PSYC2012 TWO-WAY ANOVA

Components of Variance in 2W-ANOVA

Like one-way ANOVA, variance is partitioned into systematic and unsystematic variance. However, two-way ANOVA is factorial, meaning that there are more components of systematic variance:

1. Systematic

a) Between factors/groups (for main effects)

i. IV1

ii. IV2

2. Unsystematic

a) Referred to as error/residual

In symbols...

Variation in Y = grand mean (u) + A + B + AB + Error

Df Calculation in 2W-ANOVA Summary Table

Remember J & K Levels!

> J = no. of groups of IV1, K = no. of groups of IV2

1. df(IV1) = no. of groups - 1

2. df(IV2) = no. of groups - 1

3. df(Interaction) = df(IV1) TIMES df(IV2)

4. df(Error) = N - JK

> Total sample size - no.groups IV1 TIMES no. groups IV2

5. Total = total sample size - 1!

NOTE Do not confuse df(Interaction) & df(Error) as same!

NOTE Conclusions reference df(total, error)

Interpreting 2W-ANOVA Results From Table

Interaction Difference of a difference! Look at internal cells and see if there is a vertical or horizontal difference!

Main Effects Look at column & row total!

Example No Interaction

> Same difference score across two rows

> Same difference score across two columns

NOTE Cannot conclude significance just by looking at table (but large numerical difference can indicate significance!)

Two-Way ANOVA

Determines the statistical significance of the effect of two independent categorical variables on a single continuous dependent variable as well as their interaction.

> Three analyses each with own f-ratio + p-value

Why is Two-Way ANOVA more statistically powerful?

Accounting for additional factor + interaction decreases UNEXPLAINED/ERROR variance!

Interpreting Bar Graph Results of 2W-ANOVA

Remember IV2 = Colour Segments!

Remember IV1 = Bar groups!

1. No main effect = no difference in height at all!

2. IV1 main effect = height difference between bar groups!

3. IV3 main effect = height difference between segments!

4. Interaction effect = difference of difference!*

*That is, the height difference between segments within bar groups is different!

Why Does 2W-ANOVA use Partial Eta-Squared?

Different Denominator in Calculation! Partial Eta2 DOES NOT USE SS TOTAL!

Recall Eta2 ssB/ssT

Partial Eta2 Excludes variance explained by other variables (e.g. other IV main effect or interaction)

> SS(effect)/SS(effect) + SS(within)

> Does not use SS(total) to eliminate variance represented by ss(IV1/2/Interaction)

How to tell when assumption of homogenous variance is violated?

Recall Assumption: All population variances are equal!

Identifying Violation From Descriptive Stats

> Very different SD between groups

From Graphs

> If one box-plot is dramatically wider

> If one scatterplot is much more spread

> If curve on frequency distribution is much more spread

NOTE ANOVA is robust to violations, you MUST also consider unequal sample size!