Full_Name: Marek Ancukiewicz
Version: 2.01
OS: Linux
Submission from: (NULL) (132.183.12.87)
It seems that the the standard error of prediction of the linear regression,
caclulated with predict.lm is incorrect. Consider the following example where
the standard error is first calculated with predict.lm, then using delta
method. and finally, using the formula rms*sqrt(1+1/n+(xp-x0)^2/Sxx).
Marek Ancukiewicz
> n <- 10
> x <- 1:n
> y <- x
> y[c(2,4,6)] <- y[c(2,4,6)] + 0.1
> y[c(3,5,7)] <- y[c(3,5,7)] - 0.1
> a <- lm(y~x)
> rms <- sqrt(sum(a$residuals^2)/(n-2))
> s <- covmat(a)*rms^2
> xp <- 3
> xm <- mean(x)
> summary(a)
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-0.10909 -0.07500 0.01091 0.06955 0.10182
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.020000 0.058621 0.341 0.742
x 0.996364 0.009448 105.463 7.3e-14 ***
---
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Residual standard error: 0.08581 on 8 degrees of freedom
Multiple R-Squared: 0.9993, Adjusted R-squared: 0.9992
F-statistic: 1.112e+04 on 1 and 8 DF, p-value: 7.3e-14
> print(predict(a,new=data.frame(x=xp),se.fit=T))
$fit
[1] 3.009091
$se.fit
[1] 0.0359752
$df
[1] 8
$residual.scale
[1] 0.08581163
> print(se.delta.method <- sqrt(s[1,1]+xp^2*s[2,2]+2*xp*s[1,2] + rms^2))
[1] 0.09304758
> print(se.ss.formula <- rms*sqrt(1+1/n+(xp-xm)^2/sum((x-xm)^2)))
[1] 0.09304758
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