1 Two Factor ANOVA. 2 Factorial Designs Often researchers want to study the effects of two or more independent variables at the same time Often researchers.

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Two Factor ANOVATwo Factor ANOVA

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Factorial DesignsFactorial Designs

• Often researchers want to study the Often researchers want to study the effects of two or more independent effects of two or more independent variables at the same timevariables at the same time– Does it matter where a list of words is Does it matter where a list of words is

studied, on the beach or under water?studied, on the beach or under water?– Does it matter where a list of words is Does it matter where a list of words is

recalled, on the beach or under water?recalled, on the beach or under water?

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• FactorFactor is another name for is another name for independent variableindependent variable– The preceding example has two factors: The preceding example has two factors:

where you study and where you recallwhere you study and where you recall

• In a In a factorial designfactorial design, all possible , all possible combinations of the factors are combinations of the factors are presentpresent

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A Factorial DesignA Factorial Design

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On Beach Underwater

On BeachStudy on beach;recall on beach

Study underwater;recall on beach

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UnderwaterStudy on beach;recall underwater

Study underwater;recall underwater

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Naming Factorial DesignsNaming Factorial Designs

• Factorial designs are referred to by the Factorial designs are referred to by the number the number of IVs and the number the number of IVs and the number of levels of each IVnumber of levels of each IV

• A design with two IVs is said to be an n A design with two IVs is said to be an n X m (read n by m) designX m (read n by m) design– n is replaced with the number of levels, or n is replaced with the number of levels, or

conditions, of the first IVconditions, of the first IV– m is replaced with the number of levels, or m is replaced with the number of levels, or

conditions, of the second IVconditions, of the second IV

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Naming Factorial DesignsNaming Factorial Designs

• The preceding example is a 2 X 2 The preceding example is a 2 X 2 factorial design because there are factorial design because there are two IVs, and each IV has two levelstwo IVs, and each IV has two levels

• What would you call a design that What would you call a design that had 3 IVs in which the first IV had 2 had 3 IVs in which the first IV had 2 levels, the second IV had 3 levels and levels, the second IV had 3 levels and the third IV had 4 levels?the third IV had 4 levels?

• 2 X 3 X 42 X 3 X 4

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• The number of The number of conditions in a conditions in a factorial design is factorial design is equal to the equal to the product given by product given by its nameits name

Design # Conditions

2 X 2 4

2 X 3 X 4 24

3 X 4 12

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Information From A Factorial Information From A Factorial DesignDesign

• An n X m factorial designs is very An n X m factorial designs is very powerful because it allows us to powerful because it allows us to answer three questions:answer three questions:– Is there an effect of the first IV?Is there an effect of the first IV?

•Do you recall more words when you study Do you recall more words when you study them on the beach or underwater?them on the beach or underwater?

– Is there an effect of the second IV?Is there an effect of the second IV?•Do you recall more words when you recall Do you recall more words when you recall

them on the beach or underwater?them on the beach or underwater?

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Information From A Factorial Information From A Factorial DesignDesign

• An n X m factorial designs is very An n X m factorial designs is very powerful because it allow us to powerful because it allow us to answer three questions:answer three questions:– Are the effects of the two IVs Are the effects of the two IVs

independent of each other?independent of each other?•When recalling on the beach, does it matter When recalling on the beach, does it matter

whether you studied underwater or not? whether you studied underwater or not? When recalling underwater, does it matter When recalling underwater, does it matter whether you studied underwater or not?whether you studied underwater or not?

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Main EffectsMain Effects

• Each of the first two questions (Is there an Each of the first two questions (Is there an effect of the first / second IV?) is asking effect of the first / second IV?) is asking whether there is a whether there is a main effectmain effect of that IV of that IV– A main effect occurs when an independent A main effect occurs when an independent

variable has an influence on the dependent variable has an influence on the dependent variablevariable

– If people recalled more words when they If people recalled more words when they studied them on the beach than when they studied them on the beach than when they studied them underwater, then there would be studied them underwater, then there would be a main effect of where the words are studieda main effect of where the words are studied

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• When looking at the When looking at the main effect of one main effect of one IV, you should ignore IV, you should ignore the existence of the the existence of the other IVother IV– Compare all Compare all

conditions that have conditions that have one level of the IV to one level of the IV to all conditions that all conditions that have the other level have the other level of the IVof the IV

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• For the main effect of where For the main effect of where the words were studied:the words were studied:

• Average the values of the Average the values of the left two bars (conditions in left two bars (conditions in which people studied on the which people studied on the beach)beach)– (15 + 10) / 2 = 12.5(15 + 10) / 2 = 12.5

• Average the values of the Average the values of the right two bars (conditions in right two bars (conditions in which people studied which people studied underwater)underwater)– (8 + 5) / 2 = 6.5(8 + 5) / 2 = 6.5

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• Compare the two meansCompare the two means– XXbeachbeach = 12.5 vs X = 12.5 vs Xunderwaterunderwater = =

6.56.5

• The larger the difference The larger the difference between the means, the between the means, the more likely there is a more likely there is a main effect of the main effect of the independent variableindependent variable– There probably is a main There probably is a main

effect of where the person effect of where the person studied the wordsstudied the words

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• Is there probably a Is there probably a main effect of where main effect of where the person recalled the person recalled the words?the words?

• Average the values of Average the values of the two green bars the two green bars (conditions in which (conditions in which the words were the words were recalled on the beach)recalled on the beach)– (15 + 8) / 2 = 11.5(15 + 8) / 2 = 11.5

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• Average the values of the Average the values of the two blue bars (conditions two blue bars (conditions in which the words were in which the words were recalled underwater)recalled underwater)– (10 + 5) / 2 = 7.5(10 + 5) / 2 = 7.5

• Compare the two Compare the two averagesaverages– 11.5 vs 7.511.5 vs 7.5

• There probably is a main There probably is a main effect of where the words effect of where the words are recalledare recalled

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• Main effects can be determined from Main effects can be determined from both single factor experiments and both single factor experiments and from factorial design experimentsfrom factorial design experiments

• However, factorial designs are more However, factorial designs are more efficient than single factor efficient than single factor experiments because you can experiments because you can answer the same questions with answer the same questions with fewer participantsfewer participants

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InteractionsInteractions

• The third question that we can ask about The third question that we can ask about a factorial design is:a factorial design is:– Are the effects of the two IVs independent of Are the effects of the two IVs independent of

each other?each other?

• This type of effect is called an This type of effect is called an interaction interaction effecteffect or just an or just an interactioninteraction

• Interactions are very important in Interactions are very important in research and are the real reason why research and are the real reason why factorial designs are performedfactorial designs are performed

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Definitions of Definitions of InteractionInteraction

• There are several equivalent ways of There are several equivalent ways of defining defining interaction:interaction:– An interaction occurs when the nature of the An interaction occurs when the nature of the

simple main effect of one IV depend on then simple main effect of one IV depend on then level of the other IVlevel of the other IV

– An interaction occurs when the effects of one IV An interaction occurs when the effects of one IV cannot simply be added to the effects of the cannot simply be added to the effects of the other IV in order to predict how both other IV in order to predict how both treatments will simultaneously affect the DVtreatments will simultaneously affect the DV

– An interaction occurs when the lines on a graph An interaction occurs when the lines on a graph of the results are not parallelof the results are not parallel

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Simple Main Effect Simple Main Effect Definition of Definition of InteractionInteraction• An interaction occurs An interaction occurs

when the nature of when the nature of the simple main the simple main effect of one IV effect of one IV depend on then level depend on then level of the other IVof the other IV

• Does the effect of Does the effect of whether you study whether you study dry or wet depend on dry or wet depend on whether you recall whether you recall dry or wet?dry or wet?

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Simple Main Effect Simple Main Effect Definition of Definition of InteractionInteraction• Yes Yes

• If you are recalling on If you are recalling on the beach, then the beach, then studying on the studying on the beach is better than beach is better than studying underwaterstudying underwater

• If you are recalling If you are recalling underwater, then underwater, then studying on the studying on the beach is poorer than beach is poorer than studying underwaterstudying underwater

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Simple Main Effect Simple Main Effect Definition of Definition of InteractionInteraction• The simple main effect The simple main effect

of whether you do of whether you do better studying on the better studying on the beach or underwater beach or underwater depends on whether depends on whether your recall on the your recall on the beach or underwaterbeach or underwater

• Thus, the two Thus, the two variables (where you variables (where you study and where you study and where you recall) interactrecall) interact

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Additivity Definition of Additivity Definition of InteractionInteraction• An interaction occurs when the effects of An interaction occurs when the effects of

one IV cannot simply be added to the one IV cannot simply be added to the effects of the other IV in order to predict effects of the other IV in order to predict how both treatments will simultaneously how both treatments will simultaneously affect the DVaffect the DV

• Determine what effect studying on the Determine what effect studying on the beach vs underwater hasbeach vs underwater has

• Determine what effect recalling on the Determine what effect recalling on the beach vs underwater hasbeach vs underwater has

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Additivity Definition of Additivity Definition of InteractionInteraction• The average value of The average value of

all conditions in which all conditions in which you study on the you study on the beach is (15 + 10) / 2 beach is (15 + 10) / 2 = 12.5= 12.5

• The average value of The average value of all conditions in which all conditions in which you study underwater you study underwater is (5 + 12) / 2 = 8.5is (5 + 12) / 2 = 8.5

• So studying So studying underwater leads to underwater leads to you recalling 12.5 - you recalling 12.5 - 8.5 = 4 fewer words8.5 = 4 fewer words

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Additivity Definition of Additivity Definition of InteractionInteraction• The average value of The average value of

all conditions in which all conditions in which you recall on the beach you recall on the beach is (15 + 5) / 2 = 10is (15 + 5) / 2 = 10

• The average value of The average value of all conditions in which all conditions in which you recall underwater you recall underwater is (10 + 12) / 2 = 11is (10 + 12) / 2 = 11

• So recalling underwater So recalling underwater leads to you recalling leads to you recalling 11 - 10 = 1 more word11 - 10 = 1 more word

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Additivity Definition of Additivity Definition of InteractionInteraction

• If the IVs are independent of each If the IVs are independent of each other (i.e. they do other (i.e. they do notnot interact with interact with other), then we should be able to other), then we should be able to predict the number of words recalled predict the number of words recalled in three of the conditions given the in three of the conditions given the number of words recalled in the number of words recalled in the fourth condition and the size of the fourth condition and the size of the main effectsmain effects

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Additivity Definition of Additivity Definition of InteractionInteraction• Number of words recalled when you study Number of words recalled when you study

underwater and recall underwater = number underwater and recall underwater = number of words recalled when you study on the of words recalled when you study on the beach and recall on the beach + effect of beach and recall on the beach + effect of studying underwater + effect of recalling studying underwater + effect of recalling underwaterunderwater

• XXstudy underwater, recall underwaterstudy underwater, recall underwater = X = Xstudy on beach, recall on beachstudy on beach, recall on beach + effect of studying underwater + effect of + effect of studying underwater + effect of recalling underwaterrecalling underwater

• 12 = 15 + (-4) + 1 = 1212 = 15 + (-4) + 1 = 12

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Additivity Definition of Additivity Definition of InteractionInteraction• Number of words recalled when you study Number of words recalled when you study

underwater and recall on the beach = number of underwater and recall on the beach = number of words recalled when you study on the beach and words recalled when you study on the beach and recall on the beach + effect of studying recall on the beach + effect of studying underwater underwater

• XXstudy underwater, recall on the beachstudy underwater, recall on the beach = X = Xstudy on the beach, recall on the study on the beach, recall on the

beachbeach + effect of studying underwater + effect of studying underwater

• 5 5 15 + (-4) 15 + (-4)

• Because we cannot predict, the effects are not Because we cannot predict, the effects are not additive, and the variables interactadditive, and the variables interact

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Additivity Definition of Additivity Definition of InteractionInteraction• Number of words recalled when you study on Number of words recalled when you study on

the beach and recall underwater = number of the beach and recall underwater = number of words recalled when you study on the beach words recalled when you study on the beach and recall underwater + effect of recalling and recall underwater + effect of recalling underwaterunderwater

• XXstudy on the beach, recall underwaterstudy on the beach, recall underwater = X = Xstudy on the beach, recall on the study on the beach, recall on the

beachbeach + effect of recalling underwater + effect of recalling underwater

• 10 10 15 + 1 15 + 1

• Because we cannot predict, the effects are not Because we cannot predict, the effects are not additive, and the variables interactadditive, and the variables interact

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Graphical Definition of Graphical Definition of InteractionInteraction

• When the lines or bars on a graph are not When the lines or bars on a graph are not parallel, then an interaction has occurredparallel, then an interaction has occurred

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Factorial ANOVAFactorial ANOVA

• The factorial analysis of variance answers The factorial analysis of variance answers each of the questions that can be askedeach of the questions that can be asked– Is there a main effect of the first IV?Is there a main effect of the first IV?– Is there a main effect of the second IV?Is there a main effect of the second IV?– Is there an interaction effect of the two IVs?Is there an interaction effect of the two IVs?

• The ANOVA accomplishes these goals by The ANOVA accomplishes these goals by giving us an F ratio for each of the giving us an F ratio for each of the questions that are askedquestions that are asked

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• HH00 and H and H11 for each of the main for each of the main effects takes the same form as the effects takes the same form as the single factor ANOVA Hsingle factor ANOVA H00 and H and H11::

– HH00: : 11 = = 22 = …. = = …. = nn

– HH11: not H: not H00

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• HH00 and H and H11 for the for the interaction take the interaction take the following form:following form:– HH00: : 1111 - - 2121 = = 1212 - - 2222

– HH11: : 1111 - - 2121 1212 - - 2222

1212 = mean for level 1 of = mean for level 1 of the first IV and level 2 of the first IV and level 2 of the second IVthe second IV

2121 = mean for level 2 of = mean for level 2 of the first IV and level 1 of the first IV and level 1 of the second IVthe second IV

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• Specify the alpha Specify the alpha levellevel– =.05 for all tests=.05 for all tests

• Look at a graph and Look at a graph and determine which determine which main effects and main effects and interactions are likelyinteractions are likely– No main effect of No main effect of

where you studywhere you study– No main effect of No main effect of

where you recallwhere you recall– An interactionAn interaction

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Factorial ANOVA Summary Factorial ANOVA Summary TableTable

• Source = source of varianceSource = source of variance• SS = sum of squares = top part of variance formulaSS = sum of squares = top part of variance formula• df = degrees of freedom = bottom of variance formuladf = degrees of freedom = bottom of variance formula

– Main effect df = number of levels of IV – 1Main effect df = number of levels of IV – 1– Interaction effect df = df of first IV X df of second IVInteraction effect df = df of first IV X df of second IV– Error df = Error df = ΣΣ(N(Nper conditionper condition – 1) – 1)

• MS = mean squares = variance estimateMS = mean squares = variance estimate• F = F ratio = MSF = F ratio = MSbetween-groupsbetween-groups / MS / MSwithin-groupswithin-groups

• p = probability value (significance)p = probability value (significance)

Source SS df MS F pIV1 (Study) 4 1 4 2 p=.07IV2 (Recall) 2 1 2 1 p=.50IV1 X IV2 (Study X Recall) 12 1 12 6pError (Within-groups) 88 44 2Total 106 47

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Factorial ANOVA DecisionsFactorial ANOVA Decisions

• The line labeled with the name of the first IV tells The line labeled with the name of the first IV tells you whether there is a main effect of that IVyou whether there is a main effect of that IV– When the p value is less than or equal to When the p value is less than or equal to , then you , then you

can reject Hcan reject H00 that all the means are equal that all the means are equal– That is, when p That is, when p , there is a main effect of the IV, there is a main effect of the IV


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• The line labeled with the name of the second IV The line labeled with the name of the second IV tells you whether there is a main effect of that IVtells you whether there is a main effect of that IV– When the p value is less than or equal to When the p value is less than or equal to , then you , then you

can reject Hcan reject H00 that all the means are equal that all the means are equal– That is, when p That is, when p , there is a main effect of the IV, there is a main effect of the IV

Source SS df MS F pIV1 (Study) 4 1 4 2 p=.07IV2 (Recall) 2 1 2 1 p=.50IV1 X IV2 (Study X Recall) 12 1 12 6p.02Error (Within-groups) 88 44 2Total 106 47

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• The line labeled with the name of both of the IVs The line labeled with the name of both of the IVs tells you whether there is an interaction of those IVstells you whether there is an interaction of those IVs– When the p value is less than or equal to When the p value is less than or equal to , then you can , then you can

reject Hreject H00 – That is, when p That is, when p , there is an interaction of the IVs, there is an interaction of the IVs


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Writing the ResultsWriting the Results

• The American Psychological Association The American Psychological Association has a precise format for writing the has a precise format for writing the results of ANOVA in an articleresults of ANOVA in an article

• FF(df(dfbetween-groupsbetween-groups, df, dfwithin-groupswithin-groups) = F value, ) = F value, pp = = p value, p value, MSMSerrorerror =MS =MSwithin-groupswithin-groups, , = = level level

• The MSThe MSerrorerror and and level are only with the level are only with the first F ratio reported unless they changefirst F ratio reported unless they change

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Writing the ResultsWriting the Results

• For the main effect of where you studied:For the main effect of where you studied:– FF(1, 44) = 2.00, (1, 44) = 2.00, pp = .07, = .07, MSMSerrorerror = 2.00, = 2.00,

= .05= .05

• For the main effect of where you recalled:For the main effect of where you recalled:– FF(1, 44) = 1.00, (1, 44) = 1.00, pp =.50 =.50

• For the interaction:For the interaction:– FF(1, 44) = 6.00, (1, 44) = 6.00, pp

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Other ANOVAsOther ANOVAs

• Just as in the t test and in the single factor Just as in the t test and in the single factor ANOVA, there are different types of factorial ANOVA, there are different types of factorial design ANOVAsdesign ANOVAs

• If all the IVs are between-subjects then you If all the IVs are between-subjects then you should use a between-subjects ANOVAshould use a between-subjects ANOVA

• If all the IVs are within-subjects then you If all the IVs are within-subjects then you should use a within-subjects ANOVAshould use a within-subjects ANOVA

• If at least one IV is between-subjects and and If at least one IV is between-subjects and and least one IV is within-subjects, then you least one IV is within-subjects, then you should use a mixed-design ANOVAshould use a mixed-design ANOVA

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• The primary difference between the The primary difference between the three types of factorial design three types of factorial design ANOVAs is in the number of error ANOVAs is in the number of error (within-groups) estimates of variance (within-groups) estimates of variance there arethere are– Between-subjects ANOVA has 1 MSBetween-subjects ANOVA has 1 MSerrorerror

– Mixed-design ANOVA has 2 MSMixed-design ANOVA has 2 MSerrorserrors

– Within-subjects ANOVA has 3 MSWithin-subjects ANOVA has 3 MSerrorserrors

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• ANOVA is not limited to just 2 IVs -- you ANOVA is not limited to just 2 IVs -- you can have as many IVs as is necessary to can have as many IVs as is necessary to answer your questionsanswer your questions– Because of the complexity of the design, you Because of the complexity of the design, you

will rarely see more than 3 or 4 IVs used in a will rarely see more than 3 or 4 IVs used in a single experimentsingle experiment

• When there are more than two IVs, you When there are more than two IVs, you get information on main effects, simple get information on main effects, simple interactions and higher-order interactionsinteractions and higher-order interactions

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Higher Order InteractionsHigher Order Interactions

• A A higher-order interactionhigher-order interaction occurs occurs when the nature of the lower order when the nature of the lower order interaction (such as the two variable interaction (such as the two variable interaction we have talked about) interaction we have talked about) depends on the level of a third (or depends on the level of a third (or higher) IVhigher) IV

• Higher order interactions are often Higher order interactions are often difficult to interpretdifficult to interpret

1 Two Factor ANOVA. 2 Factorial Designs Often researchers want to study the effects of two or more independent variables at the same time Often researchers.

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