```{r, message=FALSE, warning=FALSE, echo=FALSE}
require(knitr)
opts_chunk$set(eval=FALSE)
```

In this lab, we will learn how to construct and interpret confidence intervals and prediction intervals. We'll also pick up some `dplyr` along the way. 

```{r, message=FALSE, warning=FALSE}
library(tidyverse)
library(skimr)
```

#### Fuel Economy for Cars

In this example, we consider the fuel economy, as measured in miles per gallon in highway driving (`hwy`) in terms of the size of the car's engine, measured in litres (`displ`). 

This data comes from the `fueleconomy` package. You may have to install this. The data is called `vehicles`. 

```{r}
# install.packages("fueleconomy")
library(fueleconomy)
data(vehicles)
```

Note that by running the `str` command on the `vehicles` data frame, we discover that this data set is quite large (more than 33,000 rows). 

```{r}
str(vehicles)
```

`str` lets us see some of the structure of the data set, although it doesn't give us much of a summary. If we use the `skim` command, we can learn more about the data-- that it contains data on cars from 1984-2015, for example!

```{r}
skim(vehicles)
```

#### Filtering data

In this example we will consider only vehicles in the 2000 model year. To do this, we will make a smaller data set consisting of just these cars. As with almost everything in `R`, there are many ways to make a subset of data. However, we will be focusing on the `filter` command, which filters the data to ensure it meets a certain criteria. 

```{r}
myCars = vehicles %>% 
  filter(year == 2000)
nrow(myCars)
```

Notice that we used the double equals sign to mean "check if it is equal." This is a "logical expression" (meaning it returns a binary, TRUE/FALSE). Other logical expressions are `!=` for not equal, `>` for greater than, `>=` for greater than or equal to, etc. 

Also note that we are creating a new dataset called `myCars`, rather than overwriting the `vehicles` data. This is good practice.

#### Simple Linear Model

We are interested in the relationship between fuel economy and engine size. A scatterplot will help us visualize this relationship. 

```{r}
ggplot(data=myCars, aes(x=displ, y=hwy)) + geom_point()
```

* What do you see in this plot? Is the relationship linear? Are there any outliers? How can this information help you to CHOOSE the form of the model to fit? 


We proceed by fitting a simple linear regression model for `hwy` as a function of `displ`. 

```{r}
ggplot(data=myCars, aes(x=displ, y=hwy)) + geom_point() + geom_smooth(method=lm, se=FALSE)
m1 = lm(hwy ~ displ, data=myCars)
```

The `summary` command applied to the regression model object (`m1`) will give us lots of information about the model that we fit.

```{r}
summary(m1)
anova(m1)
```

#. Write down the equation for the regression line. 
#. Write down an interpretation of the coefficient for `displ` in this model. Remember to include units!
#. Does engine size appear to have a statistically significant relationship with fuel economy? Why or why not? 
#. How would you characterize the explanatory power of this model? 
#. What is the total sum of squares?

Now let's check the conditions for inference. 

#. Are the conditions for inference reasonably met? 

```{r}
plot(m1)
```

#. What kind of transformation could we make in order to improve this? We'll address this later in the lab.

Some of the observations have very large residuals (standardized residuals $> 3$). What is going on here? We can `slice()` to see what that row of the data looks like. 

```{r}
myCars %>% 
  slice(395) 
# myCars[395,] # This will also work
```

Notice that I can use `dplyr` commands without an assignment operator. If I just want to look at the data, and not re-use the results, I can just print the data to the Console (or, in the output section of my RMarkdown document) without saving it as something. `dplyr` is actually clever enough not to overwhelm your computer if the data you are printing is large-- it will automatically just show you a small selection of rows. 


Looking at that `slice()` above, it seems like this car gets *really* good highway mileage. Maybe even the best in the dataset. Let's use the `dplyr` `arrange()` command to sort the data to see if that is true

```{r}
myCars %>%
  arrange(desc(hwy))
```

#. Pick another outlier and try to determine why it might be so different from the other points. 

#### Adding information to plots

Another skill that can help you sort out what is going on with outliers, clusters of points, etc., is adding additional plotting marks. For example, we could create a plot where the points were colored differently based on whether the car took premium fuel or not. 

In order to do that, we might want to create a new variable using the `mutate()` command. 

```{r}
myCars = myCars %>%
  mutate(PremiumYN = if_else(fuel=="Premium", "Yes", "No"))
```

Notice that we used an `if_else()` statement to say, if `fuel == "Premium"` (`==` is a logical expression to check whether something is equal), set `PremiumYN` to "Yes", otherwise set it to "No." Now we can use that new variable in our plot. 

```{r}
ggplot(data=myCars, aes(x=displ, y=hwy, col=PremiumYN)) + geom_point()
```


Another option would be to create a set of plots, to compare cars with different numbers of cylinders. 

```{r}
ggplot(data=myCars, aes(x=displ, y=hwy)) + geom_point() + facet_wrap(~cyl)
```


#. What does this show you about the data?


#### Confidence and prediction intervals

To understand how a confidence interval for the mean fuel economy of cars with a specific engine size differs from a prediction interval for the fuel economy of an individual car with a specific engine size, lets create a confidence interval for cars with an engine displacement of 4 litres. 

```{r}
predict(m1, newdata=data.frame(displ=4), interval="confidence")
```

#. Interpret the confidence interval. What does it mean?

```{r}
predict(m1, newdata=data.frame(displ=4), interval="prediction")
```

#. Interpret the prediction interval. What does it mean?

We can also visualize the confidence interval over the entire model using `se=TRUE`. 


```{r}
ggplot(data=myCars, aes(x=displ, y=hwy)) + geom_point() + geom_smooth(method=lm, se=TRUE)
```

Note how the confidence interval for the **mean** fuel economy are narrowest towards the mean value of the explanatory variable (`displ`). This will always be true. 


#### Performing a transformation

When we checked the conditions above, we thought that a transformation would help the conditions not to be violated. We can use the `mutate()` command to create a new variable in our data with that transformation. 

#. Redo the analysis above, but using the transformation(s) you find appropriate!







