Two-way Independent groups ANOVA

Two-way Independent group ANOVA

We want to examine the effect of two different independent variables at once?

Two-way

2 IV (aka “factors”)

-each factor can have any # of levels

Independent

Different participants in each group

Simplest way to combine 2 or more factors: factorial design

• each level of Factor A is combined with each level of Factor B to form a cell

• each cell is a different sample made up of different participants

• Balanced design: all cells have equal ns

Factorial design (How is it represented )

• represented as a table/matrix

  • rows represent levels of one factor

  • columns represent levels of the other

  • each box in table is a cell

Factorial design (how is it named)

• named by “# of levels of one factor x # of levels of other factor”

• example 2×3 factorial design

Why not conduct 2 separate one-way ANOVAs?

• more efficient —> test effects of 2 IVs with one study

• More importantly: 2-way ANOVA allows us to examine INTERACTION between factors

Interaction

when effect of one IV changes depending on the level of the other IV

3 total effects tested for in 2-way ANOVA (each get their own F test)

• main effect of factor A

• main effect of factor B

• Interaction effect

Main effect (ME) - what is it asking?

do the means of levels of factor A differ from each other, ignoring/averaging across the levels of factor B? (and vice versa)

Main effect:

• compares marginal means of table of cells to each other

  • compares column means to each other

  • or compares row means to each other

Interaction effect: what is it asking?

Does effect of one factor differ depending on level of other factor?

• compares cell means to each other

• do differences between levels of factor A depend on which level of factor B we’re on?

Interaction effect example on table

i.e. is the pattern of differences between cell means within the same column different for different columns?

i.e. is the pattern of differences between cell means within the same row different for different rows?

If lines are parallel:

No interaction

SStot =SSbet + SSw

total variability split into two independent chunks

Assumptions of 2-way ANOVA

• independent random sampling

  • best to randomly sample

  • if not: randomly assign to conditions (if you’re doing a true experiment)

  • independence of participants across groups

• Normal Distribution of DV in populations

• Homogeneity of variance

  • in populations, all groups have equal variances

  • not an issue for balanced designs (equal ns)

SS total

each score’s deviation from the grand mean, squared

• represents total variability in DV from all causes

FOr one-way ANOVA equation

SS total = SS bet + SS w

SS bet

represents explained variance/treatment effect

SSw

represents unexplained variance/error

How is the variability in the DV broken up to analyze? (in a 2-wway ANOVA)

• as we add factors, we split the total variability in our sample (SStotal) into more and more specific components, each one explaining additional variability

• one-way anova: SStot = SSbet + SSw

• Two-way ANOVA: SStot = SSrow + SScolumn +SSinteraction +SSw

Explained variance =

three independent sources, one for each effect:

• variance between rows (main effect of task difficulty)

• variance between columns (main effect of motivation)

• variance between cells (interaction)

In 2 way ANOVA — how is the variability in the DV broken up to analyze

SSbet still represents ALL explained variability, all variability between-groups, but now is comprised of the three sources

SSbet = SSrow + SS column + SSinter

-with three sources, more variability is explained

-SS bet now will be bigger than it was with only 1 IV

With three sources, more variability is explained:

—> less variability, lef unexplained

—> SSw gets smaller

-smaller error term —> easier to reach significance

The three effects each have their own df, MS, and F:

Use these df to find the appropriate Fcrit for the test of each effect

-look up in table: Fcrit (df whichever effect you’re testing, dfw)

—> there will be 3 separate F crits

• compare calculated F ratio for each effect to its respective Fcrit

• If calculated F>Fcrit for that effect, that effect is significantHo

How to interpret significant effects

• Main effects: are there differences between the levels of one IV averaged across the levels of the second IV?

• ME of motivation: ignoring task difficulty, what is the effect of motivation?

  • are there differences between low, med, and high motivation, averaged across easy and hard tasks?

• ME of task difficulty: ignoring motivation, what is the effect of task difficulty?

  • are there differences between easy and hard tasks, averaged across motivation level?

Interpreting Significant Main effects

• if you have 3+ groups, MEs only tell you that there is a significant difference SOMEWHERE between that factor’s marginal means

  • but you don’t know where

• again, can perform follow-up tests to pinpoint which means are different from each other

Interpreting Significant Interactions

• significant interaction tells us that effect of one IV depends on what level of the second IV we are at

• pattern of means for low —> medium —> high motivation for easy tasks differs from the pattern of means for low —> medium —> high motivation for difficult tasks

Important note: significant interactions …

When interatiction is significant, significant main effects may be misleading

• interpret MEs with caution when the interaction is significant

• always look at your data visually to see what’s going on

For one-way ANOVA, we had n² (effect sizes)

• was biased

• used w² instead

• measures proportion of variability in DV that was explained by IV