# Inverse Logit Transformation

The inverse logit transformation converts parameter estimates from Logit Models into probabilities.

## Binary logit

Where $\displaystyle{ \mu }$ is the fitted value from a Binary Logit Model, the probability is computed as:

$\displaystyle{ \Pr = \frac{1}{1 + e^{-\mu}} }$

For example, $\displaystyle{ \mu = 2 \Rightarrow \Pr = 0.8807971 }$

## Multinomial logit

Where $\displaystyle{ \mu_j }$ is the utility for the $\displaystyle{ j }$th of $\displaystyle{ J }$ alternatives, the probability of choosing the $\displaystyle{ j }$th alternative is:

$\displaystyle{ \Pr_j = \frac{e^{\mu_j}}{\sum^J_{j'=1} e^{\mu_{j'}}} }$

For example, if in a MaxDiff experiment analyzed using a logit model the three alternatives, A, B and C, estimated parameters of 0, 0.5 and 0.9, the probability of choosing the first alternative is $\displaystyle{ \frac{e^{0.5}}{e^0 + e^{0.5} + e^{0.9}} = 0.3071959 }$. And, the probability of choosing B if only B and C were available is: $\displaystyle{ \frac{e^{0.5}}{e^{0.5} + e^{0.9}} = 0.3775407 }$

## Also known as

Logistic Transformation