2-Way ANOVA, 3-Way ANOVA, etc. 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+ IV’s and one DV Factorial ANOVA Notation: 2 x 3 x 4 ANOVA The number of numbers = the number of IV’s The numbers themselves = the number of levels in each IV

Factorial ANOVA 2 x 3 x 4 ANOVA = an ANOVA with 3 IV’s, 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 ANOVA’s? 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 (α), you can use fewer subjects (and time and money)

Factorial ANOVA Why use a factorial ANOVA? Why not just use multiple one-way ANOVA’s? With 3 IV’s, you’d need to run 3 one-way ANOVA’s, which would inflate your α-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 ANOVA’s 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 IV’s, 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: Note how male’s 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

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 IV’s This is what we were looking at with one-way ANOVA’s – 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: Young Medium Old Male 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 Young Medium Old Male Mean #1 Mean #2 Mean #3 Mean #10 Then we calculate means for all of these cells, and for our IV’s 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 ( ) This remains (ΣX)/N, or all of our observations added together, divided by the number of observations We can also calculate SStotal, which is also calculated the same as in a one-way ANOVA

Factorial ANOVA Next we want to calculate our SS terms for our IV’s, something new to factorial ANOVA SSIV = nxΣ( - )2 n = number of subjects per group/level of our IV x = number of groups/levels in the other IV

Factorial ANOVA SSIV = nxΣ( - )2 Subtract the grand mean from each of our levels means 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 Repeat for any IV’s 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 = Subtract the grand mean from each of our cell means Note: The number of values should equal the number of cells Square all of these values Add all of these values up Multiply this number by the number of subjects in each cell

Factorial ANOVA Now that we have SStotal, the SS’s for our IV’s, 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 F-statistics for all main effects and interactions Therefore, we need to calculate MS for all of our IV’s (our main effects) and the interaction MSIV = SSIV/dfIV We would do this for each of our IV’s MSint = SSint/dfint MSerror = SSerror/dferror

Factorial ANOVA We then divide each of our MS’s 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 F’s 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 IV’s or their interaction) and “df denominator” = dferror

Factorial ANOVA Just like in a one-way ANOVA, a significant F in factorial ANOVA doesn’t tell you which groups/levels of your IV’s are different There are several possible ways to determine where differences lie

Factorial ANOVA Multiple Comparison Techniques in Factorial ANOVA: Several one-way ANOVA’s (as many as there are IV’s) with their corresponding multiple comparison techniques probably the most common method Don’t forget the Bonferroni Method 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 IV’s 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

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 IV’s if you don’t get a nice interaction pattern the first time

Factorial ANOVA Effect Size in Factorial ANOVA: η2 (eta squared) = SSIV/SStotal (for any of the IV’s) 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 – don’t ever use it

Factorial ANOVA Effect Size in Factorial ANOVA: ω2 (omega squared) = 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: Cohen’s d = the two means can be between two IV’s, or between two groups/levels within an IV, depending on what you want to estimate Reminder: Cohen’s 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 IV’s were used it was appropriate for a one-way ANOVA because the IV’s were mushed-together. Now we will explore the same data with the IV’s unmushed. DV = Memory Performance 2 IV’s = Age – 2 levels (Young and Old); Depth of Recall – 5 levels/conditions (Counting, Rhyming, Adjective, Imagery, & Intentional) 2 x 5 Factorial ANOVA = 10 cells

Counting Rhyming Adjective Imagery Intentional Old Young 9 7 11 12 10 8 13 19 6 16 14 5 4 23 3 15 Young 20 21 18 17 22

Factorial ANOVA 10 cells Red = means of entire levels of IV’s Counting Rhyming Adjective Imagery Intentional Mean Old 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 6.75 7.25 12.9 15.5 15.65 11.61

Factorial ANOVA Critical F’s: 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 F’s: 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 = = 16,147 – 11612/100 = 2667.79 Grand Mean = ΣX/N = 1161/100 = 11.61 SSage = = (10)(5)[(10.06 – 11.61)2 + (13.16 – 11.61)2 = 240.25

Factorial ANOVA SScondition = = (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 = = 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 All 3 F’s are significant, therefore we can reject Ho in all cases

Factorial ANOVA Example #2: The previous example used data from Eysenck’s (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 Get into groups of 2 or more Identify the IV’s and the DV’s, and the number of levels of each Identify the number of cells Calculate the various df’s and the critical F’s Calculate the various F’s [two main effects (one for each IV) and one interaction] Determine the effect sizes (Cohen’s d) for the F-statistics that you’ve obtained

Factorial ANOVA IV = Age (2 levels) and Condition (2 levels) 2 x 2 ANOVA = 4 cells dage = .70 dcondition = 1.82 dint = .80