Logistic regression: odds ratio when change in the independent variable values is less than 1 unit I have one independent variable which has values less than 1, and want to see how the odds of having a disease change when the independent variable increase by 0.1.
I ran a binary logistic regression to predict an outcome (disease vs healthy) and as expected it turned out odds ratio (when the independent variable increases by 1) had a crazy high value and confidence intervals.
Therefore I took the 10th root of odds ratio and CI and now odds are interpretable, however, I don't know if what I did makes sense at all from a mathematical standpoint...
 A: It appears that your issue is that the independent variable has a small range and that, therefore, the odds ratio for a one unit increase are very high.
Rather than take the 10th root (or any root), it is much more straightforward to simply multiply the IV by 10 or 100. This changes nothing about the meaning of the regression, it just makes the output have fewer digits.
This is equivalent to changing units from (say) kilometers to meters.
There may be some situation where taking the 10th root makes sense, but I can't think of one offhand and this certainly doesn't seem like one. But the problem isn't mathematical, it's substantive.  What would the 10th root of your IV be?
A: You are right. You can take the 10-th root of the odds ratio (relative change of the odds) for a step of size 1, in order to get the odds ratio for a step of size 0.1
The reason is because a logistic model is modeling the logarithm of the odds as a linear function of the parameter/predictor. And this makes the odds an exponential function of the parameter/predictor.
$$\begin{array}{rrcl}\text{log-odds:} \qquad \qquad & \log\left( \frac{p}{1-p} \right) &=&a+bx\\
\text{odds:} \qquad  \qquad & \frac{p}{1-p}  &=&e^{a+bx}
\end{array}$$
So the changes in the odds are multiplicative when we are adding multiple effects (or increasing the size of an effect).
Consider a change of the parameter/predictor by some step $x_\Delta$. Then this is equivalent to the odds being multiplied by a factor $e^{bx_\Delta}$
$$e^{a+b(x+x_\Delta)} = e^{a+bx} \cdot e^{bx_\Delta}$$
and adding more of those steps (or only a fraction of that step) equates to more multiplications with that factor
$$e^{a+b(x+kx_\Delta)} = e^{a+bx} \cdot e^{bkx_\Delta} = e^{a+bx} \cdot \left(e^{bx_\Delta}\right)^k$$

The image below shows the different representations (probability, odds and log-odds) as function of the parameter $x$. The logistic model $p = (1+\exp(x))^{-1}$ will model the log-odds as a linear function of $x$.

