Interpretation of Log Odds in Logistic Regression

Question

$\log(\text{odds}) = \text{logit}(P)=ln \big({{P}\over{1-P}}\big)$

$ln\big({{P}\over{1-P}}\big)=\beta_0+\beta_1x$

Consider this example: $0.7=\beta_o+\beta_1(x)+\beta_2(y)+\beta_3(z)$

How can this expression be interpreted?

Answer 1

Lets first understand what does odds ratio mean : Let’s say that the probability of success of some event is .8. Then the probability of failure is 1 – .8 = .2. The odds of success are defined as the ratio of the probability of success over the probability of failure.
In our example, the odds of success are .8/.2 = 4. That is to say that the odds of success are 4 to 1. If the probability of success is .5, i.e., 50-50 percent chance, then the odds of success is 1 to 1.

Why doe we take Log of Odds: One reason is that it is usually difficult to model a variable which has restricted range, such as probability. This transformation is an attempt to get around the restricted range problem. It maps probability ranging between 0 and 1 to log odds ranging from negative infinity to positive infinity.

Now Coming to interpratation:

If this equation gives 0.7 as log of odds it means that the values of x for which it was calculated increases the odds of success by that ratio