Interpreting a log transformed Y variable when X is a binary variable

[Karen Arulsamy]

Hi,
How can I interpret the coefficient below? Cancer captures whether an individual has been diagnosed with cancer. The Y variable is log of total health expenditure.
If X is a continuous variable, then, a unit increase in X increases Y by 222% (B2*100%). If X is a binary variable and the log points are between 0 and 1, then exponentiating the log points will provide results that tally with B2*100%.
However, in this case, exponentiating 2.22 results in 9.2 and I am unsure how to interpret this. Any help much appreciated.

[Dimitriy V. Masterov]

When ln(Y)=B0 + B1*X + u, a change in X by one unit (∆X=1) is associated with a (exp(B1) - 1)*100 % change in Y.

This means it's roughly an 816% increase.

See https://davegiles.blogspot.com/2011/...r-dummies.html.

[Karen Arulsamy]

Hi Dimitry,

Thanks so much.

How do I reconcile this with this resource (https://stats.oarc.ucla.edu/other/mu...g-transformed/) that says to take the exp(B2) to obtain the percent increase when X is a binary variable.

and also https://theeffectbook.net/ch-Statist...ith-regression which says that B2 can be multiplied by 100 when X is continuous and Y is log transformed?

Appreciate your help with this.

[Dimitriy V. Masterov]

The first link deals with the effect on the geometric mean, which is different than the arithmetic mean and probably not what you care about.

Your covariate is not continuous, so the second link does not apply. This is a big change in X, not an infinitesimal one, so the approximation is not valid.

[Karen Arulsamy]

Many thanks for your help Dimitry, appreciate that.