Statistics 541: Regression introduction

Admistrivia

• Return notes to student
• Anyone figure out how to get JMP to do a trimmed mean?
• Read chapter 1 of Myers Read sections 2.1-3, 2.9

Regression introduction

• Y = beta₀ + beta₁ X₁ + ... + beta_p X_p + epsilon
• Goals
  • prediction
  • Causation (see talk this afternoon: 4:30 Paul Holland 111 Annenburg. A "big gun.")
  • The betas might actually be interesting themselves
• We will focus on how to tell if your regression is statistically legitimate
• Leave you to decide if it makes any sceintific sense

Simple linear regression (Chapter 2 from Myers)

Basic structure

• Simple means only 1 x variable (called independent variables, but I rarely use that termonology.)
• slope and intercept/centercept

• minimize sum (y - y-hat)²

  minimize sum (y - b₀ + b₁ X)²

  Now define c = b₀ + Y-bar - b₁ X-bar is the deviation from the centercept.

• b₁ = S_xy/S_xx
• c = 0
• b₀ = Y-bar - b₁ X-bar

• residuals = (y - y-hat) shouldn't be predicatable from X
• Residuals should have zero mean
• So residuals should be uncorrelated with X
• E(Y - b₀ - b₁) = 0
• Cov(Y - b₀ - b₁ X,X) = 0

• E(estimate) = true paramenter (unbiased)
• Var(slope) = sigma²/S_xx
• Var(centercept) = sigma²/n
• Var(intercept) = sigma²(1/n + x-bar²//S_xx)

Using JMP and bluging rule

• Display (A lage chain of liquor stores would like to konw how much display space in its stores to devote to a new wine. Management believes that most products generate about $50 in sales per linear shelf-foot per month.)
• Tukey's bulging rule by hand and by picture