6 Interactions and Anova

# 6 Interactions and Anova - Statistics 191 Introduction to...

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Unformatted text preview: Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Statistics 191: Introduction to Applied Statistics Qualitative Variables, Interactions & ANOVA Jonathan Taylor Department of Statistics Stanford University February 17, 2010 1 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Qualitative variables + interactions Outline Qualitative / categorical variables. Regression equations differing by group. Interactions. Analysis of Variance Models 2 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Categorical variables Categorical variables Most variables we have looked at so far were continuous: height, rating, etc. In many situations, we record a categorical variable: sex, state, country, etc. How do we include this in our model? 3 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Categorical variables A simple example One example that we have looked at does have categorical variables. Two sample problem with equal variances: suppose Y = ( Z 1 ,..., Z m , W 1 ,..., W n ) with Z j ∼ N ( μ 1 ,σ 2 ) , 1 ≤ j ≤ m and W j ∼ N ( μ 2 ,σ 2 ) , 1 ≤ j ≤ n + m . For 1 ≤ i ≤ n , let X i = ( 1 1 ≤ i ≤ m otherwise. 4 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Categorical variables A simple example Design matrix X ( n + m ) × 2 = 1 1 . . . . . . 1 1 1 0 . . . . . . 1 0 5 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Example IT salary data Outcome: S, salaries for IT staff in a corporation. Predictors: X, experience (years); E, education (3 levels): 1=Bachelor’s, 2=Master’s, 3=Ph.D; M, management (2 levels): 1=management, 0=not management. 6 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University IT salary R code 7 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University IT salary R code 8 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Two solutions Solution #1: stratification One solution is to “stratify” data set by this categorical variable. We could break data set up into groups by education and management, and fit fit model S i = β + β 1 X i + ε i in each group. Problem: this results in smaller samples in each group: lose degrees of freedom for estimating σ 2 within each group....
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6 Interactions and Anova - Statistics 191 Introduction to...

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