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Indicator variable for two categories

Indicator variable for two categories. Two categories, A and B (not A) Define an indicator variable A i = 1 for members of category A A i = 0 otherwise. Regression model with different intercepts for two categories. Consider the regression model

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Indicator variable for two categories

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  1. Indicator variable for two categories • Two categories, A and B(not A) • Define an indicator variable • Ai = 1 for members of category A • Ai = 0 otherwise

  2. Regression model with different intercepts for two categories • Consider the regression model yi = b0+ b1 xi+ d0 Ai+ ei • For Ai = 0 yi = b0+ b1 xi+ ei • ForAi = 1 yi =(b0+ d0 ) + b1 xi+ ei • Difference between the two intercepts, d0 • Test if the two intercepts are the same • H0: d0 = 0

  3. Regression model with different intercepts and Slopes yi = b0+ b1 xi+ d0 Ai+ d1 Ai xi+ ei • For Ai = 0 yi = b0+ b1 xi+ ei • ForAi = 1 yi =(b0+ d0 ) + (b1 + d1 ) xi+ ei • Difference between the two intercepts, d0 • Difference between the two slopes, d1 • Test if the two regression lines are the same • H0: d0 = d1 = 0 (Partial F)

  4. Indicator variables for three categories • Three categories, A, B, and C • Define two indicator variables • Ai = 1 for members of category A • Ai = 0 otherwise • Bi = 1 for members of category B • Bi = 0 otherwise • Members of category C are coded as • Ai = 0 (not A) and Bi = 0 (not B) • C is the base category

  5. Indicator variables for all categories • Three categories, A, B, and C • Ai = 1 for members of category A • Ai = 0 otherwise • Bi = 1 for members of category B • Bi = 0 otherwise • Ci = 1 for members of category C • Ci = 0 otherwise • Note that A + B = C = 1, hence a linear dependency if there is an intercept

  6. Different intercepts for three categories yi = b0+b1 xi+d10 Ai+ d20 Bi+ ei • For Ai = 0 (not A) and Ai = 0 (not B) yi = b0+b1 xi+ ei • ForAi = 1 (A) and Bi = 0 (not B) yi =(b0+ d10 ) +b1 xi+ ei • Difference between the intercepts for A and C, d10

  7. Different intercepts for three categories yi = b0+ b1 xi+ d01 Ai+ d02 Bi+ ei • For Ai = 0 (not A)andBi = 1 (B) yi =(b0+ d20 ) +b1 xi+ ei • Difference between the intercepts for B and C, d02 • Test if the three intercepts are the same • H0: d01 = d02= 0 • Partial F

  8. Different intercepts and slopes for three categories yi = b0+ b1 xi + d01 Ai +d02 Bi + d11 Ai xi+ d12 Bi xi+ ei • For Ai = 0 (not A) and Bi = 0 (not B) yi = b0+ b1 xi+ ei

  9. Categories A and C yi = b0+ b1 xi + d01 Ai +d02 Bi + d11 Ai xi+ d12 Bi xi+ ei • ForAi = 1 (A) and Bi = 0 (not B) yi =(b0+ d10 ) + (b1 + d11 ) xi+ ei • Difference between the intercepts for A and C, d10 • Difference between the slopes A and C, d11

  10. Group B and C yi = b0+ b1 xi + d01 Ai +d02 Bi + d11 Ai xi+ d12 Bi xi+ ei • ForAi = 0 (not A) and Bi = 1 (B) yi =(b0+d02) + (b1 +d12 ) xi+ ei • Difference between the intercepts for A and C, d0 2 • d12, Difference between the slopes for A and C

  11. Test for the same linear relationship yi = b0+ b1 xi + d01 Ai +d02 Bi + d11 Ai xi+ d12 Bi xi+ ei • For Ai = 0 (not A) and Bi = 0 (not B) yi = b0+ b1 xi+ ei • Test if the three regression lines are the same • H0: d01 = d02=d11 = d12 = 0 • Partial F

  12. Seasonality in time series • Indicator variables for the four seasons • W = 1 if winter, = 0, otherwise • SP = 1 if spring, = 0, otherwise • SU = 1 if summer, = 0, otherwise • Fall is given by W = 0, SP = 0, SU = 0 • Indicator variables for months (11 variables) • Jan = 1 if January, = 0, otherwise • etc.

  13. An alternative coding scheme • Two categories, A and B(not A) • Define a variable • Ai = 1 for members of category A • Ai = -1 otherwise

  14. Regression model with different intercepts and Slopes yi = b0+ b1 xi+ q0 Ai+ q1 Ai xi+ ei • ForAi = 1 yi =(b0+q0 ) + (b1 +q1 ) xi+ ei • For Ai = -1 yi = (b0-q0 ) + (b1 -q1 )xi+ ei • “average” intercept b0 • “average” slope, b1 • Intercept deviation for each category, q0 • Slope deviation for each category, q1

  15. Regression model with different intercepts and Slopes • Hypothesis of equality of slopes for two categories, H0: q1 = 0 • Hypothesis of equality of intercepts for two categories, H0: q0 = 0 • Hypothesis of the same regression line for two categories • H0: q0 = q1 = 0 (Partial F)

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