I have read that there are multiple ways to look at residuals for logistic regression. I am not necessarily that interested in analyzing the residuals for a logit model, but my main question is rather:
Where are residuals in the specification of the model?
For linear models, it's common to write $$y=X\beta+\varepsilon \ \ \text{ or } \ \ y = \beta_0+\beta_1x_1 + \dots + \beta_kx_k+\varepsilon$$
However, for the logit model I keep finding $$l=\log\frac{p}{1-p}=\beta_0+\beta_1x_1 + \dots + \beta_kx_k \ \ \text{ or } \ \ p = \frac{1}{1+e^{-(\beta_0+\beta_1x_1 + \dots + \beta_kx_k)}}$$
How come the disturbances are not part of the model in this case?
Edit:
I have thought of something but am not sure if this is the complete reasoning behind it. In my understanding, a linear model assumes that data have been generated according to $y=X\beta$ with random disturbances, as nothing is perfectly linear.
And for a logit model this disturbance is not needed, as the data are assumed to be generated(?) according to $p$: at every point $x_i$ there is a certain probability $p$ that the outcome will be $1$, and the points $y=0,1$ are generated according to this, rather than a set linear relation with disturbances that make it random.