I am trying to predict a sale using a logit regression. My dependent variable is isSale and my dependent variables include only binary variables such as isHighSale_City, isHighSale_Domain, etc.
I have my regression output and converted the coefficients to odds and then to probability.
predicted_probability = odds_ratio / (1 + odds_ratio)
My question is can I sum the probabilities that I generated from the odds ratio to create a combined probability for the observation?
Example:
isHighSale_City: coef: 1.19; odds_ratio: 3.52; pred_probability: 0.779
isHighSale_Domain: coef: 0.61; odds_ratio: 1.83; pred_probability: 0.65
Intercept: coef: -6.88; odds_ratio: 0.00057; pred_probability: 0.00057
If I want to score an prospective sale in a high sale city and high sale domain, would it be:
0.779 + 0.65 + 0.00057 = 1.429
Best Answer
Your formula is incorrect. Odds can be converted to probability using the equation above. However, odds ratios are ratios of two different odds representing distinct probabilities. The model estimates from a logistic regression are additive on the log-odds scale. Create predictions on this scale using the appropriate coefficients, then transform the linear predictor using the inverse logit: $$\text{expit}(\alpha + x\beta) = (1+\exp(\alpha +x\beta))^{-1}$$.
This is the probabilistic prediction equation from a logistic regression. The intercept $\alpha$ is the log odds for response when all covariates are 0. The slope $\beta$ is the log odds ratio for adjacent groups in discrete or continuous $x$.