Introduction to Factorial ANOVA Designs

Factorial Anova With factorial Anova we have more than one independent variable The terms 2-way, 3-way etc. refer to how many IVs there are in the analysis The following will discuss 2-way design but may extended to more complex designs. The analysis of interactions constitutes the focal point of factorial design

Recall the one-way Anova Total variability comes from: Differences between groups Differences within groups

Factorial Anova With factorial designs we have additional sources of variability to consider Main effects Mean differences among the levels of a particular factor Interaction Differences among cell means not attributable to main effects When the effect of one factor is influenced by the levels of another

Within-treatments var. Partition of Variability Total variability Between-treatments var. Within-treatments var. Factor A variability Factor B Interaction

Example: Arousal, task difficulty and performance Yerkes-Dodson

Example SStotal = ∑(X – grand mean)2 Df = N – 1 = 29 SSb/t =∑n(cell means – grand mean)2 = 5(3-4)2 + … 5(1-4)2 SSb/t =240 Df= K# of cells – 1 = 5 SSw/in = ∑(X – respective cell means)2 or SStotal- SSb/t SSw/in = 120 Df = N-K = 24

Sums of Squares Between SSDifficulty = n(row means –grand mean)2 = 15(6-4)2 + 15(2-4)2 = 120 Df = # of rows (levels) – 1 = 1 SSArousal = n(col means – grand mean)2 = 10(2-4)2 + 10(5-4)2 + 10(5-4)2 = 60 Df = # of columns (levels) – 1 = 2 SSDXA = SSb/t - SSDifficulty - SSArousal = 240 - 120 - 60 = 60 Df = dfb/t - dfDifficulty – dfArousal = 5-1-2 = 2 Or dfdiff X dfarous

Output Mean squares and F-statistics are calculated as before

Initial Interpretation Significant main effects of task difficulty and arousal level, as well as a significant interaction Difficulty Better performance for easy items Arousal Low worst Interaction Easy better in general but much more so with high arousal

Eta-squared is given as the effect size for B/t groups (SSeffect/SStotal) Partial eta-squared is given for the remaining factors: SSeffect/(SSeffect + SSerror) End result: significance w/ large effect sizes

Graphical display of interactions Two ways to display previous results

Graphical display of interactions What are we looking for? Do the lines behave similarly (are parallel) or not? Does the effect of one factor depend on the level of the other factor? No interaction Interaction

The general linear model Recall for the general one-way anova Where: μ = grand mean  = effect of Treatment A (μa – μ) ε = within cell error So a person’s score is a function of the grand mean, the treatment mean, and within cell error

Effects for 2-way Population main effect associated with the treatment Aj (first factor): Population main effect associated with treatment Bk (second factor): The interaction is defined as , the joint effect of treatment levels j and k (interaction of  and ) so the linear model is: Each person’s score is a function of the grand mean, the treatment means, and their interaction (plus w/in cell error).

The general linear model The interaction is a residual: Plugging in  and  leads to:

Partitioning of total sum of squares Squaring yields Interaction sum of squares can be obtained as remainder

Partitioning of total sum of squares SSA: factor A sum of squares measures the variability of the estimated factor A level means The more variable they are, the bigger will be SSA Likewise for SSB SSAB is the AB interaction sum of squares and measures the variability of the estimated interactions

GLM Factorial ANOVA Statistical Model: Statistical Hypothesis: The interaction null is that the cell means do not differ significantly (from the grand mean) outside of the main effects present, i.e. that this residual effect is zero

Interpretation: sig main fx and interaction Note that with a significant interaction, the main effects are understood only in terms of that interaction In other words, they cannot stand alone as an explanation and must be qualified by the interaction’s interpretation Some take issue with even talking about the main effects, but noting them initially may make the interaction easier for others to understand when you get to it

Interpretation: sig main fx and interaction However, interpretation depends on common sense, and should adhere to theoretical considerations Plot your results in different ways If main effects are meaningful, then it makes sense to talk about them, whether or not an interaction is statistically significant or not E.g. note that there is a gender effect but w/ interaction we now see that it is only for level(s) X of Factor B To help you interpret results, test simple effects Is simple effect of A significant within specific levels of B? Is simple effect of B significant within specific levels of A?

Simple effects Analysis of the effects of one factor at one level of the other factor Some possibilities from previous example Arousal for easy items (or hard items) Difficulty for high arousal condition (or medium or low)

Simple effects SSarousal for easy items = 5(3-6)2 + 5(6-6)2 + 5(9-6)2 = 90 SSarousal for difficult items = 5(1-2)2 + 5(4-2)2 + 5(1-2)2 = 30 SSdifficulty at lo = 5(3-2)2 + 5(1-2)2 = 10 SSdifficulty at med = 5(6-5)2 + 5(4-5)2 = 10 SSdifficulty at hi = 5(9-5)2 + 5(1-5)2 = 160

Simple effects Note that the simple effect represents a partitioning of SSmain effect and SSinteraction NOT JUST THE INTERACTION!! From Anova table: SSarousal + SSarousal by difficulty = 60 + 60 = 120 SSarousal for easy items = 90 SSarousal for difficult items = 30 90 + 30 = 120 SSdifficulty + SSarousal by difficulty = 120 + 60 = 180 SSdifficulty at lo = 10 SSdifficulty at med = 10 SSdifficulty at hi = 160 10 + 10 + 160 = 180

Pulling it off in SPSS Paste!

Pulling it off in SPSS Add /EMMEANS = tables(a*b)compare(a) /EMMEANS = tables(a*b)compare(b)

Pulling it off in SPSS Output

Test for simple fx with no sig interaction? What if there was no significant interaction, do I still test for simple effects? Maybe, but more on that later A significant simple effect suggests that at least one of the slopes across levels is significantly different than zero However, one would not conclude that the interaction is ‘close enough’ just because there was a significant simple effect The nonsig interaction suggests that the slope seen is not statistically different from the other(s) under consideration.

Multiple comparisons and contrasts For main effects multiple comparisons and contrasts can be conducted as would be normally One would have all the same considerations for choosing a particular method of post hoc analysis or weights for contrast analysis

Multiple comparisons and contrasts With interactions post hocs can be run comparing individual cell means The problem is that it rarely makes theoretical sense to compare many of the pairs of means under consideration

Contrasts for interactions We may have a specific result to look for with regard to our interaction For example, we may think based on past research moderate arousal should result in optimal performance for difficult items We would assign contrast weights to reflect this hypothesis

Pulling it off in SPSS Analyze General Linear Model Univariate Select Dependent Variable and Specify Fixed and/or Random Factor(s) (Treatment Groups and or Patient Characteristic(s), Treatment Sites, etc.) Paste Launches Syntax Window Add /LMATRIX command lines All RUN

/LMATRIX Command /LMATRIX ‘<Title for 1st Contrast>’ <Specify Weights for 1st Contrast>; ‘<Title for 2nd Contrast>’ <Specify Weights for 2nd Contrast>; … ‘<Title for Final Contrast>’ <Specify Weights for Final Contrast>

For this 3 X 2 design the weights will order as follows: A1B1 A1B2 A2B1 A2B2 A3B1 A3B2 Note for this example, SPSS is analyzing categories in alphabetical order Arousal hi lo med Task Diff Easy In other words Hi:Difficult Hi:Easy Lo:Difficult … Med:Easy

As alluded to previously it is possible to have: Sig overall F Sig contrast Nonsig posthoc Nonsig F Nonsig contrast e.g. 1 & 3 VS. 2 Sig posthoc 1 vs. 2 sig

A different model ☺ If cognitive anxiety is low, then the performance effects of physiological arousal will be low; but if it is high, the effects will be large and sudden.