The following data were collected from a repeated-measures study. a. Determine i

Question

The following data were collected from a repeated-measures study. a. Determine if there are any significant differences among the four treatments. Use the .05 level of significance. b. Compute the percentage of variance explained by the treatment effects (eta^2). The following table summarizes the results of a two-factor ANOVA evaluating an independent-measures experiment with two levels of factor A, three levels of factor B, and n = 5 subjects in each separate sample. Fill in all missing values in the table.

Explanation / Answer

Solution 1.

Null Hypothesis (Ho): There is no significant difference among the four treatment.

Alternative Hypothesis (Ha): There is significant difference among the four treatment.

SS (Total) = X2 - G^2/N

SS (Total) = 172 - 44^2/16

SS (Total) = 51

Degrees of freedom, total = N - 1 = 16 - 1 = 15

SS (within) = SS1 + SS2 + SS3 + SS4 = 4 + 2 + 6+ 4 = 16

Degrees of freedom, within = N - k = 16 - 4 = 12

SS (between) = SS (total) - SS (within) = 51 - 16 = 35

Degrees of freedom, between = k - 1 = 4 - 1 = 3

ANOVA table

Test Statistics

F = MS (between)/MS (within)

F = 11.67/1.33

F = 8.75

Using F-tables, the p-value is

F (8.75, 3, 12) = 0.00239

Since p-value is less than 0.05 significance level, we reject Ho.

Hence, we can conclude that There is significant difference among the three treatment.

b. Percentage of variation explained = SS(between)/SS (total)

Percentage of variation explained = 35/51 = 0.6863

Solution 2:

Anova table

DF (between treatment) = n - 1 = 5 - 1 = 4, Df (factor A) = nA - 1 = 2 - 1 = 1, Df (factor B) = nB - 1 = 3 - 1 = 2,

Df (A x B) = df (factor A) x df (factor B) = 1*2 = 2

Df (total) = N - 1 = 2*3*5 - 1 = 29

Df (within) = 29 - 4 -1- 2- 2 = 20

MS (factor A) = FA*MS (within) = 5*2 = 10

SS (factor A) = MS (factor A)*df (Factor A) = 10*1 = 10

SS (within) = MS (within)*df (within) = 2*20 = 40

SS (factor B) = SS (between) - SS (factorA) - SS (A x B) =60 - 10 - 30 = 20

MS (factor B) = SS(factor B)/df (factor B) = 20/2 = 10

MS (A x B) = SS(AxB)/df (A x B) = 30/2 = 15

FB = MS (factor B)/MS (within) = 10/2 = 5

FAxB = MS (AxB)/MS (within) = 15/2 = 7.5

ANOVA Source of Variation SS df MS F Between Groups 35 3 35/3 = 11.67 8.75 Within Groups 16 12 16/12 = 1.33 Total 51 15

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