# Abbildung
pfad = "E:/b/ss07/statistik/regression"
d = read.table(paste(pfad, "epg.txt", sep="/"), header=T)
attach(d)
names(d)


# Abbildung COG x F2, COG x F1, EPGSUM1278 x F1
par(mfrow=c(1,3))
plot(F2, COG)
plot(F1, COG)
temp = LAB %in% c("i", "I", "E", "a")
plot(F1[temp], SUM1278[temp])

# Kovarianz
y = F2; x = COG; n = length(y)
mx = mean(x)
my = mean(y)
dx = x - mean(x)
dy = y - mean(y)
covxy = sum(dx * dy)/(n-1)
cov(x,y)

# Korrelation
xgross = x * 1000
cov(x,y); cov(xgross,y)
r = cov(x,y)/(sd(x) * sd(y))
cor(x,y); cor(xgross,y)

# Regression
b = r * sy/sx
b = cov(x,y)/var(x)
k = my - b*mx
yhut = b*x + k
plot(x,y)
abline(k, b)

# Residuals
error =  y - yhut
SSE = sum(error^2)

# die lm() Funktion
reg = lm(y ~ x)
abline(reg)
yhut = predict(reg)
residuals(reg)

# SSY, SSR, SSE
SSY = sum( (y - my)^2)
SSR = sum((yhut - my)^2)

# R-squared
SSR/SSY
cor(x, y)^2

# Signifikanz-test
rsb = 
tstat = r/rsb
2 * (1 - pt(tstat, n-2))

fstat = tstat^2
1 - pf(fstat, 1, n-2)

# summary(), anova()
summary(reg)

MSR = SSR/1
MSE = SSE/(n-2)

anova(reg)