Bài 5: Mô hình Multinomial Logit

MULTINOMIAL LOGIT MODEL

Multinomial responses



logit/probit model: dependent variable = 0/1



What if more than 2 categories?



Example: long term effect of the exposure to

radiation may be



1 – dead of cancer

Multinomial responses



Example: choice of health care providers



1 – public hospital



2 – private hospital/clinic



3 – “lang y”



4 – self-treatment



Other examples:



choice of car (Y = Toyota, Honda, Suzuki, Mazda, KIA…)



choice of specialization at university



choice of occupation

Multinomial logistic regression model



Multinomial logit model (MNL) is used to analyze

the relationship between categorical variables and

other explanatory variables



Notice:



nominal response (MNL)



ordinal response (ordered probit model – not covered

The dependent variable



The occurrence of an alternative j for individual i



Probability of occurrence of each alternative

1, 2, 3,...,

i

Y



J

i1

i2

i3

iJ

1

probability =

2

probability =

3

probability =

...

...

probability =

i

p

p

Y

p

J

p

The logit (log-odds ratio)



Logit

i1

i1

i2

i1

i3

i1

1

log

2

log

3

log

...

...

0

i

iJ

i

iJ

i

iJ

i

iJ

p

Y

h

p

p

Y

h

p

p

Y

h

p

Y

J

h













Modelling the logits



i1

i1

1

i2

i1

2

i3

i1

3

1

log

2

log

3

log

...

...

0

i

i

iJ

i

i

iJ

i

i

iJ

i

iJ

p

Y

h

X

p

p

Y

h

X

p

p

Y

h

X

p

Y

J

h

Modeling the probabilities



i1

1

i2

1

i3

1

iJ

1

1

probability =

2

probability =

3

probability =

Maximum likelihood estimation



MNL model is estimated by maximizing the

log-likelihood function

1

1

log

ln

N

J

ij

ij

i

j

L

y

p









0

if j is NOT chosen

1

if j is chosen

Data

id

case

choice

thunhap

gioitinh

…

1

1

2

12

1

1

2

4

12

1

2

1

3

21

1

3

1

17

0

3

2

17

0

3

3

17

0

…

Choice: 1 = Commune health center; 2 = Public hospital; 3 = Private hospital; 4 = Lang y;

5 = Individual health care provider

Estimate MNL in Stata



mlogit choice thunhap gioitinh

(choice==2 is the base outcome)

_cons -2.872579 .1263704 -22.73 0.000 -3.12026 -2.624898
gioitinh .0954035 .1621446 0.59 0.556 -.222394 .413201
thunhap 1.97e-06 4.44e-07 4.43 0.000 1.10e-06 2.84e-06
5

_cons -3.795684 .3483009 -10.90 0.000 -4.478342 -3.113027
gioitinh .1885005 .3481328 0.54 0.588 -.4938273 .8708283
thunhap -8.18e-06 4.17e-06 -1.96 0.050 -.0000164 -1.22e-08
4

_cons -1.763979 .0821101 -21.48 0.000 -1.924912 -1.603046
gioitinh .2087203 .0979244 2.13 0.033 .016792 .4006485
thunhap 1.81e-06 4.19e-07 4.32 0.000 9.90e-07 2.63e-06
3

_cons -1.180521 .1060245 -11.13 0.000 -1.388325 -.9727166
gioitinh .1822304 .1061043 1.72 0.086 -.0257303 .3901911
thunhap -9.39e-06 1.29e-06 -7.29 0.000 -.0000119 -6.86e-06
1

Explain the estimation results



Suppose there are 2 persons of same sex, A’s

income is 1 mil VND higher than that of B, so



A:



B:

i1

1

1

1

1

log

_i

_i

_i

iJ

p

X

thunhap

gioitinh

p



  









1

1

1

1

log

A

_A

_A

AJ

p

thunhap

gioitinh

p



 











1

1

1

1

log

B

_A

1

_B

BJ

p

Explain the estimation results



the estimated coefficient indicates the responses

of log-odds ratio for a unit change in explanatory

variable

1

1

1

1

1

log

log

log

B

B

A

_BJ

A

BJ

AJ

AJ

p

p

p

_p

p

p

p

p

Hypothesis testing



Test the null hypothesis

1

2

1

H0:











_j

 

...



_J

_



0

Prob > chi2 = 0.0000

chi2( 4) = 82.49

( 4) [5]thunhap = 0

Hypothesis testing



Kiểm định giả thuyết

1

H0:





0

Prob > chi2 = 0.0000

chi2( 1) = 53.17

( 1) [1]thunhap = 0

Kiểm định



Test the null hypothesis: all coefs in [1] = 0

Prob > chi2 = 0.0000

chi2( 2) = 56.38

( 2) [1]gioitinh = 0

( 1) [1]thunhap = 0

. test [1]

Prob > chi2 = 0.0000

chi2( 2) = 56.38

( 2) [1]gioitinh = 0

( 1) [1]thunhap = 0

Marginal effect



What happens to the probability of choosing [1] if

income increase by 1 mil VND?

(*) dy/dx is for discrete change of dummy variable from 0 to 1

gioitinh* .0132477 .00985 1.34 0.179 -.006067 .032563 .527194

thunhap -.0009239 .0001 -8.99 0.000 -.001125 -.000723 82.9054

variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X

= .10632185

y = Pr(choice==1) (predict, p outcome(1))

Marginal effects after mlogit

Marginal effect



For a female with income 500 mil VND/year, the

probability of choosing [1] if income increase by 1

mil VND?

(*) dy/dx is for discrete change of dummy variable from 0 to 1

gioitinh* .0002118 .00023 0.91 0.364 -.000246 .000669 1

thunhap -.0000202 .00001 -2.23 0.026 -.000038 -2.4e-06 500

variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X

= .00199241

y = Pr(choice==1) (predict, p outcome(1))

Marginal effects after mlogit

Prediction



Predict the probability of choosing private

hospitals/clinic

bvtu 3475 .1502158 .0348196 .1121532 .6523073

Variable Obs Mean Std. Dev. Min Max

. sum bvtu