1
Advanced Topics in Forest
Biometrics
–
FOR6934
Multinomial Logistic Regression
What is Multinomial Logistic
Regression?
A form of logistic regression for predicting a discrete
(categorical) variable with two or more categories,
using continuous and/or discrete predictors
Addresses the same questions that discriminant
analysis and multiple regression do but without
distributional assumptions on the predictors
the independent and dependent variables need not be linearly
related
Homoscedasticity is not necessary
For a dichotomous response, you compute a single
logit function; for a multilevel response, you create
more than one logit function
Why use Multinomial Logistic
Regression?
Multinomial Logistic regression is often used
when the dependent variable is ordinal
Then,
cumulative logits
are computed
,
which are
based on the cumulative probabilities
For three response levels (high, medium, low), let:
q
1
=
p
1
q
2
=
p
1
+
p
2
1 =
p
1
+
p
2
+
p
3
And, you compute two cumulative logits:
+
=
+
=
3
2
1
2
3
2
1
1
ln
)
logit(
and
ln
)
logit(
p
p
p
q
p
p
p
q
The log odds
of high vs.
medium or low
The log odds of
high or medium
vs. low
Multinomial Logistic Regression
model
Given a set of
m
predictor variables
(categorical or continuous), the logit is:
Thus, there are separate intercept
parameters
(
b
0
)
and different sets of
regression parameters
(β
k
)
for each logit
m
mk
k
k
k
X
X
b
b
b
q
+
+
+
=
...
)
logit(
1
1
0
Questions
Can categories be correctly predicted given a set of
predictors?
Usually once this is established the predictors are
manipulated to see if the equation can be simplified.
Comparison of equation with predictors plus intercept to a
model with just the intercept
What is the strength of association between the
outcome variable and a set of predictors?
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 Spring '08
 Staff
 Regression Analysis, crown form, Multinomial Logistic

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