Residuals for Logistic Regression

Question

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.

Answer 1

You can have residuals, or not, in either case. But to start with, you don't need residuals at all to specify the model: In either case, you can write $$Y|X=x \ \ \sim \ \ p(y|x),$$ where $p(y|x)$ is some conditional distribution. In the case of logistic regression, $p(y|x)$ is a Bernoulli distribution. In the case of classical regression, $p(y|x)$ is a normal distribution, but this can be relaxed to allow other forms, like Poisson, negative binomial, multinomial, non- or semi-parametric variants, etc.

If you want residuals, you can also have them, in either case: For an observation where $X=x$, the residual is $\epsilon = Y - E(Y|X=x)$. In the case of standard regression, under the usual linearity assumption, this gives you $$\epsilon = Y - (\beta_0 + \beta_1 x).$$ In the case of logistic regression, under the usual linearity (of logit) assumption, this gives you $$\epsilon = Y - \frac{\exp{(\beta_0 + \beta_1 x)}}{1+\exp{(\beta_0 + \beta_1 x)}},$$ where $Y$ is either 0 or 1.

It is just not common to use this form in the case of logistic regression because the salient feature of the model is the form of the conditional Bernoulli distribution of $Y|X=x$.