Recap: simple linear regression

Lab 5

John Zito

Duke University
STA 199 Spring 2025

2025-03-17

Simple linear regression

Question: how do we concisely summarize the association between two variables?

Answer: simple linear regression!

Answer: simple linear regression!

mpg_wt_fit <- linear_reg() |>
  fit(mpg ~ wt, data = mtcars)

tidy(mpg_wt_fit)

# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    37.3      1.88      19.9  8.24e-19
2 wt             -5.34     0.559     -9.56 1.29e-10

\[ \widehat{mpg}=37.3 - 5.34\times weight. \]

Interpretations

We predict that a car weighing zero pounds will have 37.28 MPG on average (makes no sense);
We predict that a 1000 pound increase in weight in associated with a 5.34 decrease in MGP, on average.

Why do we care? Prediction!

Why do we care? Prediction!

You can use the fitted model to generate predictions for yet-to-be-observed subjects:

new_car <- tibble(
  wt = 3.5
)

predict(mpg_wt_fit, new_data = new_car)

# A tibble: 1 × 1
  .pred
  <dbl>
1  18.6

Part 1 of today’s lab…

Does increasing the minimum wage decrease employment?

This is a hotly debated question in economics and public policy;
ECON 101 logic says that it might: if you make something more expensive (employing people), people do less of it.
What do the actual data say?

Classic study: Card and Krueger (1994 AER)

In 1992, NJ raised minimum wage. PA did not;
Fast-food restaurants along the NJ/PA border are probably very similar. Maybe the only difference is the change in wage policy;
So PA is like control and NJ is like the treatment;
If we compare employment before and after the policy change, maybe we can give the observed differences a causal interpretation. The increase in minimum wage caused employment to go up, down, or stay the same;
This is called a natural experiment. It’s a kind of observational study where you get very lucky and “nature” does the experimental control for you.