How to understand logit results? I have some output from a binary logit regression and I have some troubles to interpret it properly. Particularly I would like to know whether the coefficient of “male” tells me about the probability of outcomes 0 and 1 and what role the intecept plays in that example
"Estimate"                      "Std. err."                          "z-score"
(Intercept) $-5.183$           $1.094$                       $-4.738$
Male $0.652$                   $0.318$                          $2.053$      
Thanks a lot
 A: Roughly speaking, in a logistic regression one has $$\log\left(\frac{p}{1-p}\right)=\beta_0+\langle \beta,x\rangle$$
where $p$ it the probability of the binary outcome r.v. $Y$, while the $x's$ are the independent variables and $\beta$'s the coefficients (to be found by MLE). The coeffcient $\beta_0$ is the intercept. Equivalently one can write
$$p=\frac{1}{1+e^{-\beta_0-\langle \beta,x\rangle}}$$
and
$$ \mathcal O(x_1,\dots,x_n;\beta):=\frac{p}{1-p}=e^{\beta_0+\langle \beta,x\rangle},$$
where the quantities $\mathcal O(x_1,\dots,x_n;\beta)$ are called odd-ratios.
In logistic regressions the odd ratios are for the interpretation of results.
Let us suppose that the MLE converges and one finds a set of ML estimates $\hat{\beta}$ of $\beta$'s for the given logistic regression model. Then
$$\frac{\mathcal O(x_1,\dots,x_i+1,\dots,x_n;\hat{\beta})}{ \mathcal O(x_1,\dots,x_n;\hat{\beta})}=e^{\hat{\beta}_i},$$
i.e. the exponential of the $i$-th estimated coefficient is equal to the relative increase of odd ratios, increasing the $i$-th regression of one unit and keeping all other regressors constant. Once the interpretation of odd ratios is clear in the given context (where the logistic regression is performed), the above formula gives a strightforward interpretation of results.
