lecture-12.Rmd

---
title: "Lecture 12"
---


### Lecture handout: 
chp8-handout.pdf

### Lecture slides (w/ answers): 
chp8.pdf

### Textbook: 
Chapter 8, Intro to Regression

### R Topics:

#### generating data for ANOVA
```{r}
x <- c(rep("a",25),rep("b",25),rep("c",25),rep("d",25))
y <- c(rnorm(25,5,10),rnorm(25,10,10),rnorm(25,15,10),rnorm(25,20,20))
yhat <- c(rep(5,25),rep(10,25),rep(15,25),rep(20,25))
res <- yhat - y
ssres <- sum(res^2)
anova(aov(y~x)) # why is sum squared residuals not the same in anova(aov()) as ssres?
ssgroup <- sum(25 * (mean(y[1:25])-mean(y))^2 + 25 * (mean(y[26:50])-mean(y))^2 + 25*(mean(y[51:75])-mean(y))^2 + 25*(mean(y[76:100])-mean(y))^2)
sstot <- sum((y-mean(y))^2)
```

#### residual plot
```{r}
eruption.lm <- lm(eruptions ~ waiting, data=faithful)
summary(eruption.lm)
yhat <- eruption.lm$coefficients["(Intercept)"] + faithful$waiting * eruption.lm$coefficients["waiting"] 
head(yhat)
head(eruption.lm$fitted.values)
head(model.matrix(eruption.lm))
eruption.res <- resid(eruption.lm)
plot(faithful$waiting, eruption.res, ylab="Residuals", xlab="Waiting Time", main="Old Faithful Eruptions")
abline(0, 0)
par(mfrow = c(2, 1))
par(mar=c(1,1,1,1))
plot(faithful$waiting, faithful$eruptions)
lines(faithful$waiting, predict(eruption.lm),col="red")
# or abline(lm(eruptions~waiting, data=faithful), col="red")
plot(faithful$waiting, eruption.res)
abline(0,0, col="red")
dev.off() # resets the graphics parameters, par()
```
#### R^2
```{r}
x <- faithful$waiting
xbar <- mean(x)
y <- faithful$eruptions
ybar <- mean(y)
R <- 1/(272-1) * sum( (x - xbar)/sd(x) * (y - ybar) / sd(y) ) # forumla from footnote p. 310
R^2
yhat <- eruption.lm$coefficients["(Intercept)"] + faithful$waiting * eruption.lm$coefficients["waiting"] 
res <- y - yhat
SSE<-sum(res^2)
SST<-sum((y-yhat)^2)
1 - SSE/SST # a way to calculate R^2
```


#### confidence interval for slope
```{r eval=FALSE}
dft <- length(faithful$waiting)-2 # degrees of freedom
z <- qt(.975, df) # note, this is close to qnorm(.975) b/c of high df
c(qt(.025, df), qt(.975, df)) # +- critical value
summary(eruption.lm) # slope is 0.075628, SE of slope is 0.002219
0.002219*c(qt(.025, df), qt(.975, df)) # margin of error
0.075628 + 0.002219*c(qt(.025, df), qt(.975, df))
# also can get the slope and standard error of slope like this
summary(eruption.lm)$coefficients[2,1]
summary(eruption.lm)$coefficients[2,2]
```

#### Additional info 

(this material will not be tested, just  FYI):

Deriving slope and intercept formulas: these formulas are  given  in  the  book but not derived.  The derivation involves some basic calculus.  Both videos cover more or less the same information but the first is a bit clearer while the second gets into linear algebra.
Youtube video 1: https://www.youtube.com/watch?v=jqoHefiIf9U
Linear Regression: https://www.youtube.com/watch?v=DSQ2plMtbLc