note14 - STAT5044 Regression and Anova Inyoung Kim Outline...

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STAT5044: Regression and Anova Inyoung Kim
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Outline 1 One regression model using data from two sources 2 Segmented regression
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Data from two sources Example Female salary Y i = α 0 + α 1 x i + σε i , j=1,...,n Male salary Y j = γ 0 + γ 1 x j + σε j , j=n+1,n+2,...,n+m and { ε i } , i=1,...,n and { ε j } , j=1,...,m, are N(0,1) Main Question: How to make one model using these two models?
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Case1: same slope between two models Consider the simplest case when slopes are same. Y i = α 0 + α 1 x i + σε i Y j = γ 0 + α 1 x i + σε i Question1: How to write the two regression as one
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Case1 Model Y 1 Y 2 . . . Y n Y n + 1 . . . Y n + M = 1 x 1 0 . . . . . . . . . 1 x n 0 1 x n + 1 1 . . . . . . . . . 1 x n + m 1 α 0 α 1 γ 0 - α 0 y j = α 0 + α 1 x j +( γ 0 - α 0 )+ ε j = [ 1 n + m x 1 x 2 ] β + ε j x 2 : indicator variable, x 2 j = ( 0 j 1 st group 1 j 2 nd group β = α 0 α 1 ( γ 0 - α 0 )
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Case1 Model Y ( n + m ) × 1 = X ( n + m ) × 3 β 3 × 1 + ε β 0 β
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