Re: st: Question regarding linktest and its implementation

[Phil Schumm <pschumm@uchicago.edu>]

On Aug 7, 2006, at 1:46 PM, Daniel Schneider wrote:
> I know the Stata manual, the Pregibon paper and Tukey's paper.   
> Nonetheless, they did not clarify the issue sufficiently for me.   
> So what you are saying is that the linear predictor (i.e. -predict,  
> xb-) should be predictive of the observed dependent variable?  I  
> believe I understand the concept, but I assumed that "the effects  
> of the covariates" should be expressed on the same scale as the  
> dependent variable not in the scale of the link-function- 
> transformation.

I'm afraid I don't really understand the question, so I must  
apologize in advance if my response is not what you are looking for.   
Suppose we have a GLM with link function g():


    g(mu) = xb


where mu is the mean of the response and xb is the linear predictor.   
Following Pregibon (1980), if we want to explore the adequacy of g(),  
we can embed it in a family indexed by one or more parameters; for  
example, let's take the one-parameter power function available with  
the -link()- option in -glm-:


    g(mu; lambda) = mu^lambda    for lambda!=0

                  = log(mu)      for lambda==0


Now, given a value of lambda (call it lambda_0), we can expand g()  
around that value to get


    g(mu; lambda) ~ g(mu; lambda_0) + (lambda - lambda_0) * g'(mu;  
lambda_0)

                  = mu^lambda_0 + (lambda - lambda_0) * (mu^lambda_0  
* log(mu))


Rewriting in terms of g(mu; lambda_0) yields the following approximation


    g(mu; lambda_0) ~ mu^lambda - (lambda - lambda_0) * (mu^lambda_0  
* log(mu))

                    = xb - (lambda - lambda_0) * (mu^lambda_0 * log 
(mu))    (1)


McCullagh and Nelder (1989; pp. 375-9) provide a nice discussion of  
this, and also of the 2-parameter case.  The important point here is  
that it leads us to fit the model initially using the link function g 
(mu; lambda_0), to obtain an estimate of mu (call it mu_hat), and  
then to add


    - (mu_hat^lambda_0 * log(mu_hat))


to the linear predictor, using the significance of the resulting  
coefficient as an indicator of the inadequacy of lambda_0.

Consider the case of the linear model (i.e., with identity link).  In  
this case, lambda_0 equals one, and thus the term we are adding becomes


    - (mu_hat * log(mu_hat))    (2)


Note that since in the linear case mu_hat = xb_hat, (2) is quite  
close to the (xb_hat)^2 term added by -linktest-, thereby providing  
one way to think about what -linktest- is doing, and showing why both  
approaches are likely to give a similar assessment of the adequacy of  
the identity link.  The advantage of (2) is that the coefficient of  
the added term provides a direct estimate of the first-order  
adjustment to lambda_0, which is helpful if we find lambda_0 to be  
inadequate.

Finally (and I hope this will begin to address your question), let's  
consider your original case of the reciprocal link.  In this case  
lambda_0 equals -1, and following equation (1) we would fit the  
augmented model


    g(mu; -1) = xb - gamma * (mu_hat^(-1) * log(mu_hat))

              = xb + gamma * (xb_hat * log(xb_hat))         (3)


where b_hat is our estimate of b from the initial fit, and gamma =  
(lambda + 1).  Once again, the second term in (3) above is not  
exactly the same as the (xb_hat)^2 used by -linktest-, but it is  
close.  More importantly, however, this should make it clear why - 
linktest- uses xb_hat and (xb_hat)^2 (rather than (xb_hat)^(-1) and  
(xb_hat)^(-2)) when conducting the test.

I hope this is helpful.


-- Phil

McCullagh, P and Nelder, JA (1989). Generalized Linear Models, 2nd  
edition.  Chapman and Hall.
Pregibon, D (1980). Goodness of link tests for generalized linear  
models. Applied Statistics 29:15-24.

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