Case Study: Pricing Homes

STAT 20: Introduction to Probability and Statistics

Agenda

• Concept Questions
• Problem Set 7.2
• Break
• Lab 7.2

Concept Questions

Consider two houses for sale, both 1,100 sqft, 2 bedroom, 1 bathroom, with a small garage, but one is in Santa Monica and the other is in Westwood. Which is true of the predicted sale prices of these two homes?

Call:
lm(formula = log_price ~ log_sqft + city, data = LA)

Coefficients:
     (Intercept)          log_sqft    cityLong Beach  citySanta Monica  
         5.46554           1.15119          -0.89345          -0.09301  
    cityWestwood  
        -0.45846  

01:00

This is a review question to assess if students can use model output to understand predictions. One way studnets may solve this is by plugging the x variables in to the corresponding formula and comparing the two predicted values. A student with stronger understanding will just compare the coefficients in front of cityWestwood and citySantaMonica. They both modify the reference level, which is Beverly Hills: SM is .1 log dollars less than the prediction for Beverly Hills and WW is .47 log dollars less than the prediction for Berkeley hills.

This is a good place to remind students that if they have transformed the respond variable prior to fitting their model, they will need to back-transform the predictions.

A simple model for price

m4 <- lm(log_price ~ bed, data = LA)

What do you expect the sign of the coefficient for bed to be?

01:00

There is no trick question here. It’s asking students to rely on their common sense to know that houses with more bedrooms tend to be more expensive, therefore the sign of the coefficient should be positive.

A simple model for price

m4 <- lm(log_price ~ bed, data = LA)

What do you expect the sign of the coefficient for bed to be?

# A tibble: 2 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)   11.8      0.0436     271.  0        
2 bed            0.532    0.0142      37.3 9.77e-220

A less simple model for price

m5 <- lm(log_price ~ bed + log_sqft, data = LA)

What do you expect the sign of the coefficient for bed and log_sqft to be?

01:00

This is a tricky question for most students because they expect that since bed and logsqft both have positive relationships with the response in a simple linear regression model, that they should still have a positive relationship in the multivariate setting. That’s not the case.

The best explanation involves asking the students a concrete question: “Imagine two houses in Long Beach next door to one another. Both are 1,500 sqrt, have two bathrooms, and a single garage. One of them, however, has three bedrooms and the other has four bedrooms. Which home would you expect to be more expensive and why?”.

Students will generally realize that if you fix the square footage, the house with 4 bedrooms will have to divide the 1500 sqft into 4 very tiny bedrooms. The 3 bedroom house, by contrast, will have larger rooms and more generous living area, characteristics that tend to be prized more than an additional bedroom.

The plots that follow this question illustrate the effect by conditioning on one of the variables by faceting and looking at the association between the other variable and the response. The final plot does the operative comparison: if we discretize the logsqft, we can see that the association in the facets looks either flat or negative.

A less simple model for price

m5 <- lm(log(price) ~ bed + log_sqft, data = LA)

What do you expect the sign of the coefficient for bed and log_sqft to be?

# A tibble: 3 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)    1.47     0.218       6.73 2.28e- 11
2 bed           -0.123    0.0164     -7.46 1.46e- 13
3 log_sqft       1.66     0.0346     47.8  2.60e-310

What is the relationship between bed and log_price?

What is the relationship between log_sqft and log_price?

What is the relationship between log_sqft and log_price, controlling for bed?

What is the relationship between bed and log_price, controlling for log_sqft?

Simpson’s Paradox

Simpson’s paradox, which also goes by several other names, is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined.

Source: Wikipedia

What can we build with data?

A prediction machine.

A summary of a data set.

A generalization to a population.

A causal explanation.

Although in this section we’re focusing on prediction, this question (and any question focusing on the coefficients of a model - their sign and magnitude) is either summary/descriptive or looking to generalize. If it is to inform an intervention (“if we add a bedroom, how much more can we wel sell the house for”) then it is a causal claim.

Model Interpretation

Question 1 What is the relationship between the number of bedrooms in a house and its price?

\[ \widehat{\textrm{log(price)}} = 11.8 + .53 \textrm{bed}\]

Question 2 After controlling for the size of a house, what is the relationship between the number of bedrooms in a house and its price?

\[ \widehat{\textrm{log(price)}} = 11.8 + -0.12 \textrm{bed} + 1.66 \textrm{log(sqft)}\]

The Tradeoff between flexility and interpretability

Fig 2.7 from An Introduction to Statistical Learning with R by James, Witten, Hastie, and Tibshirani.

Least squares linear models shows up here as not very flexible but very interpretable. That is one reason why it’s taught in Stat 20. Students likely will have heard of Deep Learning, so this demonstrates that they can be used for the same tasks. It also shows a active area of research: how to make Deep Learning models more interpretable, e.g. how can we understand why ChatGPT gives the answers that it does.

Problem Set 7.2

20:00

As a follow up to Question 5 on the handout, zillow offers the zestimate prediction service to determine the value of a home based.

The final slide is the information that they provide about how their predictions work. They use a deep learning model framework, a more flexible extension of a linear model.

This address was the property acquired by the UC to serve as the home of the president of the system (you can google it and pull up news articles).

Lab 7.2 Work

25:00