R help - Jul 2005 - predict.lm - standard error of predicted means?

Simple question.

For a simple linear regression, I obtained the "standard error of
predicted means", for both a confidence and prediction interval:

x<-1:15
y<-x + rnorm(n=15)
model<-lm(y~x)
predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="confidence")$se.fit
        1         2
0.2708064 0.7254615

predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="prediction")$se.fit
        1         2
0.2708064 0.7254615


I was surprised to find that the standard errors returned were in fact the
standard errors of the sampling distribution of Y_hat:

sqrt(MSE(1/n + (x-x_bar)^2/SS_x)),

not the standard errors of Y_new (predicted value):

sqrt(MSE(1 + 1/n + (x-x_bar)^2/SS_x)).

Is there a reason this quantity is called the "standard error of predicted
means" if it doesn't relate to the prediction distribution?

Turning to Neter et al.'s Applied Linear Statistical Models, I note that
if we have multiple observations, then the standard error of the mean of
the predicted value:

sqrt(MSE(1/m + 1/n + (x-x_bar)^2/SS_x)),

reverts to the standard error of the sampling distribution of Y-hat, as m,
the number of samples, gets large. Still, this doesn't explain the result
for small sample sizes.

Using R.2.1 for Windows

kehler at mathstat.dal.ca writes:
> Simple question.
> 
> For a simple linear regression, I obtained the "standard error of
> predicted means", for both a confidence and prediction interval:
> 
> x<-1:15
> y<-x + rnorm(n=15)
> model<-lm(y~x)
>
predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="confidence")$se.fit
>         1         2
> 0.2708064 0.7254615
> 
>
predict.lm(model,newdata=data.frame(x=c(10,20)),se.fit=T,interval="prediction")$se.fit
>         1         2
> 0.2708064 0.7254615
> 
> 
> I was surprised to find that the standard errors returned were in fact the
> standard errors of the sampling distribution of Y_hat:
> 
> sqrt(MSE(1/n + (x-x_bar)^2/SS_x)),
> 
> not the standard errors of Y_new (predicted value):
> 
> sqrt(MSE(1 + 1/n + (x-x_bar)^2/SS_x)).
> 
> Is there a reason this quantity is called the "standard error of
predicted
> means" if it doesn't relate to the prediction distribution?
Yes. Yhat is the predicted mean and se.fit is its standard deviation.
It doesn't change its meaning because you desire another kind of
prediction interval.

 
> Turning to Neter et al.'s Applied Linear Statistical Models, I note
that
> if we have multiple observations, then the standard error of the mean of
> the predicted value:
> 
> sqrt(MSE(1/m + 1/n + (x-x_bar)^2/SS_x)),
> 
> reverts to the standard error of the sampling distribution of Y-hat, as m,
> the number of samples, gets large. Still, this doesn't explain the
result
> for small sample sizes.
You can make completely similar considerations regarding the standard
errors of and about an estimated mean: sigma*sqrt(1+1/n) vs.
sigma*sqrt(1/m + 1/n) vs. sigma*sqrt(1/n). SEM is still the latter
quantity even if you are interested in another kind of prediction limit. 

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