Logistic Regression Odd Ratio I am doing logistic regression in R on a binary dependent variable with only one independent variable. I found the odd ratio as 0.99 for an outcomes. This can be shown in following. Odds ratio is defined as, $ratio.odds(H) = \frac{P(X=H)}{1-P(X=H)}$. As given earlier $ratio.odds(H) = 0.99$ which implies that $P(X=H) = 0.497$ which is close to 50% probability. This implies that the probability for having a H cases or non H cases 50% under the given condition of independent variable. This does not seem realistic from the dataset as only ~20% are found as H cases. Please give clarifications and proper explanations of this kind of cases in logistic regression. What should I do further to solve this question?
 A: It seems like your interpretation of odds ratio (OR) is not correct $\frac{P(H = 1 | X)}{1 - P(H = 1 | X)} = \frac{P(H = 1 | X)}{P(H = 0 | X)}$ is an odd [$odds(X)$] not an odd ratio.
The odds ratio would be $\frac{odds(X)}{odds(X')}$ where $X'$ is a new set of values for your predictors. It is impractical to have several predictors change values at the same time so the odds ratio is most often defined for $X'$ differing from $X$ by only one of its elements (if you see $X$ as a vector of values/predictors).
Consider the following logistic model :
$log (\frac{P(H = 1 | X)}{P(H = 0 | X)}) = \beta_0 + \beta_1x_1 +\beta_2x_2$
Then the OR of that model in regards to the variable $x_1$ would be $OR(x_1) = \frac{\frac{P(H = 1 | x_1 + 1, x_2)}{1 - P(H = 1 | x_1 + 1, x_2)}}{\frac{P(H = 1 | x_1, x_2)}{1 - P(H = 1 | x_1, x_2)}}$
You can see that $x_1$ takes two different values but every other variables stay the same. And because $x_1$ only increased by one you have the special case (actually the most common case when people are computing OR) where your OR is directly linked to your estimated parameters.
$OR(x_1) = \frac{e^{\beta_0 + \beta_1(x_1 + 1) +\beta_2x_2}}{e^{\beta_0 + \beta_1x_1 +\beta_2x_2}}$ which simplifies into $OR(x_1) = e^{\beta1}$.
That is true with all parameters of your model.
So what $OR(x) = 0.99$ actually means is that for each 1 unit increase of $x$, everything else being kept the same, your odds is multiplied by $0.99$ ($e^{\beta1}$). So $0.99* odds(x) = odds(x + 1)$ which implies that $odds(x) > odds(x + 1)$ thus meaning that high values of $x$ are more prone to be met for $H = 0$.
Hopes it helps,
A: Odds Ratios can be tricky to interpret sometimes, and I could try and explain it to you but I think this link would do the explanation better justice that I would: How do I interpret odds ratios in logistic regression?
As for the probability (which I am assuming you obtained through your log odds ratio), I think you need to delve into your data to see why you're seeing such a peak, it is most likely related to that. And also, you could calculate the change in log odds and change in odds for your independent variable to help you understand the changes per unit increase (also explained in the link above).
Hope that helps!
