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In simple logistic regression, a common interpretation of the model coefficient $\beta$ is that a 'one-unit increase in the independent variable is associated with an increase of the log-odds ratio of the outcome variable by $\beta$'). This is already answered here.

What confused me is that the logit function is not linear with respect to $x$, so how can a one-unit increase in $x$ lead to a constant change in the log-odds ratio?

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In logistic regression we have that:

$$ \ln\left({\frac{p}{1-p}}\right) = \mathbf{Xb} $$

where $\mathbf{Xb}$ is the linear predictor, $\mathbf{X}$ being the model matrix of explanatory variables and $\mathbf{b}$ the vector of coefficients. $\ln\left({\frac{p}{1-p}}\right)$ is the logit function (log of the odds).

Therefore every component of $\mathbf{X}$ (eg, an individual explanatory variable) is proportional to the log odds and so any change in one of them has a linear effect on it.

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