How to convert this log-transformed regression coefficient

Question

I am reading a paper that reports "For a 10-fold decrease in x, y increased by 0.50".

The predictor (x) has been log-transformed but no base specified (presumably natural log). There is also the question of what does a 10-fold decrease mean? Is this 100/10 = 10 = a 90% decrease?

Assuming this is the case I am getting confused in trying to work out the manipulation to convert this untransformed effect (10-fold decrease in x) back to a transformed effect (i.e. a 1 unit decrease in log x).

The best I can come up with is: $\frac{0.5}{log(0.1)}$

Any tips would be appreciated.

Answer 1

If the details of the regression model are unclear, I recommend you read more deeply into all the relevant materials; if this is still insufficient then you could contact the author to seek clarification. However, given that the author refers to a 10-fold change in the explanatory variable, I think it is more likely that the regression model used a base-10 logarithm, so it was probability a model that looked something like this:

$$Y = \beta_0 + \beta_1 \log_{10} (x) + \cdots + \varepsilon,$$

which gives the corresponding regression equation:

$$\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 \log_{10} (x) + \cdots.$$

For this model you have:

$$\begin{align} \hat{Y} - \hat{\beta}_1 &= \hat{\beta}_0 + \hat{\beta}_1 [\log_{10} (x) - 1] + \cdots \\[6pt] &= \hat{\beta}_0 + \hat{\beta}_1 [\log_{10} (x) - \log_{10} (10)] + \cdots \\[6pt] &= \hat{\beta}_0 + \hat{\beta}_1 \log_{10} (x/10) + \cdots \\[6pt] \end{align}$$

My guess is that the author got an estimated coefficient $\hat{\beta}_1 = -0.5$, which would mean that a "ten-fold reduction" in $x$ (i.e., dividing $x$ by ten) increases the predicted value of $Y$ by $-\hat{\beta}_1 = +0.5$.