Regarding the estimation of 1) a single observation logistic model, and
2) a two observation logistic model, having the binomial form with a y
being the binomial numerator and n the denominator:
When you use cii, or engage in a simple case where the estimated
coefficient or odds ratio is computed
directly from the binomial PDF you are of course more likely to get a
meaningful result. Using maximum likelihood entails assumptions which
are not met in such a situation. In fact, you cannot even get results
using exact logistic regression via the -exlogistic- command. On the
other hand, -exlogistic- estimates the second situation where you have
two observations, each with response y, binomial denominator n, and
binary predictor x. However, you do not get exact values, but rather
median unbiased estimates.
y n x
--------------
10 100 1
0 100 0
Model the above using -exlogistic-:
. input r n x
r n x
1. 10 100 1
2. 0 100 0
3. end
. exlogistic y x, binomial(n) coef estc
Enumerating sample-space combinations:
observation 1: enumerations = 11
observation 2: enumerations = 101
observation 3: enumerations = 10201
note: CMLE estimate for x is +inf; computing MUE
note: CMLE estimate for _cons is -inf; computing MUE
note: .975 quantile estimate for _cons failed to bracket the value
Exact logistic regression Number of obs =
200
Binomial variable: n Model score =
10.47368
Pr >= score =
0.0015
-------------------------------------------------------------------------
--
y | Coef. Suff. 2*Pr(Suff.) [95% Conf.
Interval]
-------------+-----------------------------------------------------------
--
x | 2.722305* 10 0.0015 .8727845
+Inf
_cons | 0* 10 0.0000 -Inf
+Inf
-------------------------------------------------------------------------
--
(*) median unbiased estimates (MUE)
I requested estimation of a constant although it is obvious that it is
not meaningful in such a situation.
Compare the above with the clearly mistaken "estimated coefficients"
that you provided in your output.
. glm r x, fam(bin n)
Generalized linear models No. of obs =
2
Optimization : ML Residual df =
0
Scale parameter =
1
Deviance = 2.00000e-08 (1/df) Deviance =
.
Pearson = 1.00000e-08 (1/df) Pearson =
.
Variance function: V(u) = u*(1-u/n) [Binomial]
Link function : g(u) = ln(u/(n-u)) [Logit]
AIC =
4.025974
Log likelihood = -2.025973987 BIC =
2.00e-08
-
-------------------------------------------------------------------------
-----
| OIM
r | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-
-------------+-----------------------------------------------------------
-----
x | 23.87722 10000 0.00 0.998 -19575.76
19623.52
_cons | -26.07444 10000 -0.00 0.998 -19625.71
19573.56
-
-------------------------------------------------------------------------
-----
These coefficients indicate a problem with convergence. Exponentiate to
obtain an odds ratio:
. di %12.0f exp(23.87722)
23428521860
We have an odds ratio here of some 23.4 billion. No surprise.
The problem is that the assumptions upon which ML estimation is based
are not met here. I tried
your examples with several other commercial applications, as well as R,
with the same results.
The bottom line is that there is nothing wrong with -glm- here.
Joseph Hilbe
*
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
