Chapter 10 Applications: Model | Introduction to Modern Statistics (1st Ed)

10.1 Case study: Houses for sale

Take a walk around your neighborhood and you’ll probably see a few houses for sale. If you find a house for sale, you can probably go online and look up its price. You’ll quickly note that the prices seem a bit arbitrary – the homeowners get to decide what the amount they want to list their house for, and many criteria factor into this decision, e.g., what do comparable houses (“comps” in real estate speak) sell for, how quickly they need to sell the house, etc.

In this case study we’ll formalize the process of figuring out how much to list a house for by using data on current home sales In November of 2020, information on 98 houses in the Duke Forest neighborhood of Durham, NC were scraped from Zillow. The homes were all recently sold at the time of data collection, and the goal of the project was to build a model for predicting the sale price based on a particular home’s characteristics. The first four homes are shown in Table 10.1, and descriptions for each variable are shown in Table 10.2.

The duke_forest data can be found in the openintro R package.

Table 10.1: Top four rows of the data describing homes for sale in the Duke Forest neighborhood of Durham, NC.
price	bed	bath	area	year_built	cooling	lot
1,520,000	3	4	6,040	1,972	central	0.97
1,030,000	5	4	4,475	1,969	central	1.38
420,000	2	3	1,745	1,959	central	0.51
680,000	4	3	2,091	1,961	central	0.84

Table 10.2: Variables and their descriptions for the `duke_forest` dataset.
Variable	Description
price	Sale price, in USD
bed	Number of bedrooms
bath	Number of bathrooms
area	Area of home, in square feet
year_built	Year the home was built
cooling	Cooling system: central or other (other is baseline)
lot	Area of the entire property, in acres

10.1.1 Correlating with `price`

As mentioned, the goal of the data collection was to build a model for the sale price of homes. While using multiple predictor variables is likely preferable to using only one variable, we start by learning about the variables themselves and their relationship to price. Figure 10.1 shows scatterplots describing price as a function of each of the predictor variables. All of the variables seem to be positively associated with price (higher values of the variable are matched with higher price values).

Figure 10.1: Scatter plots describing six different predictor variables’ relationship with the price of a home.

In Figure 10.1 there does not appear to be a correlation value calculated for the predictor variable, cooling. Why not? Can the variable still be used in the linear model? Explain.¹¹⁵

In Figure 10.1 which variable seems to be most informative for predicting house price? Provide two reasons for your answer.

The area of the home is the variable which is most highly correlated with price. Additionally, the scatterplot for price vs. area seems to show a strong linear relationship between the two variables. Note that the correlation coefficient and the scatterplot linearity will often give the same conclusion. However, recall that the correlation coefficient is very sensitive to outliers, so it is always wise to look at the scatterplot even when the variables are highly correlated.

10.1.2 Modeling `price` with `area`

A linear model was fit to predict price from area. The resulting model information is given in Table 10.3.

Table 10.3: Summary of least squares fit for price on area.
term	estimate	std.error	statistic	p.value
(Intercept)	116,652	53,302	2.19	0.0311
area	159	18	8.78	<0.0001

Adjusted R-sq = 0.4394
df = 96

Interpret the value of $b_1$ = 159 in the context of the problem.¹¹⁶

Using the output in Table 10.3, write out the model for predicting price from area.¹¹⁷

The residuals from the linear model can be used to assess whether a linear model is appropriate. Figure 10.2 plots the residuals $e_i = y_i - \hat{y}_i$ on the $y$-axis and the fitted (or predicted) values $\hat{y}_i$ on the $x$-axis.

Figure 10.2: Residuals versus predicted values for the model predicting sale price from area of home.

What aspect(s) of the residual plot indicate that a linear model is appropriate? What aspect(s) of the residual plot seem concerning when fitting a linear model?¹¹⁸

10.1.3 Modeling `price` with multiple variables

It seems as though the predictions of home price might be more accurate if more than one predictor variable was used in the linear model. Table 10.4 displays the output from a linear model of price regressed on area, bed, bath, year_built, cooling, and lot.

Table 10.4: Summary of least squares fit for price on multiple predictor variables.
term	estimate	std.error	statistic	p.value
(Intercept)	-2,910,715	1,787,934	-1.63	0.107
area	102	23	4.42	<0.0001
bed	-13,692	25,928	-0.53	0.5987
bath	41,076	24,662	1.67	0.0993
year_built	1,459	914	1.60	0.1139
coolingcentral	84,065	30,338	2.77	0.0068
lot	356,141	75,940	4.69	<0.0001

Adjusted R-sq = 0.5896
df = 90

Using Table 10.4, write out the linear model of price on the six predictor variables.

\[ \begin{aligned} \widehat{\texttt{price}} &= -2,910,715 \\ &+ 102 \times \texttt{area} - 13,692 \times \texttt{bed} \\ &+ 41,076 \times \texttt{bath} + 1,459 \times \texttt{year_built}\\ &+ 84,065 \times \texttt{cooling}_{\texttt{central}} + 356,141 \times \texttt{lot} \end{aligned} \]

The value of the estimated coefficient on $\texttt{cooling}_{\texttt{central}}$ is $b_5 = 84,065.$ Interpret the value of $b_5$ in the context of the problem.¹¹⁹

A friend suggests that maybe you do not need all six variables to have a good model for price. You consider taking a variable out, but you aren’t sure which one to remove.

Results corresponding to the full model for the housing data are shown in Table 10.4. How should we proceed under the backward elimination strategy?

Our baseline adjusted $R^2$ from the full model is 0.59, and we need to determine whether dropping a predictor will improve the adjusted $R^2$. To check, we fit models that each drop a different predictor, and we record the adjusted $R^2$:

Excluding area: 0.506
Excluding bed: 0.593
Excluding bath: 0.582
Excluding year_built: 0.583
Excluding cooling: 0.559
Excluding lot: 0.489

The model without bed has the highest adjusted $R^2$ of 0.593, higher than the adjusted $R^2$ for the full model. Because eliminating bed leads to a model with a higher adjusted $R^2$ than the full model, we drop bed from the model.

It might seem counter-intuitive to exclude information on number of bedrooms from the model. After all, we would expect homes with more bedrooms to cost more, and we can see a clear relationship between number of bedrooms and sale price in Figure 10.1. However, note that area is still in the model, and it’s quite likely that the area of the home and the number of bedrooms are highly associated. Therefore, the model already has information on “how much space is available in the house” with the inclusion of area.

Since we eliminated a predictor from the model in the first step, we see whether we should eliminate any additional predictors. Our baseline adjusted $R^2$ is now 0.593. We fit another set of new models, which consider eliminating each of the remaining predictors in addition to bed:

Excluding bed and area: 0.51
Excluding bed and bath: 0.586
Excluding bed and year_built: 0.586
Excluding bed and cooling: 0.563
Excluding bed and lot: 0.493

None of these models lead to an improvement in adjusted $R^2$, so we do not eliminate any of the remaining predictors.

That is, after backward elimination, we are left with the model that keeps all predictors except bed, which we can summarize using the coefficients from Table 10.5.

Table 10.5: Summary of least squares fit for price on multiple predictor variables, excluding number of bedrooms.
term	estimate	std.error	statistic	p.value
(Intercept)	-2,952,641	1,779,079	-1.66	0.1004
area	99	22	4.44	<0.0001
bath	36,228	22,799	1.59	0.1155
year_built	1,466	910	1.61	0.1107
coolingcentral	83,856	30,215	2.78	0.0067
lot	357,119	75,617	4.72	<0.0001

Adjusted R-sq = 0.5929
df = 91

Then, the linear model for predicting sale price based on this model is as follows:

\[ \begin{aligned} \widehat{\texttt{price}} &= -2,952,641 + 99 \times \texttt{area}\\ &+ 36,228 \times \texttt{bath} + 1,466 \times \texttt{year_built}\\ &+ 83,856 \times \texttt{cooling}_{\texttt{central}} + 357,119 \times \texttt{lot} \end{aligned} \]

The residual plot for the model with all of the predictor variables except bed is given in Figure 10.3. How do the residuals in Figure 10.3 compare to the residuals in Figure 10.2?

The residuals, for the most part, are randomly scattered around 0. However, there is one extreme outlier with a residual of -$750,000, a house whose actual sale price is a lot lower than its predicted price. Also, we observe again that the residuals are quite large for expensive homes.

Figure 10.3: Residuals versus predicted values for the model predicting sale price from all predictors except for number of bedrooms.

Consider a house with 1,803 square feet, 2.5 bathrooms, 0.145 acres, built in 1941, that has central air conditioning. What is the predicted price of the home?¹²⁰

If you later learned that the house (with a predicted price of $297,570) had recently sold for $804,133, would you think the model was terrible? What if you learned that the house was in California?¹²¹