Question

Conduct a two-factor fixed-effects ANOVA to determine if there are any effects due to A (task type), B (task difficulty), or the AB interaction (alpha = .01). Conduct Tukey HSD post-hoc comparison. The following are the scores from the individual cells of the model:

A₁B₁: 41, 39, 25, 25, 37, 51, 39, 101

A₁ B₂: 46, 54, 97, 93, 51, 36, 29, 69

A₁ B₃: 113, 135, 109, 96, 47, 49, 68, 38

A₂ B₁: 86, 38, 45, 45, 60, 106, 106, 31

A₂B₂: 74, 96, 101, 124, 48, 113, 139, 131

A₂B₃: 152, 79, 135, 144, 52, 102, 166, 155

Answer 1

Conduct a two-factor fixed-effects ANOVA to determine if there are any effects due to A (task type), B (task difficulty), or the AB interaction (alpha = .01). Conduct Tukey HSD post-hoc comparison. The following are the scores from the individual cells of the model:

A₁B₁: 41, 39, 25, 25, 37, 51, 39, 101

A₁ B₂: 46, 54, 97, 93, 51, 36, 29, 69

A₁ B₃: 113, 135, 109, 96, 47, 49, 68, 38

A₂ B₁: 86, 38, 45, 45, 60, 106, 106, 31

A₂B₂: 74, 96, 101, 124, 48, 113, 139, 131

A₂B₃: 152, 79, 135, 144, 52, 102, 166, 155

MINITAB used.

General Linear Model: score versus A, B

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
A	Fixed	2	1, 2
B	Fixed	3	1, 2, 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
A	1	14700	14700.0	14.55	0.000
B	2	18367	9183.6	9.09	0.001
A*B	2	1386	693.2	0.69	0.509
Error	42	42419	1010.0
Total	47	76872

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
31.7799	44.82%	38.25%	27.93%

Tukey Pairwise Comparisons: A

Grouping Information Using the Tukey Method and 95% Confidence

A	N	Mean	Grouping
2	24	97	A
1	24	62		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of A Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	35.00	9.17	(16.49, 53.51)	3.82	0.000

Individual confidence level = 95.00%

Tukey Pairwise Comparisons: B

Grouping Information Using the Tukey Method and 95% Confidence

B	N	Mean	Grouping
3	16	102.500	A
2	16	81.313	A	B
1	16	54.687		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of B Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	26.6	11.2	(-0.7, 54.0)	2.37	0.057
3 - 1	47.8	11.2	(20.5, 75.1)	4.26	0.000
3 - 2	21.2	11.2	(-6.1, 48.5)	1.89	0.155

Individual confidence level = 98.07%

Results:

To test the main effect A, calculated F= 14.55, P=0.000 which is < 0.05 level of significance. Main effect A is significant.

To test the main effect B, calculated F= 9.09, P=0.001 which is < 0.05 level of significance. Main effect B is significant.

To test the interaction effect, calculated F= 0.69, P=0.509 which is > 0.05 level of significance. Interaction effect is not significant.

Tukey test for factor B:

Task level 1 with 3 are significant

Other pairs of levels, 1 with 2, 2 with 3 are not significant.