, please try to clarify what you do not understand.
– Izy
May 22 '19 at 7:55
• If you had values of 0 and 100, then you must have needed to use an offset, or else some values of log(x) and logit(x) would have been undefined. What offset did you use? Have you explored the effect of different offsets on your results? Have you tried the regression with untransformed X? I do recommend the links that I put in my post, to get you thinking about the potentially unintended effects of logit transformation, and alternatives (NB. I'm not saying that logit transformation is wrong, only that you should be aware of the behaviour at the extreme ends, in case it affects your results).
– Izy
May 22 '19 at 8:04