Inverse Logit Transformation

From Displayr
Jump to navigation Jump to search

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

Binary logit

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

[math]\displaystyle{ \Pr = \frac{1}{1 + e^{-\mu}} }[/math]

For example, [math]\displaystyle{ \mu = 2 \Rightarrow \Pr = 0.8807971 }[/math]

Multinomial logit

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

[math]\displaystyle{ \Pr_j = \frac{e^{\mu_j}}{\sum^J_{j'=1} e^{\mu_{j'}}} }[/math]

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

Also known as

Logistic Transformation