The two-way analysis of variance, independent groups allows us in one experiment
ID: 3170855 • Letter: T
Question
The two-way analysis of variance, independent groups allows us in one experiment to evaluate the effect of two factors and the interaction between them. Let's analyze a basic experiment appropriate for this technique.
An investigator is interested in the effect of variable A and variable B on a particular dependent variable. She conducts an experiment in which there are two levels of variable A and two levels of variable B. Sixteen subjects are randomly selected from the population and randomly assigned to the four cells that result from this design, such that there are 4 subjects/cell. All subjects are run in their respective treatments and the dependent variable is measured. The following data result.
4
7
4
5
9
10
8
11
(b) Using = 0.05, what do you conclude?
(i) For the row effect.
(ii) For the column effect.
(iii) For the row column interaction effect.
Factor B Factor A b1 b2 a1a2 4
6
3
2
3
7
3
4
4
7
4
5
9
10
8
11
Explanation / Answer
Solution
Back-up Theory
Let xijk represent the kth observation in the kth cell of ith row-jth column, k = 1,2,3,4; I = 1,2; j = 1,2.
Then the ANOVA model is: xijk = µ + i + j + ij + ijk, where µ = common effect, i = effect of ith row, j = effect of jth column, ij = row-column interaction effect and ijk is the error component which assumed to be Normally Distributed with mean 0 and variance 2.
Now, to work out the solution,
Calculations:
Cell total = xij. sum over k of xijk
Row total = xi.. = sum over j of xij.
Column total = x.j. = sum over i of xij.
Grand total = G = sum over i of xi.. = sum over j of x.j.
Correction Factor = C = G2/ n, where n = total number of observations = 2 x 2 x 4 = 16
Total Sum of Squares: SST = (sum over i,j and k of xijk2) – C
Row Sum of Squares: SSR = {(sum over i of xi.2)/8} – C[8 = number of rows x number of observations per cell]
Column Sum of Squares: SSC = {(sum over j of x.j.2)/8} – C[8 =number of columns x number of observations per cell]
Between Sum of Squares: SSB = {(sum over i and j of xij.2)/4} – C[4 = number of observations per cell]
Interaction Sum of Squares: SSI = SSB – SSR – SSC
Error Sum of Squares: SSE = SST - SSB
ANOVA TABLE [See below the table for additional theoretical explanations of entries]
Source of
Variation
Degrees of Freedom
Sum of squares
Mean Sum of Squares
Fobs
Fcrit
Inference
Row
2 – 1 = 1
SSR:25.00
25.00
14.62
4.75
Significant
Column
2 – 1 = 1
SSC:42.25
42.25
24.71
4.75
Significant
Interaction
1
SSI:16.00
16.00
9.36
4.75
Significant
Between
4 – 1 = 3
SSB:83.25
-
Error
12
SSE20.50
1.71
Total
16 – 1 = 15
SST:103.75
-
Degrees of freedom: Number of rows – 1; Number of columns – 1; Number of cells – 1; Number of observations – 1 for row, column, between and total. Interaction is Between – Row – Column DFs and Error is Total – Between
Mean Sum of Squares = Sum of Squares/DF
Fobs = MSSR/MSSE for Row; MSSC/MSSE for Column; MSSI/MSSE for Interaction.
Fcrit = upper 5%(given level of significance, = 5%) point of F-Distribution with degrees of freedom 1 and 12.
Part (1) For Row Effect
Fobs = 14.62
Fcrit = 4.75
ANSWER
Part (2) For Column Effect
Fobs = 24.71
Fcrit = 4.75
ANSWER
Part (c) For Interaction Effect
Fobs = 9.36
Fcrit = 4.75
ANSWER
Conclusion: all three effects are significant implying difference in factor effects, level effects and intraction.
Source of
Variation
Degrees of Freedom
Sum of squares
Mean Sum of Squares
Fobs
Fcrit
Inference
Row
2 – 1 = 1
SSR:25.00
25.00
14.62
4.75
Significant
Column
2 – 1 = 1
SSC:42.25
42.25
24.71
4.75
Significant
Interaction
1
SSI:16.00
16.00
9.36
4.75
Significant
Between
4 – 1 = 3
SSB:83.25
-
Error
12
SSE20.50
1.71
Total
16 – 1 = 15
SST:103.75
-
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