---
title: "Lab 1"
author: "Your Name Here"
date: "Week 1"
output:
  html_document:
    toc: true
    toc_depth: 4
---

### 1. Changing the author field and file name.

##### (a) Change the `author:` field on the Rmd document from Your Name Here to your own name.

##### (b) Rename this file to "lab01_YourHameHere.Rmd", where YourNameHere is changed to your own name.

### 2. Installing and loading packages

Just like every other programming language you may be familiar with, R's capabilities can be greatly extended by installing additional "packages" and "libraries".

To **install** a package, use the `install.packages()` command.  You'll want to run the following commands to get the necessary packages for today's lab:

```
install.packages("ggplot2")
install.packages("MASS")
install.packages("ISLR")
install.packages("knitr")
```

You only need to install packages once.  Once they're installed, you may use them by **loading** the libraries using the `library()` command.  For today's lab, you'll want to run the following code

```{r}
library(ggplot2) # graphics library
library(MASS)    # contains data sets, including Boston
library(ISLR)    # contains code and data from the textbook
library(knitr)   # contains kable() function

options(scipen = 4)  # Suppresses scientific notation
```

### 3. Simple Linear Regression with the Boston Housing data.

> This portion of the lab gets you to carry out the Lab in &sect;3.6 of ISLR (Pages 109 - 118).  You will want to have the textbook Lab open in front you as you go through these exercises.  The ISLR Lab provides much more context and explanation for what you're doing.  

> Please run all of the code indicated in &sect;3.6 of ISLR, even if I don't explicitly ask you to do so in this document.

**Note**: You may want to use the `View(Boston)` command instead of `fix(Boston)`.

##### (a) Use the `dim()` command to figure out the number of rows and columns in the Boston housing data

```{r}
# Edit me
```

##### (b) Use the `nrow()` and `ncol()` commands to figure out the number of rows and columns in the Boston housing data.  

```{r}
# Edit me
```

##### (c) Use the `names()` command to see which variables exist in the data.  Which of these variables is our response variable?  What does this response variable refer to?  How many input variables do we have?

```{r}
# Edit me
```

##### (d) Use the `lm()` function to a fit linear regression of `medv` on `lstat`.  Save the output of your linear regression in a variable called `lm.fit`.

```{r}
# Edit me
```

##### (e) Use the `summary()` command on your `lm.fit` object to get a print-out of your regression results

```{r}
# Edit me
```

##### (f) Uncomment the line below to get a 'nice' printout of the coefficients table

```{r}
# kable(coef(summary(lm.fit)), digits = c(4, 5, 2, 4))
```

##### (g)  Call `names()` on `lm.fit` to explore what values this linear model object contains.  

```{r}
# Edit me
```

##### (h) Use the `coef()` function to get the estimated coefficients.  What is the estimated Intercept?  What is the coefficient of `lstat` in the model?  Interpret this coefficient.

```{r}
# Edit me
```


##### (i) Here's a ggplot command that overlays a linear regression line on a scatterplot of `mdev` vs. `lstat`.  Edit the `xlab` and `ylab` arguments to produce more meaningful axis labels.  Does the linear model appear to fit the data well?  Explain.

```{r}
qplot(data = Boston, x = lstat, y = medv,
      xlab = "lstat - change this!", ylab = "medv - change this!") + stat_smooth(method = "lm")
```

##### (i) Follow the ISLR examples for getting confidence intervals and prediction intervals for the regression data.

```{r}
# Fill in later
```

### 4. Multiple Linear Regression with the Boston Housing data

##### (a) Use the command `?Boston` to figure out what the `age` variable means.  What does `age` mean in the Boston Housing data?

**Your answer here**

##### (b) Following the example in part 3(i) of this lab, use the `qplot()` command to construct a scatterplot of `medv` veruses `age`.  Make sure to specify meaningful x and y axis names.  Overlay a linear regression line.  Does a linear relationship appear to hold between the two variables?

```{r}
# Edit me
```

##### (c) Use the `lm()` command to a fit a linear regression of `medv` on `lstat` and `age`.  Save your regression model in a variable called `lm.fit`.

```{r}
# Edit me
```

##### (d)  What is the coefficient of `age` in your model?  Interpret this coefficient.

```{r}
# Edit me
```

##### (e) Use `medv ~ .` syntax to fit a model regressing `medv` on all the other variables.  Use the `summary()` and `kable()` functions to produce a coefficients table in *nice* formatting.

```{r}
# Edit me
```

##### (f) Think about what the variables in the data set mean.  Do the signs of all of the coefficient estimates make sense?  Are there any that do not?  For the ones that do not, are the coefficients statistically significant (do they have p-value < 0.05)?

```{r}
# Edit me
```

### 5. Non-linear transformations of the predictors

##### (a) Perform a regression of `medv` onto a quadratic polynomial of `lstat` by using the formula `medv ~ lstat + I(lstat^2)`.  Use the `summary()` function to display the estimated coefficients.  Is the coefficient of the squared term statistically significant?

```{r}
# Edit me
```

##### (b) Try using the formula `medv ~ lstat + lstat^2` instead.  What happens?

```{r}
# Edit me
```

##### (c) Use the formula `medv ~ poly(lstat, 2)`.  Compare your results to part (a).  

```{r}
# Edit me
```

### 6. ggplot visualizations

> ggplot's `stat_smooth` command allows us to visualize simple regression models  in a really easy way.  This set of problems helps you get accustomed to specifying polynomial and step function formulas for the purpose of visualization.

> For this problem, please refer to the code posted here:  [Week 1 R code](http://www.andrew.cmu.edu/user/achoulde/95791/lectures/code/week1.html#polynomial-regression-and-step-functions)

##### (a) Use `ggplot` graphics to construct a scatterplot of `medv` vs `lstat`, overlaying a 2nd degree polynomial.  Does this appear to be a good model of the data?  Construct plots with higher degree polynomial fits.  Do any of them appear to describe the data particularly well?

```{r}
# Edit me
```

##### (b) Repeat part (a), but this time using step functions instead of polynomials. Try picking cuts to best match the trends in the data. Which functional form appears to do a better job of describing the data: polynomials, or step functions?  Explain.  

```{r}
# Edit me
```

##### (c) Repeat part (a), this time using `ptratio` as the x-axis variable, and `medv` still as the y-axis variable.

```{r}
# Edit me
```

##### (d) Repeat part (b), this time with `ptratio` instead of `lstat`.  

```{r}
# Edit me
```