Multinomial logistic regression

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In statistics , multinomial logistic regression , also known as multinomial logit regression ( MNL ), polytomous logistic regression , polychotomous logistic regression , softmax regression or maximum entropy classifier , is a regression analysis method. It “serves to estimate group membership or a corresponding probability of this.” The response variable is a nominal variable (equivalent categorical variable , ie that it falls into one of several categories and has no meaningful order). In the case of an ordinal response variable , it is called an ordered logistic regression .

Description of the procedure

This is a special form of logistic regression , in which the response variable a nominal scale level with more than two forms can have . In addition, the vector of the regressors is given. A separate regression model is output for each of the characteristics of the dependent variables (except for one reference category ) . The probability of occurrence for each category is specified as follows : ${\displaystyle Y_{i}}$ ${\displaystyle Y_{i}\in \{1,\ldots ,c+1\}}$ ${\displaystyle \mathbf {x} _{i}^{\top }=(1,x_{i1},\ldots ,x_{ik})}$ ${\displaystyle r}$

${\displaystyle \pi _{ir}=\Pr(Y_{i}=r)={\frac {\exp \left(\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{r}\right)}{1+\sum _{s=1}^{c}\exp \left(\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{s}\right)}}\quad ,r=1,\ldots ,c}$,

with the linear predictors or and the response function , ie the inverse function of the coupling function . The following applies to the reference category: ${\displaystyle \eta _{ir}=\beta _{r0}+\beta _{r1}x_{i1}+\beta _{r2}x_{i2}+\ldots +\beta _{rk}x_{ik}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{r}}$ ${\displaystyle \eta _{is}=\beta _{s0}+\beta _{s1}x_{i1}+\beta _{s2}x_{i2}+\ldots +\beta _{sk}x_{ik}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{s}}$ ${\displaystyle \pi _{ir}=h_{r}(\eta _{ir},\ldots ,\eta _{ic})\quad ,r=1,\ldots ,c}$

${\displaystyle \pi _{i,c+1}=1-\pi _{i1}-\ldots -\pi _{ic}={\frac {1}{1+\sum _{s=1}^{c}\exp \left(\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{s}\right)}}}$.

Case study

The example deals with the voting intention of a person depending on person-specific factors. A person's intention to vote according to different parties is known from survey data (dependent categorical variable ). This should be explained by various factors (the scale of which is irrelevant), for example age, gender and education.

Web links

• Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit (PDF; 92 kB)
• Chapter 6 Multinomial Response Models

Individual evidence

1. ↑ Archived copy ( Memento of the original from March 27, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ^ Ludwig Fahrmeir, Thomas Kneib , Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 330.
3. ^ Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 344.