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Re: st: Binomial regression


From   Roger Harbord <rogerharbord@bigfoot.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Binomial regression
Date   Mon, 06 Aug 2007 21:01:43 +0100

Richard Goldstein wrote:
> I have just returned from Salt Lake City to find this
> interesting discussion.  Although I have nothing to add
> for the situation where the identity link is used, I
> note that others have complained about failure to
> converge when using the log link (relative risk
> interpretation).  For those cases, there is a solution:
> use poisson regression with robust error variance;
> see Zou, G (2004), "A Modified Poisson regression
> approach to prospective studies with binary data,"
> _American Journal of Epidemiology_, 159: 702-706.
>
> Rich
Thanks for reminding me (and the list) about that Rich. And slightly
belated thanks to Bobby for posting his code for the loglogit link (i
wasn't expecting a response until after the weekend!)

A quick comparison (below) suggests that Bobby's "loglogit" link may be
a bit more efficient than poisson with robust SEs, in the same way that
Spiegelman & Hertzmark (2005) suggest the log link should be more
efficient when it converges. See also Peterson & Deddens (2005) (and
refs therein), who give yet another approach that no-one's implemented
in Stata, AFAIK (can't say i find it very attractive myself).  Comparing
them all sounds like an undergrad stats project... One obvious advantage
of Bobby's approach is that likelihood ratio tests remain available.

Spiegelman D, Hertzmark E.  Easy SAS Calculations for Risk or Prevalence
Ratios and Differences.  Am.J.Epidemiol. 2005;162:199-200.
<http://dx.doi.org/10.1093/aje/kwi188>

Petersen MR, Deddens JA.  RE: "EASY SAS CALCULATIONS FOR RISK OR
PREVALENCE RATIOS AND DIFFERENCES".  Am.J.Epidemiol. 2006;163:1158-1159.
<http://dx.doi.org/10.1093/aje/kwj162>


  poisson for price mpg, robust nolog

Poisson regression                                Number of obs
=         74
                                                   Wald chi2(2)    =
27.23
                                                   Prob > chi2     =
0.0000
Log pseudolikelihood = -43.959394                 Pseudo R2       =
0.0971

------------------------------------------------------------------------------
              |               Robust
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
-------------+----------------------------------------------------------------
        price |   .0001289   .0000593     2.17   0.030     .0000127
0002451
          mpg |   .1052061   .0201699     5.22   0.000     .0656738
1447385
        _cons |  -4.427599   .7772089    -5.70   0.000    -5.950901
-2.904298
------------------------------------------------------------------------------

  glm for price mpg, fam(bin) link(loglogit) nolog

Generalized linear models                          No. of obs
=        74
Optimization     : ML                              Residual df
=        71
                                                    Scale parameter
=         1
Deviance         =  75.68075809                    (1/df) Deviance =
1.065926
Pearson          =  66.85115827                    (1/df) Pearson  =
9415656

Variance function: V(u) = u*(1-u)                  [Bernoulli]
Link function    : g(u) = ln(u) -> logit(u)        [Log to 0.99, then logit]

                                                    AIC             =
1.103794
Log likelihood   = -37.84037905                    BIC             =
-229.9079

------------------------------------------------------------------------------
              |                 OIM
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]
-------------+----------------------------------------------------------------
        price |   .0001015   .0000442     2.30   0.022      .000015
000188
          mpg |   .1051885   .0256521     4.10   0.000     .0549112
1554657
        _cons |  -4.246603   .8992546    -4.72   0.000    -6.009109
-2.484096
------------------------------------------------------------------------------




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