### NOTE

duplicated(c(1:45, 5))

### Matrices

A <- matrix(1:10, 5, 2)
B <- matrix(21:30, nrow = 5)
C <- matrix(21:40, nrow = 2, byrow = TRUE)

nrow(A)
ncol(B)
dim(C)

A + B
A - B
A * B
A == B
A < B

t(A)

all.equal(A, B)

A %*% t(A)

# (5 * 2) (2 * 5) --> (5 * 5)

# Model-implied covariance matrix in factor analysis

# 2 Factors, three indicators each

LAMBDA <- matrix(0, 6, 2)
LAMBDA[1:3, 1] <- 0.7
LAMBDA[4:6, 2] <- 0.7

PSI <- matrix(0.5, 2, 2)
diag(PSI) <- 1

THETA <- diag(rep(0.51, 6))

CX <- LAMBDA %*% PSI %*% t(LAMBDA) + THETA

solve(CX)

# Group Statistics

aggregate(Temp ~ Month, data = airquality, FUN = mean)
aggregate(Temp ~ Month, data = airquality, FUN = sd, na.rm = TRUE)
FirstHalf <- airquality$Day <= 15
airquality2 <- cbind(airquality, FirstHalf)
aggregate(Temp ~ FirstHalf + Month, data = airquality2, FUN = mean)
aggregate(cbind(Temp, Ozone, Wind) ~ Month, data = airquality, mean)

### Basic inferential statistics

result <- t.test(airquality$Temp, mu = 70)

names(result)

# Independent t-test

summer <- (airquality$Month < 9) & (airquality$Month > 5) 
t.test(airquality$Temp ~ summer)

t.test(airquality$Temp ~ summer)

t.test(airquality$Temp ~ summer, var.equal = TRUE)

# Paired sample t-test

t.test(attitude$learning, attitude$advance, paired = TRUE)


cor.test(attitude$learning, attitude$advance)

cor(attitude)
# just the lower diagonal
as.dist(cor(attitude))


cov(attitude)

chisq.test(c(90, 83))

# Correction of continuity

chisq.test(c(90, 83), p = c(0.6, 0.4))

n <- 90 + 83
c(0.6, 0.4) * n

chisq.test(table(airquality$Month))

# Array

black <- HairEyeColor[1, , ]
black
obj <- chisq.test(black)
obj[["expected"]]
fisher.test(black)



Ozone.NA <- is.na(airquality$Ozone)	
Ozone.NA

MonthOzoneNA <- table(airquality$Month, Ozone.NA)
MonthOzoneNA

chisq.test(MonthOzoneNA)

# ANOVA

library(psych)

head(sat.act)

## Graphical checks
qqnorm(sat.act$ACT)
qqline(sat.act$ACT)

library(QuantPsyc)
plotNormX(sat.act$ACT)
# Kernel density estimation


library(psych)
skew(sat.act$ACT)
kurtosi(sat.act$ACT)
describe(sat.act)

boxplot(ACT ~ gender, data = sat.act, col = c("red", "blue"))

bartlett.test(ACT ~ gender, data = sat.act)

library(Rcmdr)
leveneTest(sat.act$ACT, group = sat.act$gender)