Mixed ANOVA Models combining between and within. Mixed ANOVA models We have examined One-way and...

Mixed ANOVA ModelsMixed ANOVA Modelscombining between and combining between and

withinwithin

Mixed ANOVA modelsMixed ANOVA models

We have examined One-way and We have examined One-way and Factorial designs that use:Factorial designs that use:– a single between-subjects IVa single between-subjects IV– multiple between-subjects IVsmultiple between-subjects IVs– a single within-subjects IVa single within-subjects IV– multiple within-subjects IVsmultiple within-subjects IVs

Mixed ANOVA modelsMixed ANOVA models

Mixed ANOVA modelsMixed ANOVA models– contain at least one between-contain at least one between-

subjects IV and one within-subjects IV and one within-subjects IVsubjects IV

– two-way, three-way, or higher two-way, three-way, or higher order factorial designs can be order factorial designs can be created using any combination created using any combination of between and within subjects of between and within subjects IVsIVs

Two-Way Mixed ANOVATwo-Way Mixed ANOVA

One between-subjects IVOne between-subjects IV One within-subjects IVOne within-subjects IV Commonly used designCommonly used design Very useful for addressing Very useful for addressing

frequently occurring research frequently occurring research questionsquestions

Often called split-plot design Often called split-plot design from origins in agricultural from origins in agricultural applicationsapplications

Split-plot designsSplit-plot designs

Two crops are comparedTwo crops are compared Each crop is exposed to three Each crop is exposed to three

fertilizer conditionsfertilizer conditions The combined effect of crop The combined effect of crop

and fertilizer is examinedand fertilizer is examined

Fertilizer I Fertilizer II Fertilizer IIICrop ACrop B


The within-subjects IV can take all The within-subjects IV can take all three forms:three forms:– the same subjects are the same subjects are

measured on 3 or more measured on 3 or more occasionsoccasions

– the same subjects are exposed the same subjects are exposed to 3 or more treatmentsto 3 or more treatments

– the same subjects provide three the same subjects provide three or more ratings that are or more ratings that are measured on the same scalemeasured on the same scale


The between-subjects IV can be:The between-subjects IV can be:– randomly assigned - treatment randomly assigned - treatment

vs. controlvs. control– attribute variable - gender, grade, attribute variable - gender, grade,

age group, etc.age group, etc. The most common use involves:The most common use involves:

– between-subjects IV – treatment between-subjects IV – treatment or control conditionor control condition

– within-subjects IV - growth over within-subjects IV - growth over timetime

ExamplesExamples

Treatment and control groups are Treatment and control groups are are assessed on pre, mid, and are assessed on pre, mid, and post treatment occasions.post treatment occasions.

Males and females are given Males and females are given three different types of three different types of medication.medication.

Tenured and non-tenured Tenured and non-tenured teachers rate three different teachers rate three different aspects of school climate.aspects of school climate.

ExamplesExamples

Children are randomly Children are randomly assigned to get the treatment assigned to get the treatment (Head Start) or not (At home & (Head Start) or not (At home & daycare), AND are assessed on daycare), AND are assessed on pre, mid, and post treatment pre, mid, and post treatment occasions.occasions.Pre Mid Post

Head StartAt homeDaycare

ExamplesExamples

Males and females rate the Males and females rate the same three reasons for same three reasons for teaching in a private teaching in a private religious school.religious school.

Values Support Rel.Bel.Males

Females


Both the between-subjects IV Both the between-subjects IV and the within-subjects IV can and the within-subjects IV can have any number of levels (2+).have any number of levels (2+).

Three research questionsThree research questions Three sets of null and Three sets of null and

alternative hypothesesalternative hypotheses Two main effects, one Two main effects, one

interactioninteraction


The question and hypotheses The question and hypotheses for the between-subjects IV for the between-subjects IV will follow the same patterns will follow the same patterns we have used before.we have used before.

The question and hypotheses The question and hypotheses for the within-subjects IV will for the within-subjects IV will also follow the same patterns also follow the same patterns we have used before. we have used before.


Interpret the interaction Interpret the interaction effect first.effect first.

Follow the same Follow the same interpretation strategies we interpretation strategies we have used for other types of have used for other types of factorial designs.factorial designs.

Graphing is particularly Graphing is particularly helpful.helpful.

Profile Analysis ApproachProfile Analysis Approach

Uses Multivariate Approach Uses Multivariate Approach No sphericity assumptionNo sphericity assumption Homogeneity of Variance - Homogeneity of Variance -

CovarianceCovariance Main Effect for Group Main Effect for Group

– HeightHeight Main Effect for Time Main Effect for Time

– SlopeSlope Group X Time InteractionGroup X Time Interaction

– ParallelismParallelism

ExamplesExamples

The Mixed ANOVA approach is the The Mixed ANOVA approach is the best way to analyze the data we best way to analyze the data we have been working with all have been working with all semester.semester.

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48

50

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Pre Mid PostHead StartAt HomeDaycare

Steps for InterpretationSteps for Interpretation

Follow the same interpretation Follow the same interpretation guidelines as other Factorial designsguidelines as other Factorial designs

Use the Tukey Spreadsheet on the Use the Tukey Spreadsheet on the webweb

Calculate the appropriate effects Calculate the appropriate effects sizes that “tell the story”sizes that “tell the story”


Step 1 – Interpret the interaction termStep 1 – Interpret the interaction term

Step 2 – Interpret the main effectsStep 2 – Interpret the main effects

Step 3 – Graph the data “both ways”, Step 3 – Graph the data “both ways”, meaning exchange the row and meaning exchange the row and column variables to determine which column variables to determine which picture is most usefulpicture is most useful


Typically it is most helpful to illustrate Typically it is most helpful to illustrate “change over time”, or whatever the “change over time”, or whatever the within-subjects variable is, on the X within-subjects variable is, on the X axisaxis

Typically it is most helpful to put the Typically it is most helpful to put the group variable, or whatever the group variable, or whatever the between-subjects term is, as the between-subjects term is, as the separate lines variable.separate lines variable.

Time on X, Groups as Time on X, Groups as LinesLines

Social Development by Schedule

1.500

2.000

2.500

3.000

3.500

4.000

Fall Winter SpringSplit Day

Split Week


Step 4 – If the interaction term is Step 4 – If the interaction term is statistically significant, qualify the statistically significant, qualify the interpretation of the main effects.interpretation of the main effects.

Step 5 – If there is a statistically Step 5 – If there is a statistically significant main effect with only two significant main effect with only two levels, no more analyses are needed levels, no more analyses are needed for that effect. Simply examine the for that effect. Simply examine the two marginal means (row or column two marginal means (row or column totals).totals).


Step 6 – If there is a main effect with Step 6 – If there is a main effect with more than two levels, perform post more than two levels, perform post hoc comparisons among the marginal hoc comparisons among the marginal means (row or column totals).means (row or column totals).

This may require running additional This may require running additional analyses as SPSS only gives you Post analyses as SPSS only gives you Post Hoc comparisons for Between-Hoc comparisons for Between-Subjects terms.Subjects terms.


Step 7 – Next, turn to the interaction Step 7 – Next, turn to the interaction effects. There is not one rule that fits effects. There is not one rule that fits all situations. The exact comparisons all situations. The exact comparisons needed to make interpretations will needed to make interpretations will vary from analysis to analysis. vary from analysis to analysis.

Look for the portion of your graphs Look for the portion of your graphs where the lines are non-parallel.where the lines are non-parallel.

Next use the Tukey spreadsheet.Next use the Tukey spreadsheet.

Steps for InterpretationSteps for InterpretationStep 8 – Consider Simple Effects first. Step 8 – Consider Simple Effects first.

This means look at the pattern of This means look at the pattern of differences with rows or columns in your differences with rows or columns in your design first. If they are different, then design first. If they are different, then you have your answer about where the you have your answer about where the interaction is coming from.interaction is coming from.

If this does not completely explain the If this does not completely explain the interaction, then consider looking at cell interaction, then consider looking at cell mean comparisons across rows and mean comparisons across rows and columns.columns.


Step 9 – Effect size calculations. Again, Step 9 – Effect size calculations. Again, there is no one rule that will fit every there is no one rule that will fit every situation. Your job is to illustrate the situation. Your job is to illustrate the findings from your study with the effect findings from your study with the effect sizes that fit the pattern in the results.sizes that fit the pattern in the results.

Within-subjects terms = Dependent CaseWithin-subjects terms = Dependent CaseBetween-subjects terms = Independent Between-subjects terms = Independent

CaseCase


Center, p<.005Center, p<.005

Time, p<.001Time, p<.001

Interaction, Interaction, p<.001p<.001

Now what?Now what?

Mixed ANOVA Models combining between and within. Mixed ANOVA models We have examined One-way and...

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