Multinomial logistic regression involves nominal response variables more than two categories. Multinomial logit models are multiequation models. A response variable with k categories will generate k-1 equations. Each of these k-1 equations is a binary logistic regression comparing a group with the reference group. Multinomial logistic regression simultaneously estimates the k-1 logits. Further, it is also the case, that the model tests all possible combinations among the k groups although it only displays coefficients for the k-1 comparisons.
Say that we had a response variable with three levels, the probabilites for each of the levels could be obtained as follows:
P(y=1) = exp(β1*x)/(exp(β1*x)+exp(β2*x)+exp(β3*x)) P(y=2) = exp(β2*x)/(exp(β1*x)+exp(β2*x)+exp(β3*x)) P(y=3) = exp(β3*x)/(exp(β1*x)+exp(β2*x)+exp(β3*x))This system of equations is unidentified, that is, there is more than one solution to the coefficients that lead to the same probabilities. To make the system identifiable, one of the coefficients is set to 0. It doesn't matter which one since they each yield the same probabilities. We will set the probability for β1 to 0, yielding:
P(y=1) = 1/(1+exp(β2*x)+exp(β3*x)) P(y=2) = exp(β2*x)/(1+exp(β2*x)+exp(β3*x)) P(y=3) = exp(β3*x)/(1+exp(β2*x)+exp(β3*x))This, in turn, leads to the following probabilities relative to the reference group, in this case, group 1.
P(y=2)/P(y=1) = exp(β2*x) P(y=3)/P(y=1) = exp(β3*x)Thus, the two coefficients, β2 and β3 represent the log odds of being in the target groups relative to the reference group.
In multinomial logistic regression the exponentiated coefficients are not odds ratios per se. The coefficients can be interpreted as relative risk ratios (RRR). Recall from the unit on contingency tables that relative risk in the 2x2 table was defined as
P(y=1|x+1)/P(y=base category|x+1)
RRR = ---------------------------------
P(y=1|x)/P(y=base category|x)
use http://www.gseis.ucla.edu/courses/data/hsb2Example 2generate honcomp=write>=60
describe
Contains data from http://www.gseis.ucla.edu/courses/data/hsb2.dta obs: 200 highschool and beyond (200 cases) vars: 12 21 Jun 2000 08:54 size: 10,400 (98.8% of memory free) ------------------------------------------------------------------------------- storage display value variable name type format label variable label ------------------------------------------------------------------------------- id float %9.0g female float %9.0g fl race float %12.0g rl ses float %9.0g sl schtyp float %9.0g scl type of school prog float %9.0g sel type of program read float %9.0g reading score write float %9.0g writing score math float %9.0g math score science float %9.0g science score socst float %9.0g social studies score honcomp float %9.0g ------------------------------------------------------------------------------- logit honcomp read math female Logit estimates Number of obs = 200 LR chi2(3) = 80.87 Prob > chi2 = 0.0000 Log likelihood = -75.209827 Pseudo R2 = 0.3496 ------------------------------------------------------------------------------ honcomp | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | .0752424 .027577 2.73 0.006 .0211924 .1292924 math | .1317117 .0324607 4.06 0.000 .06809 .1953335 female | 1.154801 .4340856 2.66 0.008 .304009 2.005593 _cons | -13.12749 1.850769 -7.09 0.000 -16.75493 -9.50005 ------------------------------------------------------------------------------
mlogit honcomp read math femaleSimplified mlogview ExampleMultinomial regression Number of obs = 200 LR chi2(3) = 80.87 Prob > chi2 = 0.0000 Log likelihood = -75.209827 Pseudo R2 = 0.3496 ------------------------------------------------------------------------------ honcomp | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1 | read | .0752424 .027577 2.73 0.006 .0211924 .1292924 math | .1317117 .0324607 4.06 0.000 .06809 .1953335 female | 1.154801 .4340856 2.66 0.008 .304009 2.005593 _cons | -13.12749 1.850769 -7.09 0.000 -16.75493 -9.50005 ------------------------------------------------------------------------------ (Outcome honcomp==0 is the comparison group) xi: mlogit prog i.female*math i.female*socst i.female _Ifemale_0-1 (naturally coded; _Ifemale_0 omitted) i.female*math _IfemXmath_# (coded as above) i.female*socst _IfemXsocst_# (coded as above) Multinomial regression Number of obs = 200 LR chi2(10) = 66.70 Prob > chi2 = 0.0000 Log likelihood = -170.7466 Pseudo R2 = 0.1634 ------------------------------------------------------------------------------ prog | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- general | _Ifemale_1 | .1312333 2.763398 0.05 0.962 -5.284927 5.547394 math | -.0752145 .0353071 -2.13 0.033 -.1444151 -.0060139 _IfemXmath_1 | .0044014 .0512273 0.09 0.932 -.0960022 .104805 socst | -.0332122 .0301033 -1.10 0.270 -.0922135 .0257891 _IfemXsocs~1 | -.0087899 .0438624 -0.20 0.841 -.0947587 .0771789 _cons | 5.004933 2.03586 2.46 0.014 1.014721 8.995146 -------------+---------------------------------------------------------------- vocation | _Ifemale_1 | -.4631782 3.09024 -0.15 0.881 -6.519938 5.593581 math | -.1010177 .0405796 -2.49 0.013 -.1805523 -.0214831 _IfemXmath_1 | -.0290723 .0586834 -0.50 0.620 -.1440896 .0859451 socst | -.0976086 .032479 -3.01 0.003 -.1612662 -.033951 _IfemXsocs~1 | .0387747 .0462958 0.84 0.402 -.0519634 .1295128 _cons | 9.369412 2.305958 4.06 0.000 4.849818 13.88901 ------------------------------------------------------------------------------ (Outcome prog==academic is the comparison group) test _IfemXmath_1 ( 1) [general]_IfemXmath_1 = 0.0 ( 2) [vocation]_IfemXmath_1 = 0.0 chi2( 2) = 0.33 Prob > chi2 = 0.8478 test _IfemXsocst_1 ( 1) [general]_IfemXsocst_1 = 0.0 ( 2) [vocation]_IfemXsocst_1 = 0.0 chi2( 2) = 1.13 Prob > chi2 = 0.5674 mlogit prog female math socst Multinomial regression Number of obs = 200 LR chi2(6) = 65.51 Prob > chi2 = 0.0000 Log likelihood = -171.34162 Pseudo R2 = 0.1605 ------------------------------------------------------------------------------ prog | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- general | female | -.0840263 .3806826 -0.22 0.825 -.8301505 .6620979 math | -.0739045 .0254512 -2.90 0.004 -.1237879 -.0240211 socst | -.0370939 .0217034 -1.71 0.087 -.0796319 .0054441 _cons | 5.130723 1.392646 3.68 0.000 2.401188 7.860258 -------------+---------------------------------------------------------------- vocation | female | -.0177488 .4085162 -0.04 0.965 -.8184258 .7829282 math | -.1127775 .0289322 -3.90 0.000 -.1694836 -.0560714 socst | -.079675 .0227946 -3.50 0.000 -.1243516 -.0349984 _cons | 9.106635 1.545711 5.89 0.000 6.077098 12.13617 ------------------------------------------------------------------------------ (Outcome prog==academic is the comparison group) test female ( 1) [general]female = 0.0 ( 2) [vocation]female = 0.0 chi2( 2) = 0.05 Prob > chi2 = 0.9745 test math ( 1) [general]math = 0.0 ( 2) [vocation]math = 0.0 chi2( 2) = 17.61 Prob > chi2 = 0.0001 test socst ( 1) [general]socst = 0.0 ( 2) [vocation]socst = 0.0 chi2( 2) = 12.23 Prob > chi2 = 0.0022 mlogit, rrr Multinomial regression Number of obs = 200 LR chi2(6) = 65.51 Prob > chi2 = 0.0000 Log likelihood = -171.34162 Pseudo R2 = 0.1605 ------------------------------------------------------------------------------ prog | RRR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- general | female | .9194071 .3500023 -0.22 0.825 .4359837 1.938856 math | .9287604 .023638 -2.90 0.004 .8835673 .9762651 socst | .9635856 .0209131 -1.71 0.087 .9234562 1.005459 -------------+---------------------------------------------------------------- vocation | female | .9824078 .4013295 -0.04 0.965 .4411255 2.18787 math | .8933494 .0258466 -3.90 0.000 .8441006 .9454716 socst | .9234164 .0210489 -3.50 0.000 .8830693 .9656069 ------------------------------------------------------------------------------ (Outcome prog==academic is the comparison group) listcoef mlogit (N=200): Factor Change in the Odds of prog Variable: female (sd= .49922) Odds comparing| Group 1 vs Group 2| b z P>|z| e^b e^bStdX ------------------+--------------------------------------------- general -vocation | -0.06628 -0.155 0.877 0.9359 0.9675 general -academic | -0.08403 -0.221 0.825 0.9194 0.9589 vocation-general | 0.06628 0.155 0.877 1.0685 1.0336 vocation-academic | -0.01775 -0.043 0.965 0.9824 0.9912 academic-general | 0.08403 0.221 0.825 1.0877 1.0428 academic-vocation | 0.01775 0.043 0.965 1.0179 1.0089 ---------------------------------------------------------------- Variable: math (sd= 9.36845) Odds comparing| Group 1 vs Group 2| b z P>|z| e^b e^bStdX ------------------+--------------------------------------------- general -vocation | 0.03887 1.273 0.203 1.0396 1.4393 general -academic | -0.07390 -2.904 0.004 0.9288 0.5004 vocation-general | -0.03887 -1.273 0.203 0.9619 0.6948 vocation-academic | -0.11278 -3.898 0.000 0.8933 0.3477 academic-general | 0.07390 2.904 0.004 1.0767 1.9984 academic-vocation | 0.11278 3.898 0.000 1.1194 2.8764 ---------------------------------------------------------------- Variable: socst (sd= 10.7358) Odds comparing| Group 1 vs Group 2| b z P>|z| e^b e^bStdX ------------------+--------------------------------------------- general -vocation | 0.04258 1.844 0.065 1.0435 1.5796 general -academic | -0.03709 -1.709 0.087 0.9636 0.6715 vocation-general | -0.04258 -1.844 0.065 0.9583 0.6331 vocation-academic | -0.07968 -3.495 0.000 0.9234 0.4251 academic-general | 0.03709 1.709 0.087 1.0378 1.4892 academic-vocation | 0.07968 3.495 0.000 1.0829 2.3523 ---------------------------------------------------------------- fitstat Measures of Fit for mlogit of prog Log-Lik Intercept Only: -204.097 Log-Lik Full Model: -171.342 D(192): 342.683 LR(6): 65.510 Prob > LR: 0.000 McFadden's R2: 0.160 McFadden's Adj R2: 0.121 Maximum Likelihood R2: 0.279 Cragg & Uhler's R2: 0.321 Count R2: 0.625 Adj Count R2: 0.211 AIC: 1.793 AIC*n: 358.683 BIC: -674.594 BIC': -33.720 mlogtest, lr wald lrcomb combine **** Likelihood-ratio tests for independent variables Ho: All coefficients associated with given variable(s) are 0. prog | chi2 df P>chi2 -------------+------------------------- female | 0.052 2 0.975 math | 20.323 2 0.000 socst | 13.373 2 0.001 --------------------------------------- **** Wald tests for independent variables Ho: All coefficients associated with given variable(s) are 0. prog | chi2 df P>chi2 -------------+------------------------- female | 0.052 2 0.975 math | 17.610 2 0.000 socst | 12.228 2 0.002 --------------------------------------- **** Wald tests for combining outcome categories Ho: All coefficients except intercepts associated with given pair of outcomes are 0 (i.e., categories can be collapsed). Categories tested | chi2 df P>chi2 ------------------+------------------------ general-vocation | 8.084 3 0.044 general-academic | 18.243 3 0.000 vocation-academic | 40.517 3 0.000 ------------------------------------------- **** LR tests for combining outcome categories Ho: All coefficients except intercepts associated with given pair of outcomes are 0 (i.e., categories can be collapsed). Categories tested | chi2 df P>chi2 ------------------+------------------------ general-vocation | 8.727 3 0.033 general-academic | 21.516 3 0.000 vocation-academic | 60.726 3 0.000 ------------------------------------------- prchange mlogit: Changes in Predicted Probabilities for prog female Avg|Chg| general vocation academic 0->1 .0101265 -.01518974 .00147069 .01371908 math Avg|Chg| general vocation academic Min->Max .49023263 -.23754089 -.49780805 .73534894 -+1/2 .01500345 -.0083954 -.01410978 .02250516 -+sd/2 .13860906 -.07673311 -.13118048 .20791358 MargEfct .04501765 -.00839781 -.01411102 .02250882 socst Avg|Chg| general vocation academic Min->Max .37775379 -.06104398 -.50558669 .56663069 -+1/2 .00918606 -.00304148 -.01073761 .0137791 -+sd/2 .09799281 -.03192489 -.11506434 .1469892 MargEfct .02755971 -.00304207 -.01073778 .01377986 general vocation academic Pr(y|x) .25754365 .19741122 .54504514 female math socst x= .545 52.645 52.405 sd(x)= .49922 9.36845 10.7358 mlogview /* sd change in math socst */ /* discrete change plot -- compare to prchange */
/* or plot -- compare to liscoef output */ /* academic vs academic has or of 1.0 */
prtab math mlogit: Predicted probabilities for prog Predicted probability of outcome 1 (general) ---------------------- math | score | Prediction ----------+----------- 33 | 0.3184 35 | 0.3230 37 | 0.3253 38 | 0.3256 39 | 0.3252 40 | 0.3242 41 | 0.3225 42 | 0.3202 43 | 0.3172 44 | 0.3135 45 | 0.3092 46 | 0.3042 47 | 0.2986 48 | 0.2924 49 | 0.2857 50 | 0.2785 51 | 0.2709 52 | 0.2629 53 | 0.2545 54 | 0.2459 55 | 0.2371 56 | 0.2281 57 | 0.2190 58 | 0.2098 59 | 0.2007 60 | 0.1916 61 | 0.1825 62 | 0.1737 63 | 0.1650 64 | 0.1565 65 | 0.1482 66 | 0.1402 67 | 0.1324 68 | 0.1249 69 | 0.1177 70 | 0.1108 71 | 0.1042 72 | 0.0980 73 | 0.0920 75 | 0.0809 ---------------------- Predicted probability of outcome 3 (vocation) ---------------------- math | score | Prediction ----------+----------- 33 | 0.5238 35 | 0.4915 37 | 0.4581 38 | 0.4410 39 | 0.4237 40 | 0.4063 41 | 0.3888 42 | 0.3712 43 | 0.3537 44 | 0.3363 45 | 0.3190 46 | 0.3019 47 | 0.2850 48 | 0.2685 49 | 0.2524 50 | 0.2366 51 | 0.2214 52 | 0.2066 53 | 0.1924 54 | 0.1788 55 | 0.1658 56 | 0.1534 57 | 0.1417 58 | 0.1306 59 | 0.1201 60 | 0.1103 61 | 0.1011 62 | 0.0925 63 | 0.0845 64 | 0.0771 65 | 0.0703 66 | 0.0639 67 | 0.0581 68 | 0.0527 69 | 0.0478 70 | 0.0433 71 | 0.0391 72 | 0.0354 73 | 0.0320 75 | 0.0260 ---------------------- Predicted probability of outcome 2 (academic) ---------------------- math | score | Prediction ----------+----------- 33 | 0.1578 35 | 0.1855 37 | 0.2166 38 | 0.2334 39 | 0.2511 40 | 0.2695 41 | 0.2887 42 | 0.3086 43 | 0.3291 44 | 0.3502 45 | 0.3719 46 | 0.3939 47 | 0.4164 48 | 0.4391 49 | 0.4619 50 | 0.4848 51 | 0.5077 52 | 0.5305 53 | 0.5530 54 | 0.5753 55 | 0.5971 56 | 0.6185 57 | 0.6393 58 | 0.6596 59 | 0.6792 60 | 0.6981 61 | 0.7163 62 | 0.7338 63 | 0.7505 64 | 0.7664 65 | 0.7815 66 | 0.7959 67 | 0.8095 68 | 0.8224 69 | 0.8345 70 | 0.8459 71 | 0.8566 72 | 0.8667 73 | 0.8761 75 | 0.8931 ---------------------- female math socst x= .545 52.645 52.405 prtab math female mlogit: Predicted probabilities for prog Predicted probability of outcome 1 (general) -------------------------- math | female score | male female ----------+--------------- 33 | 0.3268 0.3115 35 | 0.3315 0.3159 37 | 0.3340 0.3182 38 | 0.3343 0.3184 39 | 0.3340 0.3180 40 | 0.3330 0.3169 41 | 0.3314 0.3152 42 | 0.3291 0.3129 43 | 0.3261 0.3099 44 | 0.3224 0.3062 45 | 0.3180 0.3018 46 | 0.3131 0.2969 47 | 0.3074 0.2913 48 | 0.3012 0.2852 49 | 0.2945 0.2786 50 | 0.2872 0.2715 51 | 0.2795 0.2639 52 | 0.2713 0.2560 53 | 0.2628 0.2477 54 | 0.2541 0.2392 55 | 0.2451 0.2305 56 | 0.2359 0.2217 57 | 0.2266 0.2127 58 | 0.2172 0.2038 59 | 0.2079 0.1948 60 | 0.1985 0.1859 61 | 0.1893 0.1771 62 | 0.1802 0.1684 63 | 0.1712 0.1599 64 | 0.1625 0.1516 65 | 0.1540 0.1435 66 | 0.1457 0.1357 67 | 0.1377 0.1281 68 | 0.1299 0.1209 69 | 0.1225 0.1139 70 | 0.1154 0.1072 71 | 0.1086 0.1008 72 | 0.1020 0.0947 73 | 0.0958 0.0888 75 | 0.0843 0.0781 -------------------------- Predicted probability of outcome 3 (vocation) -------------------------- math | female score | male female ----------+--------------- 33 | 0.5185 0.5281 35 | 0.4866 0.4955 37 | 0.4536 0.4617 38 | 0.4367 0.4444 39 | 0.4197 0.4270 40 | 0.4025 0.4093 41 | 0.3853 0.3916 42 | 0.3680 0.3739 43 | 0.3507 0.3561 44 | 0.3336 0.3385 45 | 0.3165 0.3210 46 | 0.2997 0.3037 47 | 0.2831 0.2866 48 | 0.2668 0.2699 49 | 0.2508 0.2536 50 | 0.2353 0.2377 51 | 0.2203 0.2222 52 | 0.2057 0.2074 53 | 0.1917 0.1930 54 | 0.1782 0.1793 55 | 0.1653 0.1662 56 | 0.1531 0.1537 57 | 0.1414 0.1419 58 | 0.1304 0.1307 59 | 0.1200 0.1202 60 | 0.1103 0.1103 61 | 0.1011 0.1011 62 | 0.0926 0.0925 63 | 0.0846 0.0844 64 | 0.0773 0.0770 65 | 0.0704 0.0701 66 | 0.0641 0.0638 67 | 0.0583 0.0579 68 | 0.0529 0.0526 69 | 0.0480 0.0476 70 | 0.0435 0.0431 71 | 0.0393 0.0390 72 | 0.0356 0.0352 73 | 0.0321 0.0318 75 | 0.0261 0.0259 -------------------------- Predicted probability of outcome 2 (academic) -------------------------- math | female score | male female ----------+--------------- 33 | 0.1547 0.1604 35 | 0.1819 0.1885 37 | 0.2124 0.2201 38 | 0.2290 0.2372 39 | 0.2463 0.2551 40 | 0.2644 0.2737 41 | 0.2833 0.2931 42 | 0.3029 0.3133 43 | 0.3232 0.3340 44 | 0.3440 0.3554 45 | 0.3654 0.3772 46 | 0.3873 0.3995 47 | 0.4095 0.4221 48 | 0.4320 0.4449 49 | 0.4547 0.4679 50 | 0.4775 0.4909 51 | 0.5003 0.5138 52 | 0.5230 0.5367 53 | 0.5455 0.5592 54 | 0.5677 0.5815 55 | 0.5896 0.6033 56 | 0.6110 0.6246 57 | 0.6320 0.6454 58 | 0.6524 0.6655 59 | 0.6721 0.6850 60 | 0.6912 0.7038 61 | 0.7096 0.7219 62 | 0.7272 0.7392 63 | 0.7441 0.7557 64 | 0.7603 0.7714 65 | 0.7756 0.7864 66 | 0.7902 0.8005 67 | 0.8041 0.8139 68 | 0.8172 0.8266 69 | 0.8295 0.8385 70 | 0.8412 0.8497 71 | 0.8521 0.8602 72 | 0.8624 0.8701 73 | 0.8720 0.8793 75 | 0.8895 0.8960 -------------------------- female math socst x= .545 52.645 52.405
use hsb2
mlogit prog math, nolog
Multinomial logistic regression Number of obs = 200
LR chi2(2) = 51.96
Prob > chi2 = 0.0000
Log likelihood = -178.11423 Pseudo R2 = 0.1273
------------------------------------------------------------------------------
prog | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
general |
math | -.0920133 .0231357 -3.98 0.000 -.1373584 -.0466683
_cons | 4.055593 1.21813 3.33 0.001 1.668103 6.443084
-------------+----------------------------------------------------------------
vocation |
math | -.154975 .0267556 -5.79 0.000 -.207415 -.102535
_cons | 7.19172 1.337781 5.38 0.000 4.569717 9.813723
------------------------------------------------------------------------------
(Outcome prog==academic is the comparison group)
prchange /* output edited */
mlogit: Changes in Predicted Probabilities for prog
math
Avg|Chg| general vocation academic
-+sd/2 .18239624 -.08176914 -.19182522 .27359435
mlogview /* 1 sd discrete change in math */
listcoef /* output edited */
mlogit (N=200): Factor Change in the Odds of prog
Variable: math (sd=9.3684478)
Odds comparing|
Group 1 vs Group 2| b z P>|z| e^b e^bStdX
------------------+---------------------------------------------
general -academic | -0.09201 -3.977 0.000 0.9121 0.4223
vocation-academic | -0.15497 -5.792 0.000 0.8564 0.2341
----------------------------------------------------------------
mlogview /* 1 sd or change in math */
generate msd=math/9.3684478
mlogit prog msd, nolog /* compare with logit coefficient scale above */
Multinomial logistic regression Number of obs = 200
LR chi2(2) = 51.96
Prob > chi2 = 0.0000
Log likelihood = -178.11423 Pseudo R2 = 0.1273
------------------------------------------------------------------------------
prog | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
general |
msd | -.8620219 .2167452 -3.98 0.000 -1.286835 -.4372091
_cons | 4.055593 1.21813 3.33 0.001 1.668103 6.443084
-------------+----------------------------------------------------------------
vocation |
msd | -1.451875 .2506583 -5.79 0.000 -1.943156 -.9605938
_cons | 7.19172 1.337781 5.38 0.000 4.569717 9.813723
------------------------------------------------------------------------------
(Outcome prog==academic is the comparison group)
Example Using prgenprgen allows you to generate predicted probabilities for the response groups while holding other variables constant at specific values. These predicted probabilities can then used in scatterplots and other graphs.
use hsb2
mlogit prog math science, nolog
Multinomial logistic regression Number of obs = 200
LR chi2(4) = 58.04
Prob > chi2 = 0.0000
Log likelihood = -175.07434 Pseudo R2 = 0.1422
------------------------------------------------------------------------------
prog | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
general |
math | -.1352046 .0305449 -4.43 0.000 -.1950716 -.0753376
science | .0602744 .0254395 2.37 0.018 .0104139 .1101348
_cons | 3.166452 1.298818 2.44 0.015 .6208165 5.712088
-------------+----------------------------------------------------------------
vocation |
math | -.1690188 .0331945 -5.09 0.000 -.2340789 -.1039588
science | .0170098 .0250403 0.68 0.497 -.0320684 .0660879
_cons | 7.053851 1.37717 5.12 0.000 4.354647 9.753055
------------------------------------------------------------------------------
(Outcome prog==academic is the comparison group)
prgen math, gen(m) x(science=50) from(25) to(75) n(100)
mlogit: Predicted values as math varies from 25 to 75.
math science
x= 52.645 50
scatter mp1 mp2 mp3 mx
prgen math, gen(u) x(science=60) from(25) to(75) n(100)
mlogit: Predicted values as math varies from 25 to 75.
math science
x= 52.645 60
scatter up1 up2 up3 ux
Categorical Data Analysis Course
Phil Ender