Factorial ANOVA

Situations calling for Factorial ANOVA

The least complex factorial ANOVA design is a situation where we have only 2 IVs, and both have only 2 levels

This is called a "2x2" design ("# levels of IVA x # levels of IVB")

Example: "Researchers were interested in studying the effects of diet (high-fat vs. low-fat) and presence or absence of regular exercise on weight change over two months"

Here, we have two IVs:

(1) diet

(2) exercise

Still just one DV = "weight change"

GRAPHS INTERACTIONS

Main effects and interaction effects: definition and conceptual understanding

Next up, calculating SSA and SSB

SSA = nb∑(XbarA - GM)2

SSB = na∑(XbarB - GM)2

Subtract grand mean from row/column means, square, and add

Multiply by n (= cell size) but also by # of levels of other IV (in notation below = b or a)

Design tables and cells/ cell means / row and column means

A design table is an efficient way of graphically representing all the levels of our factors/IVs

Mean scores averaged across the rows and columns lead to the grand mean

Three different F ratios that are calculated for a 2x2 factorial ANOVA

#1 represents main effect of diet

#2 represents main effect of exercise

#3 represents interaction of diet & exercise

Breaking down SStotal into SSA and SSB

SSA = nb∑(XbarA - GM)2

SSB = na∑(XbarB - GM)2

Subtract grand mean from row/column means, square, and add

Multiply by n (= cell size) but also by # of levels of other IV (in notation below = b or a)

How to calculate SScells

We have a new term to use in factorial ANOVA

SScells: treat each individual cell as its own group

Formula:

SScells = n∑(Xbar - GM)2

Breaking down SStotal into SSAB

SSAB is also called "interaction sum of squares"

Once we get SScells, then we have all the information needed for SSAB (which is what we really want):

SSAB = SScells - SSA - SSB

Breaking down SStotal into SSerror

SSerror is calculated pretty much just like SSerror in one-way ANOVA...

...Take out all the "effects" from the total sum of squares, and it's what's left over!

In this case, SScells is a combination of all the effects, so it again provides a shortcut for calculation

SSerror = SStotal - SScells

For example data: SStotal = 943.43, SScells = 682.6

SSerror = 943.43 - 682.6 = 260.83

Calculation of Sums of Squares for Factorial ANOVA

(1) SStotal = ∑X2 - (∑X)2/N

(2) SSA = nb∑(XbarA - GM)2

(3) SSB = na∑(XbarB - GM)2

(4) SScells = n∑(Xbar - GM)2

(5) SSAB = SScells - SSA - SSB

(6) SSerror = SStotal - SScells

MEMORIZE 5 +6

Algebraic relations among various sums of squares

Calculating degrees of freedom for each main effect and for the interaction effect

MS = SS / df

(1) Degrees of freedom for main effects:

Number of levels minus 1

Here: each IV has 2 levels, therefore dfA = dfB = 1

(2) Degrees of freedom for interaction:

Always equals the product of the two main effect degrees of freedom

So...

dfAB = dfA* dfB

Here, dfAB = 1 * 1 = 1

Calculating mean squares for each of out three tests

MSA = SSA / dfA

MSB = SSB / dfB

MSAB = SSAB / dfAB

Calculating F ratios for each of our three tests

FA = MSA / MSerror

FB = MSB / MSerror

FAB = MSAB / MSerror

Interpreting SPSS ANOVA table output for a 2x2 factorial ANOVA

LOOK AT PICTURE

Distinguishing between significant main effects in graphs

Notice, from previous slide, that we have a significant interaction effect (as well as significant main effects)

For interaction: "F(1, 16) = 7.22, p < .05"

What does this mean?

"The effect of diet depends on exercise /

The effect of exercise depends on diet"

(These statements are identical in meaning)

Between-subjects effects RM-ANOVA

Similar to IVs that we've analyzed so far

Different scores represent differences among participants

Examples: Gender, Experimental Group

Multiple between-subjects effects in one analysis is okay

Within-subjects effects in RM-ANOVA

Unique to RM-ANOVA

Different scores represent differences within each participant

Usually labeled as: TIME (textbook example: "weeks")

Most cases: only one within-subjects effect

Interactions among effects are possible (beyond scope of PSYS 054)

New Notation for RM-ANOVA

t = Number of time points ( = 3)

N = number of scores ( = 15)

n = number of participants ( = 5)

Xbartime = mean score on each exam, across participants

Xbarsubj = mean score for each participant, across exams

Our means will have associated sums of squares (SStime and SSsubj), degrees of freedom, etc.

Advantages of repeated measures ANOVA

Calculations for RM-ANOVA for a single within-subjects effect

(1) Calculate SStotal (same as before)

(2) Calculate SSsubj

(3) Calculate SStime

(4) Calculate SSerror

(5) Calculate dfs and mean squares

(6) Calculate F ratio (and test hypothesis)

MOST IMPORTANT calculation point:

SStotal = SSsubj + SStime + SSerror