### 1.3.4 Don't do several simple regressions instead of multiple regression
### ------------------------------------------------------------------------

### Artificial Example

## two predictor variables $x_1, x_2$, response variable $Y$

x1 <- c( 0, 1, 2, 3, 0, 1, 2, 3)
x2 <- c(-1, 0, 1, 2, 1, 2, 3, 4)
Y  <- c( 1, 2, 3, 4,-1, 0, 1, 2)

(dat <- data.frame(x1, x2, Y))

pairs(dat) ## or, a bit nicer (for slide):
pairs(dat, pch = 20, cex = 1.5,
      main="Example: multiple or single regressions")

pdf.latex("/u/maechler/Vorl/comput-statist/Figs/ex-multi_l.pdf",
         width=7.6, height=8)
pairs(dat, pch = 20, cex = 1.5,
      main="Example: multiple or single regressions")
pdf.end()

### 3D- visualization:
library(rgl)
plot3d     (x1, x2, Y)
rgl.spheres(x1, x2, Y, radius = 0.4, col = "tomato")

rgl.clear()
rgl.close()


lm(Y ~ x1)     ## (0,     1)
lm(Y ~ x2)     ## (1.333,      0.111)
lm(Y ~ x1 + x2)## (0,     2,   -1   )

## multiple regression describes the data points exactly
## Y[i] = 2 x1[i]  - x2[i]   for all i,

summary(lm(Y ~ x1 + x2))
## hat(sigma)^2 = 0,  since all residuals "= 0"

## Y decreases when x2 increases !

## if we do a simple regression of $Y$ onto $x_2$ (while
## ignoring the values of $x_1$; and thus, we do not keep them constant), we
## obtain
## \begin{eqnarray*}
## \hat{Y}_i = \frac{1}{9} x_{i2} + \frac{4}{3}\ \mbox{for all i} \qquad
## (\hat{\sigma}^2 = 1.72).
## \end{eqnarray*}
## This least squares regression line describes how $Y$ changes when varying
## $x_2$ when ignoring $x_1$\\
## in particular: $\hat{Y}$ increases when $x_2$ increases, in contrast to
## multiple regression!