Intro to Statistics for the Behavioral Sciences PSYC 1900 Lecture 15: Interactions in Factorial ANOVA.

Intro to Statistics for the Intro to Statistics for the Behavioral SciencesBehavioral Sciences

PSYC 1900PSYC 1900

Lecture 15: Interactions in Factorial Lecture 15: Interactions in Factorial ANOVAANOVA

Factorial ANOVA ReviewFactorial ANOVA Review An example: Effects of An example: Effects of

Temperature and Gender on Temperature and Gender on AggressionAggression

Each Factor has marginal Each Factor has marginal means (i.e., means means (i.e., means averaged across the other averaged across the other iv)iv)

Main EffectMain Effect The effect of one iv The effect of one iv

averaged across the levels averaged across the levels of the otherof the other

Simply a one-way ANOVA Simply a one-way ANOVA on the marginal meanson the marginal means

Here, there are two main Here, there are two main effects.effects.

They tell us if gender They tell us if gender and/or temperature affects and/or temperature affects aggression.aggression.

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Factorial ANOVA Factorial ANOVA CalculationsCalculations

As with one-ways, we As with one-ways, we calculate SS for each effect.calculate SS for each effect.

SSSStotaltotal again captures all again captures all variance of scores around variance of scores around grand mean.grand mean.

Each main effect captures Each main effect captures variance of cell means variance of cell means around grand meanaround grand mean

Differences among all cell Differences among all cell means is captured by SSmeans is captured by SScellscells..

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2( )totalSS X G 2( )A ASS nb X G 2( )B BSS na X G

2( )cells ABSS n X G

Main EffectsMain Effects

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Testing Significance of Main Testing Significance of Main EffectsEffects

We convert each relevant We convert each relevant SS into Mean Squares (MS).SS into Mean Squares (MS). We divide SS by associated We divide SS by associated

df’s.df’s. dfdftotaltotal=N-1=N-1 dfdffactorfactor= (#conds)-1= (#conds)-1 dfdferrorerror=(a)(b)(n-1)=(a)(b)(n-1)

F is the ratio of these two F is the ratio of these two estimates of population estimates of population variance.variance.

Critical values of F are Critical values of F are found using F(dffound using F(dfgroupgroup, df, dferrorerror)),,

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A

SSMS

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errorerror

error

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group

error

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MS

InteractionsInteractions

The effects of one The effects of one independent variable independent variable depend on the level of depend on the level of another independent another independent variable.variable.

Pattern of means Pattern of means cannot be captured cannot be captured based solely on based solely on deviations form grand deviations form grand mean due to main mean due to main effects.effects.

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Interaction CalculationsInteraction Calculations

Interactions represent variation Interactions represent variation among cell means that cannot among cell means that cannot be captured by main effects be captured by main effects AND is not likely due to AND is not likely due to simple sampling error.simple sampling error.

Differences among all cell means Differences among all cell means is captured by SSis captured by SScellscells..Therefore, interaction can be Therefore, interaction can be

defined as variation defined as variation among cell means not among cell means not explained by main explained by main

effects.effects.

Degrees of freedom for Degrees of freedom for interactions are: (a-1)(b-1)interactions are: (a-1)(b-1)

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( )cells AB

A B cells A B

A BA B

A B

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error

SS n X G

SS SS SS SS

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Types of InteractionsTypes of Interactions

OrdinalOrdinal Ordinal positions of Ordinal positions of

group differences remain group differences remain constantconstant

Main effects may be Main effects may be interpretedinterpreted

DisordinalDisordinal Group differences Group differences

reverse their signs at reverse their signs at some level of the other some level of the other variablevariable

Main effects usually not Main effects usually not interpretableinterpretable

Interpreting InteractionsInterpreting Interactions

When an interaction is present, it When an interaction is present, it simply means that the effects of one simply means that the effects of one variable on the dv depend on levels variable on the dv depend on levels of the other.of the other. This implies that the “simple effects” of This implies that the “simple effects” of

one iv differ across levels of the other iv.one iv differ across levels of the other iv. Simple effect is the effect of one iv at Simple effect is the effect of one iv at

one specific level of the other.one specific level of the other.

Analysis of Simple EffectsAnalysis of Simple Effects•Analysis of simple effects proceeds by dividing the factorial design into a series of single-factor experiments.

a1

a2

b1 b2 b3

b1 b2 b3

b1 b2 b3

•Simple effects are based upon the cell means within a given level of one of the iv’s. Main effects are based upon marginal means.

•If an interaction is present, at least one of the simple effects must differ from at least one of the others.

•Multiple comparison techniques are used to determine the relations among the means at each level of one of the iv’s.

•Here, one might use Fisher’s LSD Tests to compare 3 means at each level of A.

An ExampleAn Example

A researcher conducts a study examining the A researcher conducts a study examining the effects of stress and sex (i.e., gender) on effects of stress and sex (i.e., gender) on aggression.aggression.

Men and women receive either a stressful or Men and women receive either a stressful or simple task to perform and then have their simple task to perform and then have their performance insulted by a confederate. performance insulted by a confederate. Aggression is then measured.Aggression is then measured.

Prediction was that stress should increase Prediction was that stress should increase male aggression but decrease female male aggression but decrease female aggression.aggression.

The pattern of means looks to support the claim, but we must use a factorial ANOVA to verify it.

Descriptive Statistics Dependent Variable: Aggression

Stress Sex Mean Std.

Deviation N .00 4.3000 .94868 10 1.00 2.9000 1.19722 10

.00

Total 3.6000 1.27321 20 .00 5.3000 1.15950 10 1.00 2.3000 1.15950 10

1.00

Total 3.8000 1.90843 20 .00 4.8000 1.15166 20 1.00 2.6000 1.18766 20

Total

Total 3.7000 1.60448 40

Tests of Between-Subjects Effects Dependent Variable: Aggression

Source Type III Sum of Squares df Mean Square F Sig.

Corrected Model 55.200(a) 3 18.400 14.655 .000 Intercept 547.600 1 547.600 436.142 .000 Stress .400 1 .400 .319 .576 Sex 48.400 1 48.400 38.549 .000 Stress * Sex 6.400 1 6.400 5.097 .030 Error 45.200 36 1.256 Total 648.000 40 Corrected Total 100.400 39

a R Squared = .550 (Adjusted R Squared = .512)

Simple Effects of StressSimple Effects of Stress Independent Samples Test(a)

Levene's Test for Equality of Variances t-test for Equality of Means

95% Confidence Interval of the

Difference

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference Lower Upper

Equal variances assumed .279 .604 -2.111 18 .049 -1.00000 .47376 -1.99532 -.00468

Aggression

Equal variances not assumed -2.111 17.321 .050 -1.00000 .47376 -1.99813 -.00187

a Sex = .00

Independent Samples Test(a)

Levene's Test for Equality of Variances t-test for Equality of Means

95% Confidence Interval of the

Difference

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference Lower Upper

Equal variances assumed .020 .890 1.138 18 .270 .60000 .52705 -.50728 1.70728

Aggression

Equal variances not assumed 1.138 17.982 .270 .60000 .52705 -.50736 1.70736

a Sex = 1.00

Simple Effects of SexSimple Effects of Sex Independent Samples Test(a)

Levene's Test for Equality of Variances t-test for Equality of Means

95% Confidence Interval of the

Difference

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference Lower Upper

Equal variances assumed .341 .567 2.898 18 .010 1.40000 .48305 .38516 2.41484

Aggression

Equal variances not assumed 2.898 17.106 .010 1.40000 .48305 .38134 2.41866

a Stress = .00

Independent Samples Test(a)

Levene's Test for Equality of Variances t-test for Equality of Means

95% Confidence Interval of the

Difference

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference Lower Upper

Equal variances assumed .047 .831 5.785 18 .000 3.00000 .51854 1.91058 4.08942

Aggression

Equal variances not assumed 5.785 18.000 .000 3.00000 .51854 1.91058 4.08942

a Stress = 1.00