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

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Presentation on theme: "General Linear Models -- #2 2x2 – main effects 2x2 with interactions 2x3 – main effects 2x3 with interactions 3x3 – main effects 3x3 with interactions."— Presentation transcript:

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

2 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 2-group (Tx Cx) & 2-group (Tz Cz) predictors Main Effects Model 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz = simple effects (no interaction) X = Tx vs. Cx y’ = b 0 + b 1 X + b 2 Z Z = Tz vs. Cz X C T T Z C CxCz CxTz TxCz CxCz CxTz TxTz Z Tz = 1 Cz = 0 X Tx = 1 Cx = 0

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

4 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors  2x2 ANOVA 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz X = Tx vs. Cx y’ = b 0 + b 1 X + b 2 Z + b 3 XZ Z = Tz vs. Cz TxCz CxCz CxTz TxTz Z Tz = 1 Cz = 0 X Tx = 1 Cx = 0 XZ = X * Z b 3 = dif htdifs of CxCz - TxCz & CxTz - TxTz b3b3 vs.

5 0 10 20 30 40 50 60 Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors  2x2 ANOVA 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz X = Tx vs. Cx y’ = b 0 + b 1 X + b 2 Z + b 3 XZ Z = Tz vs. Cz Z Tz = 1 Cz = 0 X Tx = 1 Cx = 0 XZ = X * Z b 3 = dif htdifs of CxCz - TxCz & CxTz - TxTz

6 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 Models with 2-group (Tx Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  ME model 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz 1 = simple effects (no interaction) y’ = b 0 + b 1 X + b 2 Z 1 + b 3 Z 2 b3b3 CxCz CxTz 1 CxTz 2 TxCz TxTz 1 TxTz 2 b 3 = htdifs of CxCz & CxTz 2 Z C T 1 T 2 C X T CxCz TxCzTxTz 1 CxTz 1 CxTz 2 TxTz 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X = Tx vs. Cx X Tx = 1 Cx = 0

7 0 10 20 30 40 50 60 Models with 2-group (Tx Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  ME model 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz 1 = simple effects (no interaction) y’ = b 0 + b 1 X + b 2 Z 1 + b 3 Z 2 b 3 = htdifs of CxCz & CxTz 2 Z C T 1 T 2 C X T CxCz TxCzTxTz 1 CxTz 1 CxTz 2 TxTz 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X = Tx vs. Cx X Tx = 1 Cx = 0

8 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 Models with 2-group (Tx Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  2x3 ANOVA 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz 1 y’ = b 0 + b 1 X + b 2 Z 1 + b 3 Z 2 +b 4 XZ 1 + b 5 XZ 2 b3b3 CxCz CxTz 1 CxTz 2 TxCz TxTz 1 TxTz 2 b 3 = htdifs of CxCz & CxTz 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X = Tx vs. Cx X Tx = 1 Cx = 0 XZ 1 = X * Z 1 XZ 2 = X * Z 2 b 4 = dif htdifs of CxCz - TxCz & CxTz 1 – TxTz 1 b 5 = dif htdifs of CxCz - TxCz & CxTz 2 – TxTz 2 b4b4 vs. b5b5

9 0 10 20 30 40 50 60 Models with 2-group (Tx Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  2x3 ANOVA 0 1  X b 0 = mean of CxCz b 1 = htdif of CxCz & TxCz b 2 = htdifs of CxCz & CxTz 1 y’ = b 0 + b 1 X + b 2 Z 1 + b 3 Z 2 +b 4 XZ 1 + b 5 XZ 2 b 3 = htdifs of CxCz & CxTz 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X = Tx vs. Cx X Tx = 1 Cx = 0 XZ 1 = X * Z 1 XZ 2 = X * Z 2 b 4 = dif htdifs of CxCz - TxCz & CxTz 1 – TxTz 1 b 5 = dif htdifs of CxCz - TxCz & CxTz 2 – TxTz 2

10 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  ME Model y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 X C T 1 T 2 T2ZT1CT2ZT1C CxTz 2 CxCz Tx 1 Cz Tx 1 Tz 2 Tx 2 Tz 2 Tx 2 Cz CxTz 1 Tx 1 Tz 1 b 0 = mean of CxCz b 1 = htdif of CxCz & Tx 1 Cz b 2 = htdifs of CxCz & Tz 2 Cz = simple effects (no interaction) b 3 = htdifs of CxCz & CxTz 1 b0b0 b1b1 b2b2 b3b3 CxCz CxTz 1 CxTz 2 Tx 2 Cz Tx 2 Tz 1 Tx 2 Tz 2 Tx 1 Cz Tx 1 Tz 1 Tx 1 Tz 2 b 4 = htdifs of CxCz & CxTz 2 b4b4 X1 Tx 1 =1 Tx 2 =0 Cx=0 X2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0

11 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors  ME Model y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 X C T 1 T 2 T2ZT1CT2ZT1C CxTz 2 CxCz Tx 1 Cz Tx 1 Tz 2 Tx 2 Tz 2 Tx 2 Cz CxTz 1 Tx 1 Tz 1 b 0 = mean of CxCz b 1 = htdif of CxCz & Tx 1 Cz b 2 = htdifs of CxCz & Tz 2 Cz = simple effects (no interaction) b 3 = htdifs of CxCz & CxTz 1 b 4 = htdifs of CxCz & CxTz 2 X1 Tx 1 =1 Tx 2 =0 Cx=0 X2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0

12 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction  3x3 ANOVA y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 + b 5 X 1 Z 1 + b 6 X 1 Z 2 + b 7 X 2 Z 1 + b 8 X 2 Z 2 b 0 = mean of CxCz b 1 = htdif of CxCz & Tx 1 Cz b 2 = htdifs of CxCz & Tz 2 Cz b 3 = htdifs of CxCz & CxTz 1 b0b0 b1b1 b2b2 b3b3 CxCz CxTz 1 CxTz 2 Tx 2 Cz Tx 2 Tz 1 Tx 2 Tz 2 Tx 1 Cz Tx 1 Tz 1 Tx 1 Tz 2 b 4 = htdifs of CxCz & CxTz 2 b4b4 X 1 Tx 1 =1 Tx 2 =0 Cx=0 X 2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X 1 Z 1 = X 1 * Z 1 X 1 Z 2 = X 1 * Z 2 X 2 Z 1 = X 2 * Z 1 X 2 Z 2 = X 2 * Z 2 Main Effects terms

13 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction  3x3 ANOVA y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 + b 5 X 1 Z 1 + b 6 X 1 Z 2 + b 7 X 2 Z 1 + b 8 X 2 Z 2 b5b5 CxCz CxTz 1 CxTz 2 Tx 2 Cz Tx 2 Tz 1 Tx 2 Tz 2 Tx 1 Cz Tx 1 Tz 1 Tx 1 Tz 2 X 1 Tx 1 =1 Tx 2 =0 Cx=0 X 2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X 1 Z 1 = X 1 * Z 1 X 1 Z 2 = X 1 * Z 2 X 2 Z 1 = X 2 * Z 1 X 2 Z 2 = X 2 * Z 2 Interaction terms b 5 = dif htdifs of CxCz - TxCz & CxTz 1 – TxTz 1 b 6 = dif htdifs of CxCz - TxCz & CxTz 2 – TxTz 2 b 7 = dif htdifs of CxCz – Tx 2 Cz & CxTz 1 – Tx 2 Tz 1 b 8 = dif htdifs of CxCz – Tx 2 Cz & CxTz 2 – Tx 2 Tz 2 vs. b6b6 b7b7 b8b8

14 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction  3x3 ANOVA y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 + b 5 X 1 Z 1 + b 6 X 1 Z 2 + b 7 X 2 Z 1 + b 8 X 2 Z 2 b 0 = mean of CxCz b 1 = htdif of CxCz & Tx 1 Cz b 2 = htdifs of CxCz & Tz 2 Cz b 3 = htdifs of CxCz & CxTz 1 b 4 = htdifs of CxCz & CxTz 2 X 1 Tx 1 =1 Tx 2 =0 Cx=0 X 2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X 1 Z 1 = X 1 * Z 1 X 1 Z 2 = X 1 * Z 2 X 2 Z 1 = X 2 * Z 1 X 2 Z 2 = X 2 * Z 2 Main Effects terms

15 0 10 20 30 40 50 60 0 1 2  X 3-group (Tx 1 Tx 2 Cx) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction  3x3 ANOVA y’ = b 0 + b 1 X + b 2 X + b 3 Z 1 + b 4 Z 2 + b 5 X 1 Z 1 + b 6 X 1 Z 2 + b 7 X 2 Z 1 + b 8 X 2 Z 2 X 1 Tx 1 =1 Tx 2 =0 Cx=0 X 2 Tx 1 =0 Tx 2 =1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 X 1 Z 1 = X 1 * Z 1 X 1 Z 2 = X 1 * Z 2 X 2 Z 1 = X 2 * Z 1 X 2 Z 2 = X 2 * Z 2 Interaction terms b 5 = dif htdifs of CxCz - TxCz & CxTz 1 – TxTz 1 b 6 = dif htdifs of CxCz - TxCz & CxTz 2 – TxTz 2 b 7 = dif htdifs of CxCz – Tx 2 Cz & CxTz 1 – Tx 2 Tz 1 b 8 = dif htdifs of CxCz – Tx 2 Cz & CxTz 2 – Tx 2 Tz 2

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