ANOVA With More Than One IV 2 way

ANOVA With More Than One IV

2 -way ANOVA n n So far, 1 -Way ANOVA, but can have 2 or more IVs aka Factors. Example: Study aids for exam q q IV 1: workbook or not IV 2: 1 cup of coffee or not Workbook (Factor A) Caffeine (Factor B) No Yes Caffeine only Both No Neither (Control) Workbook only

Main Effects N=30 per cell Workbook (Factor A) Row Means Caffeine (Factor B) No Yes Caff =80 SD=5 Both =85 SD=5 82. 5 No Control =75 SD=5 Book =80 SD=5 77. 5 Col Means 77. 5 82. 5 80

Main Effects and Interactions n n n Main effects seen by row and column means; Slopes and breaks. Interactions seen by lack of parallel lines. Interactions are a main reason to use multiple IVs

Single Main Effect for B (Coffee only)

Single Main Effect for A (Workbook only)

Two Main Effects; Both A & B Both workbook and coffee

Interaction (1) Interactions take many forms; all show lack of parallel lines. Coffee has no effect without the workbook.

Interaction (2) People with workbook do better without coffee; people without workbook do better with coffee.

Interaction (3) Coffee always helps, but it helps more if you use workbook.

Labeling Factorial Designs n Levels – each IV is referred to by its number of levels, e. g. , 2 X 2, 3 X 2, 4 X 3 designs. Two by two factorial ANOVA.

Example Factorial Design (1) n n Effects of fatigue and alcohol consumption on driving performance. Fatigue q q n Rested (8 hrs sleep then awake 4 hrs) Fatigued (24 hrs no sleep) Alcohol consumption q q q None (control) 2 beers Blood alcohol. 08 %

Cells of the Design Alcohol (Factor A) Fatigue None (Factor B) 2 beers . 08 % Cell 3 Tired Cell 1 Cell 2 Rested Cell 4 Cell 5: Rested, Cell 6 2 beers, Porsche 911 DV – closed course driving performance ratings from instructors.

Factorial Example Results Main Effects? Interactions? Both main effects and the interaction appear significant.

ANOVA Summary Table Two Factor, Between Subjects Design Source SS Df MS F A SSA a-1 SSA/df. A MSA/MSError B SSB b-1 SSB/df. B MSB/MSError Ax. B SSAx. B (a-1)(b-1) SSAx. B/df. Ax. B MSAx. B/MSError SSError ab(n-1)or N-ab SSError/df. Error Total SSTotal N-1

Review n In a 3 X 3 ANOVA q q q How many IVs are there? How many df does factor A have How many df does the interaction have

Test We can see the main effect for a variable if we examine means of the dependent variable while ____ n q q Considering the joint effects of both variables Examining a single value of a second factor Examining each cell Ignoring the other variable

Test In two-way ANOVA, the term interaction means n q q Both IVs have an impact on the DV The effect of one IV depends on the value of the other IV The on IV has no effect unless the other IV has a certain value There is a crossover – a graph of two lines shows an ‘X’.