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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

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    $\begingroup$ Could you explain how you would interpret a "probability" of $1.429$? $\endgroup$
    – whuber
    Commented Dec 13, 2017 at 20:17
  • $\begingroup$ For an observation where isHighSale_City=1 and isHighSale_Domain=1, the probability of it converting to a sale is 142% higher than if it isn't in a high sale city and a high sale domain? $\endgroup$
    – jonjon
    Commented Dec 13, 2017 at 21:40
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    $\begingroup$ @jonjon that is a probability ratio not a probability. The numbers should conform to the interpretation, not vice versa. $\endgroup$
    – AdamO
    Commented Dec 14, 2017 at 14:56

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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 $x$.

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  • $\begingroup$ Ah, so i can create an additive probability and then take the inverse logit to get a more "readable" probability for the observation? $\endgroup$
    – jonjon
    Commented Dec 15, 2017 at 16:51
  • $\begingroup$ I don't understand what a readable probability is. The probability is not added. The inverse logit of a probability is a log-odds. Using logistic regression parameters, you can add up the log odds (intercept) and log odds ratios in the fashion you've shown, then reverse transform them (with inverse logit) to obtain a probability. Or fit an additive risk model. $\endgroup$
    – AdamO
    Commented Dec 15, 2017 at 17:40
  • $\begingroup$ Apologies if I'm not completely clear. I'm relatively new to this. $\endgroup$
    – jonjon
    Commented Dec 15, 2017 at 21:25
  • $\begingroup$ The resources I'm finding online mostly stop at an explanation of the odds ratio. I'm trying to create a probabilistic score. To find the probability of a sale, I can take: a + (x)HighSale_city + (x)HighSale_domain, then take the inverse logit to find the probability? $\endgroup$
    – jonjon
    Commented Dec 15, 2017 at 21:35
  • $\begingroup$ @jonjon yes! the value you use there is the one entitled "coef" and not "odds ratio". $\endgroup$
    – AdamO
    Commented Dec 15, 2017 at 22:41

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