What is the "effects" returned by `aov` and `lm`?

Question

I would like to ask the difference between $coefficients and $effects in aov output.

Here the f1 factor and the interaction f1 * f2 are significant. I want to interpret the effect of that factor on the response and I thought that the $effects is what I needed.

Let’s consider the following simple following data set.

f1 <- c(1,1,0,0,1,1,0,0)
f2 <- c(1,0,0,1,1,0,0,1)
r <- c(80, 50, 30, 10, 87,53,29,8)
av <- aov(r ~ f1 * f2)
summary(av)
av$coefficients
av$effects
plot(f1, r)

It seems that the response is being increased by 48.25 units because of f1 mean(r[f1==1]) - mean(r[f1==0]).

But I can’t really see that on $effects output. What does the $effects output really tell me?

Answer 1

Effects are rotated response values according to the QR factorization for design matrix. Check:

all.equal(qr.qty(av$qr, r), unname(av$effects))
# [1] TRUE

Effects are useful for finding regression coefficients from QR factorization:

all.equal(backsolve(av$qr$qr, av$effects), unname(coef(av)))
# [1] TRUE

They can also be used to find fitted values and residuals:

e1 <- e2 <- av$effects
e1[(av$rank+1):length(e1)] <- 0
e2[1:av$rank] <- 0

all.equal(unname(qr.qy(av$qr, e1)), unname(fitted(av)))
# [1] TRUE

all.equal(unname(qr.qy(av$qr, e2)), unname(residuals(av)))
# [1] TRUE

So in summary, effects are representation of data at rotated domain, and is all least square regression is about.