[R] confidence interval in "predict.lm"

[ripley@stats.ox.ac.uk]

The problem is with your quote from the book.  That formula is not a
confidence interval, it is a tolerance interval, Use predict.lm with
interval="prediction" to get it.

I suggest you get a better (or at least more understandable) book!

On Fri, 15 Nov 2002, Fred Mellender wrote:

> I am studying statistics using R and a book "Understandable Statistics", by
> Brase and Brase.  The book has two
> worked examples for calculating a confidence interval around a predicted
> value from a linear model.  The answers
> to the two examples in the book differ from those I get from R.  The
> regression line, the standard error, and the
> predicted value in
> R and the book all agree for the examples.  Hence I gather that R and the
> book use different formula to calculate
> the confidence interval.  Could someone explain why the difference exists,
> and which function(s) in R I might use
> to get the answers in the book, and (perhaps) an explanation as to which
> method to use in various situations).
>
> The example:
>
> > x<-c(10,20,30,40,50,60,70)
> > y<-c(17,21,25,28,33,40,49)
> > dat <- data.frame(temp=x,amnt=y)
>   temp amnt
> 1   10   17
> 2   20   21
> 3   30   25
> 4   40   28
> 5   50   33
> 6   60   40
> 7   70   49
>
> being a table of temperatures (temp) and the corresponding amounts of copper
> sulfate that disolve in 100g of water
> at that temperature.
>
> The regression line:
>
> > mod <- lm(amnt ~ temp,dat)
> > summary(mod)
>
> Call:
> lm(formula = amnt ~ temp, data = dat)
>
> Residuals:
>       1       2       3       4       5       6       7
>  1.7857  0.7143 -0.3571 -2.4286 -2.5000 -0.5714  3.3571
>
> Coefficients:
>             Estimate Std. Error t value Pr(>|t|)
> (Intercept) 10.14286    1.98463   5.111  0.00374 **
> temp         0.50714    0.04438  11.428 8.98e-05 ***
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
>
> Residual standard error: 2.348 on 5 degrees of freedom
> Multiple R-Squared: 0.9631,     Adjusted R-squared: 0.9558
> F-statistic: 130.6 on 1 and 5 DF,  p-value: 8.985e-05
>
> The .95 confidence interval for a temperature of 45 degrees:
> >
> foo<-predict(mod,data.frame(temp=45),level=.95,interval="confidence",se.fit=
> T)
> > foo
> $fit
>           fit      lwr      upr
> [1,] 32.96429 30.61253 35.31604
>
> $se.fit
> [1] 0.9148715
>
> $df
> [1] 5
>
> $residual.scale
> [1] 2.348252
>
> The book gives the confidence interval as 26.5 <= y <= 39.5.  The book
> defines the confidence interval calculation thus:
>
>   yp - E <= y <= yp + E
>
>   Where
>    E = tc*sC *sqrt(1 + 1/n + (x-xBar)^2/SSx)
>    yp is the predicted value from the regression line
>    tc is the value from Student's t distribution for a confidence
>     level, c, using n-2 degrees of freedom,
>    sC is the standard error of estimate
>    SSx is Sum(x^2)-[Sum(x)]^2/n
>    n is the number of data pairs.
>
> So that even though the model, predicted value, standard error all agree, R
> gives a much smaller confidence
> interval than the book does.
>
> Thanks for any advice/help.
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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