I couldn't see a question addressing this. I think I'm just off the mark somewhere with my thinking. Here is a simple model:
mod <- lm(hp ~ factor(cyl), data=mtcars) summary(mod) Estimate Std. Error t value Pr(>|t|) (Intercept) 82.64 11.43 7.228 5.86e-08 *** factor(cyl)6 39.65 18.33 2.163 0.0389 * factor(cyl)8 126.58 15.28 8.285 3.92e-09 ***
This tells me the hp is significantly higher in the 6 cylinders than the baseline (4 cylinders).
Now when I run the following, I'm confused by the results.
predict(mod, data.frame(cyl=as.factor(c(4,6,8))), interval="c") fit lwr upr 1 82.63636 59.25361 106.0191 2 122.28571 92.97388 151.5975 3 209.21429 188.48769 229.9409
The fit for the first is the same as the intercept, makes sense. But the upr for the first is higher than the lwr for the second, as in, the intervals overlap. When I see confidence intervals that overlap I think, 'not significantly different'.
I don't pretend to know EVERYTHING about the ins and outs of modeling, but I thought I had a pretty solid understanding of the basics, yet I'm clearly missing something as to why these intervals overlap...
Edit: Re the confusion around the terminology of 'predicting', I merely use the word predict as that is (explicit) in the function. Am aware I am using it in the 'expansion for confidence of estimation' framework.
Note, an idea of the data:
> aggregate(hp ~ cyl, data=mtcars, mean) cyl hp 1 4 82.63636 2 6 122.28571 3 8 209.21429 > aggregate(hp ~ cyl, data=mtcars, sd) cyl hp 1 4 20.93453 2 6 24.26049 3 8 50.97689 > aggregate(hp ~ cyl, data=mtcars, length) cyl hp 1 4 11 2 6 7 3 8 14
Based on first answer and a conversation in chat. I get that the p-value 0.039 is for whether the coefficient for 6cyl is different to 0. That would say, yes, the hp is 6cyl cars is different to that of 4cyl cars. Correct interpretation right? And then you say, "okay, so what is your estimate of the hp of a 4cyl car and the hp of a 6cyl car, please include your confidence", and then you get the second set of results, 60-106 for 4cyl and 92-151 for 6cyl. But they overlap, so how can you say they're different?
So on the one hand we say the estimate of the coefficient for 6cyl cars is sufficiently far away from that of 4 cyl cars to call them different, and on the other hand we calculate a predicted value for 4cyl and 6cyl cars and they overlap therefore are not different?