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
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