# Solving for probability with negative logits

I have a multinomial logistic regression tutorial question asking to manually solve the logits and probability. When I calculate logit for both comparisons I get negative values. How do I continue and solve for P? The question part b asks, "What is the model's prediction regarding the classification of the movie?"

If my calculations give me model 1 P = 0.05599435837 and model 2 P = 0.01193858104 do I do P(Low) = 1 - (0.05599435837) - (0.01193858104) = 0.93206706059 to get the missing probability? So does 93% means the model's prediction is 93% that the movie will be classified as low revenue? 93% seems pretty skewed does it not?

I am writing my formulas as:
model 1 - Low vs Medium
Movie Success = constant + 5.316*LOpening - 0.003*Theatres + 0*Rating
Movie Success = -7.007 + 5.316*2.4893 - 0.003*3017 + 0*1
Movie Success = -7.007 + 13.2331188 - 9.051 + 0
P = -2.8248812

model 2 - Low vs High

Movie Success = constant + 8.128*LOpening - 0.002*Theatres + 0*Rating
Movie Success = -18.615 + 8.128*2.4893 - 0.002*3017 + 0*1
Movie Success = -18.615 + 20.2330304 - 6.034 + 0
P = -4.4159696

p.s. I noticed that SPSS says

b. This parameter is set to zero because it is redundant.

I was told I could therefore discard that value. That's why I have it as 0.

Logit is defined as

$$\operatorname{logit}(p) = \log\left( \frac{p}{1-p} \right)$$

where $$p$$ is a probability, logit itself is not a probability, but log-odds. It can be negative, since it potentially ranges from $$-\infty$$ to $$\infty$$. To transform logit into probability you need to use logistic function for binary classification, or softmax for multiclass classification.

• So I calculated both probabilities. P = e^-2.8248812 / (1+e^-2.8248812) = 0.05599435837 and P = e^-4.4159696 / (1+e^-4.4159696) = 0.01193858104. Do I just add those two together? 0.05599435837 + 0.01193858104 = 0.06793293941? If the final P = 67% how do I report/express that? The probability that the move will have a low chance of success is 67%? – ShibuyaShoto Oct 5 '20 at 10:25
• @TokyoToo that's not what I said. Obviously probability cannot be equal to 8.24, since probability needs to be something between 0 and 1. The values you are referring to are logits, to make probabilities of them, you need to transform them first. – Tim Oct 5 '20 at 10:25
• I know they are logits. That's why I transformed them to probabilities. See my comment :) – ShibuyaShoto Oct 5 '20 at 10:29
• You lost me in the calculations. Maybe this helps: en.wikipedia.org/wiki/Multinomial_logistic_regression ? – Tim Oct 5 '20 at 10:33
• I took the two logits -2.8248812 and -4.4159696 and used E ^ Ln[P/(1-P)] to transform them to two probabilities 0.05599435837 and 0.01193858104. Then I did P(Low) = 1 - P(Medium) - P(High) -->> P(Low) = 1 - (0.05599435837) - (0.01193858104) = 0.93206706059 -->> P(Low) = 93%. Is this correct? – ShibuyaShoto Oct 5 '20 at 23:19

A logit isn't the same as probability, see the definition here.
There's no problem with a logit being negative, that just means the probability is lower than half.
In addition, I don't think you can calculate the logit of Low w.r.t. Medium and High together, because you'd need to know the joint probabilities. And in any case, your question doesn't ask for that.
here's a link to a bunch of videos about logits and logistical regression btw, if you want a bigger refresher and background.