Solved – Odds ratio in logistic regression with multiple predictors

I think the relationship might be clearer with a different expression of the model. Consider the logistic regression model as its expressed on p. 225 of this excerpt:

Formally, the model logistic regression model is that $$logfrac{p(x)}{1-p(x)} = beta_0 + beta_1x$$

Equivalently:

$$frac{p(x)}{1-p(x)} = e^{beta_0 + beta_1x}$$

In words, logistic regression models log odds as a linear function of the predictors; the odds are the exponentiation of this linear combination. In the multivariate case, this gives:

$$frac{p(x)}{1-p(x)} = e^{beta_0 + beta_1x_1…+beta_nx_n}$$

Look at it as a product

$$frac{p(x)}{1-p(x)} = e^{beta_0}e^{beta_1x_1}…e^{beta_nx_n}$$

and the relationship between the fitted coefficients and predictors is clear: For every one-unit increase in $x_i$, the log-odds increase by a factor of $e^{beta_i}$. This seems to be the relationship that your formula seeks to stress. I believe the proof you ask for is this:

$$OR = frac{operatorname{odds}(F(hat{textbf x}))}{operatorname{odds}(F(textbf x))} = frac{frac{F(hat{textbf x})}{1 – F(hat{textbf x})}}{frac{F(textbf x)}{1 – F(textbf x)}} = frac{e^{beta_0 + beta_1 x_1 + … + beta_m(x_m+1)+beta_nx_n}}{e^{beta_0 + beta_1 x_1 + … + beta_mx_m+beta_nx_n}} = e^{beta_m}$$

with $hat{x}_i = x_i$ for $i ne m$, and $hat x_m = x_m + 1$. (For my part, I don't find this particularly intuitive or illustrative.)

What probabilities are being compared?

If $x_m$ is continuous, it's comparing the odds with a one-unit change in $x_m$, while holding all other predictors equal; if $x_m$ is binary, it's comparing two cases in which all variables are equal except the presence or absence of $x_m$.

Is it really still the OR once it changes?

You may find it more helpful to think of it as your understanding of the odds ratio changing in light of new information.