Chapter 13: Factorial ANOVA Flashcards
1
Q
Theory of Factorial ANOVA
A
- When an experiment has two or more independent variables
2
Q
Types of factorial design:
A
- Independent factorial design
- Repeated-measures (related) factorial design
- Mixed design
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)
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.
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
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.
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
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
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)
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
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
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
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
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)
15
Q
Assumptions of Factorial ANOVA:
A
- Linearity
- Normality
- Homogeneity of Variance
- Independence of Errors
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
Factorial ANOVA: Output
- Contrasts
- 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
Factorial ANOVA: Output
- Simple Effects Analysis
- 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
Factorial ANOVA: Output
- Post-Hoc Analysis
- 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
Factorial ANOVA: Interpreting Interaction Graphs
- 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
Factorial ANOVA: Calculating Effect Size
- 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
Factor ANOVA: Reporting Results
- 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.
