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
Total variability Between-treatments var. Within-treatments var. Interaction variability Factor A variability Factor B variability Partition of Variability
Example: • Arousal, task difficulty and performance • Yerkes-Dodson
Example • SStotal = ∑(X – grand mean)2 • SStotal = 360 • 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
No interaction Interaction 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?
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
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 RUN All Pulling it off in SPSS
/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.