I am trying to understand the origin of the curved shaped of confidence bands associated with an OLS linear regression and how it relates to the confidence intervals of the regression parameters (slope and intercept), for example (using R):
require(visreg) fit <- lm(Ozone ~ Solar.R,data=airquality) visreg(fit)
It appears that the band is related to the limits of the lines calculated with the 2.5% intercept, and the 97.5% slope, as well as with the 97.5% intercept, and the 2.5% slope (although not quite):
xnew <- seq(0,400) int <- confint(fit) lines(xnew, (int[1,2]+int[2,1]*xnew)) lines(xnew, (int[1,1]+int[2,2]*xnew))
What I don't understand are two things:
- What about the combination of 2.5% slope & 2.5% intercept as well as 97.5% slope and 97.5% intercept? These give lines that are clearly outside the band plotted above. Maybe I don't understand the meaning of a confidence interval, but if in 95% of the cases my estimates are within the confidence interval, these seem like a possible outcome?
- What determines the minimum distance between the upper and lower limit (i.e. close to the point where the two lines added above intercept)?
I guess both questions arise because I don't know/understand how these bands are actually calculated.
How can I calculate the upper and lower limits using the confidence intervals of the regression parameters (without relying on predict() or a similar function, i.e. by hand)? I tried to decipher the predict.lm function in R, but the coding is beyond me. I'd appreciate any pointers towards relevant literature or explanations suitable for stats beginners.