Chapter 3 Ordinal Logit
Adapted from Long and Freese (2006)
3.1 Compare Ordinal Logit and Binary Logit
First, let’s inspect the outcome of interest, which is labor force status.
(Data from 1976 PSID-T Mroz)
Paid Labor |
Force: |
1=yes 0=no | Freq. Percent Cum.
------------+-----------------------------------
NotInLF | 325 43.16 43.16
inLF | 428 56.84 100.00
------------+-----------------------------------
Total | 753 100.00Now, let’s compare our ordinal logit and binary logit coefficients when the outcome is binary.
quietly logit lfp k5 k618 age wc hc lwg inc, nolog
estimates store logit
quietly ologit lfp k5 k618 age wc hc lwg inc, nolog
estimates store ologit
estimates table logit ologit, b(%9.3f) t varlabel varwidth(30) equations(1:1) Variable | logit ologit
-------------------------------+------------------------
#1 |
# kids < 6 | -1.463 -1.463
| -7.43 -7.43
# kids 6-18 | -0.065 -0.065
| -0.95 -0.95
Wife's age in years | -0.063 -0.063
| -4.92 -4.92
Wife College: 1=yes 0=no | 0.807 0.807
| 3.51 3.51
Husband College: 1=yes 0=no | 0.112 0.112
| 0.54 0.54
Log of wife's estimated wages | 0.605 0.605
| 4.01 4.01
Family income excluding wife's | -0.034 -0.034
| -4.20 -4.20
Constant | 3.182
| 4.94
-------------------------------+------------------------
cut1 |
Constant | -3.182
| -4.94
--------------------------------------------------------
legend: b/tThe coefficients are the same between the binary logit and the ordinal logit. The difference here is that the ordinal logit has a cut point and assumes an interept of 0. The cutpoint and the intercept for binary logit are the same in absolute terms.
3.2 Example of Ordinal Dependent Variable
use "/Users/Sam/Desktop/Econ 645/Data/ordwarm2.dta", clear
describe warm yr89 male white age ed prst(77 & 89 General Social Survey)
storage display value
variable name type format label variable label
----------------------------------------------------------------------------------------------------------------------
warm byte %10.0g SD2SA Mom can have warm relations with child
yr89 byte %10.0g yrlbl Survey year: 1=1989 0=1977
male byte %10.0g sexlbl Gender: 1=male 0=female
white byte %10.0g racelbl Race: 1=white 0=not white
age byte %10.0g Age in years
ed byte %10.0g Years of education
prst byte %10.0g Occupational prestige Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
warm | 2,293 2.607501 .9282156 1 4
yr89 | 2,293 .3986044 .4897178 0 1
male | 2,293 .4648932 .4988748 0 1
white | 2,293 .8765809 .3289894 0 1
age | 2,293 44.93546 16.77903 18 89
-------------+---------------------------------------------------------
ed | 2,293 12.21805 3.160827 0 20
prst | 2,293 39.58526 14.49226 12 82Our outcome is the the survey results for people’s opinion for a working mother
Mom can |
have warm |
relations |
with child | Freq. Percent Cum.
------------+-----------------------------------
SD | 297 12.95 12.95
D | 723 31.53 44.48
A | 856 37.33 81.81
SA | 417 18.19 100.00
------------+-----------------------------------
Total | 2,293 100.00We will run our ordinal logit by regressing opinion’s on working mother’s warm onto gender, race, age, survey year, education in years, and occupational prestige.
\[ warm_{i,t} = \beta_1 male_i + \beta_2 race_i + \beta_3 age_{i,t} + \beta_4 edu_{i,t} + \beta_5 prestige_{i,t} + \beta_6 y89_t + \varepsilon_{i,t} \]
Ordered logistic regression Number of obs = 2,293
LR chi2(6) = 301.72
Prob > chi2 = 0.0000
Log likelihood = -2844.9123 Pseudo R2 = 0.0504
------------------------------------------------------------------------------
warm | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .5239025 .0798989 6.56 0.000 .3673036 .6805014
male | -.7332997 .0784827 -9.34 0.000 -.887123 -.5794765
white | -.3911595 .1183808 -3.30 0.001 -.6231816 -.1591373
age | -.0216655 .0024683 -8.78 0.000 -.0265032 -.0168278
ed | .0671728 .015975 4.20 0.000 .0358624 .0984831
prst | .0060727 .0032929 1.84 0.065 -.0003813 .0125267
-------------+----------------------------------------------------------------
/cut1 | -2.465362 .2389128 -2.933622 -1.997102
/cut2 | -.630904 .2333156 -1.088194 -.1736138
/cut3 | 1.261854 .234018 .8031871 1.720521
------------------------------------------------------------------------------We have three cutpoints \[ \tau_1 = -2.47, \tau_2 = -0.63, \tau_3 = 1.26 \]
3.3 Test Proportional Odds Assumption
We have our two tests for testing the proportional odds assumption 1. Approximate Likelihood Ratio Test 2. Wald Test
3.3.1 Approximate Likelihood Ratio Test
We use the omodel command to run an Approximate Likelihood Ratio Test
Iteration 0: log likelihood = -2995.7704
Iteration 1: log likelihood = -2846.4532
Iteration 2: log likelihood = -2844.9142
Iteration 3: log likelihood = -2844.9123
Ordered logit estimates Number of obs = 2293
LR chi2(6) = 301.72
Prob > chi2 = 0.0000
Log likelihood = -2844.9123 Pseudo R2 = 0.0504
------------------------------------------------------------------------------
warm | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .5239025 .0798988 6.56 0.000 .3673037 .6805013
male | -.7332997 .0784827 -9.34 0.000 -.8871229 -.5794766
white | -.3911595 .1183808 -3.30 0.001 -.6231815 -.1591374
age | -.0216655 .0024683 -8.78 0.000 -.0265032 -.0168278
ed | .0671728 .015975 4.20 0.000 .0358624 .0984831
prst | .0060727 .0032929 1.84 0.065 -.0003813 .0125267
-------------+----------------------------------------------------------------
_cut1 | -2.465362 .2389126 (Ancillary parameters)
_cut2 | -.630904 .2333155
_cut3 | 1.261854 .2340179
------------------------------------------------------------------------------
Approximate likelihood-ratio test of proportionality of odds
across response categories:
chi2(12) = 48.91
Prob > chi2 = 0.0000With the oparallel package
Test of the parallel regression assumption
| Chi2 df P>Chi2
-----------------+----------------------
likelihood ratio | 49.2 12 0.000We reject the null hypothesis that the coefficients are the same across \(1,...,J-1\) equations at the 0.1% level.
3.3.2 Wald Test
We use the brant command for the Wald Test
quietly use "/Users/Sam/Desktop/Econ 645/Data/ordwarm2.dta", clear
quietly ologit warm yr89 male white age ed prst, nolog
oparallel, lrWe reject the null hypothesis that the coefficients are the same across \(1,...,J-1\) equations at the 0.1% level.
3.4 Interpretation
3.4.1 Odds Ratios
Iteration 0: log likelihood = -2995.7704
Iteration 1: log likelihood = -2846.4532
Iteration 2: log likelihood = -2844.9142
Iteration 3: log likelihood = -2844.9123
Iteration 4: log likelihood = -2844.9123
Ordered logistic regression Number of obs = 2,293
LR chi2(6) = 301.72
Prob > chi2 = 0.0000
Log likelihood = -2844.9123 Pseudo R2 = 0.0504
------------------------------------------------------------------------------
warm | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | 1.688605 .1349176 6.56 0.000 1.443836 1.974868
male | .4803214 .0376969 -9.34 0.000 .4118389 .5601916
white | .6762723 .0800577 -3.30 0.001 .5362356 .8528792
age | .9785675 .0024154 -8.78 0.000 .9738449 .983313
ed | 1.06948 .0170849 4.20 0.000 1.036513 1.103496
prst | 1.006091 .003313 1.84 0.065 .9996188 1.012605
-------------+----------------------------------------------------------------
/cut1 | -2.465362 .2389128 -2.933622 -1.997102
/cut2 | -.630904 .2333156 -1.088194 -.1736138
/cut3 | 1.261854 .234018 .8031871 1.720521
------------------------------------------------------------------------------
ologit (N=2293): Factor Change in Odds
Odds of: >m vs <=m
------------------------------------------------------------------
warm | b z P>|z| e^b e^bStdX SDofX
---------+--------------------------------------------------------
yr89 | 0.52390 6.557 0.000 1.6886 1.2925 0.4897
male | -0.73330 -9.343 0.000 0.4803 0.6936 0.4989
white | -0.39116 -3.304 0.001 0.6763 0.8792 0.3290
age | -0.02167 -8.778 0.000 0.9786 0.6952 16.7790
ed | 0.06717 4.205 0.000 1.0695 1.2365 3.1608
prst | 0.00607 1.844 0.065 1.0061 1.0920 14.4923
------------------------------------------------------------------
b = raw coefficient
z = z-score for test of b=0
P>|z| = p-value for z-test
e^b = exp(b) = factor change in odds for unit increase in X
e^bStdX = exp(b*SD of X) = change in odds for SD increase in X
SDofX = standard deviation of X
ologit (N=2293): Percentage Change in Odds
Odds of: >m vs <=m
------------------------------------------------------------------
warm | b z P>|z| % %StdX SDofX
---------+--------------------------------------------------------
yr89 | 0.52390 6.557 0.000 68.9 29.2 0.4897
male | -0.73330 -9.343 0.000 -52.0 -30.6 0.4989
white | -0.39116 -3.304 0.001 -32.4 -12.1 0.3290
age | -0.02167 -8.778 0.000 -2.1 -30.5 16.7790
ed | 0.06717 4.205 0.000 6.9 23.7 3.1608
prst | 0.00607 1.844 0.065 0.6 9.2 14.4923
------------------------------------------------------------------
b = raw coefficient
z = z-score for test of b=0
P>|z| = p-value for z-test
% = percent change in odds for unit increase in X
%StdX = percent change in odds for SD increase in X
SDofX = standard deviation of XFor a 1-year increase in education , the odds of a positive attitude towards working mothers increase by a factor of 1.067 or 6.7%, ceteris paribus.
For a 1-year increase in age, the odds of a negative attitude toward working mothers increase by a factor of \(e^{-(-0.02167)}=1.0219\), ceteris paribus.
For a male, the odds of a negative attitude toward working mothers increase by a factor of \(e^{-(-0.73330)}=2.08\), ceteris paribus
3.4.2 Average Marginal Effects
ologit warm yr89 male white age ed prst, or
margins, dydx(*) predict(outcome(1))
margins, dydx(*) predict(outcome(2))
margins, dydx(*) predict(outcome(3))
margins, dydx(*) predict(outcome(4))Iteration 0: log likelihood = -2995.7704
Iteration 1: log likelihood = -2846.4532
Iteration 2: log likelihood = -2844.9142
Iteration 3: log likelihood = -2844.9123
Iteration 4: log likelihood = -2844.9123
Ordered logistic regression Number of obs = 2,293
LR chi2(6) = 301.72
Prob > chi2 = 0.0000
Log likelihood = -2844.9123 Pseudo R2 = 0.0504
------------------------------------------------------------------------------
warm | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | 1.688605 .1349176 6.56 0.000 1.443836 1.974868
male | .4803214 .0376969 -9.34 0.000 .4118389 .5601916
white | .6762723 .0800577 -3.30 0.001 .5362356 .8528792
age | .9785675 .0024154 -8.78 0.000 .9738449 .983313
ed | 1.06948 .0170849 4.20 0.000 1.036513 1.103496
prst | 1.006091 .003313 1.84 0.065 .9996188 1.012605
-------------+----------------------------------------------------------------
/cut1 | -2.465362 .2389128 -2.933622 -1.997102
/cut2 | -.630904 .2333156 -1.088194 -.1736138
/cut3 | 1.261854 .234018 .8031871 1.720521
------------------------------------------------------------------------------
Average marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==1), predict(outcome(1))
dy/dx w.r.t. : yr89 male white age ed prst
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | -.0557094 .0087921 -6.34 0.000 -.0729416 -.0384771
male | .0779757 .0088072 8.85 0.000 .0607139 .0952375
white | .0415941 .0127155 3.27 0.001 .0166721 .0665161
age | .0023038 .0002754 8.37 0.000 .0017641 .0028435
ed | -.0071428 .0017147 -4.17 0.000 -.0105036 -.0037821
prst | -.0006457 .0003514 -1.84 0.066 -.0013345 .000043
------------------------------------------------------------------------------
Average marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==2), predict(outcome(2))
dy/dx w.r.t. : yr89 male white age ed prst
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | -.0608201 .0091774 -6.63 0.000 -.0788075 -.0428328
male | .0851292 .0089196 9.54 0.000 .0676472 .1026112
white | .0454099 .0136969 3.32 0.001 .0185644 .0722554
age | .0025152 .0002817 8.93 0.000 .001963 .0030673
ed | -.0077981 .00185 -4.22 0.000 -.0114241 -.0041722
prst | -.000705 .0003818 -1.85 0.065 -.0014533 .0000433
------------------------------------------------------------------------------
Average marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==3), predict(outcome(3))
dy/dx w.r.t. : yr89 male white age ed prst
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .043531 .0069122 6.30 0.000 .0299833 .0570787
male | -.0609298 .0067839 -8.98 0.000 -.074226 -.0476336
white | -.0325014 .0099733 -3.26 0.001 -.0520487 -.0129541
age | -.0018002 .0002138 -8.42 0.000 -.0022193 -.0013811
ed | .0055814 .0013342 4.18 0.000 .0029665 .0081963
prst | .0005046 .0002752 1.83 0.067 -.0000349 .0010441
------------------------------------------------------------------------------
Average marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==4), predict(outcome(4))
dy/dx w.r.t. : yr89 male white age ed prst
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .0729985 .0111797 6.53 0.000 .0510866 .0949103
male | -.102175 .0111936 -9.13 0.000 -.1241141 -.080236
white | -.0545026 .016485 -3.31 0.001 -.0868126 -.0221926
age | -.0030188 .0003501 -8.62 0.000 -.0037051 -.0023325
ed | .0093596 .0022406 4.18 0.000 .0049681 .0137511
prst | .0008461 .0004584 1.85 0.065 -.0000522 .0017445
------------------------------------------------------------------------------3.4.3 Marginal Effects at the Average
ologit warm yr89 male white age ed prst, or
margins, dydx(*) atmeans predict(outcome(1))
margins, dydx(*) atmeans predict(outcome(2))
margins, dydx(*) atmeans predict(outcome(3))
margins, dydx(*) atmeans predict(outcome(4))Iteration 0: log likelihood = -2995.7704
Iteration 1: log likelihood = -2846.4532
Iteration 2: log likelihood = -2844.9142
Iteration 3: log likelihood = -2844.9123
Iteration 4: log likelihood = -2844.9123
Ordered logistic regression Number of obs = 2,293
LR chi2(6) = 301.72
Prob > chi2 = 0.0000
Log likelihood = -2844.9123 Pseudo R2 = 0.0504
------------------------------------------------------------------------------
warm | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | 1.688605 .1349176 6.56 0.000 1.443836 1.974868
male | .4803214 .0376969 -9.34 0.000 .4118389 .5601916
white | .6762723 .0800577 -3.30 0.001 .5362356 .8528792
age | .9785675 .0024154 -8.78 0.000 .9738449 .983313
ed | 1.06948 .0170849 4.20 0.000 1.036513 1.103496
prst | 1.006091 .003313 1.84 0.065 .9996188 1.012605
-------------+----------------------------------------------------------------
/cut1 | -2.465362 .2389128 -2.933622 -1.997102
/cut2 | -.630904 .2333156 -1.088194 -.1736138
/cut3 | 1.261854 .234018 .8031871 1.720521
------------------------------------------------------------------------------
Conditional marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==1), predict(outcome(1))
dy/dx w.r.t. : yr89 male white age ed prst
at : yr89 = .3986044 (mean)
male = .4648932 (mean)
white = .8765809 (mean)
age = 44.93546 (mean)
ed = 12.21805 (mean)
prst = 39.58526 (mean)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | -.051803 .0081383 -6.37 0.000 -.0677537 -.0358522
male | .0725079 .0081294 8.92 0.000 .0565746 .0884413
white | .0386775 .0118016 3.28 0.001 .0155468 .0618081
age | .0021423 .0002546 8.41 0.000 .0016432 .0026413
ed | -.006642 .0015972 -4.16 0.000 -.0097724 -.0035116
prst | -.0006005 .0003264 -1.84 0.066 -.0012401 .0000392
------------------------------------------------------------------------------
Conditional marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==2), predict(outcome(2))
dy/dx w.r.t. : yr89 male white age ed prst
at : yr89 = .3986044 (mean)
male = .4648932 (mean)
white = .8765809 (mean)
age = 44.93546 (mean)
ed = 12.21805 (mean)
prst = 39.58526 (mean)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | -.0772501 .0122256 -6.32 0.000 -.1012118 -.0532885
male | .1081261 .0124796 8.66 0.000 .0836666 .1325855
white | .057677 .017613 3.27 0.001 .0231561 .0921979
age | .0031946 .0003899 8.19 0.000 .0024304 .0039588
ed | -.0099047 .0023955 -4.13 0.000 -.0145999 -.0052095
prst | -.0008954 .0004869 -1.84 0.066 -.0018497 .0000588
------------------------------------------------------------------------------
Conditional marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==3), predict(outcome(3))
dy/dx w.r.t. : yr89 male white age ed prst
at : yr89 = .3986044 (mean)
male = .4648932 (mean)
white = .8765809 (mean)
age = 44.93546 (mean)
ed = 12.21805 (mean)
prst = 39.58526 (mean)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .0582103 .0096161 6.05 0.000 .0393631 .0770575
male | -.0814762 .0099979 -8.15 0.000 -.1010717 -.0618807
white | -.0434613 .0134313 -3.24 0.001 -.0697862 -.0171364
age | -.0024072 .0003126 -7.70 0.000 -.0030199 -.0017945
ed | .0074635 .0018362 4.06 0.000 .0038645 .0110625
prst | .0006747 .0003683 1.83 0.067 -.0000472 .0013967
------------------------------------------------------------------------------
Conditional marginal effects Number of obs = 2,293
Model VCE : OIM
Expression : Pr(warm==4), predict(outcome(4))
dy/dx w.r.t. : yr89 male white age ed prst
at : yr89 = .3986044 (mean)
male = .4648932 (mean)
white = .8765809 (mean)
age = 44.93546 (mean)
ed = 12.21805 (mean)
prst = 39.58526 (mean)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
yr89 | .0708428 .0109038 6.50 0.000 .0494718 .0922139
male | -.0991578 .0109073 -9.09 0.000 -.1205356 -.0777799
white | -.0528931 .0160474 -3.30 0.001 -.0843454 -.0214409
age | -.0029296 .00034 -8.62 0.000 -.0035961 -.0022632
ed | .0090832 .0021702 4.19 0.000 .0048296 .0133368
prst | .0008212 .0004455 1.84 0.065 -.0000519 .0016943
------------------------------------------------------------------------------