Re: [SPAM] RE: st: Question regarding linktest and its implementation

[Phil Schumm <pschumm@uchicago.edu>]

On Aug 5, 2006, at 7:50 PM, Daniel Schneider wrote:
> I have been using the linktest command after GLM models quite  
> frequently, and I came to wonder whether it is implemented  
> correctly or not.
<snip>

> Is it possible that the revision of GLM models for the more recent  
> stata version has changed some of the meaning of predictions and is  
> not correctly implemented in the linktest command that seems to be  
> based on an older Stata version?
On Aug 6, 2006, at 12:33 PM, Daniel Schneider wrote:
> This is the replication of the linktest.  Note that the results are  
> similar but not identical because linktest seems to rely on an  
> older version of GLM.

The test implemented by -linktest- (known as Tukey's one-degree-of- 
freedom test for non-additivity) is a test of the null hypothesis  
that the effects of the covariates are additive against alternatives  
in which they are not (the introduction of the square of the linear  
predictor to the model may be thought of as capturing a range of such  
alternatives).  Since one of the purposes of the link function in a  
GLM is to linearize the effect of the covariates, in this context, it  
may also be thought of as a test of the adequacy of the link  
function.  This is discussed in detail by Pregibon (1980), who  
proposed a slightly different approach; the entry under [R] - 
linktest- provides the full references to Pregibon (1980) and to  
Tukey's original 1949 paper, and explains why Stata uses Tukey's  
original formulation.  I strongly suggest reading Pregibon's paper,  
which provides a nice way to think about what is going on in the GLM  
context.

Your posting also raised another (minor) issue worth commenting on.   
In particular, you demonstrated the following


. sysuse auto
(1978 Automobile Data)

. glm price trunk foreign weight, fam(gam)
<output omitted>

. predict xb, xb

. gen xb2 = xb^2

. linktest, fam(gam) nolog

Residual df  =        71                                No. of obs  
=        74
Pearson X2   =  5.113241                                Deviance   =   
4.375989
Dispersion   =  .0720175                                Dispersion  
=  .0616336

Gamma distribution, power link (power = -1)
------------------------------------------------------------------------ 
------
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
        _hat |   1.333585   .3680821     3.62   0.000     . 
6121571    2.055012
      _hatsq |  -1117.429   1200.049    -0.93   0.352     
-3469.481    1234.624
       _cons |  -.0000214   .0000261    -0.82   0.412    -. 
0000726    .0000298
------------------------------------------------------------------------ 
------

. glm price xb xb2, fam(gam) nohead nolog

------------------------------------------------------------------------ 
------
             |                 OIM
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
          xb |   1.333624   .3679106     3.62   0.000     . 
6125322    2.054715
         xb2 |  -1117.528   1199.493    -0.93   0.352     
-3468.491    1233.436
       _cons |  -.0000214   .0000261    -0.82   0.412    -. 
0000726    .0000297
------------------------------------------------------------------------ 
------


noting that the results obtained by manually regressing price on xb  
and xb2 are slightly different from those obtained via -linktest-.  A  
quick look at the code for -linktest- (type -viewsource  
linktest.ado-) reveals why; namely, there is a -version 6- at the  
top, which means that when the model including _hatsq is fit, Stata  
is running under version control as if it were version 6.  Under  
version 6, -glm- used Iteratively Reweighted Least Squares (IRLS) and  
reported standard errors based on the expected information matrix.   
One of the changes made in version 7 was to make use of Stata's -ml-  
the default, as well as reporting standard errors based on the  
observed information matrix.  Thus, to replicate the results of - 
linktest-, you must use the -irls- option:


. glm price xb xb2, fam(gam) nohead nolog irls

------------------------------------------------------------------------ 
------
             |                 EIM
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
          xb |   1.333585   .3680821     3.62   0.000     . 
6121571    2.055012
         xb2 |  -1117.429   1200.049    -0.93   0.352     
-3469.481    1234.624
       _cons |  -.0000214   .0000261    -0.82   0.412    -. 
0000726    .0000298
------------------------------------------------------------------------ 
------


Given that -glm- now uses -ml- by default, this behavior seems a bit  
inconsistent (though, admittedly, it is a minor issue).


-- Phil

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