Why is logistic regression a linear model? I want to know why logistic regression is called a linear model. It uses a sigmoid function, which is not linear. So why is logistic regression a linear model?
 A: Logistic regression uses the general linear equation $Y=b_0+∑(b_i X_i)+\epsilon$. In linear regression $Y$ is a continuous dependent variable, but in logistic regression it is regressing for the probability of a categorical outcome (for example 0 and 1).
The probability of $Y=1$ is:
$$
P(Y=1) = {1 \over 1+e^{-(b_0+\sum{(b_iX_i)})}}
$$
A: The logistic regression model is of the form
$$
\mathrm{logit}(p_i) = \mathrm{ln}\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i}.
$$
It is called a generalized linear model not because the estimated probability of the response event is linear, but because the logit of the estimated probability response is a linear function of the predictors parameters.
More generally, the Generalized Linear Model is of the form
$$
\mathrm{g}(\mu_i) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i},
$$
where $\mu$ is the expected value of the response given the covariates.
Edit: Thank you whuber for the correction.
