Model Building

Backstory and Set Up

You have been recently hired to Zillow’s Zestimate product team as a junior analyst. As a part of their regular hazing, they have given you access to a small subset of their historic sales data. Your job is to present some basic predictions for housing values in a small geographic area (Ames, IA) using this historical pricing.

First, let’s load the data.

ameslist <- read.table('https://raw.githubusercontent.com/ajkirkpatrick/FS20/postS21_rev/classdata/ames.csv', 
                   header = TRUE,
                   sep = ',') 

Building a Model

We’re now ready to start playing with a model. We will start by using the lm() function to fit a simple linear regression model, with SalePrice as the response and GrLivArea as the predictor.

Recall that the basic lm() syntax is lm(y∼x,data), where y is the response, x is the predictor, and data is the data set in which these two variables are kept. Let’s quickly run this with two variables:

lm.fit = lm(SalePrice ~ GrLivArea, data = ameslist)

If we type lm.fit, some basic information about the model is output. For more detailed information, we use summary(lm.fit). This gives us p-values and standard errors for the coefficients, as well as the \(R^2\) statistic and \(F\)-statistic for the entire model.¹

Utilizing these functions hels us see some interesting results. Note that we built (nearly) the simplest possible model:

\[\text{SalePrice} = \beta_0 + \beta_1*(\text{GrLivArea}) + \epsilon.\]

But even on its own, this model is instructive. It suggest that an increase in overall living area of 1 ft \(^2\) is correlated with an expected increase in sales price of $107. (Note that we cannot make causal claims!)

Saving the model as we did above is useful because we can explore other pieces of information it stores. Specifically, we can use the names() function in order to find out what else is stored in lm.fit. Although we can extract these quantities by name—e.g. lm.fit$coefficients—it is safer to use the extractor functions like coef(lm.fit) to access them. We can also use a handy tool like plot() applied directly to lm.fit to see some interesting data that is automatically stored by the model.

Try it: Use plot() to explore the model above (it will make a sequence of plots; don’t put it in a code chunk, just use it for your own exploration). Do you suspect that some outliers have a large influence on the data? We will explore this point specifically in the future.

We can now go crazy adding variables to our model. It’s as simple as appending them to the previous code—though you should be careful executing this, as it will overwrite your previous output:

lm.fit = lm(SalePrice ~ GrLivArea + LotArea, data = ameslist)

Try it: Does controlling for LotArea change the qualitative conclusions from the previous regression? What about the quantitative results? Does the direction of the change in the quantitative results make sense to you?