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Multiple Regression Anatomy

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About This MicroSim

The equation \(\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3\) is drawn across the top with five numbered parts: the intercept, one term for each feature, and the prediction. Under each symbol is the number it stands for right now, for example 150 × 1,500 for the square feet term.

The chart below it is a waterfall. It starts with the intercept, adds each term where the last one ended, and finishes with a gold bar for the predicted price. Terms that raise the prediction are green and terms that lower it are red.

The coefficients are not typed in. The MicroSim fits a least-squares regression to 40 simulated houses and shows the result with its \(R^2\) and adjusted \(R^2\). The houses were built so that the fit comes out exactly as the chapter's model, Price = 50,000 + 150 × SqFt + 10,000 × Bedrooms − 1,000 × Age.

How to Use

  1. Read the default view. A 1,500 square foot house with 3 bedrooms that is 20 years old is predicted at $285,000, which is 50,000 + 225,000 + 30,000 − 20,000.
  2. Drag the Square feet slider. Only the square feet term and the prediction change. The bedrooms and age terms hold still.
  3. Add one bedroom and check that the prediction rises by exactly $10,000. Repeat at a different square footage and confirm that the rise is the same.
  4. Drag Age to 0 and then to 60 and watch the red bar pull the running total back.
  5. Hover over a numbered part, in the equation or in the chart, to read what it means. Click to keep it selected and click again to release it.

Iframe Embed Code

You can add this MicroSim to any web page by adding this to your HTML:

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<iframe src="https://dmccreary.github.io/data-science-course/sims/multiple-regression-anatomy/main.html"
        height="607"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

High School (11-12) and College Freshman

Duration

10-15 minutes

Prerequisites

Simple linear regression, the meaning of slope and intercept, and \(R^2\).

Bloom's Taxonomy Level

Understand (L2)

Learning Objective

Students will be able to explain how the intercept and each coefficient-times-feature term combine into one prediction, and interpret a coefficient as the change in the prediction for a one-unit increase in its feature with the other features held constant.

Activities

  1. Explore (4 min): Students move each slider in turn and record which numbers on screen change. They state in one sentence what stays the same when only one feature moves.
  2. One-unit changes (5 min): Students work out by hand the price change for 100 more square feet, 1 more bedroom, and 10 more years of age, then check each answer with the sliders.
  3. Explain (4 min): In pairs, students click each numbered part and restate its explanation in their own words, including why the intercept is not the price of a real house.

Assessment

Given the model Price = 50,000 + 150 × SqFt + 10,000 × Bedrooms − 1,000 × Age, students compute the prediction for a 2,000 square foot, 4-bedroom, 10-year-old house and explain what the coefficient −1,000 means, using the phrase "holding the other features constant".

References

  1. Wikipedia. Linear regression.
  2. scikit-learn documentation. sklearn.linear_model.LinearRegression.
  3. James, G., Witten, D., Hastie, T., and Tibshirani, R. An Introduction to Statistical Learning. Springer. Chapter 3, Linear Regression.