ANOVA With More Than One IV

2-way ANOVA So far, 1-Way ANOVA, but can have 2 or more IVs. IVs aka Factors. Example: Study aids for exam  IV 1: workbook or not  IV 2: 1 cup of coffee or not Workbook (Factor A) Caffeine (Factor B) NoYes Caffeine only Both NoNeither (Control) Workbook only

Main Effects N=30 per cell Workbook (Factor A)Row Means Caffeine (Factor B) NoYes Caff =80 SD=5 Both =85 SD= NoControl =75 SD=5 Book =80 SD= Col Means

Main Effects and Interactions 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 Levels – each IV is referred to by its number of levels, e.g., 2X2, 3X2, 4X3 designs. Two by two factorial ANOVA.

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

Cells of the Design Alcohol (Factor A) Fatigue (Factor B) None2 beers.08 % Tired Cell 1Cell 2Cell 3 Rested Cell 4Cell 5:Rested, 2 beers, Porsche 911 Cell 6 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 SourceSSDfMSF ASS A a-1SS A /df A MS A /MS Error BSS B b-1SS B /df B MS B /MS Error AxBSS AxB (a-1)(b-1)SS AxB /df AxB MS AxB /MS Error ErrorSS Error ab(n-1)or N-ab SS Error /df Error TotalSS Total N-1 Two Factor, Between Subjects Design

Review In a 3 X 3 ANOVA  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 ________  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  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’.