OLS Example With Equal Intervals
use http://www.gseis.ucla.edu/courses/data/hsb2, clear
generate oread1=read
recode oread1 20/30=1 30/40=2 40/50=3 50/60=4 60/70=5 70/80=6
tabulate oread1, gen(oread1)
oread1 | Freq. Percent Cum.
------------+-----------------------------------
1 | 1 0.50 0.50
2 | 21 10.50 11.00
3 | 61 30.50 41.50
4 | 61 30.50 72.00
5 | 47 23.50 95.50
6 | 9 4.50 100.00
------------+-----------------------------------
Total | 200 100.00
/* k-1 dummy variables */
regress write female math or12 or13 or14 or15 or16
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 7, 192) = 30.02
Model | 9343.36107 7 1334.76587 Prob > F = 0.0000
Residual | 8535.51393 192 44.4558017 R-squared = 0.5226
-------------+------------------------------ Adj R-squared = 0.5052
Total | 17878.875 199 89.843593 Root MSE = 6.6675
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5.415011 .9547621 5.67 0.000 3.531841 7.29818
math | .4352535 .0651719 6.68 0.000 .3067087 .5637983
or12 | -.7310555 6.842787 -0.11 0.915 -14.22775 12.76563
or13 | 2.840241 6.740025 0.42 0.674 -10.45376 16.13424
or14 | 5.830803 6.771746 0.86 0.390 -7.525766 19.18737
or15 | 8.876471 6.837878 1.30 0.196 -4.610536 22.36348
or16 | 8.598478 7.206724 1.19 0.234 -5.616039 22.81299
_cons | 21.86909 7.293918 3.00 0.003 7.48259 36.25559
------------------------------------------------------------------------------
test or12 or13 or14 or15 or16
( 1) or12 = 0
( 2) or13 = 0
( 3) or14 = 0
( 4) or15 = 0
( 5) or16 = 0
F( 5, 192) = 5.30
Prob > F = 0.0001
/* ordinal variable */
regress write female math oread1
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 3, 196) = 70.03
Model | 9249.65685 3 3083.21895 Prob > F = 0.0000
Residual | 8629.21815 196 44.0266232 R-squared = 0.5174
-------------+------------------------------ Adj R-squared = 0.5100
Total | 17878.875 199 89.843593 Root MSE = 6.6353
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5.427203 .9435353 5.75 0.000 3.566418 7.287988
math | .4312545 .0646041 6.68 0.000 .303846 .5586629
oread1 | 2.804904 .5652588 4.96 0.000 1.690134 3.919674
_cons | 16.46917 2.748966 5.99 0.000 11.04782 21.89052
------------------------------------------------------------------------------
/* ordinal variable and k-2 dummy variables */
regress write female math oread1 or13 or14 or15 or16
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 7, 192) = 30.02
Model | 9343.36107 7 1334.76587 Prob > F = 0.0000
Residual | 8535.51393 192 44.4558017 R-squared = 0.5226
-------------+------------------------------ Adj R-squared = 0.5052
Total | 17878.875 199 89.843593 Root MSE = 6.6675
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5.415011 .9547621 5.67 0.000 3.531841 7.29818
math | .4352535 .0651719 6.68 0.000 .3067087 .5637983
oread1 | -.7310555 6.842787 -0.11 0.915 -14.22775 12.76563
or13 | 4.302352 7.355962 0.58 0.559 -10.20652 18.81123
or14 | 8.02397 14.11177 0.57 0.570 -19.81003 35.85797
or15 | 11.80069 20.92866 0.56 0.574 -29.47892 53.08031
or16 | 12.25376 27.84588 0.44 0.660 -42.66936 67.17687
_cons | 22.60014 13.77407 1.64 0.102 -4.567782 49.76807
------------------------------------------------------------------------------
test or13 or14 or15 or16
( 1) or13 = 0
( 2) or14 = 0
( 3) or15 = 0
( 4) or16 = 0
F( 4, 192) = 0.53
Prob > F = 0.7160
/* compare to continuous predictor */
regress write female math read
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 3, 196) = 72.52
Model | 9405.34864 3 3135.11621 Prob > F = 0.0000
Residual | 8473.52636 196 43.2322773 R-squared = 0.5261
-------------+------------------------------ Adj R-squared = 0.5188
Total | 17878.875 199 89.843593 Root MSE = 6.5751
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5.44337 .9349987 5.82 0.000 3.59942 7.287319
math | .3974826 .0664037 5.99 0.000 .266525 .5284401
read | .3252389 .0607348 5.36 0.000 .2054613 .4450166
_cons | 11.89566 2.862845 4.16 0.000 6.249728 17.5416
------------------------------------------------------------------------------
Note: Dummy coded variables do not contain significant information that is not
contained in the ordinal variable.Ordered Logistic Example With Equal Intervals
generate write3=write
recode write3 30/45=1 45/60=2 60/75=3
/* k-1 dummy variables */
ologit write3 female math or12 or13 or14 or15 or16
Ordered logit estimates Number of obs = 200
LR chi2(7) = 122.89
Prob > chi2 = 0.0000
Log likelihood = -147.76514 Pseudo R2 = 0.2937
------------------------------------------------------------------------------
write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 1.485106 .3251231 4.57 0.000 .8478766 2.122336
math | .1228038 .0229207 5.36 0.000 .0778801 .1677274
or12 | -1.889068 2.056765 -0.92 0.358 -5.920255 2.142118
or13 | -1.056602 2.011659 -0.53 0.599 -4.99938 2.886176
or14 | -.2398585 2.020243 -0.12 0.905 -4.199462 3.719745
or15 | .4322685 2.043845 0.21 0.832 -3.573595 4.438132
or16 | 1.774106 2.316172 0.77 0.444 -2.765507 6.313719
-------------+----------------------------------------------------------------
_cut1 | 5.022661 2.232042 (Ancillary parameters)
_cut2 | 8.508677 2.30283
------------------------------------------------------------------------------
test or12 or13 or14 or15 or16
( 1) or12 = 0
( 2) or13 = 0
( 3) or14 = 0
( 4) or15 = 0
( 5) or16 = 0
chi2( 5) = 16.36
Prob > chi2 = 0.0059
/* ordinal variable */
ologit write3 female math oread1
Ordered logit estimates Number of obs = 200
LR chi2(3) = 120.82
Prob > chi2 = 0.0000
Log likelihood = -148.80348 Pseudo R2 = 0.2887
------------------------------------------------------------------------------
write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 1.506797 .3224937 4.67 0.000 .8747214 2.138874
math | .1245458 .0228629 5.45 0.000 .0797354 .1693562
oread1 | .7431847 .1920608 3.87 0.000 .3667524 1.119617
-------------+----------------------------------------------------------------
_cut1 | 8.371034 1.089549 (Ancillary parameters)
_cut2 | 11.84634 1.291247
------------------------------------------------------------------------------
/* ordinal variable and k-2 dummy variables */
ologit write3 female math oread1 or13 or14 or15 or16
Ordered logit estimates Number of obs = 200
LR chi2(7) = 122.89
Prob > chi2 = 0.0000
Log likelihood = -147.76514 Pseudo R2 = 0.2937
------------------------------------------------------------------------------
write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 1.485106 .3251231 4.57 0.000 .8478766 2.122336
math | .1228038 .0229207 5.36 0.000 .0778801 .1677274
oread1 | -1.889068 2.056765 -0.92 0.358 -5.920255 2.142118
or13 | 2.721535 2.252718 1.21 0.227 -1.693712 7.136781
or14 | 5.427346 4.28359 1.27 0.205 -2.968335 13.82303
or15 | 7.988542 6.332729 1.26 0.207 -4.423379 20.40046
or16 | 11.21945 8.448682 1.33 0.184 -5.339664 27.77856
-------------+----------------------------------------------------------------
_cut1 | 3.133593 4.131344 (Ancillary parameters)
_cut2 | 6.619608 4.160175
------------------------------------------------------------------------------
test or13 or14 or15 or16
( 1) or13 = 0
( 2) or14 = 0
( 3) or15 = 0
( 4) or16 = 0
chi2( 4) = 1.95
Prob > chi2 = 0.7447
/* compare to continuous predictor */
ologit write3 female math read
Ordered logit estimates Number of obs = 200
LR chi2(3) = 122.78
Prob > chi2 = 0.0000
Log likelihood = -147.82424 Pseudo R2 = 0.2934
------------------------------------------------------------------------------
write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 1.527411 .3238524 4.72 0.000 .892672 2.16215
math | .1167482 .0234653 4.98 0.000 .0707571 .1627394
read | .0855676 .0209828 4.08 0.000 .044442 .1266932
-------------+----------------------------------------------------------------
_cut1 | 9.600994 1.185526 (Ancillary parameters)
_cut2 | 13.10302 1.395534
------------------------------------------------------------------------------
Note: Again the dummy coded variables do not contain significant information that is not
contained in the ordinal variable. It should be noted, of course, that this will not always
be the case.
Categorical Data Analysis Course
Phil Ender