[R] SE of difference in fitted probabilities from logistic model.

[Rolf Turner]

I am fitting a logistic binomial model of the form

	glm(y ~ a*x,family=binomial)

where a is a factor (with 5 levels) and x is a continuous predictor.

To assess how much ``impact'' x has, I want to compare the fitted  
success probability
when x = its maximum value with the fitted probability when x = its  
mean value.
(The mean and the max are to be taken by level of the factor ``a'',  
but that's
not really an issue.)

I can of course easily calculate p.hat(x.max) - p.hat(x.mean) using  
predict()
(with type="response").  And I can get the standard error for p.hat 
(x.max) and
p.hat(x.mean) by specifiying se.fit=TRUE. No problem there.

But how can I get a handle on the standard error of the difference?

In a linear model this would just be SE(beta_1.hat)*(x.max-x.mean)  
(where
beta_1.hat is specific to the particular level of `a' being considered).
If I am not mistaken.  (Please correct me if I am!)

But in the logistic model, everything is entangled in the inverse link
function (the ``expit'' function as it is called by some), and I can see
no way of disentangling.

Is there any way of getting at this?  I figure that simulation/Monte  
Carlo inference/
parametric bootstrapping would provide a workaround, but before I go  
that route,
can anyone point me to a simpler method?  There wouldn't be anything  
built into R
or an R package, would there?  (I did a fairly basic RSiteSearch()  
and came up
empty handed.)

Thanks for any tips.

	cheers,

		Rolf Turner

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