Notes

---
title: 'Homework 5 - STAT 462: Applied Regression Analysis'
author: "Joshua Williams"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

``` {r warning=FALSE, message=FALSE}
library(lattice)
library(faraway)
```


```{r}
?read.table
data <-read.table("bears.txt",header=TRUE)
dim(data)
data[data$ID==41,]
bears = data[data$Obs.No==1,]
bears
dim(bears)
###
y = bears$Weight
x1 = bears$Neck.G

```


```{r}
lm1 = lm(y~x1)
lm1
summary(lm1)

#plot(lm1)
res1 = resid(lm1)
#res1
res1

X.design = cbind(1,x1)
X.design

```

The Regression slope is 20.59





```{r}
sigma2 = sum(res1^2)/(99-2)
sigma2

se.beta1 = sqrt(sigma2*solve(t(X.design) %*% X.design)[2,2])
se.beta1
```


The Variance is 1369.55


The Standard Error is 0.7148337






The Distribution of the test statistic is N(0,1)














The Test Statistic is equal to 28.80384
```{r}


betahat = solve(t(X.design)%*%X.design) %*% t(X.design) %*% y
betahat

beta1hat = betahat[2]
beta1hat

tvalue = beta1hat/se.beta1
tvalue
```







The P-value is equal to 0
```{r}
pvalue = 2*(1-pt(abs(tvalue),df=97))
pvalue


#pvalue = 2*(1-pt(abs(tvalue),df=97))
#pvalue
```

We reject the null hypothesis










```{r}

x1 = bears$Neck.G
x2 = bears$Head.W
y = bears$Weight
X = cbind(1, x1, x2)
lm_fit <- lm(y~x1+x2)
n = dim(X)[1]
p = dim(X)[2]

betahat2 = solve(t(X)%*%X, t(X)%*%y)
yhat = X%*%betahat2
eps_hat = y - yhat
sigma2 = sum(eps_hat^2)/(n-p)
sigma2
cv_mat = sigma2*solve(t(X)%*%X)
round(cv_mat,digits=5)

se = sqrt(diag(cv_mat))
mytable<-cbind(betahat2,se)
colnames(mytable)<-c("Est","se")
mytable

tvalue = betahat2/se
tvalue
pval = 2*pt(abs(tvalue), df=n-p, lower.tail = FALSE)

mytable<-cbind(mytable,tvalue,pval)
colnames(mytable)<-c("Est","se","t-stat","p-val")
mytable


#
# lm2 = lm(y~x1+x2)
# res2 = resid(lm2)
# RSS2 = sum(res2^2)
# RSS2
# TSS= sum((y-mean(y))^2)
# TSS
# SS_reg = TSS - RSS2
# SS_reg
# summary(lm2)





#summary(lm1)

#summary(lm1)$sigma^2
```



The RSS is equal to 132756.1
The TSS is equal to 1269109
The SS_reg is equal to 1136353









The Distribution of the test statistic is N(0,1)



```{r}


# res2 = resid(lm2)
#
# X.design = cbind(1,x2,x1)
# #X.design
#
#
# sigma2 = sum(res2^2)/(99-2)
# sigma2
#
# se.beta1 = sqrt(sigma2*solve(t(X.design) %*% X.design)[2,2])
# #se.beta1
#
#
#
# #betahatt = solve(t(X.design)%*%X.design) %*%t(X.design) %*% y
# #betahat
#
# beta1hat = betahatt[2]
# #beta1hat
#
# tvalue = beta1hat/se.beta1
# tvalue
```






```