Modeling Categorical Outcome in Stata
Giovanni Minchio giovanni.minchio@unitn.it
Yuxin Zhang yuxin.zhang@unitn.it
Quantitative Methods Lab, Lesson 8.1
19
Nov. 2024
Outline
- Binomial logistic regression — binary response *
Bonus:
Problem with LPMs
Unbounded Y prediction in linear regression:
Image source
here
So, it is necessary to restrict the prediction value range between 0
and 1.
Logistic (Sigmoid) function: s-curve
For binary outcome, we want \(X \in
\mathbb{R}\) and \(p(X) \in [0,
1]\), and logistic regression can “squeeze” the output to be
between 0 and 1.
Image source
here
\[
P(Y_i = 1 | X_i) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 X_{i1} + \beta_2
X_{i2} + \dots + \beta_k X_{ik})}}
\] \[
P(Y_i = 0 | X_i) = 1- \frac{1}{1 + e^{-(\beta_0 + \beta_1 X_{i1} +
\beta_2 X_{i2} + \dots + \beta_k X_{ik})}}
\]
E.g., probability of voter turnout:
\[
P(vote) = \frac{1}{1 + e^{-(\beta_0 + \beta_1age + \beta_2gender +
\beta_3education + ...)}}
\]
Key terms
Odds are defined as the ratio of the probability of
an event occurring to the probability of it not occurring, where \(X \in [0, +\infty]\): \[odds = \frac{p}{1-p} = e^{a + bx}\]
Logit / log odds are logarithmic odds (logit
function), where \(X \in [-\infty,
+\infty]\). Logit is assumed to be linear, so in logistic
regression, the log odds of the dependent variable are modeled as a
linear combination of the independent variables and the intercept.
\[logit = ln(odds) = ln(\frac{p}{1-p}) =
ln(e^{a + bx}) = a + bx\]
Probability can also be calculated, where \(X \in [0, 1]\):
\[P(Y = 1) = exp(logit) =
\frac{odds}{1+odds} = exp(ln(e^{a + bx})) = \frac{e^{(a + bx)}}{1 +
e^{(a + bx)}} = \frac{1}{1 + e^{-(a + bx)}} \]
1. Binary logistic regression
Let’s use lesson8.dta which can be downloaded from Moodle
cd ""
use "lesson8.dta", clear
- first investigate variables to be used
-> tabulation of work
Working |
condition | Freq. Percent Cum.
------------+-----------------------------------
Unemployed | 4,452 57.41 57.41
Employed | 3,303 42.59 100.00
------------+-----------------------------------
Total | 7,755 100.00
work:
0 Unemployed
1 Employed
First a simple LPM.
Linear regression Number of obs = 7,755
F(1, 7753) = 51791.44
Prob > F = 0.0000
R-squared = 0.7198
Root MSE = .26178
------------------------------------------------------------------------------
| Robust
work | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0670447 .0002946 227.58 0.000 .0664672 .0676222
_cons | -1.171042 .0072912 -160.61 0.000 -1.185334 -1.156749
------------------------------------------------------------------------------
predict yhat
summarize yhat
(option xb assumed; fitted values)
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
yhat | 9,009 .4258633 .393403 -.1653713 1.108478
twoway (scatter work age) || (lfit work age)
- plot observations and predicted line using yhat
twoway scatter yhat work age, connect(l i) msymbol(i) sort ylabel(0 1)
Now let’s try to fit a logit model.
Iteration 0: Log likelihood = -5289.9229
Iteration 1: Log likelihood = -1573.3867
Iteration 2: Log likelihood = -1458.7
Iteration 3: Log likelihood = -1456.7051
Iteration 4: Log likelihood = -1456.6995
Iteration 5: Log likelihood = -1456.6995
Logistic regression Number of obs = 7,755
LR chi2(1) = 7666.45
Prob > chi2 = 0.0000
Log likelihood = -1456.6995 Pseudo R2 = 0.7246
------------------------------------------------------------------------------
work | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .6590841 .0154463 42.67 0.000 .62881 .6893582
_cons | -16.66854 .3943523 -42.27 0.000 -17.44145 -15.89562
------------------------------------------------------------------------------
We see Iteration 1, 2, 3, …, they indicate how quickly the model
converged. Plus, log likelihood (-1456.6995) can be used in comparisons
of nested models. Then, the likelihood ratio (LR) chi-square of 7666.45
with a p-value < .001 tells us that our model as a whole fits
significantly better than an empty model (i.e., a model with no
predictors).
A one-unit increase in the variable age is associated with a .66
increase in the relative log odds of being employed
vs. unemployed, and it is statistically significant at a level of <
0.001.
predict yhat1
twoway scatter yhat1 work age, connect(l i) msymbol(i) sort ylabel(0 1)
(option pr assumed; Pr(work))
Interpreting outputs
Let’s go back to the same example from the LPM we saw last week.
cd ""
use "ESS10.dta", clear
tab1 vote agea gndr edulvlb cntry netustm
-> tabulation of vote
Voted last national |
election | Freq. Percent Cum.
---------------------+-----------------------------------
Yes | 26,794 72.12 72.12
No | 7,764 20.90 93.02
Not eligible to vote | 2,594 6.98 100.00
---------------------+-----------------------------------
Total | 37,152 100.00
-> tabulation of agea
Age of |
respondent, |
calculated | Freq. Percent Cum.
--------------+-----------------------------------
15 | 99 0.27 0.27
16 | 267 0.72 0.98
17 | 360 0.96 1.95
18 | 402 1.08 3.02
19 | 458 1.23 4.25
20 | 385 1.03 5.28
21 | 457 1.22 6.51
22 | 452 1.21 7.72
23 | 400 1.07 8.79
24 | 389 1.04 9.83
25 | 419 1.12 10.95
26 | 423 1.13 12.09
27 | 401 1.07 13.16
28 | 451 1.21 14.37
29 | 448 1.20 15.57
30 | 486 1.30 16.87
31 | 492 1.32 18.19
32 | 517 1.39 19.58
33 | 536 1.44 21.01
34 | 488 1.31 22.32
35 | 555 1.49 23.81
36 | 584 1.56 25.37
37 | 550 1.47 26.85
38 | 568 1.52 28.37
39 | 566 1.52 29.89
40 | 598 1.60 31.49
41 | 680 1.82 33.31
42 | 600 1.61 34.92
43 | 593 1.59 36.51
44 | 567 1.52 38.03
45 | 598 1.60 39.63
46 | 656 1.76 41.39
47 | 609 1.63 43.02
48 | 623 1.67 44.69
49 | 649 1.74 46.43
50 | 653 1.75 48.18
51 | 687 1.84 50.02
52 | 672 1.80 51.82
53 | 662 1.77 53.59
54 | 621 1.66 55.26
55 | 653 1.75 57.01
56 | 715 1.92 58.92
57 | 664 1.78 60.70
58 | 687 1.84 62.54
59 | 660 1.77 64.31
60 | 687 1.84 66.15
61 | 728 1.95 68.10
62 | 673 1.80 69.91
63 | 634 1.70 71.60
64 | 598 1.60 73.21
65 | 630 1.69 74.89
66 | 677 1.81 76.71
67 | 657 1.76 78.47
68 | 609 1.63 80.10
69 | 587 1.57 81.67
70 | 630 1.69 83.36
71 | 630 1.69 85.05
72 | 573 1.54 86.59
73 | 521 1.40 87.98
74 | 477 1.28 89.26
75 | 506 1.36 90.62
76 | 444 1.19 91.81
77 | 393 1.05 92.86
78 | 326 0.87 93.73
79 | 310 0.83 94.56
80 | 312 0.84 95.40
81 | 307 0.82 96.22
82 | 278 0.74 96.97
83 | 213 0.57 97.54
84 | 182 0.49 98.03
85 | 159 0.43 98.45
86 | 150 0.40 98.85
87 | 98 0.26 99.12
88 | 86 0.23 99.35
89 | 93 0.25 99.60
90 | 151 0.40 100.00
--------------+-----------------------------------
Total | 37,319 100.00
-> tabulation of gndr
Gender | Freq. Percent Cum.
------------+-----------------------------------
Male | 17,463 46.43 46.43
Female | 20,148 53.57 100.00
------------+-----------------------------------
Total | 37,611 100.00
-> tabulation of edulvlb
Highest level of education | Freq. Percent Cum.
----------------------------------------+-----------------------------------
Not completed ISCED level 1 | 317 0.85 0.85
ISCED 1, completed primary education | 2,252 6.01 6.86
Vocational ISCED 2C < 2 years, no acces | 8 0.02 6.88
General/pre-vocational ISCED 2A/2B, acc | 223 0.60 7.48
General ISCED 2A, access ISCED 3A gener | 4,358 11.64 19.11
Vocational ISCED 2C >= 2 years, no acce | 44 0.12 19.23
Vocational ISCED 2A/2B, access ISCED 3 | 341 0.91 20.14
Vocational ISCED 2, access ISCED 3 gene | 44 0.12 20.26
Vocational ISCED 3C < 2 years, no acces | 508 1.36 21.62
General ISCED 3A/3B, access ISCED 5B/lo | 93 0.25 21.86
General ISCED 3A, access upper tier ISC | 5,304 14.16 36.03
Vocational ISCED 3C >= 2 years, no acce | 4,078 10.89 46.92
Vocational ISCED 3A, access ISCED 5B/ l | 666 1.78 48.69
Vocational ISCED 3A, access upper tier | 5,386 14.38 63.08
General ISCED 4A/4B, access ISCED 5B/lo | 17 0.05 63.12
General ISCED 4A, access upper tier ISC | 19 0.05 63.17
ISCED 4 programmes without access ISCED | 836 2.23 65.40
Vocational ISCED 4A/4B, access ISCED 5B | 91 0.24 65.65
Vocational ISCED 4A, access upper tier | 996 2.66 68.31
ISCED 5A short, intermediate/academic/g | 203 0.54 68.85
ISCED 5B short, advanced vocational qua | 1,626 4.34 73.19
ISCED 5A medium, bachelor/equivalent fr | 1,665 4.45 77.64
ISCED 5A medium, bachelor/equivalent fr | 3,133 8.37 86.00
ISCED 5A long, master/equivalent from l | 792 2.11 88.12
ISCED 5A long, master/equivalent from u | 3,961 10.58 98.69
ISCED 6, doctoral degree | 408 1.09 99.78
Other | 81 0.22 100.00
----------------------------------------+-----------------------------------
Total | 37,450 100.00
-> tabulation of cntry
Country | Freq. Percent Cum.
------------+-----------------------------------
BE | 1,341 3.57 3.57
BG | 2,718 7.23 10.79
CH | 1,523 4.05 14.84
CZ | 2,476 6.58 21.42
EE | 1,542 4.10 25.52
FI | 1,577 4.19 29.72
FR | 1,977 5.26 34.97
GB | 1,149 3.05 38.03
GR | 2,799 7.44 45.47
HR | 1,592 4.23 49.70
HU | 1,849 4.92 54.62
IE | 1,770 4.71 59.33
IS | 903 2.40 61.73
IT | 2,640 7.02 68.75
LT | 1,659 4.41 73.16
ME | 1,278 3.40 76.55
MK | 1,429 3.80 80.35
NL | 1,470 3.91 84.26
NO | 1,411 3.75 88.01
PT | 1,838 4.89 92.90
SI | 1,252 3.33 96.23
SK | 1,418 3.77 100.00
------------+-----------------------------------
Total | 37,611 100.00
-> tabulation of netustm
Internet use, |
how much time |
on typical |
day, in |
minutes | Freq. Percent Cum.
---------------+-----------------------------------
0 | 43 0.16 0.16
1 | 7 0.03 0.18
2 | 7 0.03 0.21
3 | 1 0.00 0.21
5 | 21 0.08 0.29
6 | 62 0.22 0.51
7 | 24 0.09 0.60
8 | 72 0.26 0.86
9 | 28 0.10 0.96
10 | 142 0.51 1.47
14 | 1 0.00 1.48
15 | 167 0.61 2.08
18 | 1 0.00 2.09
20 | 126 0.46 2.54
25 | 10 0.04 2.58
28 | 1 0.00 2.58
30 | 1,034 3.75 6.33
31 | 1 0.00 6.33
35 | 8 0.03 6.36
38 | 2 0.01 6.37
40 | 61 0.22 6.59
45 | 229 0.83 7.42
50 | 52 0.19 7.61
55 | 8 0.03 7.64
59 | 2 0.01 7.65
60 | 3,401 12.32 19.97
61 | 6 0.02 19.99
63 | 2 0.01 20.00
64 | 2 0.01 20.01
65 | 18 0.07 20.07
68 | 4 0.01 20.08
69 | 1 0.00 20.09
70 | 55 0.20 20.29
71 | 2 0.01 20.29
72 | 1 0.00 20.30
74 | 1 0.00 20.30
75 | 65 0.24 20.54
78 | 1 0.00 20.54
80 | 80 0.29 20.83
85 | 10 0.04 20.87
88 | 2 0.01 20.87
90 | 1,393 5.05 25.92
95 | 7 0.03 25.95
98 | 2 0.01 25.95
99 | 1 0.00 25.96
100 | 20 0.07 26.03
105 | 49 0.18 26.21
110 | 43 0.16 26.36
115 | 4 0.01 26.38
118 | 1 0.00 26.38
119 | 5 0.02 26.40
120 | 4,432 16.06 42.46
121 | 3 0.01 42.47
122 | 6 0.02 42.49
123 | 5 0.02 42.51
125 | 5 0.02 42.53
128 | 9 0.03 42.56
130 | 34 0.12 42.68
132 | 1 0.00 42.69
133 | 1 0.00 42.69
135 | 43 0.16 42.85
138 | 1 0.00 42.85
140 | 50 0.18 43.03
143 | 1 0.00 43.04
145 | 1 0.00 43.04
150 | 1,106 4.01 47.05
155 | 3 0.01 47.06
158 | 4 0.01 47.07
160 | 24 0.09 47.16
165 | 29 0.11 47.26
168 | 1 0.00 47.27
170 | 13 0.05 47.32
175 | 1 0.00 47.32
177 | 1 0.00 47.32
180 | 3,147 11.40 58.73
181 | 3 0.01 58.74
182 | 2 0.01 58.74
183 | 5 0.02 58.76
185 | 14 0.05 58.81
188 | 2 0.01 58.82
189 | 1 0.00 58.82
190 | 24 0.09 58.91
192 | 2 0.01 58.92
195 | 17 0.06 58.98
196 | 1 0.00 58.98
198 | 1 0.00 58.99
200 | 32 0.12 59.10
202 | 1 0.00 59.11
204 | 1 0.00 59.11
205 | 5 0.02 59.13
210 | 537 1.95 61.07
215 | 3 0.01 61.08
220 | 3 0.01 61.10
225 | 12 0.04 61.14
230 | 21 0.08 61.21
240 | 2,207 8.00 69.21
242 | 1 0.00 69.22
243 | 1 0.00 69.22
244 | 3 0.01 69.23
245 | 1 0.00 69.23
246 | 1 0.00 69.24
248 | 1 0.00 69.24
250 | 9 0.03 69.27
255 | 8 0.03 69.30
256 | 1 0.00 69.31
260 | 21 0.08 69.38
264 | 1 0.00 69.39
265 | 4 0.01 69.40
270 | 353 1.28 70.68
271 | 1 0.00 70.68
275 | 2 0.01 70.69
276 | 1 0.00 70.69
278 | 1 0.00 70.70
280 | 10 0.04 70.73
285 | 6 0.02 70.76
290 | 4 0.01 70.77
299 | 1 0.00 70.77
300 | 1,948 7.06 77.83
301 | 4 0.01 77.85
302 | 1 0.00 77.85
304 | 1 0.00 77.85
305 | 3 0.01 77.86
308 | 3 0.01 77.88
310 | 11 0.04 77.92
311 | 1 0.00 77.92
315 | 6 0.02 77.94
320 | 12 0.04 77.98
325 | 2 0.01 77.99
328 | 1 0.00 77.99
330 | 234 0.85 78.84
333 | 1 0.00 78.85
338 | 1 0.00 78.85
340 | 4 0.01 78.86
345 | 3 0.01 78.88
350 | 11 0.04 78.92
359 | 1 0.00 78.92
360 | 1,205 4.37 83.29
361 | 2 0.01 83.29
362 | 1 0.00 83.30
363 | 1 0.00 83.30
365 | 2 0.01 83.31
368 | 2 0.01 83.31
370 | 5 0.02 83.33
375 | 4 0.01 83.35
377 | 1 0.00 83.35
380 | 7 0.03 83.38
390 | 126 0.46 83.83
400 | 3 0.01 83.84
405 | 6 0.02 83.86
410 | 6 0.02 83.89
420 | 482 1.75 85.63
425 | 4 0.01 85.65
430 | 4 0.01 85.66
435 | 2 0.01 85.67
440 | 2 0.01 85.68
445 | 1 0.00 85.68
450 | 62 0.22 85.90
460 | 1 0.00 85.91
470 | 2 0.01 85.92
480 | 1,315 4.76 90.68
481 | 1 0.00 90.68
485 | 1 0.00 90.69
488 | 6 0.02 90.71
489 | 1 0.00 90.71
490 | 5 0.02 90.73
492 | 1 0.00 90.73
495 | 3 0.01 90.75
500 | 5 0.02 90.76
505 | 1 0.00 90.77
510 | 95 0.34 91.11
520 | 4 0.01 91.13
525 | 2 0.01 91.13
530 | 2 0.01 91.14
533 | 1 0.00 91.14
540 | 406 1.47 92.62
545 | 1 0.00 92.62
555 | 1 0.00 92.62
560 | 1 0.00 92.63
570 | 53 0.19 92.82
580 | 1 0.00 92.82
585 | 1 0.00 92.83
590 | 4 0.01 92.84
595 | 6 0.02 92.86
599 | 2 0.01 92.87
600 | 1,126 4.08 96.95
601 | 1 0.00 96.95
602 | 1 0.00 96.96
608 | 1 0.00 96.96
609 | 1 0.00 96.96
610 | 3 0.01 96.97
615 | 2 0.01 96.98
620 | 4 0.01 97.00
630 | 29 0.11 97.10
640 | 1 0.00 97.10
650 | 2 0.01 97.11
660 | 108 0.39 97.50
665 | 1 0.00 97.51
690 | 5 0.02 97.53
720 | 428 1.55 99.08
732 | 1 0.00 99.08
735 | 1 0.00 99.08
740 | 2 0.01 99.09
745 | 1 0.00 99.09
750 | 13 0.05 99.14
765 | 1 0.00 99.14
780 | 31 0.11 99.26
810 | 1 0.00 99.26
840 | 58 0.21 99.47
870 | 1 0.00 99.47
899 | 2 0.01 99.48
900 | 60 0.22 99.70
930 | 2 0.01 99.71
940 | 1 0.00 99.71
960 | 37 0.13 99.84
990 | 1 0.00 99.85
1020 | 5 0.02 99.87
1038 | 1 0.00 99.87
1080 | 13 0.05 99.92
1140 | 1 0.00 99.92
1200 | 12 0.04 99.96
1380 | 3 0.01 99.97
1440 | 7 0.03 100.00
---------------+-----------------------------------
Total | 27,598 100.00
Recode variables
- we keep countries
GB: United Kingdom, NO:
Norway, FR: France, IT: Italy
keep if cntry == "GB" | cntry == "NO" | cntry == "FR" | cntry == "IT"
tab cntry
(30,434 observations deleted)
Country | Freq. Percent Cum.
------------+-----------------------------------
FR | 1,977 27.55 27.55
GB | 1,149 16.01 43.56
IT | 2,640 36.78 80.34
NO | 1,411 19.66 100.00
------------+-----------------------------------
Total | 7,177 100.00
- recode country variable to numeric for later regression, with new
label starting from 0
label def cntry_4 0 "NO" 1 "IT" 2 "FR" 3"GB"
encode cntry, gen(cntry_4) label(cntry_4)
tab cntry_4
label list cntry_4
Country | Freq. Percent Cum.
------------+-----------------------------------
NO | 1,411 19.66 19.66
IT | 2,640 36.78 56.44
FR | 1,977 27.55 83.99
GB | 1,149 16.01 100.00
------------+-----------------------------------
Total | 7,177 100.00
cntry_4:
0 NO
1 IT
2 FR
3 GB
label list vote
drop if vote == 3
recode vote (1 = 1) (2 = 0), gen(vote_bi)
tab vote_bi
vote:
1 Yes
2 No
3 Not eligible to vote
.a Refusal
.b Don't know
.c No answer
(742 observations deleted)
(1,480 differences between vote and vote_bi)
RECODE of |
vote (Voted |
last |
national |
election) | Freq. Percent Cum.
------------+-----------------------------------
0 | 1,480 23.47 23.47
1 | 4,825 76.53 100.00
------------+-----------------------------------
Total | 6,305 100.00
keep if agea >= 25 & agea <= 60
(2,725 observations deleted)
- recode gender, so 0 = male, 1 = female
label list gndr
recode gndr (1 = 0) (2 = 1), gen(gndr_bi)
tab gndr_bi
gndr:
1 Male
2 Female
.a No answer
(3,710 differences between gndr and gndr_bi)
RECODE of |
gndr |
(Gender) | Freq. Percent Cum.
------------+-----------------------------------
0 | 1,797 48.44 48.44
1 | 1,913 51.56 100.00
------------+-----------------------------------
Total | 3,710 100.00
- recode education, so 0 = no tertiary, 1 = tertiary
label list edulvlb
recode edulvlb (0/520 = 0) (610/888 = 1) (5555 = .), gen(edu_bi)
edulvlb:
0 Not completed ISCED level 1
113 ISCED 1, completed primary education
129 Vocational ISCED 2C < 2 years, no access ISCED 3
212 General/pre-vocational ISCED 2A/2B, access ISCED 3 vocational
213 General ISCED 2A, access ISCED 3A general/all 3
221 Vocational ISCED 2C >= 2 years, no access ISCED 3
222 Vocational ISCED 2A/2B, access ISCED 3 vocational
223 Vocational ISCED 2, access ISCED 3 general/all
229 Vocational ISCED 3C < 2 years, no access ISCED 5
311 General ISCED 3 >=2 years, no access ISCED 5
312 General ISCED 3A/3B, access ISCED 5B/lower tier 5A
313 General ISCED 3A, access upper tier ISCED 5A/all 5
321 Vocational ISCED 3C >= 2 years, no access ISCED 5
322 Vocational ISCED 3A, access ISCED 5B/ lower tier 5A
323 Vocational ISCED 3A, access upper tier ISCED 5A/all 5
412 General ISCED 4A/4B, access ISCED 5B/lower tier 5A
413 General ISCED 4A, access upper tier ISCED 5A/all 5
421 ISCED 4 programmes without access ISCED 5
422 Vocational ISCED 4A/4B, access ISCED 5B/lower tier 5A
423 Vocational ISCED 4A, access upper tier ISCED 5A/all 5
510 ISCED 5A short, intermediate/academic/general tertiary below bachelor
520 ISCED 5B short, advanced vocational qualifications
610 ISCED 5A medium, bachelor/equivalent from lower tier tertiary
620 ISCED 5A medium, bachelor/equivalent from upper/single tier tertiary
710 ISCED 5A long, master/equivalent from lower tier tertiary
720 ISCED 5A long, master/equivalent from upper/single tier tertiary
800 ISCED 6, doctoral degree
5555 Other
.a Refusal
.b Don't know
.c No answer
(3,678 differences between edulvlb and edu_bi)
Log odds: beta
logit vote_bi agea i.edu_bi netustm i.gndr_bi i.cntry_4
Iteration 0: Log likelihood = -1780.0819
Iteration 1: Log likelihood = -1553.2508
Iteration 2: Log likelihood = -1538.6147
Iteration 3: Log likelihood = -1538.5512
Iteration 4: Log likelihood = -1538.5512
Logistic regression Number of obs = 3,279
LR chi2(7) = 483.06
Prob > chi2 = 0.0000
Log likelihood = -1538.5512 Pseudo R2 = 0.1357
------------------------------------------------------------------------------
vote_bi | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | .0529852 .0046327 11.44 0.000 .0439052 .0620651
1.edu_bi | .8685538 .1041379 8.34 0.000 .6644474 1.07266
netustm | .000117 .0002564 0.46 0.648 -.0003855 .0006195
1.gndr_bi | .0008793 .0899158 0.01 0.992 -.1753525 .1771111
|
cntry_4 |
IT | -.6177332 .1549437 -3.99 0.000 -.9214172 -.3140492
FR | -1.961983 .1484614 -13.22 0.000 -2.252962 -1.671004
GB | -1.082758 .1690285 -6.41 0.000 -1.414048 -.7514684
|
_cons | -.3336569 .2559958 -1.30 0.192 -.8353994 .1680856
------------------------------------------------------------------------------
E.g., beta(agea) = .05 means that one unit increase in age is related
to a 0.05 increase in the log odds of voting, holding
all other variables constant.
You can use display exp(beta) to use Stata as the
calculator and check the odds ratios.
Odds ratios: exp(beta)
- use the option
or (odds ratios) with logit
command
logit vote_bi agea i.edu_bi netustm i.gndr_bi i.cntry_4, or
Iteration 0: Log likelihood = -1780.0819
Iteration 1: Log likelihood = -1553.2508
Iteration 2: Log likelihood = -1538.6147
Iteration 3: Log likelihood = -1538.5512
Iteration 4: Log likelihood = -1538.5512
Logistic regression Number of obs = 3,279
LR chi2(7) = 483.06
Prob > chi2 = 0.0000
Log likelihood = -1538.5512 Pseudo R2 = 0.1357
------------------------------------------------------------------------------
vote_bi | Odds ratio Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | 1.054414 .0048848 11.44 0.000 1.044883 1.064032
1.edu_bi | 2.383462 .2482086 8.34 0.000 1.943416 2.923146
netustm | 1.000117 .0002564 0.46 0.648 .9996146 1.00062
1.gndr_bi | 1.00088 .0899949 0.01 0.992 .8391612 1.193764
|
cntry_4 |
IT | .5391652 .0835402 -3.99 0.000 .3979547 .7304831
FR | .1405794 .0208706 -13.22 0.000 .1050875 .1880582
GB | .3386602 .0572432 -6.41 0.000 .243157 .4716735
|
_cons | .7162995 .1833697 -1.30 0.192 .4337012 1.183038
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.
logistic vote_bi agea i.edu_bi netustm i.gndr_bi i.cntry_4
Logistic regression Number of obs = 3,279
LR chi2(7) = 483.06
Prob > chi2 = 0.0000
Log likelihood = -1538.5512 Pseudo R2 = 0.1357
------------------------------------------------------------------------------
vote_bi | Odds ratio Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | 1.054414 .0048848 11.44 0.000 1.044883 1.064032
1.edu_bi | 2.383462 .2482086 8.34 0.000 1.943416 2.923146
netustm | 1.000117 .0002564 0.46 0.648 .9996146 1.00062
1.gndr_bi | 1.00088 .0899949 0.01 0.992 .8391612 1.193764
|
cntry_4 |
IT | .5391652 .0835402 -3.99 0.000 .3979547 .7304831
FR | .1405794 .0208706 -13.22 0.000 .1050875 .1880582
GB | .3386602 .0572432 -6.41 0.000 .243157 .4716735
|
_cons | .7162995 .1833697 -1.30 0.192 .4337012 1.183038
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.
odds(if the variable is incremented by 1 unit)/odds(if variable stays
at base)
Probabilities
logit vote_bi agea i.edu_bi netustm i.gndr_bi i.cntry_4
Iteration 0: Log likelihood = -1780.0819
Iteration 1: Log likelihood = -1553.2508
Iteration 2: Log likelihood = -1538.6147
Iteration 3: Log likelihood = -1538.5512
Iteration 4: Log likelihood = -1538.5512
Logistic regression Number of obs = 3,279
LR chi2(7) = 483.06
Prob > chi2 = 0.0000
Log likelihood = -1538.5512 Pseudo R2 = 0.1357
------------------------------------------------------------------------------
vote_bi | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | .0529852 .0046327 11.44 0.000 .0439052 .0620651
1.edu_bi | .8685538 .1041379 8.34 0.000 .6644474 1.07266
netustm | .000117 .0002564 0.46 0.648 -.0003855 .0006195
1.gndr_bi | .0008793 .0899158 0.01 0.992 -.1753525 .1771111
|
cntry_4 |
IT | -.6177332 .1549437 -3.99 0.000 -.9214172 -.3140492
FR | -1.961983 .1484614 -13.22 0.000 -2.252962 -1.671004
GB | -1.082758 .1690285 -6.41 0.000 -1.414048 -.7514684
|
_cons | -.3336569 .2559958 -1.30 0.192 -.8353994 .1680856
------------------------------------------------------------------------------
With margins
- predicted probabilities of the outcome
margins edu_bi
margins gndr_bi
margins cntry_4
Predictive margins Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
edu_bi |
0 | .7202895 .0093642 76.92 0.000 .7019361 .7386429
1 | .8465596 .0100391 84.33 0.000 .8268833 .866236
------------------------------------------------------------------------------
Predictive margins Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
gndr_bi |
0 | .766934 .0097441 78.71 0.000 .7478359 .786032
1 | .7670676 .0095499 80.32 0.000 .7483502 .7857851
------------------------------------------------------------------------------
Predictive margins Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
cntry_4 |
NO | .8984267 .011336 79.25 0.000 .8762087 .9206448
IT | .8307957 .0108654 76.46 0.000 .8095 .8520914
FR | .586057 .0157509 37.21 0.000 .5551857 .6169282
GB | .7605729 .0189654 40.10 0.000 .7234013 .7977444
------------------------------------------------------------------------------
- average marginal effects (AMEs)
margins, dydx(agea)
margins, dydx(netustm)
margins, dydx(cntry_4)
Average marginal effects Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
dy/dx wrt: agea
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | .0080538 .0006605 12.19 0.000 .0067592 .0093483
------------------------------------------------------------------------------
Average marginal effects Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
dy/dx wrt: netustm
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
netustm | .0000178 .000039 0.46 0.648 -.0000586 .0000941
------------------------------------------------------------------------------
Average marginal effects Number of obs = 3,279
Model VCE: OIM
Expression: Pr(vote_bi), predict()
dy/dx wrt: 1.cntry_4 2.cntry_4 3.cntry_4
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
cntry_4 |
IT | -.067631 .0159567 -4.24 0.000 -.0989056 -.0363565
FR | -.3123698 .019558 -15.97 0.000 -.3507028 -.2740368
GB | -.1378539 .0218305 -6.31 0.000 -.1806409 -.0950669
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
Compare to the LPM
regress vote_bi agea i.edu_bi netustm i.gndr_bi i.cntry_4, robust
Linear regression Number of obs = 3,279
F(7, 3271) = 77.11
Prob > F = 0.0000
R-squared = 0.1412
Root MSE = .39223
------------------------------------------------------------------------------
| Robust
vote_bi | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | .0082132 .0007075 11.61 0.000 .006826 .0096003
1.edu_bi | .1274084 .0143858 8.86 0.000 .0992024 .1556144
netustm | .0000223 .0000386 0.58 0.563 -.0000534 .0000981
1.gndr_bi | -.0016086 .0137437 -0.12 0.907 -.0285558 .0253385
|
cntry_4 |
IT | -.0549815 .0162725 -3.38 0.001 -.0868868 -.0230762
FR | -.3062807 .0194974 -15.71 0.000 -.344509 -.2680524
GB | -.1251441 .0203418 -6.15 0.000 -.1650281 -.0852602
|
_cons | .4760632 .0397109 11.99 0.000 .3982024 .5539239
------------------------------------------------------------------------------
Seems the LPM was efficient enough.
2. Multinomial logistic regression
For unordered categorical outcomes.
Let’s say we are interested in predicting party preference
prtcleit in Italy using a set of covariates.
Recode variables
(2,368 observations deleted)
- check the outcome variable
Which party feel closer to, Italy | Freq. Percent Cum.
-------------------------------------+-----------------------------------
Movimento 5 Stelle | 64 21.05 21.05
Partido Democratico | 88 28.95 50.00
Lega | 37 12.17 62.17
Forza Italia | 30 9.87 72.04
Fratelli d'Italia con Giorgia Meloni | 52 17.11 89.14
Liberi e Uguali (LEU) | 5 1.64 90.79
+ Europa | 2 0.66 91.45
Noi con l'Italia - UDC | 4 1.32 92.76
Potere al popolo | 6 1.97 94.74
SVP-PATT | 6 1.97 96.71
Altro | 2 0.66 97.37
Partito Comunista | 3 0.99 98.36
Partito Socialista | 1 0.33 98.68
Italexit | 2 0.66 99.34
Azione di Calenda | 2 0.66 100.00
-------------------------------------+-----------------------------------
Total | 304 100.00
keep if prtcleit < 6
tab prtcleit
(1,071 observations deleted)
Which party feel closer to, Italy | Freq. Percent Cum.
-------------------------------------+-----------------------------------
Movimento 5 Stelle | 64 23.62 23.62
Partido Democratico | 88 32.47 56.09
Lega | 37 13.65 69.74
Forza Italia | 30 11.07 80.81
Fratelli d'Italia con Giorgia Meloni | 52 19.19 100.00
-------------------------------------+-----------------------------------
Total | 271 100.00
-> tabulation of agea
Age of |
respondent, |
calculated | Freq. Percent Cum.
--------------+-----------------------------------
25 | 6 2.21 2.21
26 | 3 1.11 3.32
27 | 9 3.32 6.64
28 | 5 1.85 8.49
29 | 6 2.21 10.70
30 | 3 1.11 11.81
31 | 5 1.85 13.65
32 | 4 1.48 15.13
33 | 3 1.11 16.24
34 | 5 1.85 18.08
35 | 8 2.95 21.03
36 | 9 3.32 24.35
37 | 6 2.21 26.57
38 | 5 1.85 28.41
39 | 4 1.48 29.89
40 | 4 1.48 31.37
41 | 10 3.69 35.06
42 | 6 2.21 37.27
43 | 7 2.58 39.85
44 | 5 1.85 41.70
45 | 9 3.32 45.02
46 | 8 2.95 47.97
47 | 12 4.43 52.40
48 | 13 4.80 57.20
49 | 9 3.32 60.52
50 | 9 3.32 63.84
51 | 8 2.95 66.79
52 | 12 4.43 71.22
53 | 3 1.11 72.32
54 | 15 5.54 77.86
55 | 3 1.11 78.97
56 | 14 5.17 84.13
57 | 10 3.69 87.82
58 | 11 4.06 91.88
59 | 10 3.69 95.57
60 | 12 4.43 100.00
--------------+-----------------------------------
Total | 271 100.00
-> tabulation of gndr_bi
RECODE of |
gndr |
(Gender) | Freq. Percent Cum.
------------+-----------------------------------
0 | 134 49.45 49.45
1 | 137 50.55 100.00
------------+-----------------------------------
Total | 271 100.00
-> tabulation of edu_bi
RECODE of |
edulvlb |
(Highest |
level of |
education) | Freq. Percent Cum.
------------+-----------------------------------
0 | 193 72.28 72.28
1 | 74 27.72 100.00
------------+-----------------------------------
Total | 267 100.00
Log odds
mlogit prtcleit agea i.gndr_bi i.edu_bi
Iteration 0: Log likelihood = -411.73463
Iteration 1: Log likelihood = -400.95048
Iteration 2: Log likelihood = -400.57344
Iteration 3: Log likelihood = -400.57044
Iteration 4: Log likelihood = -400.57044
Multinomial logistic regression Number of obs = 267
LR chi2(12) = 22.33
Prob > chi2 = 0.0340
Log likelihood = -400.57044 Pseudo R2 = 0.0271
--------------------------------------------------------------------------------------------------
prtcleit | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------------------+----------------------------------------------------------------
Movimento_5_Stelle |
agea | -.0166718 .0168768 -0.99 0.323 -.0497497 .0164061
1.gndr_bi | -.3622246 .3402236 -1.06 0.287 -1.029051 .3046013
1.edu_bi | -.8921583 .3913603 -2.28 0.023 -1.65921 -.1251063
_cons | .8451113 .8128062 1.04 0.298 -.7479597 2.438182
---------------------------------+----------------------------------------------------------------
Partido_Democratico | (base outcome)
---------------------------------+----------------------------------------------------------------
Lega |
agea | .0157313 .0209567 0.75 0.453 -.0253431 .0568056
1.gndr_bi | .0304104 .4043588 0.08 0.940 -.7621182 .822939
1.edu_bi | -1.896445 .6472133 -2.93 0.003 -3.16496 -.6279307
_cons | -1.196412 1.042321 -1.15 0.251 -3.239324 .8464987
---------------------------------+----------------------------------------------------------------
Forza_Italia |
agea | -.0117892 .0211426 -0.56 0.577 -.053228 .0296496
1.gndr_bi | -.1336221 .4262747 -0.31 0.754 -.9691051 .701861
1.edu_bi | -.2554771 .4555397 -0.56 0.575 -1.148319 .6373643
_cons | -.372004 1.022091 -0.36 0.716 -2.375266 1.631258
---------------------------------+----------------------------------------------------------------
Fratelli_d_Italia_con_Giorgia_Me |
agea | .0224098 .0183421 1.22 0.222 -.0135402 .0583597
1.gndr_bi | .0633409 .3557715 0.18 0.859 -.6339585 .7606403
1.edu_bi | -.3498695 .3860904 -0.91 0.365 -1.106593 .4068539
_cons | -1.448272 .9203001 -1.57 0.116 -3.252027 .3554831
--------------------------------------------------------------------------------------------------
For each one-unit increase in age, the log odds of preferring
“Movimento 5 Stelle” over “Partido Democratico” would decrease by
approximately 0.017, holding other variables constant.
Risk ratios: exp(beta)
The ratio of the probability of one outcome category over the
probability of the baseline category.
(P.s., relative risk ratio (RRR): \(r_1 =
\frac{P(y = 1)}{P(y = basecategory)}\), \(r_2 = \frac{P(y = 2)}{P(y =
basecategory)}\) …)
- add
rrr option for relative risk ratios
mlogit prtcleit agea i.gndr_bi i.edu_bi, rrr
Iteration 0: Log likelihood = -411.73463
Iteration 1: Log likelihood = -400.95048
Iteration 2: Log likelihood = -400.57344
Iteration 3: Log likelihood = -400.57044
Iteration 4: Log likelihood = -400.57044
Multinomial logistic regression Number of obs = 267
LR chi2(12) = 22.33
Prob > chi2 = 0.0340
Log likelihood = -400.57044 Pseudo R2 = 0.0271
--------------------------------------------------------------------------------------------------
prtcleit | RRR Std. err. z P>|z| [95% conf. interval]
---------------------------------+----------------------------------------------------------------
Movimento_5_Stelle |
agea | .9834664 .0165978 -0.99 0.323 .9514675 1.016541
1.gndr_bi | .696126 .2368385 -1.06 0.287 .3573461 1.356084
1.edu_bi | .4097704 .1603679 -2.28 0.023 .1902892 .8824031
_cons | 2.328237 1.892406 1.04 0.298 .4733313 11.4522
---------------------------------+----------------------------------------------------------------
Partido_Democratico | (base outcome)
---------------------------------+----------------------------------------------------------------
Lega |
agea | 1.015856 .021289 0.75 0.453 .9749754 1.05845
1.gndr_bi | 1.030878 .4168444 0.08 0.940 .4666769 2.277183
1.edu_bi | .1501012 .0971475 -2.93 0.003 .0422158 .533695
_cons | .3022767 .3150693 -1.15 0.251 .0391904 2.331469
---------------------------------+----------------------------------------------------------------
Forza_Italia |
agea | .98828 .0208948 -0.56 0.577 .9481638 1.030094
1.gndr_bi | .8749206 .3729565 -0.31 0.754 .3794224 2.017504
1.edu_bi | .7745469 .3528368 -0.56 0.575 .3171696 1.891489
_cons | .6893515 .70458 -0.36 0.716 .0929898 5.110297
---------------------------------+----------------------------------------------------------------
Fratelli_d_Italia_con_Giorgia_Me |
agea | 1.022663 .0187578 1.22 0.222 .9865511 1.060096
1.gndr_bi | 1.06539 .3790354 0.18 0.859 .5304877 2.139646
1.edu_bi | .7047801 .2721089 -0.91 0.365 .3306837 1.502085
_cons | .234976 .2162484 -1.57 0.116 .0386957 1.42687
--------------------------------------------------------------------------------------------------
Note: _cons estimates baseline relative risk for each outcome.
The relative risk ratio for a one-unit increase in the variable agea
is .9834664 (i.e., exp(-.0166718)) from the output of the original
mlogit command before) for being in category “Movimento 5 Stelle”
vs. “Partido Democratico”.
The relative risk ratio for a one-unit increase in the variable
edu_bi is .4097704 for being in category “Movimento 5 Stelle”
vs. “Partido Democratico”. In other words, the expected risk of staying
in “Movimento 5 Stelle” is lower than in “Partido Democratico” for those
who attained a tertiary education.
Probabilities
mlogit prtcleit agea i.gndr_bi i.edu_bi
Iteration 0: Log likelihood = -411.73463
Iteration 1: Log likelihood = -400.95048
Iteration 2: Log likelihood = -400.57344
Iteration 3: Log likelihood = -400.57044
Iteration 4: Log likelihood = -400.57044
Multinomial logistic regression Number of obs = 267
LR chi2(12) = 22.33
Prob > chi2 = 0.0340
Log likelihood = -400.57044 Pseudo R2 = 0.0271
--------------------------------------------------------------------------------------------------
prtcleit | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------------------+----------------------------------------------------------------
Movimento_5_Stelle |
agea | -.0166718 .0168768 -0.99 0.323 -.0497497 .0164061
1.gndr_bi | -.3622246 .3402236 -1.06 0.287 -1.029051 .3046013
1.edu_bi | -.8921583 .3913603 -2.28 0.023 -1.65921 -.1251063
_cons | .8451113 .8128062 1.04 0.298 -.7479597 2.438182
---------------------------------+----------------------------------------------------------------
Partido_Democratico | (base outcome)
---------------------------------+----------------------------------------------------------------
Lega |
agea | .0157313 .0209567 0.75 0.453 -.0253431 .0568056
1.gndr_bi | .0304104 .4043588 0.08 0.940 -.7621182 .822939
1.edu_bi | -1.896445 .6472133 -2.93 0.003 -3.16496 -.6279307
_cons | -1.196412 1.042321 -1.15 0.251 -3.239324 .8464987
---------------------------------+----------------------------------------------------------------
Forza_Italia |
agea | -.0117892 .0211426 -0.56 0.577 -.053228 .0296496
1.gndr_bi | -.1336221 .4262747 -0.31 0.754 -.9691051 .701861
1.edu_bi | -.2554771 .4555397 -0.56 0.575 -1.148319 .6373643
_cons | -.372004 1.022091 -0.36 0.716 -2.375266 1.631258
---------------------------------+----------------------------------------------------------------
Fratelli_d_Italia_con_Giorgia_Me |
agea | .0224098 .0183421 1.22 0.222 -.0135402 .0583597
1.gndr_bi | .0633409 .3557715 0.18 0.859 -.6339585 .7606403
1.edu_bi | -.3498695 .3860904 -0.91 0.365 -1.106593 .4068539
_cons | -1.448272 .9203001 -1.57 0.116 -3.252027 .3554831
--------------------------------------------------------------------------------------------------
With margins
Predictive margins Number of obs = 267
Model VCE: OIM
1._predict: Pr(prtcleit==Movimento_5_Stelle), predict(pr outcome(1))
2._predict: Pr(prtcleit==Partido_Democratico), predict(pr outcome(2))
3._predict: Pr(prtcleit==Lega), predict(pr outcome(3))
4._predict: Pr(prtcleit==Forza_Italia), predict(pr outcome(4))
5._predict: Pr(prtcleit==Fratelli_d_Italia_con_Giorgia_Me), predict(pr outcome(5))
----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
_predict#gndr_bi |
1 0 | .2641582 .0380342 6.95 0.000 .1896126 .3387038
1 1 | .2006323 .0343254 5.85 0.000 .1333557 .2679089
2 0 | .3087973 .039933 7.73 0.000 .23053 .3870647
2 1 | .3346888 .0399921 8.37 0.000 .2563058 .4130719
3 0 | .1308903 .0283682 4.61 0.000 .0752897 .1864909
3 1 | .1466289 .0304812 4.81 0.000 .0868869 .2063709
4 0 | .1152284 .0279522 4.12 0.000 .0604431 .1700137
4 1 | .1096237 .026717 4.10 0.000 .0572594 .1619881
5 0 | .1809258 .0333325 5.43 0.000 .1155954 .2462562
5 1 | .2084262 .0349256 5.97 0.000 .1399733 .2768791
----------------------------------------------------------------------------------
- average marginal effects (AMEs)
Average marginal effects Number of obs = 267
Model VCE: OIM
dy/dx wrt: 1.gndr_bi
1._predict: Pr(prtcleit==Movimento_5_Stelle), predict(pr outcome(1))
2._predict: Pr(prtcleit==Partido_Democratico), predict(pr outcome(2))
3._predict: Pr(prtcleit==Lega), predict(pr outcome(3))
4._predict: Pr(prtcleit==Forza_Italia), predict(pr outcome(4))
5._predict: Pr(prtcleit==Fratelli_d_Italia_con_Giorgia_Me), predict(pr outcome(5))
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
0.gndr_bi | (base outcome)
-------------+----------------------------------------------------------------
1.gndr_bi |
_predict |
1 | -.0635259 .051312 -1.24 0.216 -.1640956 .0370439
2 | .0258915 .0566176 0.46 0.647 -.0850769 .1368599
3 | .0157386 .0416924 0.38 0.706 -.0659769 .0974542
4 | -.0056046 .0387318 -0.14 0.885 -.0815175 .0703082
5 | .0275004 .0483734 0.57 0.570 -.0673097 .1223104
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
- specify the outcome category
margins, dydx(gndr_bi) predict(outcome(2))
Average marginal effects Number of obs = 267
Model VCE: OIM
Expression: Pr(prtcleit==Partido_Democratico), predict(outcome(2))
dy/dx wrt: 1.gndr_bi
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
1.gndr_bi | .0258915 .0566176 0.46 0.647 -.0850769 .1368599
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
- plot predicted probabilities
margins edu_bi
marginsplot, by(edu_bi)
- plot predicted probabilities
margins edu_bi, atmeans predict(outcome(1))
marginsplot, name(Movimento_5_Stelle)
margins edu_bi, atmeans predict(outcome(2))
marginsplot, name(Partido_Democratico)
margins edu_bi, atmeans predict(outcome(3))
marginsplot, name(Lega)
margins edu_bi, atmeans predict(outcome(4))
marginsplot, name(Forza_Italia)
margins edu_bi, atmeans predict(outcome(5))
marginsplot, name(Fratelli_d_Italia)
graph combine Movimento_5_Stelle Partido_Democratico Lega Forza_Italia Fratelli_d_Italia, ycommon
- plot AMEs (probability differences)
margins, dydx(edu_bi)
marginsplot, recast(scatter)
More on marginsplot see the documentation
here
3. Ordinal logistic regression
For ordered categorical outcomes.
The proportional odds assumption: in ordered logistic regression, the
coefficients representing the relationship between the lowest category
and all higher categories of the outcome variable are the same as those
representing the relationship between the next lowest category and all
higher categories, and so on.
Let’s say we are interested in predicting political efficacy
cptppola (confidence in own ability to participate in
politics) in Italy using a set of covariates.
- check variable to be used
tab cptppola
tab cptppola, nolabel
Confident in own |
ability to |
participate in |
politics | Freq. Percent Cum.
---------------------+-----------------------------------
Not at all confident | 29 10.78 10.78
A little confident | 107 39.78 50.56
Quite confident | 102 37.92 88.48
Very confident | 27 10.04 98.51
Completely confident | 4 1.49 100.00
---------------------+-----------------------------------
Total | 269 100.00
Confident |
in own |
ability to |
participate |
in politics | Freq. Percent Cum.
------------+-----------------------------------
1 | 29 10.78 10.78
2 | 107 39.78 50.56
3 | 102 37.92 88.48
4 | 27 10.04 98.51
5 | 4 1.49 100.00
------------+-----------------------------------
Total | 269 100.00
- recode variable,soL categories 1/2 are combined into 0, category 3 =
1, and categories 4/5 into 2
recode cptppola (1/2=0) (3=1) (4/5=2), gen(cptppola3)
tab cptppola3, nolabel
(269 differences between cptppola and cptppola3)
RECODE of |
cptppola |
(Confident |
in own |
ability to |
participate |
in |
politics) | Freq. Percent Cum.
------------+-----------------------------------
0 | 136 50.56 50.56
1 | 102 37.92 88.48
2 | 31 11.52 100.00
------------+-----------------------------------
Total | 269 100.00
Log odds
ologit cptppola3 agea i.gndr_bi i.edu_bi
Iteration 0: Log likelihood = -254.42882
Iteration 1: Log likelihood = -239.22197
Iteration 2: Log likelihood = -239.08618
Iteration 3: Log likelihood = -239.08593
Iteration 4: Log likelihood = -239.08593
Ordered logistic regression Number of obs = 265
LR chi2(3) = 30.69
Prob > chi2 = 0.0000
Log likelihood = -239.08593 Pseudo R2 = 0.0603
------------------------------------------------------------------------------
cptppola3 | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | -.0065224 .0121352 -0.54 0.591 -.0303069 .0172621
1.gndr_bi | -.4741244 .2442656 -1.94 0.052 -.9528763 .0046274
1.edu_bi | 1.402793 .2786091 5.03 0.000 .8567294 1.948857
-------------+----------------------------------------------------------------
/cut1 | -.153444 .5886419 -1.307161 1.000273
/cut2 | 2.093408 .6103984 .8970495 3.289767
------------------------------------------------------------------------------
A one unit increase in age is related to an expected 0.007 decrease
in the log odds of political efficacy cptppola3, given all
of the other variables held constant. However, it is not statistically
significant at the 0.05 level.
Odds ratios: exp(beta)
ologit cptppola3 agea i.gndr_bi i.edu_bi, or
Iteration 0: Log likelihood = -254.42882
Iteration 1: Log likelihood = -239.22197
Iteration 2: Log likelihood = -239.08618
Iteration 3: Log likelihood = -239.08593
Iteration 4: Log likelihood = -239.08593
Ordered logistic regression Number of obs = 265
LR chi2(3) = 30.69
Prob > chi2 = 0.0000
Log likelihood = -239.08593 Pseudo R2 = 0.0603
------------------------------------------------------------------------------
cptppola3 | Odds ratio Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | .9934988 .0120563 -0.54 0.591 .9701477 1.017412
1.gndr_bi | .6224298 .1520382 -1.94 0.052 .3856303 1.004638
1.edu_bi | 4.066543 1.132976 5.03 0.000 2.355444 7.020659
-------------+----------------------------------------------------------------
/cut1 | -.153444 .5886419 -1.307161 1.000273
/cut2 | 2.093408 .6103984 .8970495 3.289767
------------------------------------------------------------------------------
Note: Estimates are transformed only in the first equation to odds ratios.
Probabilities
ologit cptppola3 agea i.gndr_bi i.edu_bi
Iteration 0: Log likelihood = -254.42882
Iteration 1: Log likelihood = -239.22197
Iteration 2: Log likelihood = -239.08618
Iteration 3: Log likelihood = -239.08593
Iteration 4: Log likelihood = -239.08593
Ordered logistic regression Number of obs = 265
LR chi2(3) = 30.69
Prob > chi2 = 0.0000
Log likelihood = -239.08593 Pseudo R2 = 0.0603
------------------------------------------------------------------------------
cptppola3 | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea | -.0065224 .0121352 -0.54 0.591 -.0303069 .0172621
1.gndr_bi | -.4741244 .2442656 -1.94 0.052 -.9528763 .0046274
1.edu_bi | 1.402793 .2786091 5.03 0.000 .8567294 1.948857
-------------+----------------------------------------------------------------
/cut1 | -.153444 .5886419 -1.307161 1.000273
/cut2 | 2.093408 .6103984 .8970495 3.289767
------------------------------------------------------------------------------
With margins
Predictive margins Number of obs = 265
Model VCE: OIM
1._predict: Pr(cptppola3==0), predict(pr outcome(0))
2._predict: Pr(cptppola3==1), predict(pr outcome(1))
3._predict: Pr(cptppola3==2), predict(pr outcome(2))
----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
_predict#gndr_bi |
1 0 | .4490069 .0399203 11.25 0.000 .3707645 .5272493
1 1 | .5564695 .0398033 13.98 0.000 .4784564 .6344826
2 0 | .4143869 .034125 12.14 0.000 .3475031 .4812706
2 1 | .3517536 .0330428 10.65 0.000 .2869908 .4165163
3 0 | .1366062 .0253045 5.40 0.000 .0870103 .1862021
3 1 | .0917769 .0189291 4.85 0.000 .0546765 .1288774
----------------------------------------------------------------------------------
- average marginal effects (AMEs)
margins, dydx(agea)
margins, dydx(gndr_bi)
Average marginal effects Number of obs = 265
Model VCE: OIM
dy/dx wrt: agea
1._predict: Pr(cptppola3==0), predict(pr outcome(0))
2._predict: Pr(cptppola3==1), predict(pr outcome(1))
3._predict: Pr(cptppola3==2), predict(pr outcome(2))
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
agea |
_predict |
1 | .0014758 .0027411 0.54 0.590 -.0038966 .0068482
2 | -.0008604 .0015988 -0.54 0.590 -.003994 .0022732
3 | -.0006154 .0011477 -0.54 0.592 -.0028648 .001634
------------------------------------------------------------------------------
Average marginal effects Number of obs = 265
Model VCE: OIM
dy/dx wrt: 1.gndr_bi
1._predict: Pr(cptppola3==0), predict(pr outcome(0))
2._predict: Pr(cptppola3==1), predict(pr outcome(1))
3._predict: Pr(cptppola3==2), predict(pr outcome(2))
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
0.gndr_bi | (base outcome)
-------------+----------------------------------------------------------------
1.gndr_bi |
_predict |
1 | .1074626 .0547045 1.96 0.049 .0002437 .2146815
2 | -.0626333 .032445 -1.93 0.054 -.1262243 .0009577
3 | -.0448293 .0236709 -1.89 0.058 -.0912234 .0015648
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
- plot predicted probabilities
margins edu_bi
marginsplot, by(edu_bi)
- plot AMEs (probability differences)
margins, dydx(edu_bi)
marginsplot, recast(scatter)
Graph Editor in Stata!
“With Stata’s Graph Editor, you can change how your graph looks. You
can add. You can remove. You can move. You can modify. Anything…” See
the documentation
here
It helps us tidy up our graphs and make them pop! So please use it
for your projects.
We’ve noticed that you tend to hold off on in-class assignments when
they aren’t required.
Image source
here
So… We have to “force” you to take a step forward and finish up the
work you haven’t completed.
Last in-class assignment (mandatory!)
Due by 23:59 tomorrow, 20/11/2024.
The outcome is binary: you either uploaded or did
not, and no room for further negotiation.
- Complete and review your assignment from last week, and upload your
do-file along with your interpretations of the model outputs (just
inline comments within your do-file). I expect to see your
comments on the interaction terms.
You should have obtained similar graphs by the end:
- Upload your DAGs following the instructions provided in last week’s
DAGs slides:
