Group comparisons and dummy variables in regression models

Use the following commands in R to load the SaratogaHouses dataset from the mosaic package:

library(mosaic)
data(SaratogaHouses)

These data were originally collected to estimate how much value a fireplace adds to a home’s value. Answering this question turns out to be tricky! We’ll start by examining the association between the number of fireplaces and the sale price of a house.

Part A

Create a scatterplot of price (y) against fireplaces (x). What do you see?
Treating fireplaces as a numeric variable, fit a simple linear regression model for price as a function of the number of fireplaces. What is the estimated increase in value for each additional fireplace?
The model we fit in part 2 assumes that the expected sale price of a house increases the same amount with each additional fireplace – that is, the difference in expected sale price between a house with 1 vs. 0 fireplaces is the same as the expected diffrence in sale price for a house with 3 vs. 2 fireplaces (or 2 vs. 1, or 4 vs. 3). Do you think this is a good assumption? Why or why not?

Part B

Here we’ll treat the number of fireplaces as a categorical variable. In general it’s better to avoid discretizing continuous or numeric variables, but here the number of fireplaces is almost always 0, 1, or 2, so it is essentially a categorical variable anyway (it would be hard to learn much about the relationship between price and fireplaces above 3 anyway, since we only have four data points there).

To begin let’s compare houses with any fireplaces to houses with none. To do this we can create a new categorical variable (and inspect it using the summary() command) using the R code below:

  SaratogaHouses$any_fireplaces = factor(SaratogaHouses$fireplaces>0)
  summary(SaratogaHouses$any_fireplaces)

Unpacking this code, remember that SaratogaHouses$fireplaces>0 will create a new vector with the value TRUE when the house has at least one fireplace and FALSE when it does not. The factor() command tells R to treat this new variable as a categorical variable, instead of a logical/Boolean or character vector.

Compute the sample mean sale price for houses with and without fireplaces using the code below:
```
 mean(price~any_fireplaces, data=SaratogaHouses)
```
Ignoring uncertainty in these estimates for now, do houses with fireplaces tend to have higher sale prices on average? If so, by how much?
Fit a simple linear regression model predicting price from any_fireplaces. How do the estimated coefficients relate to the two subgroup means you computed above?

Now let’s make a new categorical variable indicating whether a house has zero, one, or more than one fireplace. We can do this using the R code below:

 SaratogaHouses$fireplace_categories = SaratogaHouses$fireplaces # Make a copy
 # Then replace the 2, 3, and 4's with a label 2+
 SaratogaHouses$fireplace_categories[SaratogaHouses$fireplaces>1] = "2+"
 # Make a factor
 SaratogaHouses$fireplace_categories = factor(SaratogaHouses$fireplace_categories)
 summary(SaratogaHouses$fireplace_categories)

(You can also accomplish this using the cut() function; type ?cut into your R console to learn more). Compute the average sale price for each fireplace category by modifying the code in part B #1.

Fit a linear regression model predicting price using our new categorical variable. How do the estimated coefficients relate to the means you computed above?
Bootstrap the regression model in part B #4. Is there strong evidence in the data that houses with one fireplace have higher average prices than houses with no fireplaces? Draw a histogram of the bootstrap-estimated sampling distribution for the relevant coefficient, and compute a 95% confidence interval.
Briefly explain why the model we just fit doesn’t tell us directly whether there is evidence for a difference in average prices between houses with one and two or more fireplaces.
We can fit a model that does include this contrast by changing the reference category using the R code below:
```
 SaratogaHouses$fireplace_categories = relevel(SaratogaHouses$fireplace_categories, ref = "1")
```
After running this command, refit the linear model from part B #4 above. How do the new estimated coefficients relate to the sample means you computed in part B #3? Using the bootstrap, do you find strong evidence in the data that houses with 2+ fireplaces have higher average sale prices than houses with 1 fireplace?
Say adding a fireplace to a house would cost $15,000. Based on our analysis here, if you were selling a house in Saratoga (at the time these data were collected) should you add a fireplace to a house with no fireplace? With one fireplace? (That is, do you think the differences in average sale prices here are caused by the varying numbers of fireplaces?) Why or why not?