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Introduction to the General Linear Model (GLM)

Introduction to the General Linear Model (GLM). 1 quantitative variable & 1 2-group variable 1a  main effects model with no interaction 1b  interaction model 1 quantitative variable & 1 3-group variable 2a  main effects model with no interaction 2b  interaction model.

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Introduction to the General Linear Model (GLM)

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  1. Introduction to the General Linear Model (GLM) • 1 quantitative variable & 1 2-group variable • 1a  main effects model with no interaction • 1b  interaction model • 1 quantitative variable & 1 3-group variable • 2a  main effects model with no interaction • 2b  interaction model

  2. There are two important variations of each of these models • Main effects model • Centered or coded terms for each variable • No interaction – assumes regression slope homogeneity • b-weights for binary & quant variables each represent main effect of that variable • 2. Interaction model • Centered or coded terms for each variable • Term for interaction - does not assume reg slp homogen !! • b-weights for binary & quant variables each represent the simple effect of that variable when the other variable = 0 • b-weight for the interaction term represented how the simple effect of one variable changes with changes in the value of the other variable (e.g., the extent and direction of the interaction)

  3. #1a  centered quant variable & dummy coded 2-grp variable y’ = b0+ b1x+ b2z “X” is a centered quantitative variable X  X – Xmean “Z” is a dummy-coded 2-group variable (Cz = 0 & Tx = 1) Z Tz = 1 Cz = 0

  4. #1a  centered quant variable & dummy coded 2-grp variable y’ = b0+ b1x+ b2z • b0 mean of those in Cz with X=0 (mean) • b1 slope of Y-X regression line for Cz (=0) • - slope same for both groups  no interaction • b2  group difference for X=mean (=0) • - group different same for all values of X  no interaction

  5. #1a quantitative (Xcen) & 2-group (Tz=1 Cz=0) y’ = b0 + b1X + b2Z 20 5 10 b0 = ht of Cz line b1 = slp of Cz line 0 10 20 30 40 50 60 b2 = htdif Cz & Tz Tz Z-lines have same slp (no interaction) Cz -2 -1 0 1 2  Xcen

  6. #1b  centered quant var, dummy coded 2-group var & their product term/interaction y’ = b0+ b1x+ b2z+ b3xz “X” is a centered quantitative variable X  X – Xmean “Z” is a dummy-coded 2-group variable Z Tz = 1 Cz = 0 “XZ” represents the interaction of “X” and “Z” XZX*Z

  7. #1b  centered quant var, dummy coded 2-group var & their product term/interaction y’ = b0+ b1x+ b2z+ b3xz • b0 mean of those in Cz with X= 0 (mean) • b1 slope of Y-X regression line for Cz (=0)* • b2  group difference for X=0 (mean)* • b3  how slope of y-x reg line for Tz (=1) differs from slope of y-x reg line for Cz (=0) • * Because the interaction is included, slopes may be different for different grps • * Because the interaction is included, group differences may be different for different X values

  8. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ 30 15 15 -5 b0 = ht of Cz line Tz b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz Cz -2 -1 0 1 2  Xcen

  9. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  10. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  11. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  12. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  13. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  14. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  15. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  16. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  17. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  18. #1b quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b0 + b1X + b2Z + b3XZ b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -2 -1 0 1 2  Xcen

  19. #2a  centered quant var & dummy coded 3-grp var y’ = b0+ b1x+ b2z1+ b3z2 “X” is centered quantitative variable X  X – Xmean “Z1” & “Z2” are dummy-codes for the 3-group variable Z1 Tz1 = 1 Tz2 = 0 Cz = 0 Z2  Tz1 = 0 Tz2 = 1 Cz = 0

  20. #2a  centered quant var & dummy coded 3-grp var y’ = b0+ b1x+ b2z1+ b3z2 • b0 mean of those in Cz with X=0 (mean) • b1 slope of Y-X regression line for Cz (=0) • - slope same for all groups  no interaction • b2  Tz1 - Cz difference for X=mean (=0) • - group different same for all values of X  no interaction • b3  Tz2 - Cz difference for X=mean (=0) • - group different same for all values of X  no interaction

  21. #2a quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 y’ = b0 + b1X+ b2Z1 + b3Z2 35 5 5 -15 b0 = ht of Cz line Tz2 b1 = slp of Cz line Cz b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 Tz1 b3 = htdif Cz & Tz2 Z-lines have same slp (no interaction) -2 -1 0 1 2  X

  22. #2b  centered quant var, dummy coded 3-group var & their product terms/interaction y’ = b0+ b1x+ b2z1+ b3z2+ b4xz1+ b5xz2 “X” is centered quantitative variable X  X – Xmean “Z1” & “Z2” are dummy-codes for the 3-group variable Z1 Tz1 = 1 Tz2 = 0 Cz = 0 Z2  Tz1 = 0 Tz2 = 1 Cz = 0 “XZ1” & “XZ2” represent the interaction of “X” and “Z” XZ1X*Z1 XZ2X*Z2

  23. #2b  centered quant var, dummy coded 3-group var & their product terms/interaction y’ = b0+ b1x+ b2z1+ b3z2+ b4xz1+ b5xz2 • b0 mean of those in Cz with X= 0 (mean) • b1 slope of Y-X regression line for Cz • b2  Tz1 - Cz difference for X=0 (mean)* • b3  Tz2 - Cz difference for X=0 (mean)* • b4  how slope of y-x reg line for Tz1 differs from slope of y-x reg line for Cz * • b4  how slope of y-x reg line for Tz2 differs from slope of y-x reg line for Cz * • *Because the interaction is included, group differences may be different for different X values • * Because the interaction is included, slopes may be different for different grps

  24. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 -8 2 5 30 15 5 b0 = ht of Cz line b1 = slp of Cz line Tx2 b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 Tx1 b3 = htdif Cz & Tz2 Cx b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  25. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  26. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  27. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  28. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  29. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  30. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  31. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  32. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  33. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

  34. #2b quantitative (Xcen) & 3-group Z1 Tz1 = 1 Tz2 = 0 Cz = 0 • Z2  Tz1=0 Tz2 = 1 Cz = 0 • and interactions XZ1 = X*Z1 XZ2 = X*Z2 y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -2 -1 0 1 2  Xcen

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