TWO-WAY BETWEEN SUBJECTS ANOVA

Also called: Two-Way Randomized ANOVA

Purpose: Measure main effects and interaction of two independent variables

Design: factorial Assumptions: same as one-way BS

ANOVA

Dividing the Variance Total = A + B + AxB + Within

Groups A: differences between levels of A B: differences between levels of B AxB: other between group

differences Within Groups: differences within

groups

A Variance affected by:– effect of Factor A (systematic)– individual differences (non-

systematic)– measurement error (non-systematic)

B Variance affected by:– effect of Factor B (systematic)– individual differences (non-

systematic)– measurement error (non-systematic)

AxB Variance affected by:– AxB interaction (systematic)– individual differences (non-

systematic)– measurement error (non-systematic)

Within Groups variance affected by:– individual differences (non-

systematic)– measurement error (non-systematic)

Comparing the Variance

FA MSA

MS Within

FB MSB

MS Within

FAxB MSAx B

MS Within

ANOVA Summary Table

Source SS df MS F p

Factor AFactor BAxBWithinTotal

EXAMPLE: An oral or written spelling test was given in one of three noise levels. Determine whether there were significant effects of test type, noise level, and the interaction of test type with noise level. (See data on next page)

Computation of Two-Way BS ANOVA

Noise LevelNo Low High

oral15 15 12

Test 17 19 10

written 18 14 1014 12 12

Noise LevelNo Low High

oral 16 17 11 14.67Test

written 16 13 11 13.33

16 15 11

Overall mean = 14

Means

Source SS df MS F pTest TypeNoise LevelTest x NoiseWithin Total

ANOVA Summary Table

STEP 1: SS Between = S(xc-x)2

condition mean

xc xc-x (xc-x)2

16 2 416 2 417 3 917 3 911 -3 911 -3 916 2 416 2 413 -1 113 -1 111 -3 9 11 -3 9 SS Between = 72

STEP 2: SS A = S(xa-x)2 (A is Test Type) mean for level of A

xa xa-x (xa-x)2

14.67 .67 .4514.67 .67 .4514.67 .67 .4514.67 .67 .4514.67 .67 .4514.67 .67 .4513.33 -.67 .4513.33 -.67 .4513.33 -.67 .4513.33 -.67 .4513.33 -.67 .45 13.33 -.67 .45 SS Test Type = 5.40

STEP 3: SS B = S(xb-x)2 (B is Noise Level) mean for level of B

xb xb-x (xb-x)2

16 2 416 2 416 2 416 2 415 1 115 1 115 1 115 1 111 -3 911 -3 911 -3 9 11 -3 9 SS Noise Level = 56

STEP 4: SS AxB = SS Between - SS A - SS B

SS Test Type x Noise Level =72 - 5.40 - 56 = 10.60

STEP 5: SS Within = S(x- xc)2

x x-xc (x-xc)2

15 -1 117 1 118 2 414 -2 415 -2 419 2 414 1 112 -1 112 1 110 -1 110 -1 1 12 1 1 SS Within = 24

Source SS df MS F pTest Type 5.40Noise Level 56.00Test x Noise 10.60Within 24.00Total 96.00

ANOVA Summary Table

STEP 6: Calculate degrees of freedom.df A = a-1 a= # levels of A

df Test Type = 2-1 = 1

df B = b-1 b= # levels of Bdf Noise Level = 3-1 = 2

df AxB = (a-1)(b-1) df Test x Noise = (1)(2) = 2

df Within = (a)(b)(n-1) n = # per group df Within = (2)(3)(1) = 6

Source SS df MS F pTest Type 5.40 1Noise Level 56.00 2Test x Noise 10.60 2Within 24.00 6Total 96.00 11

ANOVA Summary Table

STEP 7: Calculate Mean Squares.

MS Test Type = 5.40/1 = 5.40MS Noise Level = 56/2 = 28.00MS Test x Noise = 10.60/2 = 5.30MS Within = 24/6 = 4.00

Source SS df MS F pTest Type 5.40 1 5.40Noise Level 56.00 2 28.00Test x Noise 10.60 2 5.30Within 24.00 6 4.00Total 96.00 11

ANOVA Summary Table

STEP 8: Calculate F-ratios.F(Test Type) = 5.40 /4.00 = 1.35F(Noise Level) = 28/4.00 = 7.00F(Test x Noise) = 5.30/4.00 = 1.32

STEP 9: Look up critical values of F.Test Type F-crit (1,6) = 5.99Noise Level F-crit (2,6) = 5.14Test x Noise F-crit (2,6) = 5.14

STEP 10: Compare F to F-crit. If F is equal to or greater than F-crit, reject the Null Hypothesis.

Test Type 1.35 < 5.99 Not sig.Noise Level 7.00 > 5.14 Sig.Test x Noise 1.32 < 5.14 Not sig.

Source SS df MS F pTest Type 5.40 1 5.40 1.35 >.05Noise Level 56.00 2 28.00 7.00 <.05Test x Noise 10.60 2 5.30 1.32 >.05Within 24.00 6 4.00Total 96.00 11

ANOVA Summary Table

APA Format Sentence

A Two-Way Between Subjects ANOVA showed a significant main effect of Noise Level, F (2,6) = 7.00, p < .05, a nonsignificant main effect of Test Type, F (1,6) = 5.40, p > .05, and a nonsignificant interaction, F (2,6) = 1.32, p > .05.

Computing Effect Size

Compute 2 for each effect

2 =SS Effect

SS Total

2 for A =SS A

SS Total

2 for B =SS B

SS Total

2 for AxB =SS AxB

SS Total

2 for Test = 5.40

96.00 = .06

2 for Noise =56.00

96.00 = .58

2 for Test x Noise =10.60

96.00 = .11