6 anov as+combined

Essence of ANOVA

1

Criteria for One-Way ANOVA

2

Criteria for Repeated Measures ANOVA

3

Criteria for Two-Way ANOVA

4

One-Way Example

5

One-Way SSBetween treatments

• SSB = 60

6

ErrorSSWithin groups

• SSw = 300

7

SSTotal

• SST = 360

• SST = SSB + SSW = 360

8

One-Way F

• SSB = 300

• dfB = 2

• MSB = 30

• dfW = 27

• MSW = 11.11

• F = 2.70• Decision rule: Reject H0 if F > 3.36

• Decision: Fail to reject H0; 2.70 < 3.36

9

Source Table

10

Statistical Conclusion

• F(dfb, dferror) = obtained value, p value.

• Conclusion about theoretical hypothesis, F(2, 27) = 2.70, p > .05.

• Means, SDs, and effect size extra on exam.

11

Repeated Measures Example

Same data 12

Repeated Measures Calculations

• SSB = 60 (same as one-way)

• SSW = 300 (same as one-way)

same formulas as one-way

13

SSbetween subjects

• SSsubjects = 150

14

SSerror

• SSerror = SSW – SSsubjects

• SSerror = 150

15

SSTotal

• Same as one-way• SST = 360

16

Repeated Measures F

• MSB = 30 (same as one-way)

• SSerror = 150; SSsubjects = 150

• dferror = 18

• MSerror = 8.33

• FB = 3.60

• Decision rule: Reject H0 if F > 3.56 Decision: Reject H0; 3.60 > 3.56; α = .05

• Note: A larger F than with a one-way (F = 2.70)

• Mssubjects = 16.67

• Fsubjects = 2.00

• Fsubjects is used in ω2 (measure of effect size)

17

Summary Table

Source SS df MS FBetween treatments 60 2 30.00 3.60 *Within treatments 300 27 Between subjects 150 9 16.67 2.00 Error 150 18 8.33Total 360 29

*p < .05

18

Two-Way ANOVA Example

Same data

Factor A

Lev 1 Lev 2 Lev3

3 2 9

1 5 9

Lev 1 1 9 13 90

6 7 6

Factor B 4 7 80 3 0

Lev 2 2 8 0

0 3 0 30

0 3 5

3 3 0

20 50 50

19

SSA = Between A treatmentsSSB = Between B treatments

• SSA = between treatment for factor A

• SSB = between treatment for factor B

• SSA = 60 (same as one-way SSB & repeated measures)

• SSB = [902/15 + 302/15] – 1202/30 = 120

Factor ALev 1 Lev 2 Lev3

3 2 91 5 9

Lev 1 1 9 13 906 7 6

Factor B 4 7 80 3 0

Lev 2 2 8 00 3 0 300 3 53 3 020 50 50

Factor BLev 1 Lev2

3 01 2

Lev 1 1 0 206 04 3

Factor A 2 35 8

Lev 2 9 3 507 37 39 09 0

Lev 3 13 0 506 58 090 30 20

Interaction

• SSAxB = 60

Factor ALev 1 Lev 2 Lev3

3 2 91 5 9

Lev 1 1 9 13 906 7 6

Factor B 4 7 80 3 0

Lev 2 2 8 00 3 0 300 3 53 3 020 50 50

21

ErrorSSWithin groups

• SSW = 120

Factor ALev 1 Lev 2 Lev3

3 2 91 5 9

Lev 1 1 9 13 906 7 6

Factor B 4 7 80 3 0

Lev 2 2 8 00 3 0 300 3 53 3 020 50 50

22

Two-Way Fs

• SSW = 120; SSA = 60; SSB = 120; SSAxB = 60

• dfW = = 24

• MSW = 5.00

• MSA = 6.00

• Reject H0; 6.00 > 3.40

• MSB = 120; F = 24.00

• Reject H0; 24.00 > 4.26

• MSAxB = 30.00; F = 6.00

• Reject H0; 6.00 > 3.40

23

SST

• SST = 360 (same as one-way & repeated)

24

Source Table

25

SSTotal

One-Way

Repeated Measures

2-Way

26

SSBetween treatments

One-Way

Repeated Measures

Two-Way

Repeated Measures

27

Error

One-Way

Repeated Measures

Two-Way

– SSsubjectsSSerror =

28

SSAxB

Two-Way

29

dfT

dfT = N – 1

One-Way

Repeated Measures

Two-Way

dfT = N – 1

Total number of subjects minus 1

dfT = N – 1

30

dfB

dfB = k – 1

dfA = j – 1dfB = k – 1

One-Way

Repeated Measures

Two-Way

dfsubjects = n – 1 Repeated Measures

dfB = k – 1

Each level minus 1

31

dfw

dfW = N – k

dfw = N – jk

One-Way

Repeated Measures

Two-Way

dfw = N – k

Number of subjects in each treatment minus 1

32

Additional dfs

dfA X B = (j – 1)(k – 1) Two-Way

dferror = dfw − dfsubjects Repeated Measures

33

F Ratios

• Divide MSbetween by the error term

• For one-way and two-way the error term is MSwithin

• For repeated measures, the Mserror = (SSwithin − SSbetween subjects)/dferror

34

Null Hypothesis

• H0: µ1 = µ2 = µ3; one-way

• H0: µ1 = µ2 = µ3; repeated measures

• H0: µA1 = µA2

; two-way (2x2)

• H0: µA x µB = 0; two-way (2x2)

35

Alternative Hypothesis

• H1: Not H0; one-way

• H1: Not H0; repeated measures

• H1: µA1 ≠ µA2

two-way (2x2)

• H1: µA x µB 0; two-way (2x2)≠

36

Decision Rules

• dfB = numerator

• dferror = denominator

37

One-Way ANOVA

38

Repeated Measures ANOVA

39

2-Way ANOVA

40

The Essential Formulas

41

Use for one-way, repeated measures (between treatments and between subjects), both factors of two-way

Use for all ANOVAs

The total SS for all ANOVAS.