#==========================================================
#                 P R I K L A D    11.1
#==========================================================

X <- c(6, 7, 1, 8, 4, 2.5, 9, 12, 10, 2.5, 5, 11) # 78
Y <- c(4, 5, 2, 10, 6, 1, 7, 11, 8, 3, 12, 9) # 78

library(car)
dataEllipse(X, Y, level = 0.95, xlim = c(-10, 20),
            ylim = c(-10, 20), col = 'black',
            pch = 21, bg = 'red', lwd = 1, las = 1) # teckovy diagram s 95% elipsou spolehlivosti
Ri <- rank(X) # poradi hodnoceni programatoru
Qi <- rank(Y) # porade hodnoceni antropologu
n <- 12 
(Rs <- 1 - 6 / (n * (n^2 - 1)) * sum((Ri-Qi)^2)) # Spearmanuv koeficient poradove korelace - vypocet vzorcem
cor(X, Y, method = 'spearman') # Spearmanuv koeficient poradove korelace - vypocet funkci

#--------------------------------------------
# H0: rho = 0
# H1: rho != 0
#--------------------------------------------

# Exaktni test (KO)
t0 <- abs(Rs) # testovaci statistika
# W = < 0.584 ; 1>

# Asymptoticky test (KO)
(T0 <- Rs * sqrt(n - 2) / sqrt(1 - Rs^2)) # testovaci statistika
alpha <- 0.05
qt(alpha / 2, n-2) 
qt(1 - alpha/2, n-2) 
# W = (-infty ; -2.22> U < 2.22 ; infty)

# (p-hohnota)
2 *min (pt(T0, n - 2), 1 - pt(T0, n - 2)) # p-hodnota

#==========================================================
#                 P R I K L A D    11.2
#==========================================================
X <- c(40, 64, 34, 15, 57, 45)
Y <- c(33, 46, 23, 12, 56, 40)

dataEllipse(X, Y, level = 0.95, col = 'black',
            pch = 21, bg = 'red', las= 1,
            xlim = c(-20, 100), ylim = c(-20, 100),
            xlab = 'mnozstvi kys. mlecne v tele matky',
            ylab = 'mnozstvi kys. mlecne v tele novorozence') # teckovy diagram s 95% elipsou spolehlivosti

#--------------------------------------------
# H0: rho = 0
# H1: rho != 0
#--------------------------------------------

# Exaktni test (KO)
(R12 <- cor(X, Y, method = 'pearson')) # Pearsonuv koeficient poradove korelace - vypocet funkci
n <- 6
(T0 <- R12 / sqrt(1 - R12^2)*sqrt(n-2)) # testovaci statistika
alpha <- 0.05
qt(alpha / 2, n-2)
qt(1 - alpha / 2, n-2)
# W = (-infty ; -2.77> U < 2.77 ; infty)

# (IS)
(dh <- qt(alpha / 2, n - 2) / sqrt(qt(alpha / 2, n - 2)^2 + n - 2))
(hh <- qt(1 - alpha / 2, n - 2) / sqrt(qt(1 - alpha / 2, n - 2)^2 + n - 2))
### !!! VYJIMKA: R12 nenalezi IS -> H0 zamitame (obvykle kontrolujeme, zda do IS patri c z H0, zde ale vyjimka)


# (p-hodnota)
2 * min (pt(T0, n - 2), 1 - pt(T0, n - 2))

# Asymptoticky test (IS)
R12 # Pearsonuv koeficient poradove korelace - vypocet funkci
Z <- 1 / 2 * log((1 + R12) / (1 - R12)) # Z-transformace
(dh.Z <- Z - qnorm(1-alpha / 2) / sqrt(n-3)) # dolni hranice IS pro Z-transformaci
(hh.Z <- Z + qnorm(1-alpha / 2) / sqrt(n-3)) # horni hranice IS pro Z-transformaci

(dh <- tanh(dh.Z)) # finalni dolni hranice
(hh <- tanh(hh.Z)) # finalni horni hranice
#c = 0 nenalezi do IS -> H0 zamitame

#==========================================================
#                 P R I K L A D    11.3
#==========================================================
n1 <- 426 # pocet kluku
n2 <- 430 # pocet holek
R1 <- 0.6033 # korelacni koeficient pro kluky
R2 <- 0.5833 # korelacni koeficient pro holky
Z1 <- 1 / 2 * log((1 + R1) / (1-R1)) # Z-transformace pro kluky
Z2 <- 1 / 2 * log((1 + R2) / (1 - R2)) # Z-transformace pro holky

#--------------------------------------------
# H0: rho1 - rho2 = 0
# H1: rho1 - rho2 != 0
#--------------------------------------------

# (KO)
(U <- (Z1 - Z2) / sqrt (1 / (n1 - 3) + 1 / (n2 - 3))) # testovaci statistika
alpha <- 0.05
qnorm(alpha / 2)
qnorm(1 - alpha / 2)
# W = (-infty ; -1.96> U <1.96 ; infty)

# p-hodnota
2 * min(pnorm(U), 1 - pnorm(U))