I am trying to proof that logit is a linear model using the asumption that logisitic regresion is to a linear model?
We start from that asumption:
$$P(Y=1|X_1,\cdots X_p) = \frac{1}{1+e^{-(\beta_0+\beta_1X_1+ \cdots +\beta_pX_p)}} = \frac{1}{1+e^{-z}}$$
Proof
We can define logit like:
$$logit = log(\frac{p(X)}{1-p(X)})$$
Usin the first asumption $$\frac{1}{1+e^{-z}}$$
We have:
$$logit = log(\frac{\frac{1}{1+e^{-z}}}{1-\frac{1}{1+e^{-z}}}) = log(\frac{\frac{1}{1+e^{-z}}}{\frac{1+e^{-z}-1}{1+e^{-z}}}) = log(\frac{1}{e^{-Z}}) = log(e^{z}) = Z = \beta_0+\beta_1X_1+ \cdots +\beta_pX_p$$
So logit is linear
Is this a correct way, or how can i proof that using the first asumption?
Thanks