6 anov as+combined
41
Essence of ANOVA 1
-
Upload
mattiemayes -
Category
Documents
-
view
34 -
download
0
Transcript of 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.
