Question

The following table summarizes the results of a two-factor ANOVA evaluating an independent-measures experiment with 2 levels of factor A, 3 levels of factor B, and n = 6 participants in each treatment condition.

A. Fill in all missing values in the table. Show your work (i.e., all computational steps for finding the missing values). Hint: start with the df values.

B.  Do these data indicate any significant effects (assume p < .05 for hypothesis testing of all three effects)?

• the null hypothesis H₀ & the alternative hypothesis H₁
• the critical F value used for the decision about H₀
• if the difference is statistically significant, the computed effect size, η²
• the conclusion in APA style format

Between treatments: SS=75 df=? MS=?
Factor A: SS=? df=? MS=? Fa=
Factor B: SS=? df= ? MS= 15 Fb=

AXB: SS=? df=? MS=? Faxb=6.00

within treatments: SS=? df=? MS=?

total: SS=165 df=?

Answer 1

Solution

Part (A)

ANOVA Table: Significance Level 0.05

[Given figures are in bold font. Other figures are derived. Explanations follow the table.

Source	SS	df	MS	F	Fcrit
Factor A	9	1	9	3	4.17
Factor B	30	2	15	5	3.22
A x B Interaction	36	2	18	6	3.22
Between Treatments	75	5
Within Treatments	90	30	3
Total	165	35

Explanations

Additional information given

Factor A – 2 levels

Factor B – 3 levels

Replication - 6

The above imply:

Total number of observations N = 2 x 3 x 6 = 36

Number of cells C = 2 x 3 = 6.

df

Factor A	Number of levels – 1 = 2 – 1 = 1
Factor B	Number of levels – 1 = 3 – 1 = 2
A x B Interaction	Between – A – B = 5 – 1 – 2 = 2
Between Treatments	Number of cells – 1 = 6 – 1 = 5
Within Treatments	Total – Between = 35 – 5 = 30
Total	N – 1 = 36 – 1 = 35

For filling other figures, the following relationships are employed:

MS = SS/df, which also implies, SS = MS x df

F = MS[A/B/A x B}÷ MSW

Fcrit = Fυ₁, υ₂, where υ₁, υ₂ are the df of the MS in the numerator and denominator of the respective F.

ANOVA Table with details of computations [with order of computations in ()]

Source	SS	df	MS	F	Fcrit
A	9 = 75 – 36 – 30 (6)	1	9 = 9/1 (7)	3 = 9/3 (8)	4.17
B	30 = 15 x 2 (1)	2	15	5 = 15/3 (9)	3.22
A x B	36 = 18 x 2 (5)	2	18 = 6 x 3 (4)	6	3.22
Between	75	5
Within	90 = 165 – 75 (2)	30	3 = 90/30 (3)
Total	165	35

Fcrit is obtained from standard F-Table [can also be obtained using Excel Function: Statistical FINV ]

Answer 1 DONE

Part (B)

Hypotheses

Null H₀₁: Mean effect of Factor A is the same for both levels Vs Alternative H_A1: Mean effect of Factor A differs between the two levels

Null H₀₂: Mean effect of Factor B is the same for all three levels Vs Alternative H_A2: Mean effect of Factor B differs for at least one level

Null H₀₃: Mean interaction AxB effect is zero Vs Alternative H_A3: Mean interaction AxB effect is not zero

Answer 2

Fcrit values are given in the above ANOVA Table. Answer 3

Decision

Since F > Fcrit for Factor B and AxB Interaction, these two effects are significant.

Since F < Fcrit for Factor A, its effect is not significant. Answer 4

Conclusion:

There is sufficient evidence to suggest that Factor B and AxB Interaction effects exist, but evidence is not sufficient to support the effect of Factor A.

Hence we conclude that Factor B impacts the response variable, but not Factor A. However, their interaction does affect the response variable. Answer 5

DONE