## Define the directory where the data set is
setwd("C:/UConn/242_3115/R/")

## Load the data set and prepare for use
data <- read.table("Multiple_regress.txt")
dimnames(data)[[2]] <- c("X1","X2","X3","X4","X5","X6","X7","Y")
data <- as.data.frame(data)

## Fit a multiple linear regression with the data
fit <- lm(Y~X2+X3+X4+X5+X6+X7, data=data)
X <- model.matrix(fit)
beta <- coefficients(fit)
e <- residuals(fit)
n <- nrow(X)
p <- ncol(X)

##F test to add an extra variable (e.g. X7) given all others are in the model
## Fit a multiple linear regression with the data
fit <- lm(Y~X2+X3+X4+X5+X6+X7, data=data)

## Fit a multiple linear regression with the data
fit2 <- lm(Y~X2+X3+X4+X5+X6, data=data)

##anova() function perform the F-test for the addition of 1 extra variable
anova(fit2,fit)

##Do the F-test by hand
q <- 1
C <- c(rep(0,6),1)
Q <- t((t(C) %*% beta)) %*% solve(t(C) %*% solve(t(X)%*%X) %*% C) %*% (t(C) %*% beta)
SSE <- t(e) %*% e

F <- (Q/q) / (SSE/(n-p))
1-pf(F,q,n-p)
2*(1-pt(sqrt(F),n-p))
anova(fit2,fit)

##F test to add a group variables (e.g. X2,X5,X7) given all others are in the model
## Fit a multiple linear regression with the data
fit <- lm(Y~X2+X3+X4+X5+X6+X7, data=data)

## Fit a multiple linear regression with the data
fit2 <- lm(Y~X3+X4+X6, data=data)

##anova() function perform the F-test for the addition of 1 extra variable
anova(fit2,fit)

##Do the F-test by hand
q <- 3
C <- matrix(0,nrow=p,ncol=q)
C[2,1] = C[5,2] = C[7,3] <- 1
d <- c(0,0,0)
Q <- t((t(C) %*% beta - d)) %*% solve(t(C) %*% solve(t(X)%*%X) %*% C) %*% (t(C) %*% beta -d)
SSE <- t(e) %*% e

F <- (Q/q) / (SSE/(n-p))
1-pf(F,q,n-p)

anova(fit2,fit)