Conditional Distribution in logistic regression

Question

In a logistic regression, given that $\pi_i=Pr(Y_i=1|X_i=x_i)=\dfrac{\text{exp}(\beta_0+\beta_1 x_i)}{1+\text{exp}(\beta_0+\beta_1 x_i)}$, is it correct to say that logistic regression assumes that the conditional distribution $f(Y|X) \sim Logistic$?

Answer 1

I am going to flesh out some of the comments to this post into an answer (with thanks to Glen_b and AlexR for their excellent comments). The first thing to note is that the response variable in a logistic regression is binary, with allowable values $y_i = 0,1$, so any distribution that does not accord with that support is incorrect. The distribution for the response variable in a logistic regression is Bernoulli, with a logistic mean.

Response distribution: The response distribution for a logistic regression is:

$$Y_i | \mathbf{x} \sim \text{Bern}(\pi(x_i)) \quad \quad \quad \pi(x_i) = \frac{\exp(\beta_0+\beta_1 x_i)}{1 + \exp(\beta_0+\beta_1 x_i)}.$$

The function $\pi$ is a logistic function of an affine transformation of the argument value $x_i$.

The logistic regression model can be represented using various (equivalent) model forms that yield the above response distribution. One of these model formulations uses a pseudo-error term with a standard logistic distribution. Taking $\varepsilon_i \sim \text{Logistic}(0, 1)$ we have:

$$\begin{equation} \begin{aligned} \mathbb{P}(\beta_0+\beta_1 x_i + \varepsilon_i > 0) &= \mathbb{P}(\varepsilon_i > -(\beta_0+\beta_1 x_i)) \\[16pt] &= 1- F_{\varepsilon_i}(-(\beta_0+\beta_1 x_i)) \\[12pt] &= 1 - \frac{1}{1 + \exp(\beta_0+\beta_1 x_i)} \\[6pt] &= \frac{\exp(\beta_0+\beta_1 x_i)}{1 + \exp(\beta_0+\beta_1 x_i)} \\[8pt] &= \pi(x_i). \\[6pt] \end{aligned} \end{equation}$$

Since this is the probability of a positive response outcome in the above distribution, we can formulate the logistic regression model in a way that is similar to a standard linear regression model, but with a pseudo-error term that is only used to classify the response into two categories:

$$Y_i = \mathbb{I}(\beta_0+\beta_1 x_i + \varepsilon_i > 0) \quad \quad \quad \varepsilon_1, ..., \varepsilon_n | \mathbf{x} \sim \text{IID Logistic}(0, 1).$$