Bài 5: Mô hình Multinomial Logit

 ordinal response (ordered probit model – not covered. in this lecture).[r]

MULTINOMIAL LOGIT MODEL

Multinomial responses

 logit/probit model: dependent variable = 0/1  What if more than 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: = Commune health center; = Public hospital; = Private hospital; = 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

_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

_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

_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

Explain the estimation results

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

income is 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] =

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 mil VND?

(*) dy/dx is for discrete change of dummy variable from 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 mil VND?

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