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STA 235H - Multiple Regression: Overview and Statistical Adjustment

Fall 2023

McCombs School of Business, UT Austin

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Today

  • Quick multiple regression review

    • How does OLS work?

  • What can we say using regressions?

    • Interpreting coefficients

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Nothing "Ordinary" about OLS

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What do you understand about regressions?

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Remembering Regressions

  • Linear Regression is a very useful tool.

    • Simple supervised learning approach.
    • Many fancy methods are generalizations or extensions of linear regression!

  • It's a way to (partially) describe a data generating process (DGP).

Y=β0+β1X1+β2X2+ε

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Essential Parts of a Regression

Y

Outcome Variable

Response Variable

Dependent Variable

Thing you want to explain or predict

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Essential Parts of a Regression

Y

Outcome Variable

Response Variable

Dependent Variable

Thing you want to explain or predict

X

Explanatory Variable

Predictor Variable

Independent Variable

Thing you use to explain or predict Y

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Identify the variables

A study examines the effect of smoking on lung cancer

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Identify the variables

A study examines the effect of smoking on lung cancer
Fantasy football fanatics predict the performance of a player based on past performance, health status, and characteristics of the opposite team

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Identify the variables

A study examines the effect of smoking on lung cancer
Fantasy football fanatics predict the performance of a player based on past performance, health status, and characteristics of the opposite team

You want to see if taking more AP classes in high school improves college grades

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Identify the variables

A study examines the effect of smoking on lung cancer
Fantasy football fanatics predict the performance of a player based on past performance, health status, and characteristics of the opposite team

You want to see if taking more AP classes in high school improves college grades
Netflix uses your past viewing history, the day of the week, and the time of the day to guess which show you want to watch next

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Two Purposes of Regression

Prediction

Forecast the future

Focus is on Y

Netflix trying to guess your next show

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Two Purposes of Regression

Prediction

Forecast the future

Focus is on Y

Netflix trying to guess your next show

Explanation

Explain the effect of X on Y

Focus is on X

Netflix looking at the effect of time of the day on show selection

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What do we want to estimate in a regression?

  • When we run a regression we have an outcome Y and explanatory variables or covariates X.

  • We want to estimate the β's

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What do we want to estimate in a regression?

  • When we run a regression we have an outcome Y and explanatory variables or covariates X.

  • We want to estimate the β's

  • One important distinction:

    • β's are the population parameters we want to estimate.
    • ˆβ are the estimates of those parameters.
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How do we estimate the coefficients in a regression ?

  • Ordinary Least Squares is the most popular way.

minβ∑[Yi−(p∑j=1βjXij)]2

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How do we estimate the coefficients in a regression ?

  • Ordinary Least Squares is the most popular way.

minβ∑[Yi−(p∑j=1βjXij)]2

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How do we estimate the coefficients in a regression ? (cont.)

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Let's get into some data

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Let's introduce an example: The Bechdel Test

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Let's introduce an example: The Bechdel Test

  • Three criteria:

    1. At least two named women
    2. Who talk to each other
    3. About something besides a man

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Do movies pass the test?

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Is it convenient for my movie to pass the Bechdel test?

  • I'm a profit-maximizing investor and want to know whether it's in my best interest to switch a male for a female character.

    • What is the simplest model you could fit?
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Is it convenient for my movie to pass the Bechdel test?

  • I'm a profit-maximizing investor and want to know whether it's in my best interest to switch a male for a female character.

    • What is the simplest model you could fit?

Revenue=α+βBechdel+ε

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Let's analyze some models

Summary Statistics
Variable N Mean Std. Dev. Min Pctl. 25 Pctl. 75 Max
Year 2087 2004.963 6.755 1990 1999 2011 2014
Adj_Revenue 2087 66.254 92.07 0 4.36 86.936 968.41
Adj_Budget 1369 61.498 57.784 0.02 19.3 88.47 470.839
Metascore 1755 5.663 1.66 1.1 4.5 6.8 9.7
imdbRating 2085 6.546 0.979 1.5 6 7.2 9.3
bechdel_test 2087 0.571 0.495 0 0 1 1
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Let's analyze some models

summary(lm(Adj_Revenue ~ bechdel_test, data = bechdel))
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)   76.4553     3.0641 24.9521        0
## bechdel_test -17.8616     4.0544 -4.4055        0
  • How do you interpret these results?
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Let's analyze some models

summary(lm(Adj_Revenue ~ bechdel_test, data = bechdel))
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)   76.4553     3.0641 24.9521        0
## bechdel_test -17.8616     4.0544 -4.4055        0

  • ˆβ0 is the average adjusted revenue (in millions of dollars) for movies that do not pass the Bechdel test.

  • On average, movies that pass the Bechdel test have an adjusted revenue that is |ˆβ1| million dollars less than a movie that doesn't pass the Bechdel test.



Negative effect of including more women?

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What gives?

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More variables

  • Bechdel test could be capturing the effect of other variables:

    • What type of movies are the ones that pass the test?

    • What is their budget?

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More variables

lm(Adj_Revenue ~ bechdel_test + Adj_Budget + Metascore + imdbRating, data=bechdel)
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -127.0710    17.0563 -7.4501   0.0000
## bechdel_test   11.0009     4.3786  2.5124   0.0121
## Adj_Budget      1.1192     0.0367 30.4866   0.0000
## Metascore       7.0254     1.9058  3.6864   0.0002
## imdbRating     15.4631     3.3914  4.5595   0.0000


Positive and significant!

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More variables

lm(Adj_Revenue ~ bechdel_test + Adj_Budget + Metascore + imdbRating, data=bechdel)
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -127.0710    17.0563 -7.4501   0.0000
## bechdel_test   11.0009     4.3786  2.5124   0.0121
## Adj_Budget      1.1192     0.0367 30.4866   0.0000
## Metascore       7.0254     1.9058  3.6864   0.0002
## imdbRating     15.4631     3.3914  4.5595   0.0000


Positive and significant!

  • How do we interpret the relevant coefficient now?
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Main takeaway points

  • Regressions are super useful...

    • But you need to know how to interpret them.

  • Be sure not to overstate your claims!

  • Remember the magic words for interpretation

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Next class

  • Continue with multiple regression models:

    • Interactions and how to interpret them

  • "Nonlinear" models

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References

  • Heiss, A. (2020). "Course: Program Evaluation for Public Service". Slides for Regression and Inference.

  • Ismay, C. & A. Kim. (2021). “Statistical Inference via Data Science”. Chapter 10.

  • Keegan, B. (2018). "The Need for Openess in Data Journalism". Github Repository

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Today

  • Quick multiple regression review

    • How does OLS work?

  • What can we say using regressions?

    • Interpreting coefficients

2 / 28
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