Transformation of a regression coefficient when independent variable was log-transdormed

In the context of a linear regression model where the independent variable ($X$) was log-transformed, like:

$Y = \alpha + \beta·ln(X)$

Is there a straightforward way to transform a regression coefficient ($\beta$) that was estimated using log-transformed independent variable $X$?

I know that in this model, an increase of 1% in $X$ would mean an increase of $\beta·ln(X)$ in the value of $Y$.

Is there a way to transform $\beta$ so I could say that an increase of 1% in $X$ means an increase of $\beta'·X$ in variable $Y$?

• Possible duplicate of Back-transformation of regression coefficients Commented Jul 5, 2016 at 13:05
• Since $X$ was transformed, not $\beta$, a "back-transformation" of $\beta$ would seem to make no sense. Could you clarify what you are hoping to achieve?
– whuber
Commented Jul 5, 2016 at 13:15
• $\beta$ is estimated here using $ln(X)$, so interpretation dependens on $ln(X)$ [as in the main text: an increase of 1% in $X$ results in an increase of $\beta·ln(X)$ for $Y$]. What I wonder to achieve (if possible) is an expression of the change in $Y$ due to $X$ in terms not of $ln(X)$ but $X$ itself, using $ln(X)$ in regression analysis. Commented Jul 5, 2016 at 13:21
• But of course: it's right in front of us. If $X$ changes to $\alpha X$ (for positive $\alpha$, of course) then $\log(X)$ becomes $\log(\alpha)+\log(X)$, whence $Y$ becomes $Y + \beta\log\alpha$. But there is no "back-transformation" of any sort needed in this analysis.
– whuber
Commented Jul 5, 2016 at 13:51
• @whuber, why not develop that into an answer? I don't see an answer beyond that, and this isn't really a duplicate of the linked thread as it is about the transformation of Y whereas this is about the transformation of X. Commented Jul 5, 2016 at 14:35

You can't transform in the sense you'd wish in the question, i.e. preserving the interpretation of the coefficient $\beta$. In the log transformed regression $\beta$ becomes the sensitivity of the dependent variable to the percentage change in the independent variable (IV), instead of the sensitivity to the simple change in IV. In other words, $\beta$ in the log transformed case isthe measure of the relative sensitivity rather than absolute one.

You can get some intuition for small changes in $X$ by considering

\begin{align} Y+\delta Y&=\alpha + \beta \ln(X + \delta X)\\ &=\underbrace{\alpha + \beta \ln X}_Y + \beta\ln\left(1+\frac{\delta X}{X}\right)\\ \therefore\delta Y&\approx \beta\frac{\delta X}{X}, \end{align}

where the approximation is obtained by a Taylor series expansion of the logarithm to first order in $\delta X$.

In other words, a 1% increase in $X$ gives you an increase of $0.01\beta$ in the dependent variable. But this only holds for small changes in $X$.

• When $\log(X)$ is used in a model, the natural way to represent changes to $X$ is multiplicative, not additive. The question is clear about that: it refers to a "1% change," which is multiplicative.
– whuber
Commented Jul 5, 2016 at 13:53