Re: st: Discrepancy between logit and log links in glm

["roger webb" <roger.webb@man.ac.uk>]

Robert,

I found your post about the reasons for and solutions to  convergence problems in log-binomial models very enlightening. I  was particularly interested in the point you raise about the  discrepancy between the Poisson model (with robust variance) and  the log-binomial model increasing as the probability of the outcome  event increases (see below):

When I began this reply, I had three alternatives in mind.  Since then, Roger
Webb <roger.webb@man.ac.uk> has mentioned the first to the list:  use
-poisson- with the -robust- option.  He states that the results are the same
as -glm, fam(binomial) link(log)-.  They are not exactly the same, but nearly
the same in most cases.  The discrepancy comes from the fact that Poisson
models allow for counts bigger than 0/1, but when your data are all 0/1, the
probability of counts bigger than 1 is usually going to be small, and
oftentimes too small to make a difference.  Note, however, that the difference
between binomial/log and Poisson increases with the probability of an event,
so the discrepancy can be higher in cases where binomial/log would fail to
converge in the first place.  Be cautious.

In a data set of mine, the probability of poor outcome overall is  about 50%, 82% in the exposed group and 36% in the unexposed  group, i.e.:

. tab rwbstat2 rwillpn2, col

|       rwillpn2
rwbstat2 |         0          1 |     Total
-----------+----------------------+----------
0 |       103          8 |       111 
|     63.58      17.78 |     53.62 
-----------+----------------------+----------
1 |        59         37 |        96 
|     36.42      82.22 |     46.38 
-----------+----------------------+----------
Total |       162         45 |       207 
|    100.00     100.00 |    100.00

The Poisson model (with robust variance) gives an almost identical  risk ratio (and SE) to that given by 'binreg' or 'glm, link(log)  family(binomial)' (please see below). So I'm now a little confused as  when the Poisson model is inappropriate and in what  circumstances it becomes significantly discrepant from the log- binomial. I'd be very grateful for you advice on this matter.

Roger Webb
University of Manchester (UK)



i.rwillpn2        _Irwillpn2_0-1      (naturally coded; _Irwillpn2_0 omitted)

Iteration 1 : deviance =  365.2877
Iteration 2 : deviance =  257.9924
Iteration 3 : deviance =  254.6188
Iteration 4 : deviance =  254.5983
Iteration 5 : deviance =  254.5983
Iteration 6 : deviance =  254.5983

Residual df  =       205                                No. of obs =       207
Pearson X2   =       207                                Deviance   =  254.5983
Dispersion   =  1.009756                                Dispersion =  1.241943

Bernoulli distribution, log link
------------------------------------------------------------------------------
|                 EIM
rwbstat2 | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Irwillpn2_1 |   2.257627   .2818068     6.52   0.000     1.767671    2.883387
------------------------------------------------------------------------------

. xi: glm rwbstat2 i.rwillpn2, link(log) family(binomial) eform
i.rwillpn2        _Irwillpn2_0-1      (naturally coded; _Irwillpn2_0 omitted)

Iteration 0:   log likelihood = -182.64385  
Iteration 1:   log likelihood = -133.29222  
Iteration 2:   log likelihood = -127.46681  
Iteration 3:   log likelihood = -127.29986  
Iteration 4:   log likelihood = -127.29913  
Iteration 5:   log likelihood = -127.29913  

Generalized linear models                          No. of obs      =       207
Optimization     : ML: Newton-Raphson              Residual df     =       205
Scale parameter =         1
Deviance         =  254.5982565                    (1/df) Deviance =  1.241943
Pearson          =          207                    (1/df) Pearson  =  1.009756

Variance function: V(u) = u*(1-u)                  [Bernoulli]
Link function    : g(u) = ln(u)                    [Log]
Standard errors  : OIM

Log likelihood   = -127.2991283                    AIC             =  1.249267
BIC              = -838.6090961

------------------------------------------------------------------------------
rwbstat2 | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Irwillpn2_1 |   2.257627   .2818068     6.52   0.000     1.767671    2.883387
------------------------------------------------------------------------------

. xi: poisson rwbstat2 i.rwillpn2, robust irr
i.rwillpn2        _Irwillpn2_0-1      (naturally coded; _Irwillpn2_0 omitted)

Iteration 0:   log pseudo-likelihood =  -162.8367  
Iteration 1:   log pseudo-likelihood = -162.83602  
Iteration 2:   log pseudo-likelihood = -162.83602  

Poisson regression                                Number of obs   =        207
Wald chi2(1)    =      42.35
Prob > chi2     =     0.0000
Log pseudo-likelihood = -162.83602                Pseudo R2       =     0.0408

------------------------------------------------------------------------------
|               Robust
rwbstat2 |        IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Irwillpn2_1 |   2.257627     .28249     6.51   0.000     1.766623    2.885097
------------------------------------------------------------------------------

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