Topics

• Introduce multiple linear regression

• Interpret a coefficient ˆβj

• Use the model to calculate predicted values and the corresponding interval

House prices in Levittown

The data set contains the sales price and characteristics of 85 homes in Levittown, NY that sold between June 2010 and May 2011.

Levittown was built right after WWII and was the first planned suburban community built using mass production techniques.

The article "Levittown, the prototypical American suburb – a history of cities in 50 buildings, day 25" gives an overview of Levittown's controversial history.

Analysis goals

We would like to use the characteristics of a house to understand variability in the sales price.

To do so, we will fit a multiple linear regression model

Using our model, we can answers questions such as

• What is the relationship between the characteristics of a house in Levittown and its sale price?
• Given its characteristics, what is the expected sale price of a house in Levittown?

The data

## # A tibble: 10 x 7
##    bedrooms bathrooms living_area lot_size year_built property_tax sale_price
##       <dbl>     <dbl>       <dbl>    <dbl>      <dbl>        <dbl>      <dbl>
##  1        4       1          1380     6000       1948         8360     350000
##  2        4       2          1761     7400       1951         5754     360000
##  3        4       2          1564     6000       1948         8982     350000
##  4        5       2          2904     9898       1949        11664     375000
##  5        5       2.5        1942     7788       1948         8120     370000
##  6        4       2          1830     6000       1948         8197     335000
##  7        4       1          1585     6000       1948         6223     295000
##  8        4       1           941     6800       1951         2448     250000
##  9        4       1.5        1481     6000       1948         9087     299990
## 10        3       2          1630     5998       1948         9430     375000

Variables

Predictors

• bedrooms: Number of bedrooms
• bathrooms: Number of bathrooms
• living_area: Total living area of the house (in square feet)
• lot_size: Total area of the lot (in square feet)
• year_built: Year the house was built
• property_tax: Annual property taxes (in U.S. dollars)

Response

• sale_price: Sales price (in U.S. dollars)

EDA: Response variable

EDA: Predictor variables

EDA: Response vs. Predictors

Multiple linear regression (MLR)

Based on the analysis goals, we will use a multiple linear regression model of the following form

Similar to simple linear regression, this model assumes that at each combination of the predictor variables, the values sale_price follow a Normal distribution

Regression Model

• Recall: The simple linear regression model assumes

Y|X∼N(β0+β1X,σ2ϵ)

• Similarly: The multiple linear regression model assumes

Y|X1,X2,…,Xp∼N(β0+β1X1+β2X2+⋯+βpXp,σ2ϵ)

Regression Model

• At any combination of the predictors, the mean value of the response Y, is μY|X1,…,Xp=β0+β1X1+β2X2+⋯+βpXp

  
  

• Using multiple linear regression, we can estimate the mean response for any combination of predictors

ˆY=ˆβ0+ˆβ1X1+ˆβ2X2+⋯+ˆβpXp

Home price model

Interpreting ˆβj

• The estimated coefficient ˆβj is the expected change in the mean of y when xj increases by one unit, holding the values of all other predictor variables constant.

• Example: The estimated coefficient for living_area is 65.90. This means for each additional square foot of living area, we expect the sale price of a house in Levittown, NY to increase by $65.90, on average, holding all other predictor variables constant.

Prediction

Example: What is the predicted sale price for a house in Levittown, NY with 3 bedrooms, 1 bathroom, 1050 square feet of living area, 6000 square foot lot size, built in 1948 with $6306 in property taxes?

-7148818.957 - 12291.011 * 3 + 51699.236 * 1 + 
  65.903 * 1050 - 0.897 * 6000 + 3760.898 * 1948 + 
  1.476 * 6306

## [1] 265360.4

The predicted sale price for a house in Levittown, NY with 3 bedrooms, 1 bathroom, 1050 square feet of living area, 6000 square foot lot size, built in 1948 with $6306 in property taxes is $265,360.

Intervals for predictions

Just like with simple linear regression, we can use the predict function in R to calculate the appropriate intervals for our predicted values

x0 <- data.frame(bedrooms = 3, bathrooms = 1, 
                 living_area = 1050, lot_size = 6000, 
                 year_built = 1948, 
                 property_tax = 6306)

Confidence interval for ˆμy

Calculate a 95% confidence interval for the estimated mean price of houses in Levittown, NY with 3 bedrooms, 1 bathroom, 1050 square feet of living area, 6000 square foot lot size, built in 1948 with $6306 in property taxes:

predict(price_model, x0, interval = "confidence", 
        level = 0.95)

##        fit      lwr      upr
## 1 265360.2 238481.7 292238.7

Prediction interval for ˆy

Calculate a 95% prediction interval for an individual house in Levittown, NY with 3 bedrooms, 1 bathroom, 1050 square feet of living area, 6000 square foot lot size, built in 1948 with $6306 in property taxes:

predict(price_model, x0, interval = "prediction", 
        level = 0.95)

##        fit      lwr      upr
## 1 265360.2 167276.8 363443.6

🛑 Cautions

• Do not extrapolate! Because there are multiple predictor variables, there is the potential to extrapolate in many directions

• The multiple regression model only shows association, not causality

  • To show causality, you must have a carefully designed experiment or carefully account for confounding variables in an observational study

Recap

• Introduced multiple linear regression

• Interpreted a coefficient ˆβj

• Used the model to calculate predicted values and the corresponding interval