Inverse Logit Transformation

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

Binary logit

Where \(\mu\) is the fitted value from a Binary Logit Model, the probability is computed as:

\[\begin{align} \Pr = \frac{1}{1 + e^{-\mu}}\end{align}\]

For example, \(\mu = 2 \Rightarrow \Pr = 0.8807971\)

Multinomial logit

Where \(\mu_j\) is the utility for the \(j\)th of \(J\) alternatives, the probability of choosing the \(j\)th alternative is:

\[\begin{align} \Pr_j = \frac{e^{\mu_j}}{\sum^J_{j'=1} e^{\mu_{j'}}}\end{align}\]

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 \(\frac{e^{0.5}}{e^0 + e^{0.5} + e^{0.9}} = 0.3071959\). And, the probability of choosing B if only B and C were available is: \(\frac{e^{0.5}}{e^{0.5} + e^{0.9}} = 0.3775407\).

Also known as

Logistic Transformation