---
title: "STAT 516 Lec 04 in class"
format: html
---



SC apartment data:
```{r}
link <- url("https://gregorkb.github.io/data/scapts.csv")
scapts <- read.csv(link)
head(scapts)
```



Overall F test:

```{r}
lm_out <- lm(log(price) ~ city + nbed + nbath + log(sqft) + pets + log(sqft):city, data = scapts)
summary(lm_out)

```

How to get the critical value for the F test:

```{r}
alpha <- 0.05
p <- 10
n <- nrow(scapts)
Fcrit <- qf(1-alpha, p,n-(p+1)) # quantile function for F distributions
pval <- 1 - pf(15.31, p, n-(p+1))
```



Do a full-reduced model F test to see if number of bedrooms/bathrooms makes any difference.

```{r}

lm_full <- lm(log(price) ~ city + nbed + nbath + log(sqft) + pets + log(sqft):city, data = scapts)
lm_red <- lm(log(price) ~ city + log(sqft) + pets + log(sqft):city, data = scapts)

SSE_full <- sum(lm_full$residuals^2)
SSE_red <- sum(lm_red$residuals^2)

Ftest <- ( (SSE_red - SSE_full) / 2 ) / (SSE_full / (n - (p + 1)))
Ftest
Fcrit <- qf(1-alpha,2,n - (p+1))
Fcrit
pval <- 1 - pf(Ftest,2, n - (p+1))
pval
```


Is "City" important?

```{r}
lm_red <- lm(log(price) ~ nbed + nbath + log(sqft) + pets, data = scapts)
# same steps, here we removed 6 betas from the full model. 
```


Suppose I want the VIF for number of bedrooms in the full model.  Fit a model with number of bedrooms as the response:

```{r}
lm_nbed <- lm(nbed ~ city + nbath + log(sqft) + pets + log(sqft):city, data = scapts)
summary(lm_nbed)
vif <- 1/(1 - 0.7482)
```



Do an experiment:  Add a spurious predictor to the model, which is highly related to another variable.

```{r}
scapts$x <- scapts$sqft + rnorm(n,0,50)
plot(scapts)
```

Now fit a model with the new variable:

```{r}
lmx_out <- lm(log(price) ~ city + nbed + nbath + log(sqft) + pets + log(sqft):city + log(x), data = scapts)

summary(lmx_out)

```

Get VIFs for all variables easily:
```{r}
library(car) # first time you must run install.packages("car")

vif(lm_out)
vif(lmx_out)
```


Read in the SC apartments data again.
```{r}
link <- url("https://gregorkb.github.io/data/scapts.csv")
scapts <- read.csv(link)
head(scapts)

```

Choose a model using backward selection with AIC:

```{r}
lm_all <- lm(log(price) ~ city + nbed + nbath + log(sqft) + pets + log(sqft):city, data = scapts)
step_back <- step(lm_all,direction = "backward", scope = formula(lm_all))
summary(step_back)
```

For forward selection, fit a model with no predictors first.
```{r}
lm_none <- lm(log(price)~1, data=scapts)
step_forw <- step(lm_none, direction = "forward", scope=formula(lm_all))
summary(step_forw)
```