Exam 01 review

Announcements

• HW 02 due TODAY at 11:59pm

• Exam 01: Tuesday, February 18 (in-class + take-home)

  • Go directly to assigned room (emailed Wednesday evening)

• Friday’s lab: Exam 01 review - Graded on attendance and participation

• No office hours February 18 - 20

Exam 01

• 50 points total

  • in-class: 35 points

  • take-home: 15 points

• In-class: 75 minutes during February 18 lecture

• Take-home: due February 20 at 9pm (no lecture on Thursday)

• If you miss any part of the exam for an excused absence (with academic dean’s note), your Exam 02 score will be counted twice

Outline of in-class portion

• Closed-book, closed-note.

• Question types:

  • Short answer (show work / explain response)
  • True/ False.
    • If false, write 1 - 2 sentence justification about why it is false.
  • Derivations

• Will be provided all relevant R output and a page of matrix calculus and probability rules

• Can use any results from class or assignments without reproving them (e.g., $\mathbf{H}$ is symmetric and idempotent)

• Just need a pencil or pen. No calculator permitted on exam.

Outline of take-home portion

• Released: Tuesday, February 18 right after class
• Due: Thursday, February 20 at 9pm (no lecture February 20)
• Similar in format to a lab/ HW
  • Will receive Exam questions in README of GitHub repo
  • Formatting + using a reproducible workflow will be part of grade
• Submit a PDF of responses to GitHub

Tips for studying

• Rework derivations from assignments and lecture notes
• Review exercises in AEs and assignments, asking “why” as you review your process and reasoning
  • e.g., Why do we include “holding all else constant” in interpretations?
• Focus on understanding not memorization
• Explain concepts / process to others
• Ask questions in office hours
• Review lecture recordings as needed

Content: Weeks 1 - 6

• Exploratory data analysis

• Fitting and interpreting linear regression models

• Model assessment and comparison

• ANOVA

• Categorical + interaction terms

• Inference for model coefficients

• Matrix representation of regression

• Hat matrix

• Finding the least-squares estimator

• Assumptions for least-squares regression

Population-level vs. sample-level models

Statistical model (population-level model)

$$\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathit{ϵ},ϵ\sim N(\mathbf{0},{\sigma }_{ϵ}^2\mathbf{I})$$

Estimated regression model (sample-level model)

$$\hat{\mathbf{y}}=\mathbf{X}\hat{\mathit{\beta }}\mathbf{e}=\mathbf{y}-\hat{\mathbf{y}}$$

Model in matrix form

$$\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathit{ϵ}$$

1. What are the dimensions of $\mathbf{y},\mathbf{X},\mathit{\beta },\mathit{ϵ}$ ?
2. What assumption do we make about the columns of $\mathbf{X}$? Why is that important?

Model in matrix form

$$\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathit{ϵ}$$

1. What assumptions do we make about $\mathit{ϵ}$ making given this model?
2. What does this model tell us about the distribution of $\mathbf{y}$ ?

Find least-squares estimator $\hat{\mathit{\beta }}$

Expected value of $\hat{\mathit{\beta }}$

Variance of $\hat{\mathit{\beta }}$

SSR

Show

$$SSR={\mathbf{y}}^{\mathsf{T}}\mathbf{y}-\hat{\mathit{\beta }}{\mathbf{X}}^{\mathsf{T}}\mathbf{y}$$