---
title: "ViF"
author: "Gustavo Cepparo and Milica Cudina"
output:
  pdf_document: default
---

## Car efficiency [revisited]

First, let's import the textbook's library. 

```{r}
library(ISLR2)
attach(Auto)
#View(Auto)
```

Let's do some exploratory analysis. 

```{r}
pairs(Auto,
      pch=20, col="lightblue")
round(cor(Auto[,-c(8,9)]),4)
```

We notice some pretty sizable correlations. What about a multiple linear regression?

```{r}
lm.fit=lm(mpg~cylinders+horsepower+weight,data=Auto)
summary(lm.fit)
```
We should import the `car` library (nothing to do with vehicles; it's short for *Companion to Applied Regression.*). 

```{r}
library(car)
vif(lm.fit)
```
Our textbook says: *"...a VIF value that exceeds 5 or 10 indicates a problematic amount of collinearity..."


## Seat Position 

Let's consider another data set. This one is from the `faraway` library. 

```{r}
#install.packages("faraway")
library(faraway)
```

The data set `seatpos` is used to predict the carseat position (`hipcenter`) based on biometric data of the driver. 

```{r}
data(seatpos)
```

When we look at the documentation, we see that one of the variables is `HtShoes`, i.e., height in shoes, and another is `Ht`, i.e., height barefoot. These are bound to be incredibly correlated. Similarly, there is the `Seated`, i.e., the seated height, `Weight`, and others that should be heavily positively correlated. Let's do some exploratory data analysis:

```{r}
pairs(seatpos,
      pch=20, col="lightblue")
round(cor(seatpos),4)
```

`Age` appears to be the only predictor not linked with other predictors. 

Let's try a multiple linear regression.

```{r}
lm.fit=lm(hipcenter~.,data=seatpos)
summary(lm.fit)
```

As anticipated, we get much nonsense. So, let's check out the variance inflation factors.

```{r}
vif(lm.fit)
```
It seems that we really should take a pick between height in shoes and height without. 

```{r}
lm.fit.1=lm(hipcenter~. - Ht,data=seatpos)
summary(lm.fit.1)
vif(lm.fit.1)
```


What if we get rid of height in shoes as well?

```{r}
lm.fit.2=lm(hipcenter~. - Ht-HtShoes,data=seatpos)
summary(lm.fit.2)
vif(lm.fit.2)
```

We have not sacrificed anything in terms of $R^2$ by eliminating the two height variables. Just the `Leg` is still statistically significant with `Age` in the surprising second place. 

Just for laughs, how about a simple linear regression on `Leg`?

```{r}
lm.fit.s<-lm(hipcenter ~ Leg, data=seatpos)
summary(lm.fit.s)
```
So, more predictors are not always better?