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Linear regression is one of the key concepts in statistics [wikipedia1, wikipedia2]. However, people are often confuse the meaning of parameters of linear regression – the intercept tells us the average value of y at x=0, while the slope tells us how much change of y can we expect on average when we change x for one unit – exactly the same as in the linear function, though we use averages here due to noise.
Today colleague got confused with the meaning of adjusting covariate (x variable) and the effect of parameter estimates. By shifting the x scale, we also shift the point at which intercept is estimated. I made the following graph to demonstrate this point in the case of nested regression of y on x within a group factor having two levels. R code to produce this plots is shown on bottom.
```library(package="MASS")

x1 <- mvrnorm(n=100, mu=50, Sigma=20, empirical=TRUE)
x2 <- mvrnorm(n=100, mu=70, Sigma=20, empirical=TRUE)

mu1 <- mu2 <- 4
b1 <- 0.300
b2 <- 0.250

y1 <- mu1 + b1 * x1 + rnorm(n=100, sd=1)
y2 <- mu2 + b2 * x2 + rnorm(n=100, sd=1)

x <- c(x1, x2)
xK <- x - 60
y <- c(y1, y2)
g <- factor(rep(c(1, 2), each=100))

par(mfrow=c(2, 1), pty="m", bty="l")

(fit1n <- lm(y ~ g + x + x:g))
## (Intercept)           g2            x         g2:x
##     3.06785      2.32448      0.31967     -0.09077
beta <- coef(fit1n)

plot(y ~ x, col=c("blue", "red")[g], ylim=c(0, max(y)), xlim=c(0, max(x)), pch=19, cex=0.25)
points(x=mean(x1), y=mean(y1), pch=19)
points(x=mean(x2), y=mean(y2), pch=19)
abline(v=c(mean(x1), mean(x2)), lty=2, col="gray")
abline(h=c(mean(y1), mean(y2)), lty=2, col="gray")

points(x=0, y=beta["(Intercept)"],              pch=19, col="blue")
points(x=0, y=beta["(Intercept)"] + beta["g2"], pch=19, col="red")

z <- 0:max(x)
lines(y= beta["(Intercept)"]               +  beta["x"] * z                , x=z, col="blue")
lines(y=(beta["(Intercept)"] + beta["g2"]) + (beta["x"] + beta["g2:x"]) * z, x=z, col="red")

(fit2n <- lm(y ~ g + xK + xK:g))
## (Intercept)           g2           xK        g2:xK
##    22.24824     -3.12153      0.31967     -0.09077
beta <- coef(fit2n)

plot(y ~ x, col=c("blue", "red")[g], ylim=c(0, max(y)), xlim=c(0, max(x)), pch=19, cex=0.25)
points(x=mean(x1), y=mean(y1), pch=19)
points(x=mean(x2), y=mean(y2), pch=19)
abline(v=c(mean(x1), mean(x2)), lty=2, col="gray")
abline(h=c(mean(y1), mean(y2)), lty=2, col="gray")

abline(v=60, lty=2, col="gray")

points(x=60, y=beta["(Intercept)"],              pch=19, col="blue")
points(x=60, y=beta["(Intercept)"] + beta["g2"], pch=19, col="red")

z <- 0:max(x) - 60
lines(y= beta["(Intercept)"]               +  beta["xK"] * z                 , x=z + 60, col="blue")
lines(y=(beta["(Intercept)"] + beta["g2"]) + (beta["xK"] + beta["g2:xK"]) * z, x=z + 60, col="red")```