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Factorial ANOVA Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, 2-Way ANOVA, 3-Way ANOVA, etc. 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 One-Way ANOVA = ANOVA with

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Page 1: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

2-Way ANOVA, 3-Way ANOVA, 2-Way ANOVA, 3-Way ANOVA, etc.etc.

Page 2: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• One-Way ANOVA = ANOVA with one One-Way ANOVA = ANOVA with one IV with 1+ levels and one DVIV with 1+ levels and one DV

• Factorial ANOVA = ANOVA with 2+ Factorial ANOVA = ANOVA with 2+ IV’s and one DVIV’s and one DV– Factorial ANOVA Notation:Factorial ANOVA Notation:

•2 x 3 x 4 ANOVA2 x 3 x 4 ANOVA•The number of numbers = the number of IV’sThe number of numbers = the number of IV’s•The numbers themselves = the number of The numbers themselves = the number of

levels in each IVlevels in each IV

Page 3: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• 2 x 3 x 4 ANOVA = an ANOVA with 3 IV’s, one of 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 which has 2 levels, one of which has 3 levels, and the last of which has 4 levelsthe last of which has 4 levels

• Why use a factorial ANOVA? Why not just Why use a factorial ANOVA? Why not just use multiple one-way ANOVA’s?use multiple one-way ANOVA’s?

1.1. Increased power – with the same sample size Increased power – with the same sample size and effect size, a factorial ANOVA is more and effect size, a factorial ANOVA is more likely to result in the rejection of Hlikely to result in the rejection of Hoo

– aka with equal effect size and probability of aka with equal effect size and probability of rejecting Hrejecting Hoo if it is true ( if it is true (αα), you can use fewer ), you can use fewer subjects (and time and money)subjects (and time and money)

Page 4: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Why use a factorial ANOVA? Why not Why use a factorial ANOVA? Why not just use multiple one-way ANOVA’s?just use multiple one-way ANOVA’s?

2.2. With 3 IV’s, you’d need to run 3 one-way With 3 IV’s, you’d need to run 3 one-way ANOVA’s, which would inflate your ANOVA’s, which would inflate your αα-level-level– However, this could be corrected with a However, this could be corrected with a

Bonferroni CorrectionBonferroni Correction

3.3. The The bestbest reason is that a factorial ANOVA reason is that a factorial ANOVA can detect can detect interactionsinteractions, something that , something that multiple one-way ANOVA’s cannot domultiple one-way ANOVA’s cannot do

Page 5: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Interaction:Interaction:– when the effects of one independent variable when the effects of one independent variable

differ according to levels of another independent differ according to levels of another independent variablevariable

– Ex. We are testing two IV’s, Gender (male and Ex. We are testing two IV’s, Gender (male and female) and Age (young, medium, and old) and female) and Age (young, medium, and old) and their effect on performancetheir effect on performance• If males performance differed as a function of age, i.e. If males performance differed as a function of age, i.e.

males performed better or worse with age, but females males performed better or worse with age, but females performance was the same across ages, we would say performance was the same across ages, we would say that Age and Gender that Age and Gender interactinteract, or that we have an, or that we have an Age x Age x Gender interactionGender interaction

Page 6: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Interaction:Interaction:– Presented graphically:Presented graphically:

• Note how male’s Note how male’s performance changes performance changes as a function of age as a function of age while females does notwhile females does not

• Note also that the lines Note also that the lines cross one another, this cross one another, this is the hallmark of an is the hallmark of an interaction, and why interaction, and why interactions are interactions are sometimes called sometimes called cross-over or disordinal cross-over or disordinal interactionsinteractions

AGE

OldMediumYoung

Pe

rfo

rma

nce

40

30

20

10

0

GENDER

Male

Female

Page 7: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Interactions:Interactions:– However, it is not necessary that the However, it is not necessary that the

lines cross, only that the slopes differ lines cross, only that the slopes differ from one anotherfrom one another• I.e. one line can be flat, and the other I.e. one line can be flat, and the other

sloping upward, but not cross – this is still an sloping upward, but not cross – this is still an interactioninteraction

•See Fig. 17.2 on page 410 in the text for See Fig. 17.2 on page 410 in the text for more examplesmore examples

Page 8: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• As opposed to interactions, we have As opposed to interactions, we have what are called what are called main effectsmain effects::– the effect of an IV independent of any the effect of an IV independent of any

other IV’sother IV’s•This is what we were looking at with one-way This is what we were looking at with one-way

ANOVA’s – if we have a significant main effect ANOVA’s – if we have a significant main effect of our IV, then we can say that the mean of at of our IV, then we can say that the mean of at least one of the groups/levels of that IV is least one of the groups/levels of that IV is different than at least one of the other different than at least one of the other groups/levelsgroups/levels

Page 9: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Finally, we also have Finally, we also have simple effectssimple effects::– the effect of one group/level of our IV at the effect of one group/level of our IV at

one group/level of another IVone group/level of another IV•Using our example earlier of the effects of Using our example earlier of the effects of

Gender (Men/Women) and Age Gender (Men/Women) and Age (Young/Medium/Old) on Performance, to say (Young/Medium/Old) on Performance, to say that young women outperformed other that young women outperformed other groups would be to talk about a simple groups would be to talk about a simple effecteffect

Page 10: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Calculating a Factorial ANOVA:Calculating a Factorial ANOVA:– First, we have to divide our data into First, we have to divide our data into cellscells

• the data represented by our simple effectsthe data represented by our simple effects• If we have a 2 x 3 ANOVA, as in our Age and Gender If we have a 2 x 3 ANOVA, as in our Age and Gender

example, we have 3 x 2 = 6 cellsexample, we have 3 x 2 = 6 cells

YoungYoung MediumMedium OldOld

MaleMale Cell #1Cell #1 Cell #2Cell #2 Cell #3Cell #3

FemaleFemale Cell #4Cell #4 Cell #5Cell #5 Cell #6Cell #6

Page 11: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Then we calculate means for all of these cells, and for our IV’s across cellsThen we calculate means for all of these cells, and for our IV’s across cells– Mean #1 = Mean for Young Males onlyMean #1 = Mean for Young Males only– Mean #2 = Mean for Medium Males onlyMean #2 = Mean for Medium Males only– Mean #3 = Mean for Old MalesMean #3 = Mean for Old Males– Mean #4 = Mean for Young FemalesMean #4 = Mean for Young Females– Mean #5 = Mean for Medium FemalesMean #5 = Mean for Medium Females– Mean #6 = Mean for Old FemalesMean #6 = Mean for Old Females– Mean #7 = Mean for all Young people (Male and Female)Mean #7 = Mean for all Young people (Male and Female)– Mean #8 = Mean for all Medium 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 #9 = Mean for all Old people (Male and Female)– Mean #10 = Mean for all Males (Young, Medium, and Old)Mean #10 = Mean for all Males (Young, Medium, and Old)– Mean #11 = Mean for all Females (Young, Medium, and Old)Mean #11 = Mean for all Females (Young, Medium, and Old)

Young Young MediumMedium OldOld

Male Male Mean #1Mean #1 Mean #2Mean #2 Mean #3Mean #3 Mean #10Mean #10

FemaleFemale Mean #4Mean #4 Mean #5Mean #5 Mean #6Mean #6 Mean #11Mean #11

Mean #7Mean #7 Mean #8Mean #8 Mean #9Mean #9

Page 12: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• We then calculate the Grand Mean ( We then calculate the Grand Mean ( ) )– This remains (This remains (ΣΣX)/N, or all of our X)/N, or all of our

observations added together, divided by observations added together, divided by the number of observationsthe number of observations

• We can also calculate SSWe can also calculate SStotaltotal, which is , which is also calculated the same as in a one-also calculated the same as in a one-way ANOVAway ANOVA

..X

N

XX

22

Page 13: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Next we want to calculate our SS Next we want to calculate our SS terms for our IV’s, something new to terms for our IV’s, something new to factorial ANOVAfactorial ANOVA– SSSSIVIV = nx = nxΣΣ( - )( - )22

•n = number of subjects per group/level of n = number of subjects per group/level of our IVour IV

•x = number of groups/levels in the x = number of groups/levels in the otherother IV IV

IVX ..X

Page 14: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

– SSSSIVIV = nx = nxΣΣ( - )( - )22

1.1. Subtract the grand mean from each of our levels Subtract the grand mean from each of our levels meansmeans• For SSFor SSgendergender, this would involve subtracting the mean , this would involve subtracting the mean

for males from the grand mean, and the mean for for males from the grand mean, and the mean for females from the grand meanfemales from the grand mean

• Note: The number of values should equal the number Note: The number of values should equal the number of levels of your IVof levels of your IV

2.2. Square all of these valuesSquare all of these values3.3. Add all of these values upAdd all of these values up4.4. Multiply this number by the number of subjects in Multiply this number by the number of subjects in

each cell x the number of levels of the each cell x the number of levels of the otherother IV IV5.5. Repeat for any IV’sRepeat for any IV’s

• Using the previous example, we would have both Using the previous example, we would have both SSSSgendergender and SS and SSageage

IVX ..X

Page 15: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Next we want to calculate SSNext we want to calculate SScellscells, which has a , which has a formula similar to SSformula similar to SSIVIV

– SSSScells cells ==

1.1. Subtract the grand mean from each of our cell meansSubtract the grand mean from each of our cell means• Note: The number of values should equal the number of cellsNote: The number of values should equal the number of cells

2.2. Square all of these valuesSquare all of these values

3.3. Add all of these values upAdd all of these values up

4.4. Multiply this number by the number of subjects in each cellMultiply this number by the number of subjects in each cell

2..XXn cell

Page 16: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Now that we have SSNow that we have SStotaltotal, the SS’s for , the SS’s for our IV’s, and SSour IV’s, and SScellscells, we can find SS, we can find SSerrorerror and the SS for our interaction term, and the SS for our interaction term, SSSSintint

– SSSSintint = SS = SScellscells – SS – SSIV1IV1 – SS – SSIV2IV2 – etc… – etc…•Going back to our previous example, Going back to our previous example,

SSSSintint = SS = SScellscells – SS – SSgendergender – SS – SSageage

– SSSSerrorerror = SS = SStotaltotal – SS – SScellscells

Page 17: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Similar to a one-way ANOVA, factorial Similar to a one-way ANOVA, factorial ANOVA uses df to obtain MSANOVA uses df to obtain MS– dfdftotaltotal = N – 1 = N – 1

– dfdfIVIV = k – 1 = k – 1

• Using the previous example, dfUsing the previous example, dfageage = 3 = 3 (Young/Medium/Old) – 1 = 2 and df(Young/Medium/Old) – 1 = 2 and dfgendergender = 2 = 2 (Male/Female) – 1 = 1(Male/Female) – 1 = 1

– dfdfintint = df = dfIV1IV1 x df x dfIV2IV2 x etc… x etc…

• Again, using the previous example, dfAgain, using the previous example, df intint = 2 x 1 = 2 = 2 x 1 = 2

– dfdferrorerror = df = dftotaltotal – df – dfintint - df - dfIV1IV1 – df – dfIV2IV2 – etc… – etc…

Page 18: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Factorial ANOVA provides you with Factorial ANOVA provides you with FF--statistics for all main effects and statistics for all main effects and interactionsinteractions– Therefore, we need to calculate MS for all of Therefore, we need to calculate MS for all of

our IV’s (our main effects) and the our IV’s (our main effects) and the interactioninteraction

– MSMSIVIV = SS = SSIVIV/df/dfIVIV

•We would do this for each of our IV’sWe would do this for each of our IV’s

– MSMSintint = SS = SSintint/df/dfintint

– MSMSerrorerror = SS = SSerrorerror/df/dferrorerror

Page 19: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• We then divide each of our MS’s by MSWe then divide each of our MS’s by MSerrorerror to obtain our to obtain our FF-statistics-statistics

• Finally, we compare this with our critical Finally, we compare this with our critical FF to determine if we accept or reject Hto determine if we accept or reject Hoo– All of our main effects and our interaction have All of our main effects and our interaction have

their own critical F’stheir own critical F’s– Just as in the one-way ANOVA, use table E.3 or Just as in the one-way ANOVA, use table E.3 or

E.4 depending on your alpha level (.05 or .01)E.4 depending on your alpha level (.05 or .01)– Just as in the one-way ANOVA, “df numerator” Just as in the one-way ANOVA, “df numerator”

= the df for the term in question (the IV’s or = the df for the term in question (the IV’s or their interaction) and “df denominator” = dftheir interaction) and “df denominator” = dferrorerror

Page 20: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Just like in a one-way ANOVA, a Just like in a one-way ANOVA, a significant significant FF in factorial ANOVA in factorial ANOVA doesn’t tell you which groups/levels doesn’t tell you which groups/levels of your IV’s are differentof your IV’s are different– There are several possible ways to There are several possible ways to

determine where differences liedetermine where differences lie

Page 21: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Multiple Comparison Techniques in Multiple Comparison Techniques in Factorial ANOVA:Factorial ANOVA:

1.1. Several one-way ANOVA’s (as many as there Several one-way ANOVA’s (as many as there are IV’s) with their corresponding multiple are IV’s) with their corresponding multiple comparison techniquescomparison techniques– probably the most common methodprobably the most common method– Don’t forget the Bonferroni MethodDon’t forget the Bonferroni Method

2.2. Analysis of Simple EffectsAnalysis of Simple Effects– Calculate MS for each cell/simple effect, obtain an Calculate MS for each cell/simple effect, obtain an FF

for each one and determine its associated for each one and determine its associated pp-value-value– See pages 411-413 in your text – you should be See pages 411-413 in your text – you should be

familiar with the theory of the technique, but you familiar with the theory of the technique, but you will not be asked to use it on the Final Examwill not be asked to use it on the Final Exam

Page 22: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Multiple Comparison Techniques in Multiple Comparison Techniques in Factorial ANOVA:Factorial ANOVA:– In addition, interactions must be In addition, interactions must be

decomposed to determine what they meandecomposed to determine what they mean•A significant interaction between two variables A significant interaction between two variables

means that one IV’s value changes as a function means that one IV’s value changes as a function of the other, but gives no specific informationof the other, but gives no specific information

•The most simple and common method of The most simple and common method of interpreting interactions is to look at a graphinterpreting interactions is to look at a graph

Page 23: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

• Interpreting Interactions:Interpreting Interactions:– In the example above, you can see that for Males, as age In the example above, you can see that for Males, as age

increases, Performance increases, whereas for Females there is increases, Performance increases, whereas for Females there is no relation between Age and Performanceno relation between Age and Performance

– To interpret an interaction, we graph the DV on the y-axis, place 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 IVone 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 You may have to try switching the IV’s if you don’t get a nice

interaction pattern the first timeinteraction pattern the first time

AGE

OldMediumYoung

Pe

rfo

rma

nce

40

30

20

10

0

GENDER

Male

Female

Page 24: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Effect Size in Factorial ANOVA:Effect Size in Factorial ANOVA:– ηη22 (eta squared) = SS (eta squared) = SSIVIV/SS/SStotal total (for any of (for any of

the IV’s)the IV’s) or SSor SSintint/SS/SStotal total (for the (for the interaction)interaction)•tells you the percent of variability in the DV tells you the percent of variability in the DV

accounted for by the IV/interactionaccounted for by the IV/interaction

• like the one-way ANOVA, very easily like the one-way ANOVA, very easily computed and commonly used, but also computed and commonly used, but also very biased – don’t ever use itvery biased – don’t ever use it

Page 25: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Effect Size in Factorial ANOVA:Effect Size in Factorial ANOVA:– ωω22 (omega squared) = (omega squared) =

•oror

•also provides an estimate of the percent of also provides an estimate of the percent of variability in the DV accounted for by the variability in the DV accounted for by the IV/interaction, but is not biasedIV/interaction, but is not biased

errortotal

errorIVIV

MSSS

MSdfSS

errortotal

error

MSSS

MSdfSS

intint

Page 26: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Effect Size in Factorial ANOVA:Effect Size in Factorial ANOVA:– Cohen’s d = Cohen’s d =

• the two means can be between two IV’s, or between the two means can be between two IV’s, or between two groups/levels within an IV, depending on what two groups/levels within an IV, depending on what you want to estimateyou want to estimate

• Reminder: Cohen’s conventions for d – small = .3, Reminder: Cohen’s conventions for d – small = .3, medium = .5, large = .8medium = .5, large = .8

– Your text says that d = .5 corresponds to a large effect Your text says that d = .5 corresponds to a large effect (pg. 415), but is mistaken – check the Cohen article on (pg. 415), but is mistaken – check the Cohen article on the top of pg. 157the top of pg. 157

ps

XX 21

Page 27: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Example #1:Example #1:– Remember the example we used in one-way Remember the example we used in one-way

ANOVA of the study by Eysenck (1974) looking ANOVA of the study by Eysenck (1974) looking at the effects of Age/Depth of Recall on at the effects of Age/Depth of Recall on Memory Performance? Recall how I said that Memory Performance? Recall how I said that although 2 IV’s were used it was appropriate although 2 IV’s were used it was appropriate for a one-way ANOVA because the IV’s were for a one-way ANOVA because the IV’s were mushed-together. Now we will explore the mushed-together. Now we will explore the same data with the IV’s unmushed.same data with the IV’s unmushed.• DV = Memory PerformanceDV = Memory Performance• 2 IV’s = Age – 2 levels (Young and Old); Depth of 2 IV’s = Age – 2 levels (Young and Old); Depth of

Recall – 5 levels/conditions (Counting, Rhyming, Recall – 5 levels/conditions (Counting, Rhyming, Adjective, Imagery, & Intentional)Adjective, Imagery, & Intentional)

• 2 x 5 Factorial ANOVA = 10 cells2 x 5 Factorial ANOVA = 10 cells

Page 28: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

CountingCounting RhymingRhyming AdjectiveAdjective ImageryImagery IntentionalIntentional

OldOld 99 77 1111 1212 1010

88 99 1313 1111 1919

66 66 88 1616 1414

88 66 66 1111 55

1010 66 1414 99 1010

44 1111 1111 2323 1111

66 66 1313 1212 1414

55 33 1313 1010 1515

77 88 1010 1919 1111

77 77 1111 1111 1111

YoungYoung 88 1010 1414 2020 2121

66 77 1111 1616 1919

44 88 1818 1616 1717

66 1010 1414 1515 1515

77 44 1313 1818 2222

66 77 2222 1616 1616

55 1010 1717 2020 2222

77 66 1616 2222 2222

99 77 1212 1414 1818

77 77 1111 1919 2121

Page 29: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• 10 cells10 cells

• RedRed = means of entire levels of IV’s = means of entire levels of IV’s

CountinCountingg

RhyminRhymingg

AdjectivAdjectivee

ImagerImageryy

IntentionIntentionalal

MeanMean

OldOld 7.07.0 6.96.9 11.011.0 13.413.4 12.012.0 10.0610.06

YoungYoung 6.56.5 7.67.6 14.814.8 17.617.6 19.319.3 13.1613.16

MeanMean 6.756.75 7.257.25 12.912.9 15.515.5 15.6515.65 11.6111.61

Page 30: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

– dfdftotaltotal = N – 1 = 100 – 1 = 99 = N – 1 = 100 – 1 = 99– dfdfageage = k – 1 = 2 – 1 = 1 = k – 1 = 2 – 1 = 1– dfdfconditioncondition = 5 – 1 = 4 = 5 – 1 = 4– dfdfintint = df = dfageage x df x dfconditioncondition = 4 x 1 = 4 = 4 x 1 = 4– dfdferrorerror = df = dftotal total –– dfdfage age – df– dfcondition condition -- dfdfintint = 99 – 4 – 4 – 1 = 99 – 4 – 4 – 1

= 90= 90• Critical F’s:Critical F’s:

– For Age – FFor Age – F.05.05(1, 90) = 3.96(1, 90) = 3.96– For Condition – FFor Condition – F.05.05(4, 90) = 2.49(4, 90) = 2.49– For the Age x Condition Interaction - FFor the Age x Condition Interaction - F.05.05(4, 90) = (4, 90) =

2.492.49

Page 31: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• SSSStotaltotal = = 16,147 – 1161 = = 16,147 – 116122/100/100

= 2667.79= 2667.79

• Grand Mean = Grand Mean = ΣΣX/N = 1161/100 = 11.61X/N = 1161/100 = 11.61

• SSSSageage = =

= (10)(5)[(10.06 – 11.61)= (10)(5)[(10.06 – 11.61)22 + + (13.16 (13.16 – 11.61)– 11.61)2 2 = 240.25= 240.25

N

XX

22

2..XXnc age

Page 32: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• SSSSconditioncondition = =

= (10)(2)[(6.75 – 11.61)= (10)(2)[(6.75 – 11.61)22 + + (7.25 – 11.61)(7.25 – 11.61)22 + (12.9 + (12.9

– 11.61)– 11.61)22 + (15.5 – 11.61)+ (15.5 – 11.61)2 2 + + (15.65 – (15.65 – 11.61)11.61)22 = 1514.94 = 1514.94

2..XXna condition

Page 33: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• SSSScellscells = =

= 10 [(7.0 – 11.61)= 10 [(7.0 – 11.61)22 + (6.9 – + (6.9 – 11.61)11.61)22 + (11.0 – 11.61) + (11.0 – 11.61)22 + (13.4 – + (13.4 –

11.61)11.61)22 + (12.0 – 11.61) + (12.0 – 11.61)22 + (6.5 – + (6.5 – 11.61)11.61)22 + (7.6 – 11.61)+ (7.6 – 11.61)2 2 + (14.8 – + (14.8 – 11.61)11.61)22 + + (17.6 – 11.61)(17.6 – 11.61)22 + (19.3 – + (19.3 – 11.61)11.61)22 = = 1945.491945.49

• SSSSintint = SS = SScellscells – SS – SSageage – SS – SSconditioncondition = 1945.49 – = 1945.49 – 240.25 – 1514.94 = 190.30240.25 – 1514.94 = 190.30

2..XXn cell

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Factorial ANOVAFactorial ANOVA

• SSSSerrorerror = SS = SStotaltotal – SS – SScellscells = 2667.79 – = 2667.79 – 1945.49 = 722.301945.49 = 722.30

• MSMSageage = 240.25/1 = 240.25 = 240.25/1 = 240.25

• MSMSconditioncondition = 1514.94/4 = 378.735 = 1514.94/4 = 378.735

• MSMSintint = 190.30/4 = 47.575 = 190.30/4 = 47.575

• MSMSerrorerror = 722.30/90 = 8.026 = 722.30/90 = 8.026

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Factorial ANOVAFactorial ANOVA

• FF (Age) = 240.25/8.026 = 29.94 (Age) = 240.25/8.026 = 29.94– Critical FCritical F.05.05(1, 90) = 3.96(1, 90) = 3.96

• FF (Condition) = 378.735/8.026 = 47.19 (Condition) = 378.735/8.026 = 47.19– Critical FCritical F.05.05(4, 90) = 2.49(4, 90) = 2.49

• FF (Interaction) = 47.575/8.026 = 5.93 (Interaction) = 47.575/8.026 = 5.93– Critical FCritical F.05.05(4, 90) = 2.49(4, 90) = 2.49

• All 3 F’s are significant, therefore we All 3 F’s are significant, therefore we can reject Hcan reject Hoo in all cases in all cases

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Factorial ANOVAFactorial ANOVA

• Example #2:Example #2:– The previous example used data from The previous example used data from

Eysenck’s (1974) study of the effects of age Eysenck’s (1974) study of the effects of age and various conditions on memory and various conditions on memory performance. Another aspect of this study performance. Another aspect of this study manipulated depth of processing more directly manipulated depth of processing more directly by placing the participants into conditions that by placing the participants into conditions that directly elicited High or Low levels of directly elicited High or Low levels of processing. Age was maintained as a variable processing. Age was maintained as a variable and was subdivided into Young and Old groups. and was subdivided into Young and Old groups. The data is as follows:The data is as follows:

Page 37: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• Young/Low: 8 6 4 6 7 6 5 7 9 7Young/Low: 8 6 4 6 7 6 5 7 9 7• Young/High: 21 19 17 15 22 16 22 22 18 21Young/High: 21 19 17 15 22 16 22 22 18 21• Old/Low: 9 8 6 8 10 4 6 5 7 7Old/Low: 9 8 6 8 10 4 6 5 7 7• Old/High: 10 19 14 5 10 11 14 15 11 11Old/High: 10 19 14 5 10 11 14 15 11 11

1.1. Get into groups of 2 or moreGet into groups of 2 or more2.2. Identify the IV’s and the DV’s, and the number of levels of Identify the IV’s and the DV’s, and the number of levels of

eacheach3.3. Identify the number of cellsIdentify the number of cells4.4. Calculate the various df’s and the critical F’sCalculate the various df’s and the critical F’s5.5. Calculate the various F’s [two main effects (one for each Calculate the various F’s [two main effects (one for each

IV) and one interaction]IV) and one interaction]6.6. Determine the effect sizes (Cohen’s d) for the F-statistics Determine the effect sizes (Cohen’s d) for the F-statistics

that you’ve obtainedthat you’ve obtained

Page 38: Factorial ANOVA 2-Way ANOVA, 3-Way ANOVA, etc.. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV One-Way ANOVA = ANOVA with

Factorial ANOVAFactorial ANOVA

• IV = Age (2 IV = Age (2 levels) and levels) and Condition (2 Condition (2 levels)levels)

• 2 x 2 ANOVA 2 x 2 ANOVA = 4 cells= 4 cells

• ddageage = .70 = .70

• ddconditioncondition = 1.82 = 1.82

• ddintint = .80 = .80

Between-Subjects Factors

Young 20

Old 20

Low 20

High 20

.00

1.00

AGE

.00

1.00

CONDITIO

Value Label N

Descriptive Statistics

Dependent Variable: MEMPERF

6.5000 1.43372 10

19.3000 2.66875 10

12.9000 6.88935 20

7.0000 1.82574 10

12.0000 3.74166 10

9.5000 3.84571 20

6.7500 1.61815 20

15.6500 4.90193 20

11.2000 5.76995 40

CONDITIOLow

High

Total

Low

High

Total

Low

High

Total

AGEYoung

Old

Total

Mean Std. Deviation N

Tests of Between-Subjects Effects

Dependent Variable: MEMPERF

1059.800a

3 353.267 53.301 .000 .816

5017.600 1 5017.600 757.056 .000 .955

115.600 1 115.600 17.442 .000 .326

792.100 1 792.100 119.512 .000 .769

152.100 1 152.100 22.949 .000 .389

238.600 36 6.628

6316.000 40

1298.400 39

SourceCorrected Model

Intercept

AGE

CONDITIO

AGE *CONDITIO

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .816 (Adjusted R Squared = .801)a.