DOCTORAL SEMINAR, SPRING SEMESTER 2007

Experimental Design & Analysis

Two-Factor Experiments

February 20, 2007

Two-Factor Experiments

Two advantagesEconomyDetection of interaction effects

Economy

a1 a2 a3

n=30 n=30n=30

b1 b2 b3

n=30 n=30n=30

b1 b2 b3

a1 n=10 n=10 n=10

a2 n=10 n=10 n=10

a3 n=10 n=10 n=10

Compare N for 2 one-factor experiments

with 1 two-factor experiment

N=180 N=90

Detection of Interactive Effects

Factors may have multiplicative effect, rather than an additive one

Interactions suggest important boundary conditions for hypothesized relationships, giving clues to nature of explanation

Two-Factor Analysis

Sources of variance when A and B are independent variables A B AxB S/AxB

The model is Yij = μ + αi + βj + (αβ)ij +εij

Overall grand mean

Average effect of α

Average effect of β

Interaction effect of α, β (effect left in data

after subtracting offlower-order effects)

Error term, alsoknown as S/AxB,or randomness

Two-Factor Analysis

Yijk = μ + αi + βj + (αβ)ij +εijk

We want to test 3 main hypotheses Main effect of A

H0: α1 = α2 = …= αa = 0 vs. H1: at least one α ≠ 0

Main effect of B H0: β1 = β2 = …= βb = 0 vs. H1: at least one β ≠ 0

Interaction effect of AB H0: αβij = 0 for all ij vs. H1: at least one αβ ≠ 0

Two-Factor Analysis

Sources of variance in two-factor design Total sum of squares: Difference between each score

and grand mean is squared and then summed The deviation of a score from the grand mean can be

divided into 4 independent components 1st component - deviation of row mean from grand mean 2nd component - deviation of column mean from grand

mean 3rd component - deviation of an individual's score from its

corresponding cell mean (only affected by random variation) If we take these 3 components and subtract them from SST

we can find a remaining 4th source of variation, which is interaction effect

Two-Factor Analysis

Sum of SquaresTotal = (Xijk – X…)2

Sum of SquaresB = an(X.j – X…)2

Sum of SquaresA = bn(Xi. – X…)2

Sum of SquaresS/AxB = n(Xijk – Xij)2

.

Two-Factor Analysis

Computations in two-way ANOVA involves 4 steps 1. Examining the model for sources of variance when A and B are

independent variables A (with a levels) B (with b levels) AxB (interaction effect of A, B) S/AxB (subjects nested within factors A, B)

2. Determine degrees of freedom A: a-1 B: b-1 AxB: (a-1)(b-1) = ab - a - b +1 S/AxB: ab(n-1) = abn - ab Total: abn - 1

Two-Factor Analysis

3. Construct formulas for sums of squares using bracket terms [A], [B], [AB], [Y], [T]

Sums and means [A] = ΣAj

2 /bn [A] = bnΣYAj2

[B] = ΣBk2 /an [B] = anΣYBk

2

[AB] = ΣABjk2 /n [AB] = nΣYijk

2

[Y] = ΣYijk2 [Y] = ΣYijk

2

[T] = T2 /abn [T] = abnYT2

Bracket terms SSA = [A] – [T] SSB = [B] – [T] SSAxB = [AB] – [A] – [B] + [T] SSS/AB = [Y] – [T]

See Keppel and Wickens, p. 217-218, for summary table of computational formulas

Two-Factor Analysis

See Keppel and Wickens, p. 217-218, for summary table of computational formulas

Source SS computation df MS f

A [A]-[T] a-1 SSA/dfA MSA/MSS/AB

B [B]-[T] b-1 SSB/dfB MSB/MSS/AB

AxB [AB]-[A]-[B]+[T] (a-1)(b-1)

= ab-a-b+1

SSAxB

dfAxB

MSAxB/MSS/AB

S/AB [Y]-[AB] ab(n-1)

= abn-ab

SSS/AB

dfS/AB

Total [Y]-[T] abn-1

4. Specify mean squares and F ratios for analysis

Numerical Example

See Keppel and Wickens, p. 221 Control Drug X Drug Y Control Drug X Drug Y

a1b1 a2b1 a3b1 a1b2 a2b2 a2b2

1 13 9 15 6 14

4 5 16 6 18 7

0 7 18 10 9 6

7 15 13 13 15 13

1-hour deprivation 24-hour deprivation

ABjk 12 40 56 44 48 40

ΣY2 66 468 830 530 666 450

Mean 3 10 14 11 12 10

Std dev 3.16 4.76 3.92 3.92 5.48 4.08

Std error 1.58 2.38 1.96 1.96 2.74 2.04of mean

Numerical Example

What is the total sum? What are the marginal

sums?

1hour 24hour Sum

Control 12 44 56

Drug X 40 48 88

Drug Y 56 40 96

Sum 108 132 240

Two-Factor Analysis

[T] = T2/abn = 2402/(3)(2)(4) = 2,400

[A] = ΣAj2/bn = 562 + 882 + 962/(2)(4) = 2,512

[B] = ΣBk2/an = 1082 + 1322/(3)(4) = 2,424

[AB] = ΣABjk2/n = 122 + 402 + … + 482 + 402/4 = 2,680

[Y] = ΣYijk2 = 66 + 468 + 830 + 530 + 666 + 450 = 3,010

Numerical Example

0

10

20

30

40

50

60

Control Drug X Drug Y

1-hr deprivation

24-hr deprivation

Main Effects and Interactionsa1

a2 a2a2

a1

a1

b1 b2 b1 b2 b1 b2

a2a2

a2

a1

a1

a1

b1 b2 b1 b2 b1 b2

What’s the Story?

Excitement ad Nutrition ad

Children

Adults

Cerealrating

What’s the Story?

“Not easy to use” “Not difficult to use”

10 seconds

45 seconds

Productevaluation

What’s the Story?

No advertising Advertising

Milk

Soft drink

Grossmargins

What’s the Story?

Exceededexpectations

Did not meetexpectations

Low expectations

High expectations

Satisfaction

Metexpectations

What’s the Story?

Think of 2 reasons Think of 10 reasons

Novices

Experts

BMWevaluation

Ceiling Effect

Effect of Time on Word Memory

02

468

10

1214

15 minutes 25 minutes

Wo

rds

rem

emb

ered

6 year olds

10 year olds

Ordinal Interactions

Effect of Caffeine, Exercise on Calories Consumed

1500

2000

2500

3000

No exercise Exercise

Cal

ori

es c

on

sum

ed

No caffeine

Caffeine

Ordinal Interactions

Effect of Caffeine, Exercise on Hunger

1

3

5

7

9

11

No exercise Exercise

Rat

ing

s o

f h

un

ger

No caffeine

Caffeine