Relationship between logit and odds ratios

Question

Alright so I am trying to sort out exactly what a logit is in terms of a ratio...

I understand that :

$logit(p) = log(\frac{p}{1-p}) = \beta$

and that

$exp(\beta) =$ odds ratio =$ \frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$

I guess what's not coming across is how $\beta$, not being a ratio of odds, converts to the odds ratio metric, when taken out of logarithmic space.

To provide a bit more, if this is the logistic regression equation for the constant

$log(\frac{p}{1-p}) = \beta + \beta_1*0 + \beta_2*0 + \beta_3*0 + \beta_4*0 + \beta_5*0$.

then $exp(\beta)$ = odds ratio

so likewise for

$log(\frac{p}{1-p}) = \beta + \beta_1*1$

then $exp(\beta+\beta_1*1)$ = odds ratio for a one unit increase in $\beta_1$

BUT, how does

$$exp(\beta+\beta_1*1) = \frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}\tag{1}$$

because doesn't the $+$ in log terms (e.g. $\beta+\beta_1*1$) equal a multiplication and not a division?

Answer 1

The misunderstanding is in (1). In fact $exp(\beta+\beta_1*1)\neq \frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$

You already know $log(\frac{p_1}{1-p_1}) = \beta + \beta_1*1$ then

$exp(\beta+\beta_1*1)=\frac{p_1}{1-p_1}$ it is an odds not an odds ratio.

while the $exp(\beta_1)$ itself indeed is an odds ration, since $OR=\frac{p_1}{1-p_1}/\frac{p_0}{1-p_0}=\frac{exp(\beta+\beta_1*1)}{exp(\beta+\beta_1*0)}=e^{\beta+\beta_1-\beta-0*\beta_1}=exp{(\beta_1)}$ suppose you have a binary(dummy) predictor variable or when it is a continuous variable you are talking about one unit change of the variable..

Also note $exp(\beta) =$ odds ratio =$ \frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$ is not correct either.