Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV Factorial ANOVA = ANOVA with 2+ IVs and one DV Factorial ANOVA Notation: 2 x 3 x 4 ANOVA The number of numbers = the number of IVs The numbers themselves = the number of levels in each IV Factorial ANOVA 2 x 3 x 4 ANOVA = an ANOVA with 3 IVs, one of which has 2 levels, one of which has 3 levels, and the last of which has 4 levels Why use a factorial ANOVA? Why not just use multiple one-way ANOVAs? 1. Increased power with the same sample size and effect size, a factorial ANOVA is more likely to result in the rejection of Ho aka with equal effect size and probability of

rejecting Ho if it is true (), you can use fewer subjects (and time and money) Factorial ANOVA Why use a factorial ANOVA? Why not just use multiple one-way ANOVAs? 2. With 3 IVs, youd need to run 3 one-way ANOVAs, which would inflate your -level However, this could be corrected with a Bonferroni Correction 3. The best reason is that a factorial ANOVA can detect interactions, something that multiple one-way ANOVAs cannot do Factorial ANOVA Interaction: when the effects of one independent variable differ according to levels of another independent variable Ex. We are testing two IVs, Gender (male and female) and Age (young, medium, and old) and their effect on performance If males performance differed as a function of age, i.e. males performed better or worse with age, but females performance was the same across ages, we would say that Age and Gender interact, or that we have an Age x Gender interaction

Factorial ANOVA Interaction: Presented graphically: 40 Note how males 30 20 Performance performance changes as a function of age while females does not Note also that the lines cross one another, this is the hallmark of an interaction, and why interactions are sometimes called cross-over or disordinal interactions 10 GENDER

Male 0 Female Young AGE Medium Old Factorial ANOVA Interactions: However, it is not necessary that the lines cross, only that the slopes differ from one another I.e. one line can be flat, and the other sloping upward, but not cross this is still an interaction See Fig. 17.2 on page 410 in the text for more examples Factorial ANOVA As opposed to interactions, we have what are called main effects:

the effect of an IV independent of any other IVs This is what we were looking at with one-way ANOVAs if we have a significant main effect of our IV, then we can say that the mean of at least one of the groups/levels of that IV is different than at least one of the other groups/levels Factorial ANOVA Finally, we also have simple effects: the effect of one group/level of our IV at one group/level of another IV Using our example earlier of the effects of Gender (Men/Women) and Age (Young/Medium/Old) on Performance, to say that young women outperformed other groups would be to talk about a simple effect Factorial ANOVA Calculating a Factorial ANOVA: First, we have to divide our data into cells the data represented by our simple effects If we have a 2 x 3 ANOVA, as in our Age and Gender example, we have 3 x 2 = 6 cells

Young Medium Old Male Cell #1 Cell #2 Cell #3 Female Cell #4 Cell #5 Cell #6 Factorial ANOVA Then we calculate means for all of these cells, and for our IVs across cells Mean #1 = Mean for Young Males only Mean #2 = Mean for Medium Males only

Mean #3 = Mean for Old Males Mean #4 = Mean for Young Females Mean #5 = Mean for Medium Females Mean #6 = Mean for Old Females Mean #7 = Mean for all Young people (Male and Female) Mean #8 = Mean for all Medium people (Male and Female) Mean #9 = Mean for all Old people (Male and Female) Mean #10 = Mean for all Males (Young, Medium, and Old) Mean #11 = Mean for all Females (Young, Medium, and Old) Young Medium Old Male Mean #1 Mean #2 Mean #3 Mean #10 Female Mean #4

Mean #5 Mean #6 Mean #11 Mean #7 Mean #8 Mean #9 Factorial ANOVA We then calculate the Grand Mean X .. ( ) This remains (X)/N, or all of our observations added together, divided by the number of observations We can also calculate SStotal, which is 2 in a onealso calculated the same as X 2 way ANOVAX

N Factorial ANOVA Next we want to calculate our SS terms for our IVs, something new to factorial ANOVA SSIV = nx(X IV -X )2 .. n = number of subjects per group/level of our IV x = number of groups/levels in the other IV Factorial ANOVA SSIV = nx(X IV - X .. )2 1. Subtract the grand mean from each of our levels means

2. 3. 4. For SSgender, this would involve subtracting the mean for males from the grand mean, and the mean for females from the grand mean Note: The number of values should equal the number of levels of your IV Square all of these values Add all of these values up Multiply this number by the number of subjects in each cell x the number of levels of the other IV 5. Repeat for any IVs Using the previous example, we would have both SSgender and SSage Factorial ANOVA Next we want to calculate SScells, which has a formula similar to SSIV SScells =

n X cell X .. 2 1. Subtract the grand mean from each of our cell means Note: The number of values should equal the number of cells 2. Square all of these values 3. Add all of these values up 4. Multiply this number by the number of subjects in each cell Factorial ANOVA Now that we have SStotal, the SSs for our IVs, and SScells, we can find SSerror and the SS for our interaction term, SSint SSint = SScells SSIV1 SSIV2 etc Going back to our previous example, SSint = SScells SSgender SSage SSerror = SStotal SScells Factorial ANOVA Similar to a one-way ANOVA, factorial ANOVA uses df to obtain MS

dftotal = N 1 dfIV = k 1 Using the previous example, dfage = 3 (Young/Medium/ Old) 1 = 2 and dfgender = 2 (Male/Female) 1 = 1 dfint = dfIV1 x dfIV2 x etc Again, using the previous example, dfint = 2 x 1 = 2 dferror = dftotal dfint - dfIV1 dfIV2 etc Factorial ANOVA Factorial ANOVA provides you with Fstatistics for all main effects and interactions Therefore, we need to calculate MS for all of our IVs (our main effects) and the interaction MSIV = SSIV/dfIV We would do this for each of our IVs MSint = SSint/dfint MSerror = SSerror/dferror Factorial ANOVA We then divide each of our MSs by MSerror to obtain our F-statistics Finally, we compare this with our critical F to determine if we accept or reject Ho All of our main effects and our interaction have

their own critical Fs Just as in the one-way ANOVA, use table E.3 or E.4 depending on your alpha level (.05 or .01) Just as in the one-way ANOVA, df numerator = the df for the term in question (the IVs or their interaction) and df denominator = dferror Factorial ANOVA Just like in a one-way ANOVA, a significant F in factorial ANOVA doesnt tell you which groups/levels of your IVs are different There are several possible ways to determine where differences lie Factorial ANOVA Multiple Comparison Techniques in Factorial ANOVA: 1. Several one-way ANOVAs (as many as there are IVs) with their corresponding multiple comparison techniques probably the most common method Dont forget the Bonferroni Method 2. Analysis of Simple Effects Calculate MS for each cell/simple effect, obtain an F

for each one and determine its associated p-value See pages 411-413 in your text you should be familiar with the theory of the technique, but you will not be asked to use it on the Final Exam Factorial ANOVA Multiple Comparison Techniques in Factorial ANOVA: In addition, interactions must be decomposed to determine what they mean A significant interaction between two variables means that one IVs value changes as a function of the other, but gives no specific information The most simple and common method of interpreting interactions is to look at a graph 40 30 Performance 20 10 GENDER

Male 0 Female Young Medium Old AGE Interpreting Interactions: In the example above, you can see that for Males, as age increases, Performance increases, whereas for Females there is no relation between Age and Performance To interpret an interaction, we graph the DV on the y-axis, place one IV on the x-axis, and define the lines by the other IV You may have to try switching the IVs if you dont get a nice interaction pattern the first time Factorial ANOVA Effect Size in Factorial ANOVA: 2 (eta squared) = SSIV/SStotal (for any of the IVs) or SSint/SStotal (for the interaction)

tells you the percent of variability in the DV accounted for by the IV/interaction like the one-way ANOVA, very easily computed and commonly used, but also very biased dont ever use it Factorial ANOVA Effect Size in Factorial ANOVA: SS IV df IV MS error 2 (omega squared) = SS int df int MS error or SS total MS error SS total MS error also provides an estimate of the percent of variability in the DV accounted for by the IV/interaction, but is not biased Factorial ANOVA Effect Size in Factorial ANOVA: Cohens d = X X 1 2 sp the two means can be between two IVs, or between

two groups/levels within an IV, depending on what you want to estimate Reminder: Cohens conventions for d small = .3, medium = .5, large = .8 Your text says that d = .5 corresponds to a large effect (pg. 415), but is mistaken check the Cohen article on the top of pg. 157 Factorial ANOVA Example #1: Remember the example we used in one-way ANOVA of the study by Eysenck (1974) looking at the effects of Age/Depth of Recall on Memory Performance? Recall how I said that although 2 IVs were used it was appropriate for a one-way ANOVA because the IVs were mushed-together. Now we will explore the same data with the IVs unmushed. DV = Memory Performance 2 IVs = Age 2 levels (Young and Old); Depth of Recall 5 levels/conditions (Counting, Rhyming, Adjective, Imagery, & Intentional) 2 x 5 Factorial ANOVA = 10 cells Old Young

Counting Rhyming Adjective Imagery Intentional 9 7 11 12 10 8 9 13 11

19 6 6 8 16 14 8 6 6 11 5 10 6 14 9

10 4 11 11 23 11 6 6 13 12 14 5 3 13

10 15 7 8 10 19 11 7 7 11 11 11 8 10 14

20 21 6 7 11 16 19 4 8 18 16 17 6 10

14 15 15 7 4 13 18 22 6 7 22 16 16 5 10

17 20 22 7 6 16 22 22 9 7 12 14 18 7

7 11 19 21 Factorial ANOVA 10 cells Red = means of entire levels of IVs Countin Rhymin Adjectiv g g e Imager Intention y al Mean 7.0 6.9 11.0

13.4 12.0 10.06 Young 6.5 7.6 14.8 17.6 19.3 13.16 Mean 7.25 12.9 15.5 15.65 11.61

Old 6.75 Factorial ANOVA dftotal = N 1 = 100 1 = 99 dfage = k 1 = 2 1 = 1 dfcondition = 5 1 = 4 dfint = dfage x dfcondition = 4 x 1 = 4 dferror = dftotal dfage dfcondition - dfint = 99 4 4 1 = 90 Critical Fs: For Age F.05(1, 90) = 3.96 For Condition F.05(4, 90) = 2.49 For the Age x Condition Interaction - F.05(4, 90) = 2.49 Factorial ANOVA SStotal

2 X = X 2 = 16,147 11612/100 N = 2667.79 Grand Mean = X/N = 1161/100 = 11.61 2 SSage = nc X age X .. = (10)(5)[(10.06 11.61)2 + (13.16 11.61)2 = 240.25 Factorial ANOVA SScondition =na X condition X .. 2

= (10)(2)[(6.75 11.61)2 + (7.25 11.61)2 + (12.9 11.61)2 + (15.5 11.61)2 + (15.65 11.61)2 = 1514.94 Factorial ANOVA SScells = n X cell X .. 2 = 10 [(7.0 11.61)2 + (6.9 11.61)2 + (11.0 11.61)2 + (13.4 11.61)2 + (12.0 11.61)2 + (6.5 11.61)2 + (7.6 11.61)2 + (14.8 11.61)2 + (17.6 11.61)2 + (19.3 11.61)2 = 1945.49 SSint = SScells SSage SScondition = 1945.49

240.25 1514.94 = 190.30 Factorial ANOVA SSerror = SStotal SScells = 2667.79 1945.49 = 722.30 MSage = 240.25/1 = 240.25 MScondition = 1514.94/4 = 378.735 MSint = 190.30/4 = 47.575 MSerror = 722.30/90 = 8.026 Factorial ANOVA F (Age) = 240.25/8.026 = 29.94 Critical F.05(1, 90) = 3.96 F (Condition) = 378.735/8.026 = 47.19 Critical F.05(4, 90) = 2.49 F (Interaction) = 47.575/8.026 = 5.93 Critical F.05(4, 90) = 2.49 All 3 Fs are significant, therefore we can reject Ho in all cases Factorial ANOVA Example #2: The previous example used data from Eysencks (1974) study of the effects of age and various conditions on memory performance. Another

aspect of this study manipulated depth of processing more directly by placing the participants into conditions that directly elicited High or Low levels of processing. Age was maintained as a variable and was subdivided into Young and Old groups. The data is as follows: Factorial ANOVA Young/Low: 8 6 4 6 7 6 5 7 9 7 Young/High: 21 19 17 15 22 16 22 22 18 21 Old/Low: 9 8 6 8 10 4 6 5 7 7 Old/High: 10 19 14 5 10 11 14 15 11 11 1. 2. Get into groups of 2 or more Identify the IVs and the DVs, and the number of levels of each Identify the number of cells Calculate the various dfs and the critical Fs Calculate the various Fs [two main effects (one for each IV) and one interaction] Determine the effect sizes (Cohens d) for the F-statistics

that youve obtained 3. 4. 5. 6. Factorial ANOVA Descriptive Statistics IV = Age (2 levels) and Condition (2 levels) 2 x 2 ANOVA = 4 cells dage = .70 dcondition = 1.82 dint = .80 Dependent Variable: MEMPERF Between-Subjects Factors

AGE CONDITIO .00 1.00 .00 1.00 Value Label Young Old Low High AGE Young N 20 20 20 20 Old Total CONDITIO Low

High Total Low High Total Low High Total Mean 6.5000 19.3000 12.9000 7.0000 12.0000 9.5000 6.7500 15.6500 11.2000 Std. Deviation 1.43372 2.66875 6.88935 1.82574 3.74166 3.84571 1.61815 4.90193

5.76995 Tests of Between-Subjects Effects Dependent Variable: MEMPERF Source Corrected Model Intercept AGE CONDITIO AGE * CONDITIO Error Total Corrected Total Type III Sum of Squares a df Mean Square F Sig. Partial Eta Squared

1059.800 3 353.267 53.301 .000 .816 5017.600 115.600 792.100 1 1 1 5017.600 115.600 792.100 757.056 17.442 119.512

.000 .000 .000 .955 .326 .769 152.100 1 152.100 22.949 .000 .389 238.600 6316.000 1298.400 36 40 39 6.628

a. R Squared = .816 (Adjusted R Squared = .801) N 10 10 20 10 10 20 20 20 40