Given a vector of covariates x and unknown parameters b, the odds for
the event y <= j is k_j*exp(-x'b), j=1,..,m. That is, the odds of
observing y less than or equal to category j is proportional to
exp(-x'b). The reported cut points from Stata are cut_j = log(k_j).
Using the notation of the Reference Manual we have odds(j) =
exp(cut_j-x'b). Given x, the ratio of odds y<=j and y<=i is then
odds(j)/odds(i) = k_j/k_i which is independent of x. I believe this
addresses the question about the Reference Manual.
On calculating the likelihood, McCullagh (1980) has a formulation that
uses P(Y < j) = invlogit(cut_j + x'b), j=1,..m.
Another approach I can do by example
. ologit rep78 foreign
Iteration 0: log likelihood = -93.692061
Iteration 1: log likelihood = -79.696089
Iteration 2: log likelihood = -79.044933
Iteration 3: log likelihood = -79.029267
Iteration 4: log likelihood = -79.029243
Ordered logit estimates Number of obs
= 69
LR chi2(1) =
29.33
Prob > chi2 =
0.0000
Log likelihood = -79.029243 Pseudo R2 =
0.1565
------------------------------------------------------------------------------
rep78 | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
foreign | 2.98155 .6203637 4.81 0.000 1.76566
4.197441
-------------+----------------------------------------------------------------
_cut1 | -3.158382 .7224269 (Ancillary parameters)
_cut2 | -1.362642 .3557343
_cut3 | 1.232161 .3431227
_cut4 | 3.246209 .5556646
------------------------------------------------------------------------------
For domestic (foreign = 0) cars we have
. scalar P1 = invlogit(_b[_cut1])
. scalar P2 = invlogit(_b[_cut2])
. scalar P3 = invlogit(_b[_cut3])
. scalar P4 = invlogit(_b[_cut4])
. di "poor " P1 " fair " P2-P1 " average " P3-P2 " good " P4-P3
"excellent " 1-P4
poor .04076226 fair .16304903 average .57038536 good .18834001 excellent
.03746334
For foreign cars we have
. scalar P1 = invlogit(_b[_cut1]-_b[foreign])
. scalar P2 = invlogit(_b[_cut2]-_b[foreign])
. scalar P3 = invlogit(_b[_cut3]-_b[foreign])
. scalar P4 = invlogit(_b[_cut4]-_b[foreign])
. di "poor " P1 " fair " P2-P1 " average " P3-P2 " good " P4-P3 "
excellent " 1-P4
poor .00215043 fair .01066519 average .13530867 good .41765691 excellent
.4342188
Or
. predict poor fair average good excellent
(option p assumed; predicted probabilities)
. table foreign, c(mean poor mean fair mean average mean good mean
excellent)
--------------------------------------------------------------------------------
Car type | mean(poor) mean(fair) mean(aver~e) mean(good)
mean(exce~t)
----------+---------------------------------------------------------------------
Domestic | .0407623 .163049 .5703853 .18834
.0374633
Foreign | .0021504 .0106652 .1353087 .4176569
.4342188
--------------------------------------------------------------------------------
I hope this helps.
McCullagh (1980) Regression models for ordinal data. JRSS B. 42 109-142
-Rich
[email protected]
On Sun, 2004-02-01 at 16:35, Jose Maria wrote:
> Dear Stata people:
> I am at loss to understand better the output of the command -ologit-:
> 1. is it possible to show me, numerically, using the example of page 475-476 of version6 manual, the property given in the third paragraph of Methods and Formulas, on page 479: odds(k)= P(Y<=k)/ P((Y>k) for odds (k1) and odds(k2)?
> 2. how can I construct the likelihood function to obtain the ki (_cut1, _cut2, _cut3 etc)?
> I am not sure whether these are statistical or Stata questions, but I thank any guidance.
> Cheers,
> Jose Maria
>
> Jose Maria Pacheco de Souza, Professor Titular
> Departamento de Epidemiologia
> Faculdade de Saude Publica/Universidade de Sao Paulo
> Av. Dr. Arnaldo, 715 cep 01246-904
> Sao Paulo Brasil
> fones (11)3082-3886 (11)3714-2403 (11)3768-8612
> fax (11)3082-2920 (11)3714-2403
> www.fsp.usp.br/~jmpsouza
> [email protected]
>
>
>
>
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