Re: st: Confidence intervals for predictions after GLMs

[Steven Samuels <ssamuels@albany.edu>]

Redundant Line

Allan is referring to the fact that -predictnl- returns confidence  
intervals (CI's) that are symmetrical about the predicted value.  
Allan would prefer that -predictnl- compute asymmetric intervals:   
predict the linear function,, compute the SE of this function,  
estimate CI's for the linear function, and then invert the CI  
endpoints.  This process will produce a "better" interval then that  
produced by -predictnl-; for example the inverted interval will more  
closely match one obtained by the bootstrap.

However I agree with Stata Support: Alan is wrong to expect this  
behavior from -predictnl-.  It ts such a general command that it  
computes a standard error & CI  for functions in which there is NO  
invertible linear form.
 For example: predictnl p2 = _b[weight]*weight/_b[height]*height


I would not expect -predictnl- to diagnose the RHS expression to  
determine if it contains an invertible linear form. However it is  
reasonable to ask StataCorp to add options to -predict- to compute  
the inverted CI after commands like -logistic-, where -predict-   
inverts a linear form to obtain the estimated probability.

-Steve



On Jun 15, 2007, at 11:17 AM, Allan Reese ((Cefas)) wrote:


> Stata boasts many commands for estimation following model fits, so  
> it was a surprise to find a gap for generalized linear models  
> (glm's).  Point estimates are available through predict, but the  
> confidence interval from predictnl is clearly wrong when the error  
> distribution has been specified as non-normal.
>
> I've raised this with Tech Desk, using a simple example with well- 
> known data:
> . use auto
> . logit foreign weight
> . predict mu
> -----Tech desk Message-----
> The issue here is that -predictnl- is a very flexible command that  
> will allow you to create various types of nonlinear predictions.   
> In the command you used
>
>   predictnl mu2= exp( _b[_cons] + _b[weight]*weight ) /
>        ( 1 + exp( _b[_cons] + _b[weight]*weight) ), ci(lo hi)
>
> [inverse link for a logit example]
>
> we know that _b[_cons] + _b[weight]*weight follows a normal  
> distribution.  However, -predictnl- has no way of discerning this  
> or knowing that this particular nonlinear combination creates a  
> predicted probability.  All confidence intervals created by - 
> predictnl- are based on the nonlinear combination being  
> asymptotically normal which is fine even in the case of the  
> nonlinear expression that you used above.  The difference in this  
> case is that we know that this is a predicted probability and can  
> create a more precise confidence interval manually transforming the  
> endpoints of the confidence interval for xb.
>
> Thus, there is not a problem with the confidence intervals that - 
> predictnl- creates for nonlinear functions.  However, in certain  
> cases there may be a more direct way to calculate the intervals  
> that is not based on asymptotic normality.  This is true in your  
> situation.
> --------------------------
>
> Would you be happy with an "asymptotic" approach?  I don't follow,  
> as the error distribution doesn't change and this is for individual  
> predictions.  Asymptotic with respect to what?  As I read it, they  
> are claiming that *any* function can have a CI predicted as if it  
> is normal, even when it is explicitly a transformation of a normal  
> variate!
>
> The GLM approach is well-defined: the linear predictor is a  
> combination of weighted parameter estimates, so has a normal  
> distribution.  Define the CI for the linear predictor, then back- 
> transform using the inverse-link function.  This must be done by  
> predict when evaluating the mean.
>
> Are other users being misled by this approximate approach to CIs?
> Allan
>
>
>
>
>
>
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