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Statistics > Chapter 13: Factorial ANOVA > Flashcards

Chapter 13: Factorial ANOVA Flashcards

1
Q

Theory of Factorial ANOVA

A
  • When an experiment has two or more independent variables
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2
Q

Types of factorial design:

A
  • Independent factorial design
  • Repeated-measures (related) factorial design
  • Mixed design
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3
Q

Independent factorial design

A
  • In this type of experiment there are several independent variables or predictors and each has been measured using different entities (between groups)
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4
Q

Repeated-measures (related) factorial design

A
  • This is an experiment in which several independent variables or predictors have been measured, but the same entities have been used in all conditions.
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5
Q

Mixed design:

A
  • This is a design in which several independent variables or predictors have been measured; some have been measured with different entities whereas others used the same entities
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6
Q

Naming ANCOVA

A

Number of Independent Variables-way-how these variables were measured

  • one-way independent ANOVA;
  • two-way repeated-measures ANOVA;
  • two-way mixed ANOVA;
  • three-way independent ANOVA.
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7
Q

Factorial ANOVA as linear model: Equation

A

Outcome= bo + b1predictor1 + b2predictor2 + b3interaction

  • we need dummy variables
  • interaction variable is simply value of gender dummy variable multiplied by value of alcohol dummy variable

Mean of men who drank no alcohol=bo+b1(0)+b2(0)+b3(0)

  • constant bo: mean of the group for which all variables are coded as 0
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8
Q

Breaking down Variance in two-way ANOVA

A
  • SSa: variance explained by variable a
  • SSb: variance explained by variable b
  • SSaxb: variance explained by interaction of both variables
  • SSR: unexplained variance
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9
Q

Two-way ANOVA: SST

A
  • SST= sum(obs.data-grand mean)^2
  • SST= grand variance (N-1)
    —> SST = s^2 (grand) x (N-1)
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10
Q

Two-way ANOVA: Grand Variance

A
  • variance of all scores when we ignore the group to which they belong
  • grand variance s^(grand)=SST/dfT
  • dfT=N-1
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11
Q

Two-way ANOVA:

  • Model Sum of Squares (SSM)
A
  • total variation explained by model
  • SSM=sum[nk (group mean - grand mean)^2]
  • nk: no. of people in each group
  • How many groups in total= no. of categories of IV1 x no. of Categories of IV2
  • dfM= (no. of groups in total) - 1
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12
Q

Two-way ANOVA: Main Effect of Variable 1

-SSM1

A
  • ignore the effect of other variable
  • SSM1=sum[nk(group mean-grand mean)

Example:
Variable 1: Gender
Categories of gender: female and male

—> SSMg=10(60-58)^2 +10(62-58)^2
~ Multiply the number of people in each category by the squared difference of category mean and grand mean
~ Add the values from each category

~ dfM= (no. of categories)-1

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13
Q

Two-way ANOVA: Interaction effect

  • SSMaxb
A
  • SSMaxb= SSM-SSM1-SSM2
    —> Total model-main effect of variable 1- main effect of variable 2
  • dfaxb= dfM-dfM1-dfM2
  • dfaxb= dfM1 x dfM2
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14
Q

Two-way ANOVA: SSR

A
  • unexplained variance
  • SSR=sum[s^2(nk-1)]
    —> variance of each group x (nk-1)
  • dfR= (overall no. of categories) x (nk-1)
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15
Q

Assumptions of Factorial ANOVA:

A
  • Linearity
  • Normality
  • Homogeneity of Variance
  • Independence of Errors
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16
Q

How to correct for violations of assumptions of Factorial ANOVA:

A
  • if Homogeneity of variance is violated use Welch but only if it’s not more complicated than 2x2 design
  • good option: use post-Hoc or bootstrap because these are robust
  • they won’t help with the main analysis though
  • for robust versions of factorial ANOVA: use R
17
Q

Factorial ANOVA: SPSS

A
  • Analyze
  • General Linear Model
  • Univariate
  • Enter dependent variable on ‘dependent variable’
  • Enter all independent variables on ‘fixed factor’
18
Q

Factorial ANOVA: Graphing Interactions

A
  • Plots
  • Enter predictor variable 1 on ‘Horizontal axis’
  • Enter predictor variable 2 on ‘Separate Lines’
  • Click ‘Add’
  • Click ‘Continue’
19
Q

Factorial ANOVA: Contrasts

A
  • for factorial ANOVA: we cannot enter our own codes
  • instead of using planned contrasts: use standard contrasts
  • use contrasts only for predictors that have more than 2 categories
  • Click ‘Contrasts’
  • Click on the Factor
  • Choose Contrast
  • Click ‘Continue’
20
Q

Factorial ANOVA: Post-Hoc Tests

A
  • Click ‘Post-Hoc’
  • do Post-Hoc only for predictors with more than 2 categories
  • choose the right post hoc tests for your analysis
  • Click ‘Continue’
21
Q

Factorial ANOVA: Bootstrap and other Options

A
  • Click ‘Options’
  • Enter main effects and interaction on ‘Display Means for’
  • Bootstrap: main use when we do Post-Hoc tests
22
Q

Factorial ANOVA: Output

  • Levene’s Test
A
  • Levene’s test: be careful not to depend on its results
23
Q

Factorial ANOVA: Output

  • Main ANOVA table
A
  • tells us if main effects and interaction are significant
    —> did this predictor affect the outcome?
  • for the main effect of one predictor: ignore the effect of other predictors
24
Q

Factorial ANOVA: Output

  • What shouldn’t I do?
A
  • you should not interpret main effects in the presence of a significant interaction effect involving that main effect
25
Q

Factorial ANOVA: Output

  • Contrasts
A
  • in reality we wouldn’t look at this effect if the interaction involving that predictor was significant
  • be careful about how you interpret these contrasts: you need to have a look at the remaining contrast as well
26
Q

Factorial ANOVA: Output

  • Simple Effects Analysis
A
  • a technique used to break interaction effects
  • This analysis looks at the effect of one independent variable at individual levels of the other independent variable
  • Unfortunately, simple effects analyses can’t be done through the dialog boxes and instead you have to use SPSS syntax
  • GLM Attractiveness by gender alcohol
    /EMMEANS = TABLES(gender*alcohol) COMPARE(gender)
    —> This syntax for looking at the effect of gender at different levels of alcohol
27
Q

Factorial ANOVA: Output

  • Post-Hoc Analysis
A
  • You don’t need to interpret main effects if an interaction effect involving that variable is significant
  • If you do interpret main effects then consult Post-Hoc tests to see which groups differ: significance is shown by values in the columns labelled Sig. smaller than .05, and bootstrap confidence intervals that do not contain zero.
28
Q

Factorial ANOVA: Interpreting Interaction Graphs

A
  • Non-parallel lines on an interaction graph show up significant interactions. However, this doesn’t mean that non-parallel lines always reflect significant interaction effects: it depends on how non- parallel the lines are
  • If the lines on an interaction graph cross then obviously they are not parallel and this can be a dead give-away that you have a possible significant interaction. However, if the lines of the interaction graph cross it isn’t always the case that the interaction is significant
  • bar charts: interaction if groups have different patterns
29
Q

Factorial ANOVA: Calculating Effect Size

A
  • SPSS calculates partial eta squared
  • don’t use that
  • use omega squared instead
  • omega squared: effect size is the variance estimate for the effect in which you’re interested divided by the total variance estimate
  • variance of the effect of a= (a-1) (MSa-MSR)/ nab
  • variance of the effect of b= (b-1) (MSb-MSR)/ nab
  • variance of interaction =(a-1)(b-1) (MSaxb-MSR)/ nab
  • total variance= Va+ Vb+ Vaxb +MSR
  • Effect sizes for Simple Effect Analysis

—> r= √(F/ (F+dfR)

30
Q

Factor ANOVA: Reporting Results

A
  • There was a significant main effect of the amount of alcohol consumed in the nightclub on the
    attractiveness of the mate selected, F (2, 42) = 20.07, p < .001, ω2 = .35. Bonferroni post hoc tests revealed that the attractiveness of selected dates was significantly lower after 4 pints than both after 2 pints and no alcohol (both ps < .001). There was no significant difference in the attractiveness of dates after 2 pints and no alcohol, p = 1
  • There was a non-significant main effect of gender on the attractiveness of selected mates, F(1, 42) = 2.03, p = .161, ω2 = .009.
  • There was a significant interaction between the amount of alcohol consumed and the gender of the person selecting a mate, on the attractiveness of the partner selected, F(2, 42) = 11.91, p <
    .001, ω2= .20. This effect indicates that males and females were affected differently by alcohol.
  • Specifically, the attractiveness of partners was similar in males (M = 66.88, SD = 10.33) and females (M = 60.63, SD = 4.96) after no alcohol and 2 pints (males, M = 66.88, SD = 12.52; females, M = 62.50, SD = 6.55); however, attractiveness of partners selected by males (M = 35.63, SD = 10.84) was significantly lower than those selected by females (M = 57.50, SD = 7.07) after 4 pints.

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