#=========================================================================================
# Pr. 11.4 -- CVICENI 11 -- Test podilem sanci
#=========================================================================================

data <- data.frame(muzi = c(20, 30),
                   zeny = c(30, 20),
                   row.names = c('A', 'B'))

chisq.test(data)$expected # podminka dobre aproximace

a <- 20
b <- 30
c <- 30
d <- 20

(OR <- (a/c) / (b/d)) # podil sanci

log(OR) # logaritmus podilu sanci

(t0 <- log(OR) / sqrt(1/a + 1/b + 1/c + 1/d)) # testovaci statistika

alpha <- 0.05

qnorm(alpha/2)   # kriticky obor cast 1
qnorm(1 - alpha/2) # kriticky obor cast 2



(dh <- log(OR) - sqrt(1/a + 1/b + 1/c + 1/d) * qnorm(1 - alpha/2)) # IS dolni hranice
(hh <- log(OR) - sqrt(1/a + 1/b + 1/c + 1/d) * qnorm(alpha/2))   # IS horni hranice


#=========================================================================================
# Priklad 12.1
#=========================================================================================

X <- c(6, 7, 1,  8, 4, 2.5, 9, 12, 10, 2.5,  5, 11) # poradi odborniku
Y <- c(4, 5, 2, 10, 6, 1.0, 7, 11,  8, 3.0, 12,  9)     # poradi antropologu

cor(X,Y, method='spearman') # Spearmanuv korelacni koeficient

library(car)
# Test dvourozmerne normality -- graficky -- elipsa spolehlivosti
dataEllipse(X, Y, level = 0.95, xlim = c(-7, 20), ylim = c(-7, 20))

#1. exaktni test
n <- length(X)
(RS <- cor(X, Y, method = 'spearman')) # Spearmanuv korelacni koeficient
# kriticky obor; hledame v tabulkach

# 2. asymptoticky test
(t0 <- RS*sqrt(n-2)/sqrt(1-RS^2))  # testovaci statistika


alpha <- 0.05
qt(1 - alpha/2, n - 2) # kriticky obor cast 1
qt(alpha/2, n - 2)   # kriticky obor cast 2

2 * min(pt(t0, n - 2), 1 - pt(t0, n - 2)) # p-hdonota


#=========================================================================================
# Priklad 12.2
#=========================================================================================

X <- c(40, 64, 34, 15, 57, 45) # mnozstvi kys. mlecne v krvi matky
Y <- c(33, 46, 23, 12, 56, 40) # mnozstvi kys. mlecne v krvi ditete

cor(X, Y, method = 'pearson' ) # Pearsonuv korelacni koeficient

# Test dvourozmerne normality -- graficky -- elipsa spolehlivosti
dataEllipse(X, Y, xlim = c(-20, 100), ylim = c(-20, 100),
            level = 0.95)

#-------------------------------------------------------------------------------------

# 1. Exaktni test
n   <- length(X)
R12 <- cor(X, Y, method = 'pearson' ) # Pearsonuv korelacni koeficient

(t0 <- R12 / sqrt(1 - R12^2) * sqrt(n - 2)) # testovaci statistika

alpha <- 0.05
qt(1 - alpha / 2, n - 2) # kriticky obor cast 1
qt(alpha / 2, n - 2)   # kriticky obor cast 2

(dh <- qt(alpha/2, n - 2) / sqrt(qt(alpha/2, n - 2) ^ 2 + n - 2)) # IS - dolni hranice
(hh <- qt(1 - alpha/2, n - 2) / sqrt(qt(1 - alpha/2, n - 2) ^ 2 + n - 2)) # IS - horni hranice


2 * min(pt(t0, n - 2), 1 - pt(t0, n - 2)) # p-hodnota

#2. Asymptoticky interval spolehlivosti
Z <- 1 / 2 * log((1 + R12) / (1 - R12)) # Z-transformace koeficientu R12

DH <- Z - qnorm(1 - alpha/2) / sqrt(n - 3) # dolni hranice IS pro Z-transformaci
HH <- Z + qnorm(1 - alpha/2) / sqrt(n - 3) # horni hranice IS pro Z-transformaci

tanh(DH) # hyperbolicky tangens dolni hranice -> zpetna transformace IS do skaly korelacniho koeficientu R12
tanh(HH) # hyperbolicky tangens horni hranice -> zpetna transformace IS do skaly korelacniho koeficientu R12