Bài 5: Mô hình Multinomial Logit
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MULTINOMIAL LOGIT MODEL
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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
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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
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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
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The dependent variable
The occurrence of an alternative j for individual i
Probability of occurrence of each alternative
1, 2, 3,...,
<i>i</i>
<i>Y</i>
<i>J</i>
i1
i2
i3
iJ
1
probability =
2
probability =
3
probability =
...
...
probability =
<i>i</i>
<i>p</i>
<i>p</i>
<i>Y</i>
<i>p</i>
<i>J</i>
<i>p</i>
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The logit (log-odds ratio)
Logit
i1
i1
i2
i1
i3
i1
1
log
2
log
3
log
...
...
0
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>iJ</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>p</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>p</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>p</i>
<i>Y</i>
<i>J</i>
<i>h</i>
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Modelling the logits
i1
i1
1
i2
i1
2
i3
i1
3
1
log
2
log
3
log
...
...
0
<i>i</i>
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>i</i>
<i>iJ</i>
<i>i</i>
<i>iJ</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>X</i>
<i>p</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>X</i>
<i>p</i>
<i>p</i>
<i>Y</i>
<i>h</i>
<i>X</i>
<i>p</i>
<i>Y</i>
<i>J</i>
<i>h</i>
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Modeling the probabilities
1
2
3
i1
1
i2
1
i3
1
iJ
1
1
probability =
2
probability =
3
probability =
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Maximum likelihood estimation
MNL model is estimated by maximizing the
log-likelihood function
1
1
log
ln
<i>N</i>
<i>J</i>
<i>ij</i>
<i>ij</i>
<i>i</i>
<i>j</i>
<i>L</i>
<i>y</i>
<i>p</i>
0
if j is NOT chosen
1
if j is chosen
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Data
<b>id</b>
<b>case</b>
<b>choice</b>
<b>thunhap</b>
<b>gioitinh</b>
<b>…</b>
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
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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
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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><sub>i</sub></i>
<i><sub>i</sub></i>
<i><sub>i</sub></i>
<i>iJ</i>
<i>p</i>
<i>X</i>
<i>thunhap</i>
<i>gioitinh</i>
<i>p</i>
1
1
1
1
log
<i>A</i>
<i><sub>A</sub></i>
<i><sub>A</sub></i>
<i>AJ</i>
<i>p</i>
<i>thunhap</i>
<i>gioitinh</i>
<i>p</i>
1
1
1
1
log
<i>B</i>
<i><sub>A</sub></i>
1
<i><sub>B</sub></i>
<i>BJ</i>
<i>p</i>
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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
<i>B</i>
<i>B</i>
<i>A</i>
<i><sub>BJ</sub></i>
<i>A</i>
<i>BJ</i>
<i>AJ</i>
<i>AJ</i>
<i>p</i>
<i>p</i>
<i>p</i>
<i><sub>p</sub></i>
<i>p</i>
<i>p</i>
<i>p</i>
<i>p</i>
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Hypothesis testing
Test the null hypothesis
1
2
1
H0:
<i><sub>j</sub></i>
...
<i><sub>J</sub></i>
<sub></sub>
0
Prob > chi2 = 0.0000
chi2( 4) = 82.49
( 4) [5]thunhap = 0
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Hypothesis testing
Kiểm định giả thuyết
1
H0:
0
Prob > chi2 = 0.0000
chi2( 1) = 53.17
( 1) [1]thunhap = 0
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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
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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
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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
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Prediction
Predict the probability of choosing private
hospitals/clinic
bvtu 3475 .1502158 .0348196 .1121532 .6523073
Variable Obs Mean Std. Dev. Min Max
. sum bvtu
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