---
title:  "STAT340: Discussion 9: Multiple linear regression"
author: "Names"
date:   "`r format(Sys.time(), '%d %B, %Y')`" # autogenerate date as date of last knit
documentclass: article
classoption: letterpaper
output:
  html_document:
    highlight: tango
    fig_caption: false
---

```{r setup, include=FALSE}
# if sourced, set working directory to file location
# added tryCatch in case knitting runs into error
tryCatch({
  if(Sys.getenv('RSTUDIO')=='1'){
    setwd(dirname(rstudioapi::getActiveDocumentContext()$path))
  }}, error = function(e){}
)

# install necessary packages
if(!require(pacman)) install.packages("pacman")
pacman::p_load(knitr,tidyverse)

knitr::opts_chunk$set(tidy=FALSE,strip.white=FALSE,fig.align="center",comment=" #")
```

---

<style>
a:link {
  text-decoration: underline;
}
</style>

[Link to source file](ds09.Rmd)



## Just for fun (1 min)

<center><a href="https://xkcd.com/2400/"><img id="comic" src="https://imgs.xkcd.com/comics/statistics.png" title="We reject the null hypothesis based on the 'hot damn, check out this chart' test." style="width:222px;"></a></center>

<br/>



## Discuss together (5-10 min)

As an entire section, discuss these together:

1. What are the advantages of using multiple linear vs single linear regression, especially regarding interpreting the estimated coefficients?
2. What are the main assumptions you need to check for multiple linear regression that could suggest a poor fit, and how do these compare to simple linear regression?

<br/>



## Exercise

Today we're going to revisit the `mtcars` dataset and analyze it using multiple linear regression. Note this is a built-in dataset provided as part of the `datasets` package in `R`.

As usual, break off into groups of 3-4 students. In your group, nominate one person to share their screen.



### Background

Run `?(mtcars)` in the console (do NOT add it to this `Rmd` file) and briefly read the help page. Specifically, take note of the following:

1. What is the source of this data?
2. What is this dataset measuring? (i.e. what is the response variable?)
3. What predictors are available and what do they mean?

Feel free to also run `head(mtcars, 10)` or `View(mtcars)` to inspect the data frame briefly before moving on.



### Fitting

Uncomment the line below and finish it. Specifically, use `lm` to run a regression of `mpg` on all other predictors (an easy way to do this is to use `mpg ~ .` as the first argument). Make sure to also include `data = mtcars` as an argument or it won't know where to get the variable names from.

```{r}
# lm.mtcars = lm(...)
```

View a summary of the regression by uncommenting and running the line below

```{r}
# summary(lm.mtcars)
```


Briefly inspect the residuals plot by running `plot(lm.mtcars,which=1:2)` . What do you observe, and what does it mean?

> _REPLACE TEXT WITH RESPONSE_



### Interpretation

Uncomment the line below to get the estimated coefficients along with their standard errors.

```{r}
# summary(lm.mtcars)$coefficients[,1:2]
```

Give an interpretation of the estimate and standard error for one of these predictor variables. Be careful in your wording of the interpretation.

> _REPLACE TEXT WITH RESPONSE_

What does the intercept here mean? (Except for special situations, we generally don't care much about the intercept, but you should still understand what it means.)

> _REPLACE TEXT WITH RESPONSE_

Karl doesn't like the R² statistic, but what is the R² for this model? (Hint: look at the output of `summary`) Give an interpretation of this value.

> _REPLACE TEXT WITH RESPONSE_

Briefly read about the [adjusted R² here](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/adjusted-r2/). What is the adjusted R² of this model and how does this differ from the normal R² value? (Hint: again, look at the output of `summary`).

> _REPLACE TEXT WITH RESPONSE_

Generate $95\%$ confidence intervals for the coefficients using the `confint` function. Give an interpretation of these confidence intervals.

```{r}
# confint(...)
```

> _REPLACE TEXT WITH RESPONSE_



### Prediction

According to the model, what mileage would I expect on average with a car that has 6 cylinders, 200 displacement, 120 horsepower, 3.4 rear axle ratio, 2500 pounds, 17.5 1/4 mile time, a straight engine, automatic transmission, 4 forward gears, and 3 carburetors? (Be careful of your units and how you denote the engine and transmission variables. Again, the help page `?mtcars` may be very helpful here.)

<br/>



## Submission

As usual, make sure the names of everyone who worked on this with you is included in the header of this document. Then, knit this document and submit both this file and the HTML output on Canvas under Assignments ⇒ Discussion 9.