Analysis of Variance (Two Factors)

Two Factor Analysis of VarianceMain effect

The effect of a single factor when any other factor is ignored.

Example page 386 – Responsibility in crowdsMain effects

Crowd size Gender

Table 18.1

Two Factor Analysis of VarianceInteraction effect

Combination or interaction of crowd size and gender on reaction time

Interaction occurs whenever the effects of one factor on the dependent variable are not consistent for all values (or levels) of the second factor.

Two factor hypothesesThree different null hypotheses are tested

one at a time, with three different F tests.Fcolumn

Variability between columns (crowd size)Frow

Variability between rows (gender)Finteraction

Any remaining variability between cells not attributed to either of the other types of variability

Simple effectThis represents the effect of one factor on the

dependent variable at a single level of the second factor.

Page 392 – two simple effects of crowd size, one for males, one for females

Example C - since both simple effects are in the same direction the main effect can be interpreted without referring to its two simple effects.

Interaction can be viewed as the product of inconsistent simple effects.

SStotal = SScolumn + SSrow + SSinteraction +Sswithin

SStotal = SSbetween +Sswithin

SSbetween = SScolumn + SSrow + Ssinteraction

SSinteraction = SSbetween - (SScolomn + SSrow )

Computation formulas T2

cell G2

SSbetween = Σ n N

T2cell

SSwithin = Σ X2 – Σ n

T2column G2

SScolumn = Σ rn N

Computation formulas

T2row G2

SSrow = Σ cn N

Degrees of freedomdftotal = N – 1 N = all data elements

dfcolumn = c – 1 c = columns or groups

dfrow = r – 1 r = rows or categories

dfinteraction = (c – 1)(r – 1)

dfwithin = N – (c)(r)

Estimating Effect Size _________SScolumn___________ ____SScolumn_______

η2p(column)= SStotal – (SSrow + SSinteraction) = SScolumn + SSwithin

____SSrow___

η2p(row)= SSrow + SSwithin

____SSinteraction___

η2p(interaction)= SSinteraction + SSwithin

Each η2 is a partial eta squared, accounting for only part of the total variance.

Calculating Simple effectsPage 404

T2se G2

se

SSse = Σ n Nse

Where SSse signifies the sum of squares for the simple effect, or a single row (or column)

Tukey’s HSD for Multiple ComparisonsPage 405Use Tukey’s HSD test to find differences

between pairs of means.

Tukey’s “honestly significant difference” test

MSwithin

HSD = q√ n

Where q (studentized range statistic) comes from Table G, Appendix C, page 529

Estimating effect SizeUse Cohen’s d (defined on page 355,

Equation 16.10)

X1 – X2

D = √ MSwithin

Interpreting Two Factor ANOVAPage 406If interaction is significant,

Estimate its effect size with η2p and conduct Fse

tests for at least one set of simple effectsFurther analyze any significant simple effects

with HSD tests and any significant HSD test with an estimate of its effect size, d.

Interpreting Two Factor ANOVAPage 406If interaction is NOT significant,

Estimate its effect size with η2p and conduct F

testFurther analyze any significant effects with

HSD tests and any significant HSD test with an estimate of its effect size, d.

(See flow chart in Word format)