ANOVA II (Part 2)

ANOVA II (Part 2)

Class 16

Implications of Interaction

1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts. 3. Analysis of findings, when an interaction is present, will focus on individual treatment means rather than on overall factor (IV) means.

4. Interaction indicates moderation.

Interactions are Non-Additive Relationships Between Factors

1. Additive: When presence of one factor changes the expression of another factor consistently, across all levels.

2. Non-Additive: When the presence of one factor changes the expression of another factor differently, at different levels.

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5

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North South

Democrat

Republican

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North South

Democrat

Republican

Ordinal and Disordinal Interactions

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B1 B2

A1

A2

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B1 B2

A1

A2

XXXX Interaction

YYY Interaction

Ordinal and Disordinal Interactions

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B1 B2

A1

A2

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B1 B2

A1

A2

Ordinal Interaction

Disordinal Interaction

Birth Order and Gender Effects on Ratings of Help Seeker

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First Born Last Born

Rat

ing Male

Female

Birth Order Main Effect: NO

Gender Main Effect: NO

Interaction: NO


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Rat

ing Male

Female

Birth Order Main Effect: YES


Interaction: NO


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Rat

ing Male

Female


Gender Main Effect: YES

Interaction: N0


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Rat

ing Male

Female



Interaction: NO


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ing Male

Female



Interaction: YES


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Rat

ing Male

Female



Interaction: YES


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Rat

ing Male

Female



Interaction: YES


0123456


Rat

ing Male

Female



Interaction: YES

Birth Order Means


01234567

Male Female

Ratin

g First BornLast Born

Development of ANOVA Analytic Components

1. Individual scores Condition (cell) sums 2. Condition sums Condition means 3. Cond. means – ind. scores Deviations Deviations2

4. Deviations2 Sums of squares (SS between, SS within)

5. Sum Sqrs / df Mean squares (Between and Within) 6. MS Between F Ratio MS Within F (X, Y df) Probability of null (p) p Accept null, or accept alt.

Birth Order and Ratings of “Activity” Deviation Scores

AS Total Between Within (AS – T) = (A – T) + (AS –A)

1.33 (-2.97) = (-1.17) + (-1.80) 2.00 (-2.30) = (-1.17) + (-1.13) 3.33 (-0.97) = (-1.17) + ( 0.20) 4.33 (0.03) = (-1.17) + ( 1.20) 4.67 (0.37) = (-1.17) + ( 1.54)

Level a1: Oldest Child

Level a2: Youngest Child4.33 (0.03) = (1.17) + (-1.14) 5.00 (0.07) = (1.17) + (-0.47) 5.33 (1.03) = (1.17) + (-0.14) 5.67 (1.37) = (1.17) + ( 0.20) 7.00 (2.70) = (1.17) + ( 1.53)

Sum: (0) = (0) + (0)

Mean scores: Oldest = 3.13 Youngest = 5.47 Total = 4.30

Sum of Squared Deviations

Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations

SSTotal = SSBetween + SSWithin

Computing Sums of Squares from Deviation Scores

Birth Order and Activity Ratings (continued)

SS = Sum of squared diffs, AKA “sum of squares”

SST = Sum of squares., total (all subjects)

SSA = Sum of squares, between groups (treatment)

SSs/A = Sum of squares, within groups (error)

SST = (2.97)2 + (2.30)2 + … + (1.37)2 + (2.70)2 = 25.88

SSA = (-1.17)2 + (-1.17)2 + … + (1.17)2 + (1.17)2 = 13.61

SSs/A = (-1.80)2 + (-1.13)2 + … + (0.20)2 + (1.53)2 = 12.27Total (SSA + SSs/A) = 25.88

Variance

Code Calculation Meaning

Mean Square Between Groups

MSA SSA

dfA

Between groups variance

Mean Square Within Groups

MSS/A SSS/A

dfS/A

Within groups variance

Variance

Code Calculation Data Result

Mean Square Between Groups

MSA SSA

dfA

13.61 1

13.61

Mean Square Within Groups

MSS/A SSS/A

dfS/A

12.278

1.53

Mean Squares Calculations

F Ratio Computation

F = 13.611.51

= 8.78

F = MSA = Between Group Variance

MSS/A Within Group Variance

Conceptual Approach to Two Way ANOVA

SS total = SS between groups + SS within groups

Oneway ANOVA

SS between groups =

Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA

SS between groups =

Factor A and its levels (e.g., birth order; older/younger)

XXXX

YYYY

Conceptual Approach to Two Way ANOVA

SS total = SS between groups + SS within groups

Oneway ANOVA

SS between groups =

Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA

SS between groups =

Factor A and its levels (e.g., birth order; older/younger)

Factor B and its levels (e.g., gender; male / female)

The interaction between Factors A and B (e.g., how ratingsof help seeker are jointly affected by birth order and gender)

Total Mean (4.32)

Distributions of All Four ConditionsBirth Order and Gender Effects on Ratings of

Help Seeker

01234567

Male Female

Ratin


Total Mean (4.32)

Gender Effect (collapsing across birth order)Birth Order and Gender Effects on Ratings of

Help Seeker

01234567

Male Female

Ratin


Total Mean (4.32)

Birth Order Effect (collapsing across gender)Birth Order and Gender Effects on Ratings of

Help Seeker

01234567

Male Female

Ratin


Understanding Effects of Individual Treatment Groups

How much can the variance of any particular treatment group be explained by:Factor AFactor B

The interaction of Factors A and B

Quantification of AB Effects

AB - T = (A effect) + (B effect) + ?????

AB - T = (A - T) + (B - T) + (AB - A - B + T)

(AB - A - B + T) = ??? AKA "????"

(AB - T) - (? - T) - (? - T) = Interaction

Error Term in Two-Way ANOVA

Error = (ABS - AB)

Understanding Effects of Individual Treatment Groups

How much can the variance of any particular treatment group be explained by:Factor AFactor B

The interaction of Factors A and B

Quantification of AB Effects

AB - T = (A effect) + (B effect) + (A x B Interaction)

AB - T = (A - T) + (B - T) + (AB - A - B + T)

(AB - A - B + T) = Interaction AKA "residual"

(AB - T) - (A - T) - (B - T) = Interaction

Error Term in Two-Way ANOVA

Error = (ABS - AB)

Deviation of an Individual Score in Two Way ANOVA

ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij)

Ind. score

Total Mean

??? Effect ??? Effect ??? Effect ??? (w’n Effect)

Deviation of an Individual Score in Two Way ANOVA

ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij)

Ind. score

Total Mean

Factor A Effect

Factor B Effect

Interaction AXB Effect

Error (w’n Effect)

Degrees of Freedom in 2-Way ANOVA

Between Groups

Factor A df A = a - 1 2 – 1 = 1

Factor B df B = b – 1 2 – 1 = 1

Interaction Effect

Factor A X Factor B dfA X B = (a –1) (b – 1) (2-1) x (2-1) = 1

Error Effect

Subject Variance df s/AB = ab(s – 1)

df s/AB = n - ab 20 – (2 x 2) = 16

Total Effect

Variance for All Factors df Total = abs – 1

df Total = n – 1 20 – 1 = 19

Conceptualizing Degrees of Freedom (df) in Factorial ANOVA

Factor A

Factor B a1 a2 Sum

b1 # X B1

b2 X X X

Sum A1 X T

Factor A = Birth Order

Factor B = Gender # = Known quantity


Birth Order

Gender Youngest Oldest Sum

Males

Sum

Females

4.50

5.50

9.00

11.00

4.50

5.50

20.0010.0010.00

NOTE: “Fictional sums” for demonstration.


Factor A

Factor B a1 a2 a3 Sum

b1 # # X B1

b2 # # X B2

b3 X X X X

Sum A1 A2 X T

A, B, T, # = free to vary; X = determined by #s

Once # are established, Xs are known

Analysis of Variance Summary Table:

Two Factor (Two Way) ANOVA

A SSA a - 1 SSA

dfA

MSA

MSS/AB

B SSB b - 1 SSb

dfb

MSB

MSS/AB

A X B SSA X B (a - 1)(b - 1) SSAB

dfA X B

MSA X B

MSS/AB

Within(S/AB)

SSS/A ab (s- 1) SSS/AB

dfS/AB

Total SST abs - 1

Source of Variation Sum of Squares df Mean Square F Ratio

(SS) (MS)

Effect of Multi-Factorial Design on Significance Levels

MeanMen

MeanWomen

Sum of Sqrs.

Betw'n

dt Betw'n

MSBetw'n

Sum of Sqrs. Within

df Within

MS Within

F p

One Way

4.78 3.58 3.42 1 3.42 22.45 8 2.81 1.22 .30

Two Way

4.78 3.58 3.42 1 3.42 5.09 6 .85 4.03 .09

ONEWAY ANOVA AND GENDER MAIN EFFECT

Source Sum of Squares

df Mean Square

F Sig.

Gender 3.42 1 3.42 1.22 .34

Error 22.45 8 2.81

Source Sum of Squares

df Mean Square

F Sig.

Gender 3.42 1 3.42 4.03 .09

Birth Order 16.02 1 16.02 18.87 .005

Interaction 3.75 1 3.75 4.42 .08

Error 5.09 6 0.85

Total 9

TWOWAY ANOVA AND GENDER MAIN EFFECT

Oneway F: 3.42 = 1.22 Twoway F: 3.42 = 4.42 2.81 .85

Documents

ANOVA II (Part 2)