From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>

Date: Thu 21 Jul 2005 - 02:09:03 EST

Date: Thu 21 Jul 2005 - 02:09:03 EST

kehler@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.

-- O__ ---- Peter Dalgaard ุster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk) FAX: (+45) 35327907 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Thu Jul 21 02:12:53 2005

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