Multinomial Logit Models – Page 1
Multinomial Logit Models - Overview
This is adapted heavily from Menard’s Applied Logistic Regression analysis; also, Borooah’s
Logit and Probit: Ordered and Multinomial Models; Also, Hamilton’s Statistics with Stata,
Updated for Version 7.
When categories are unordered, Multinomial Logistic regression is one often-used strategy.
Mlogit models are a straightforward extension of logistic models.
Suppose a DV has M categories. One value (typically the first, the last, or the value with the
highest frequency) of the DV is designated as the reference category. The probability of
membership in other categories is compared to the probability of membership in the reference
category.
For a DV with M categories, this requires the calculation of M-1 equations, one for each
category relative to the reference category, to describe the relationship between the DV and the
IVs.
Hence, if the first category is the reference, then, for m = 2, …, M,
mi
K
k
ik
mk
m
Z
X
Yi
P
m
P
1
)
1
(
)
(
ln
Hence, for each case, there will be M-1 predicted log odds, one for each category relative to the
reference category. (Note that when m = 1 you get ln(1) = 0 = Z
11
, and exp(0) = 1.)
When there are more than 2 groups, computing probabilities is a little more complicated than it
was in logistic regression. For m = 2, …, M,
M
h
hi
mi
i
Z
Z
m
Y
P
2
)
exp(
1
)
exp(
)
(
For the reference category,
M
h
i
Z
Y
P
2
)
exp(
1
1
)
1
(
In other words, you take each of the M-1 log odds you computed and exponentiate it. Once you
have done that the calculation of the probabilities is straightforward.
Note that, when M = 2, the mlogit and logistic regression models (and for that matter the ordered
logit model) become one and the same.