st: Mixed ordered probit (Stata 13) varying intercept when there is no intercept

[rraciborski@stata.com (Rafal Raciborski, StataCorp)]

Sam <lucaselastic@gmail.com> has some questions about the -meoprobit-
command:

> How do I specify a varying intercept in the -meoprobit- model when the
> model does not have an intercept?

> How do I specify a level-2 explanatory covariate for variance in that
> intercept in the -meoprobit- model when the model does not have an
> intercept?

In Stata's implementation of ordered outcomes models, the effect of the
constant term in the fixed-effects equation is absorbed into the cutpoints.
Therefore, there is no -noconstant- option for the fixed-effects equation.
This is explained in the following FAQ:

Why is there no constant term reported in ologit and oprobit? 
http://www.stata.com/support/faqs/statistics/ordered-logistic-and-probit-models/

Essentially, from the model, we cannot identify all the cutpoints and the
constant. Introducing a constant equal to c, would result in the same model
with the constant added to all the cutpoints, for example:

. sysuse auto, clear
(1978 Automobile Data)

. oprobit rep mpg disp turn, nolog

Ordered probit regression                         Number of obs   =         69
                                                  LR chi2(3)      =      19.65
                                                  Prob > chi2     =     0.0002
Log likelihood = -83.865955                       Pseudo R2       =     0.1049

------------------------------------------------------------------------------
       rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |   .0206013   .0374653     0.55   0.582    -.0528293    .0940318
displacement |   .0002449   .0026009     0.09   0.925    -.0048528    .0053426
        turn |  -.1231891   .0555626    -2.22   0.027    -.2320897   -.0142885
-------------+----------------------------------------------------------------
       /cut1 |  -6.562678   2.523843                     -11.50932   -1.616037
       /cut2 |  -5.677153    2.49473                     -10.56674   -.7875718
       /cut3 |   -4.19813   2.444281                     -8.988833    .5925736
       /cut4 |  -3.215816   2.437516                     -7.993258    1.561627
------------------------------------------------------------------------------

. generate one = 1

. constraint 1  _b[one] = 5

. oprobit rep mpg disp turn one, constraint(1) nolog collinear

Ordered probit regression                         Number of obs   =         69
                                                  Wald chi2(3)    =      18.66
Log likelihood = -83.865955                       Prob > chi2     =     0.0003

 ( 1)  [rep78]one = 5
------------------------------------------------------------------------------
       rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |   .0206013   .0374653     0.55   0.582    -.0528293    .0940318
displacement |   .0002449   .0026009     0.09   0.925    -.0048528    .0053426
        turn |  -.1231891   .0555626    -2.22   0.027    -.2320897   -.0142885
         one |          5  (constrained)
-------------+----------------------------------------------------------------
       /cut1 |  -1.562678   2.523843                      -6.50932    3.383963
       /cut2 |  -.6771535    2.49473                     -5.566735    4.212428
       /cut3 |   .8018702   2.444281                     -3.988833    5.592574
       /cut4 |   1.784184   2.437516                     -2.993258    6.561627
------------------------------------------------------------------------------

Therefore, if we want to introduce the equivalent to different 'constants'
for boys and girls, we just need to include a dummy variable for one
of the two groups; the coefficient for this dummy will tell us how
much this group is shifted with respect to the other, for example:

  clear
  use http://www.stata-press.com/data/r13/childweight
  gen weightcat = int(weight/4)

  oprobit weightcat girl c.age c.age#c.age

Notice that if we try to include both dummies, boy and girl, one of
them will be dropped for the same reason that the model does not have
a constant.

If we fit a two-level model, the option -noconstant- is valid at the second
level. For example, in the model

  oprobit y girl || school:

The 'constant' at the school level models a random shift in the cutpoints
from their overall values.  If, in addition, we want to introduce a different
random slope for boys and girls at the school level, we just include the two
dummies at the school level, with the option -noconstant-, as in the following
artificial example:

  clear
  set seed 13597
  webuse childweight
  egen v = group(id)
  gen school =int(v)
  gen sch_eff = rnormal()
  bysort school: replace sch_eff = sch_eff[1]
  replace weight = weight + sch_eff
  gen weightcat = int(weight/5)
  gen boy = 1-girl
  
  meoprobit weightcat girl c.age c.age#c.age || school: boy girl, nocons

Sam may have a look at the "Heteroskedastic random effects" section of the
-mixed- entry of the Multilevel Mixed-Effects Reference Manual for more
examples.  The manual is available on-line at
http://www.stata-press.com/manuals/multilevel-mixed-effects-reference-manual.

-- Isabel			-- Rafal
icanette@stata.com   		rraciborski@stata.com

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