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Data analysis markus brauer


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Orthogonal contrasts


Contrasts for a 2 x 2 ANOVA:
Main Main Inter-

Variable 1 Variable 2 effect effect action

Group (V1) (V2) V1 V2 V1 X V2

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[1] 1st level 1st level –1 –1 +1

[2] 1st level 2nd level –1 +1 –1

[3] 2nd level 1st level +1 –1 –1

[4] 2nd level 2nd level +1 +1 +1



Contrasts for a 2 x 2 x 2 ANOVA:
Main Main Main Inter- Inter- Inter- Inter-

Var. 1 Var. 2 Var. 3 effect effect effect action action action action

Gr. (V1) (V2) (V3) V1 V2 V3 V1 X V2 V1 X V3 V2 X V3 V1 X V2 X V3

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[1] 1st lev. 1st lev. 1st lev. –1 –1 –1 +1 +1 +1 –1

[2] 1st lev. 1st lev. 2nd lev. –1 –1 +1 +1 –1 –1 +1

[3] 1st lev. 2nd lev. 1st lev. –1 +1 –1 –1 +1 –1 +1

[4] 1st lev. 2nd lev. 2nd lev. –1 +1 +1 –1 –1 +1 –1

[5] 2nd lev. 1st lev. 1st lev. +1 –1 –1 –1 –1 +1 +1

[6] 2nd lev. 1st lev. 2nd lev. +1 –1 +1 –1 +1 –1 –1

[7] 2nd lev. 2nd lev. 1st lev. +1 +1 –1 +1 –1 –1 –1

[8] 2nd lev. 2nd lev. 2nd lev. +1 +1 +1 +1 +1 +1 +1



Contrasts for a 2 x 3 ANOVA:
Main V2 V2 Inter- Inter-

Variable 1 Variable 2 effect Contrast 1 Contrast 2 action action

Group (V1) (V2) V1 (C1) (C2) V1 X C1 V1 X C2

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[1] 1st lev. 1st lev. –1 –1 –1 +1 +1

[2] 1st lev. 2nd lev. –1 –1 +1 +1 –1

[3] 1st lev. 3rd lev. –1 +2 0 –2 0

[4] 2nd lev. 1st lev. +1 –1 –1 –1 –1

[5] 2nd lev. 2nd lev. +1 –1 +1 –1 –1

[6] 2nd lev. 3rd lev. +1 +2 0 +2 0


The main effect of Variable 2 is the following comparison (2 df):
C: Y = b0 + b1V1 + b2(V1*C1) + b3(V1*C2)

A: Y = b0 + b1V1 + b2C1 + b3C2 + b4(V1*C1) + b5(V1*C2)


The interaction effect between Variables 1 et 2 is the following comparison (2 df):
C: Y = b0 + b1V1 + b2C1 + b3C2

A: Y = b0 + b1V1 + b2C1 + b3C2 + b4(V1*C1) + b5(V1*C2)



Polynomial contrasts (3 groups):
Linear Quadratic

Group trend trend

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[1] –1 –1

[2] 0 +2

[3] +1 –1



Polynomial contrasts (4 groups):
Linear Quadratic Cubic

Group trend trend trend

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[1] –3 +1 –1

[2] –1 –1 +3

[3] +1 –1 –3

[4] +3 +1 +1

Polynomial contrasts (5 groups):
Linear Quadratic Cubic Quartic

Group trend trend trend trend

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[1] –2 +2 –1 +1

[2] –1 –1 +2 –4

[3] 0 –2 0 6

[4] +1 –1 –2 –4

[5] +2 +2 +1 +1




Helmert contrasts (3 groups):
Group [1] vs. [2]-[3] [2] vs. [3]

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[1] –2 0

[2] +1 –1

[3] +1 +1

Helmert contrasts (4 groups):
Group [1] vs. [2]-[4] [2] vs. [3]-[4] [3] vs. [4]

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[1] –3 0 0

[2] +1 –2 0

[3] +1 +1 –1

[4] +1 +1 +1



Helmert contrasts (5 groups):
Group [1] vs. [2]-[5] [2] vs. [3]-[5] [3] vs. [4]-[5] [4] vs. [5]

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[1] –4 0 0 0

[2] +1 –3 0 0

[3] +1 +1 –2 0

[4] +1 +1 +1 –1



[5] +1 +1 +1 +1



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