Complementary log-log models repesent a third altenative to logistic regression and probit analysis for binary response variables. Complementary log-log models are fequently used when the probability of an event is very small or very large. Unlike logit and probit the complementary log-log function is asymmetrical. A graph of the complementary log-log fuanction is given below.
Examples follow a similar pattern that the ones in logit and probit analyses.
Example 1
set matsize 100
use http://www.gseis.ucla.edu/courses/data/honors
tab ses, gen(ses)
ses | Freq. Percent Cum.
------------+-----------------------------------
low | 47 23.50 23.50
middle | 95 47.50 71.00
high | 58 29.00 100.00
------------+-----------------------------------
Total | 200 100.00
cloglog honors lang math female ses1 ses2
Complementary log-log regression Number of obs = 200
Zero outcomes = 147
Nonzero outcomes = 53
LR chi2(5) = 87.46
Log likelihood = -71.91256 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
honors | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lang | .0480773 .0213486 2.25 0.024 .0062348 .0899197
math | .1075815 .024252 4.44 0.000 .0600485 .1551145
female | .8893274 .332183 2.68 0.007 .2382606 1.540394
ses1 | .0151648 .4334635 0.03 0.972 -.834408 .8647376
ses2 | -.8427618 .357412 -2.36 0.018 -1.543276 -.1422472
_cons | -10.05438 1.401321 -7.17 0.000 -12.80092 -7.307837
------------------------------------------------------------------------------
listcoef
cloglog (N=200): Unstandardized and Standardized Estimates
Observed SD: .4424407
-------------------------------------------------------------
honors | b z P>|z| bStdX SDofX
-------------+-----------------------------------------------
lang | 0.04808 2.252 0.024 0.4929 10.2529
math | 0.10758 4.436 0.000 1.0079 9.3684
female | 0.88933 2.677 0.007 0.4440 0.4992
ses1 | 0.01516 0.035 0.972 0.0064 0.4251
ses2 | -0.84276 -2.358 0.018 -0.4219 0.5006
-------------------------------------------------------------
prchange
cloglog: Changes in Predicted Probabilities for honors
min->max 0->1 -+1/2 -+sd/2
lang 0.3564 0.0007 0.0068 0.0699
math 0.8223 0.0001 0.0152 0.1453
female 0.1235 0.1235 0.1276 0.0629
ses1 0.0021 0.0021 0.0021 0.0009
ses2 -0.1187 -0.1187 -0.1207 -0.0598
0 1
Pr(y|x) 0.8465 0.1535
lang math female ses1 ses2
x= 52.23 52.645 .545 .235 .475
sd(x)= 10.2529 9.36845 .49922 .425063 .500628
mfx compute
Marginal effects after cloglog
y = Pr(honors) (predict)
= .15352122
------------------------------------------------------------------------------
variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X
---------+--------------------------------------------------------------------
lang | .0067829 .00301 2.26 0.024 .00089 .012675 52.2300
math | .0151779 .00338 4.49 0.000 .00855 .021806 52.6450
female*| .1234855 .046 2.68 0.007 .033327 .213644 .545000
ses1*| .0021467 .06156 0.03 0.972 -.118504 .122798 .235000
ses2*| -.1186566 .05132 -2.31 0.021 -.219241 -.018072 .475000
------------------------------------------------------------------------------
(*) dy/dx is for discrete change of dummy variable from 0 to 1
prtab math
cloglog: Predicted probabilities of positive outcome for honors
----------------------
math |
score | Prediction
----------+-----------
33 | 0.0199
35 | 0.0247
37 | 0.0305
38 | 0.0339
39 | 0.0377
40 | 0.0419
41 | 0.0465
42 | 0.0516
43 | 0.0573
44 | 0.0636
45 | 0.0706
46 | 0.0783
47 | 0.0868
48 | 0.0962
49 | 0.1065
50 | 0.1179
51 | 0.1303
52 | 0.1440
53 | 0.1590
54 | 0.1754
55 | 0.1932
56 | 0.2127
57 | 0.2338
58 | 0.2566
59 | 0.2812
60 | 0.3077
61 | 0.3360
62 | 0.3662
63 | 0.3982
64 | 0.4319
65 | 0.4672
66 | 0.5040
67 | 0.5420
68 | 0.5808
69 | 0.6203
70 | 0.6598
71 | 0.6990
72 | 0.7374
73 | 0.7744
75 | 0.8422
----------------------
lang math female ses1 ses2
x= 52.23 52.645 .545 .235 .475
prtab female
cloglog: Predicted probabilities of positive outcome for honors
----------------------
female | Prediction
----------+-----------
male | 0.0976
female | 0.2210
----------------------
lang math female ses1 ses2
x= 52.23 52.645 .545 .235 .475
prtab math female
cloglog: Predicted probabilities of positive outcome for honors
--------------------------
math | female
score | male female
----------+---------------
33 | 0.0123 0.0297
35 | 0.0153 0.0367
37 | 0.0189 0.0454
38 | 0.0210 0.0504
39 | 0.0234 0.0559
40 | 0.0260 0.0621
41 | 0.0289 0.0689
42 | 0.0321 0.0764
43 | 0.0357 0.0847
44 | 0.0397 0.0939
45 | 0.0441 0.1039
46 | 0.0490 0.1150
47 | 0.0544 0.1272
48 | 0.0604 0.1406
49 | 0.0670 0.1553
50 | 0.0743 0.1713
51 | 0.0824 0.1888
52 | 0.0913 0.2079
53 | 0.1012 0.2286
54 | 0.1120 0.2510
55 | 0.1239 0.2752
56 | 0.1369 0.3012
57 | 0.1513 0.3291
58 | 0.1669 0.3588
59 | 0.1840 0.3904
60 | 0.2027 0.4237
61 | 0.2229 0.4587
62 | 0.2448 0.4951
63 | 0.2686 0.5328
64 | 0.2941 0.5715
65 | 0.3215 0.6108
66 | 0.3507 0.6504
67 | 0.3818 0.6897
68 | 0.4146 0.7283
69 | 0.4492 0.7657
70 | 0.4853 0.8013
71 | 0.5227 0.8346
72 | 0.5611 0.8652
73 | 0.6003 0.8926
75 | 0.6793 0.9372
--------------------------
lang math female ses1 ses2
x= 52.23 52.645 .545 .235 .475
Example 2
use http://www.gseis.ucla.edu/courses/data/retain
describe
Contains data from http://www.gseis.ucla.edu/courses/data/retain.dta
obs: 200
vars: 6 8 Feb 2001 13:20
size: 5,600 (99.9% of memory free)
-------------------------------------------------------------------------------
storage display value
variable name type format label variable label
-------------------------------------------------------------------------------
id float %9.0g
female float %9.0g fl
read float %9.0g reading test
math float %9.0g math test
retain float %9.0g not promoted
work float %9.0g work 1/2 day
-------------------------------------------------------------------------------
summarize
Variable | Obs Mean Std. Dev. Min Max
-------------+-----------------------------------------------------
id | 200 100.5 57.87918 1 200
female | 200 .545 .4992205 0 1
read | 200 52.23 10.25294 28 76
math | 200 52.645 9.368448 33 75
retain | 200 .045 .2078243 0 1
work | 200 .26 .439735 0 1
tab1 female work retain
-> tabulation of female
female | Freq. Percent Cum.
------------+-----------------------------------
male | 91 45.50 45.50
female | 109 54.50 100.00
------------+-----------------------------------
Total | 200 100.00
-> tabulation of work
work 1/2 |
day | Freq. Percent Cum.
------------+-----------------------------------
0 | 148 74.00 74.00
1 | 52 26.00 100.00
------------+-----------------------------------
Total | 200 100.00
-> tabulation of retain
not |
promoted | Freq. Percent Cum.
------------+-----------------------------------
0 | 191 95.50 95.50
1 | 9 4.50 100.00
------------+-----------------------------------
Total | 200 100.00
cloglog retain read math female work
Complementary log-log regression Number of obs = 200
Zero outcomes = 191
Nonzero outcomes = 9
LR chi2(4) = 21.54
Log likelihood = -25.936448 Prob > chi2 = 0.0002
------------------------------------------------------------------------------
retain | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | -.0778436 .0470081 -1.66 0.098 -.1699778 .0142906
math | -.064375 .0595326 -1.08 0.280 -.1810567 .0523067
female | -2.396835 1.067861 -2.24 0.025 -4.489804 -.3038662
work | .7195881 .8009147 0.90 0.369 -.8501759 2.289352
_cons | 4.075455 3.195718 1.28 0.202 -2.188038 10.33895
------------------------------------------------------------------------------
cloglog retain read female work
Complementary log-log regression Number of obs = 200
Zero outcomes = 191
Nonzero outcomes = 9
LR chi2(3) = 20.26
Log likelihood = -26.57175 Prob > chi2 = 0.0001
------------------------------------------------------------------------------
retain | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | -.0952269 .0448223 -2.12 0.034 -.1830769 -.0073769
female | -2.270178 1.06383 -2.13 0.033 -4.355246 -.1851093
work | .91947 .8013326 1.15 0.251 -.651113 2.490053
_cons | 1.698534 2.215278 0.77 0.443 -2.643331 6.040398
------------------------------------------------------------------------------
linktest
Complementary log-log regression Number of obs = 200
Zero outcomes = 191
Nonzero outcomes = 9
LR chi2(2) = 32.25
Log likelihood = -20.578686 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
retain | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_hat | -6.417454 3.23661 -1.98 0.047 -12.76109 -.0738161
_hatsq | -1.834949 .8286093 -2.21 0.027 -3.458994 -.2109047
_cons | -6.203267 3.018963 -2.05 0.040 -12.12033 -.286208
------------------------------------------------------------------------------
generate rxw = read*work
cloglog retain read female work rxw
Complementary log-log regression Number of obs = 200
Zero outcomes = 191
Nonzero outcomes = 9
LR chi2(4) = 24.60
Log likelihood = -24.403675 Prob > chi2 = 0.0001
------------------------------------------------------------------------------
retain | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read | -.2215494 .0816473 -2.71 0.007 -.3815751 -.0615236
female | -2.578335 1.087218 -2.37 0.018 -4.709244 -.4474262
work | -7.298563 3.960604 -1.84 0.065 -15.0612 .4640787
rxw | .1916727 .0948582 2.02 0.043 .0057541 .3775913
_cons | 7.258715 3.360687 2.16 0.031 .6718901 13.84554
------------------------------------------------------------------------------
linktest
Complementary log-log regression Number of obs = 200
Zero outcomes = 191
Nonzero outcomes = 9
LR chi2(2) = 25.67
Log likelihood = -23.868815 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
retain | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_hat | -.2517141 1.407084 -0.18 0.858 -3.009548 2.50612
_hatsq | -.2616563 .2978952 -0.88 0.380 -.8455202 .3222076
_cons | -1.109333 1.411149 -0.79 0.432 -3.875134 1.656468
------------------------------------------------------------------------------
mfx compute
Marginal effects after cloglog
y = Pr(retain) (predict)
= .00489781
------------------------------------------------------------------------------
variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X
---------+--------------------------------------------------------------------
read | -.0010824 .0009 -1.20 0.230 -.00285 .000685 52.2300
female*| -.0182972 .01746 -1.05 0.295 -.052517 .015923 .545000
work*| -.0321962 .02854 -1.13 0.259 -.088138 .023745 .260000
rxw | .0009365 .00078 1.20 0.232 -.000599 .002471 11.9950
------------------------------------------------------------------------------
(*) dy/dx is for discrete change of dummy variable from 0 to 1
Categorical Data Analysis Course
Phil Ender