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Comprehensive Guide to General Linear Models: Main Effects & Interactions in Various Designs

Understanding general linear models with main effects and interactions in 2x2, 2x3, and 3x3 designs. Learn to interpret predictors and simple effects using ANOVA methodology.

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Comprehensive Guide to General Linear Models: Main Effects & Interactions in Various Designs

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  1. General Linear Models -- #2 • 2x2 – main effects • 2x2 with interactions • 2x3 – main effects • 2x3 with interactions • 3x3 – main effects • 3x3 with interactions

  2. 2-group (Tx Cx) & 2-group (Tz Cz) predictors Main Effects Model Z Tz = 1 Cz = 0 X Tx = 1 Cx = 0 X C T X = Tx vs. Cx Z = Tz vs. Cz TxTz CxTz T Z C TxCz CxCz y’ = b0 + b1X + b2Z TxTz b0 = mean of CxCz CxTz b1 = htdif of CxCz & TxCz TxCz b2 0 10 20 30 40 50 60 b1 b2 = htdifs of CxCz & CxTz CxCz b0 = simple effects (no interaction) 0 1  X

  3. 2-group (Tx Cx) & 2-group (Tz Cz) predictors Main Effects Model Z Tz = 1 Cz = 0 X Tx = 1 Cx = 0 X C T X = Tx vs. Cx Z = Tz vs. Cz TxTz CxTz T Z C TxCz CxCz y’ = b0 + b1X + b2Z b0 = mean of CxCz b1 = htdif of CxCz & TxCz 0 10 20 30 40 50 60 b2 = htdifs of CxCz & CxTz = simple effects (no interaction) 0 1  X

  4. Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors  2x2 ANOVA Z Tz = 1 Cz = 0 XZ = X * Z X Tx = 1 Cx = 0 X = Tx vs. Cx Z = Tz vs. Cz y’ = b0 + b1X + b2Z + b3XZ b0 = mean of CxCz b1 = htdif of CxCz & TxCz CxTz b3 TxTz vs. b2 = htdifs of CxCz & CxTz b2 0 10 20 30 40 50 60 TxCz b1 b3 = dif htdifs of CxCz - TxCz & CxTz - TxTz CxCz b0 0 1  X

  5. Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors  2x2 ANOVA Z Tz = 1 Cz = 0 XZ = X * Z X Tx = 1 Cx = 0 X = Tx vs. Cx Z = Tz vs. Cz y’ = b0 + b1X + b2Z + b3XZ b0 = mean of CxCz b1 = htdif of CxCz & TxCz b2 = htdifs of CxCz & CxTz 0 10 20 30 40 50 60 b3 = dif htdifs of CxCz - TxCz & CxTz - TxTz 0 1  X

  6. Z C T1 T2 Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors  ME model CxTz2 CxTz1 CxCz C X T TxCz TxTz1 TxTz2 X Tx = 1 Cx = 0 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz X = Tx vs. Cx y’ = b0 + b1X + b2Z1 + b3Z2 b0 = mean of CxCz TxTz2 b1 = htdif of CxCz & TxCz TxTz1 CxTz2 b3 b2 = htdifs of CxCz & CxTz1 CxTz1 TxCz 0 10 20 30 40 50 60 b1 b2 b3 = htdifs of CxCz & CxTz2 CxCz = simple effects (no interaction) b0 0 1  X

  7. Z C T1 T2 Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors  ME model CxTz2 CxTz1 CxCz C X T TxCz TxTz1 TxTz2 X Tx = 1 Cx = 0 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz X = Tx vs. Cx y’ = b0 + b1X + b2Z1 + b3Z2 b0 = mean of CxCz b1 = htdif of CxCz & TxCz b2 = htdifs of CxCz & CxTz1 0 10 20 30 40 50 60 b3 = htdifs of CxCz & CxTz2 = simple effects (no interaction) 0 1  X

  8. Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors  2x3 ANOVA X Tx = 1 Cx = 0 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 XZ1 = X * Z1 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz XZ2 = X * Z2 X = Tx vs. Cx b0 = mean of CxCz y’ = b0 + b1X + b2Z1 + b3Z2 +b4XZ1 + b5XZ2 b1 = htdif of CxCz & TxCz TxTz2 b2 = htdifs of CxCz & CxTz1 b5 CxTz2 vs. b3 = htdifs of CxCz & CxTz2 b3 TxTz1 b4 CxTz1 TxCz vs. 0 10 20 30 40 50 60 b4 = dif htdifs of CxCz - TxCz & CxTz1 – TxTz1 b2 b1 CxCz b5 = dif htdifs of CxCz - TxCz & CxTz2 – TxTz2 b0 0 1  X

  9. Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors  2x3 ANOVA X Tx = 1 Cx = 0 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 XZ1 = X * Z1 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz XZ2 = X * Z2 X = Tx vs. Cx b0 = mean of CxCz y’ = b0 + b1X + b2Z1 + b3Z2 +b4XZ1 + b5XZ2 b1 = htdif of CxCz & TxCz b2 = htdifs of CxCz & CxTz1 b3 = htdifs of CxCz & CxTz2 0 10 20 30 40 50 60 b4 = dif htdifs of CxCz - TxCz & CxTz1 – TxTz1 b5 = dif htdifs of CxCz - TxCz & CxTz2 – TxTz2 0 1  X

  10. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors  ME Model X C T1 T2 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx Tx1Tz2 CxTz2 Tx2Tz2 T2 Z T1 C Tx1Tz1 Tx1Tz1 CxTz1 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz CxCz Tx1Cz Tx2Cz y’ = b0 + b1X + b2X + b3Z1 + b4Z2 b0 = mean of CxCz Tx1Tz2 Tx2Tz2 b1 = htdif of CxCz & Tx1Cz CxTz2 Tx1Tz1 Tx2Tz1 b4 b2 = htdifs of CxCz & Tz2Cz Tx1Cz CxTz1 Tx2Cz 0 10 20 30 40 50 60 b3 = htdifs of CxCz & CxTz1 b3 b1 b2 b4 = htdifs of CxCz & CxTz2 CxCz b0 = simple effects (no interaction) 0 1 2  X

  11. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors  ME Model X C T1 T2 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx Tx1Tz2 CxTz2 Tx2Tz2 T2 Z T1 C Tx1Tz1 Tx1Tz1 CxTz1 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz CxCz Tx1Cz Tx2Cz y’ = b0 + b1X + b2X + b3Z1 + b4Z2 b0 = mean of CxCz b1 = htdif of CxCz & Tx1Cz b2 = htdifs of CxCz & Tz2Cz 0 10 20 30 40 50 60 b3 = htdifs of CxCz & CxTz1 b4 = htdifs of CxCz & CxTz2 = simple effects (no interaction) 0 1 2  X

  12. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction  3x3 ANOVA X1Z1 = X1 * Z1 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx X1Z2 = X1 * Z2 X2Z1 = X2 * Z1 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 X2Z2 = X2 * Z2 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz y’ = b0 + b1X + b2X + b3Z1 + b4Z2 + b5X1Z1 + b6X1Z2 + b7X2Z1 + b8X2Z2 Main Effects terms Tx1Tz2 Tx2Tz2 b0 = mean of CxCz CxTz2 Tx2Tz1 b1 = htdif of CxCz & Tx1Cz b4 Tx1Tz1 Tx2Cz CxTz1 0 10 20 30 40 50 60 b2 = htdifs of CxCz & Tz2Cz Tx1Cz b3 b1 b2 b3 = htdifs of CxCz & CxTz1 CxCz b0 b4 = htdifs of CxCz & CxTz2 0 1 2  X

  13. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction  3x3 ANOVA X1Z1 = X1 * Z1 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx X1Z2 = X1 * Z2 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 X2Z1 = X2 * Z1 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz X2Z2 = X2 * Z2 y’ = b0 + b1X + b2X + b3Z1 + b4Z2 + b5X1Z1 + b6X1Z2 + b7X2Z1 + b8X2Z2 Interaction terms b5 = dif htdifs of CxCz - TxCz & CxTz1 – TxTz1 Tx1Tz2 Tx2Tz2 CxTz2 vs. vs. b6 = dif htdifs of CxCz - TxCz & CxTz2 – TxTz2 Tx2Tz1 b6 b8 Tx1Tz1 CxTz1 Tx2Cz 0 10 20 30 40 50 60 Tx1Cz vs. vs. b7 = dif htdifs of CxCz – Tx2Cz & CxTz1 – Tx2Tz1 b5 b7 CxCz b8 = dif htdifs of CxCz – Tx2Cz & CxTz2 – Tx2Tz2 0 1 2  X

  14. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction  3x3 ANOVA X1Z1 = X1 * Z1 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx X1Z2 = X1 * Z2 X2Z1 = X2 * Z1 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 X2Z2 = X2 * Z2 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz y’ = b0 + b1X + b2X + b3Z1 + b4Z2 + b5X1Z1 + b6X1Z2 + b7X2Z1 + b8X2Z2 Main Effects terms b0 = mean of CxCz b1 = htdif of CxCz & Tx1Cz 0 10 20 30 40 50 60 b2 = htdifs of CxCz & Tz2Cz b3 = htdifs of CxCz & CxTz1 b4 = htdifs of CxCz & CxTz2 0 1 2  X

  15. 3-group (Tx1 Tx2 Cx) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction  3x3 ANOVA X1Z1 = X1 * Z1 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx X1Z2 = X1 * Z2 Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0 X2Z1 = X2 * Z1 Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz X2Z2 = X2 * Z2 y’ = b0 + b1X + b2X + b3Z1 + b4Z2 + b5X1Z1 + b6X1Z2 + b7X2Z1 + b8X2Z2 Interaction terms b5 = dif htdifs of CxCz - TxCz & CxTz1 – TxTz1 b6 = dif htdifs of CxCz - TxCz & CxTz2 – TxTz2 0 10 20 30 40 50 60 b7 = dif htdifs of CxCz – Tx2Cz & CxTz1 – Tx2Tz1 b8 = dif htdifs of CxCz – Tx2Cz & CxTz2 – Tx2Tz2 0 1 2  X

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