Models + SLR

Lecture 12

Dr. Elijah Meyer

Duke University
STA 199 - Spring 2023

Feburary 22nd, 2023

Checklist

– Clone ae-11

Announcements

– HW-2 Due Friday at 11:59

– Lab 4 Due Tuesday at 11:59

Data Fest 2023 at Duke

– data analysis competition where teams of up to five students attack a large, complex, and surprise dataset over a weekend

– DataFest is a great opportunity to gain experience that employers are looking for

– Each team will give a brief presentation of their findings that will be judged by a panel of judges comprised of faculty and professionals from a variety of fields.

Lab 4

– Team Submission

– Attach ALL team members to submission on Gradescope

– Communicate!

– “It was my responsibility to turn the lab in and I forgot….”

Warm up: Simpson’s Paradox

What is it?

Why is it important?

Warm up

Where we are going

\(\checkmark\) Data Viz

\(\checkmark\) Probability

– Modeling Data

Goals

– Introduce the idea of modeling

– Why we model data?

— Modeling with single predictors

— How to write equations

— Interpret Slopes

— Interpret Intercepts

Warm up 2

– What is the relationship?

– What is your best guess for a car’s MPG that weighs 5000 pounds? 3000 pounds?

What is a statistical model?

– Statistical modeling is the process of applying statistical analysis to a data set.

– A statistical model is a mathematical representation of observed data.

Why

– Interpretation

– Prediction

What is linear regression?

– Model data using a straight line

– Quantitative response

– Quantitative or categorical explanatory

Vocab - Response variable

Vocab - Explanatory variable

How are Models Fit?

What about when X is categorical?

In R: SLR

linear_reg() |>

     set_engine("lm") |>

     fit(y ~ x , data = data-set ) |>

     tidy()

ae-11

Model Notation - Population

\[ Y_i = \beta\_o + \beta\_1x_i +\epsilon\_i\]

\[ Y\; - True\; mean\;response \]

\[\beta\_o\; -True\; intercept\]

\[\beta\_1\; - True\; slope\; coefficient\]

\[\epsilon\_i\; - Error\; term\; for\; each\; observation\; i\]

Model Notation - Estimated

\[\hat{Y} = b + b_1x\]

\[\hat{Y} = \hat{\beta_o} + \hat{\beta_1}x\]

\[\hat{Y} - estimated\; (predicted)\; mean \;response\]

\[\hat{\beta_o} - estimated\; intercept\]

\[\hat{\beta_1} - estimated\; slope\] We assume that our error term is normally distributed and has a mean of 0. Thus, it does not show up in our model.

Model Notes

Things to note:

X is our explanatory variable and is not random. We know the value of X.

Y is our response variable. For a fixed X, Y will be a random variable (have a random outcome).

This random outcome is observed based on a random draw from a distribution we assume.

When using this model for prediction, we expect Y to take on the most likely value for a given X… which is the center of the distribution.

Let’s draw it out…

Takeaway

So, for an observed X….. we are modeling the mean of the distribution of Y

Or, Y is a mean response

“we estimate a mean change in Y”

“we estimate, on average…..”

This is extremely important when we think about interpretation

Summary

– Y is a random variable.

– We assume that observations from this random variable are normally distributed.

– Because of this distributional assumption, we are modeling the mean of Y and not just Y.

Wrap up: Check yourself

– What is a model?

– What are the 2 main reasons we fit models?

– Slope coefficients? Intercepts?

– How does linear regression change when x is quantitative vs categorical?