How is the doubling amount for an input variable calculated in logistic regression?

In logistic regression, the coefficient (b1) associated with an input variable indicates how the odds of the outcome change with a one-unit increase in that variable. Specifically, the relationship between the odds and the input variable can be expressed using the exponential function.

When analyzing how much the odds of the outcome increase when the input variable is doubled, we need to consider the properties of the logistic function. Doubling an input variable means substituting (x) with (2x). The odds of the outcome, when evaluated at (x) and (2x), can be expressed as follows:

• For (x), the odds = (e^{b1 \cdot x})

• For (2x), the odds = (e^{b1 \cdot (2x)} = e^{2 \cdot b1 \cdot x})

The increase in odds can be calculated as the ratio of the two odds:

[

\text{Odds at } 2x / \text{Odds at } x = e^{2 \cdot b1 \cdot x} / e^{b1 \cdot x} = e^{(2b1 - b1) \cd