252anovaex4 1 24 07 Open this document in page layout view Example of 3 way ANOVA We have measurements that describe the mpg gotten by 8 drivers Factor 1 using 6 cars Factor 2 during 4 seasons Factor C For each combination of factors there are 10 measurements The SS1 is 700 SS2 is 400 and SS3 is 300 For the interactions SS12 50 SS13 60 and SS23 70 The within error or residual sum of squares is 100 and the total sums of squares is 1700 Set up an ANOVA table showing all interactions Assume a 5 significance level Solution The sums of squares have been copied into the table below If we multiply the levels of the factors together and then multiply by the number of measurements per cell we find a total of 8 6 4 10 1920 measurements There are a total of 1920 1 1919 degrees of freedom Since drivers has 8 levels there are 7 degrees of freedom Similarly for cars there are 6 1 5 degrees of freedom and for seasons there are 3 degrees of freedom These are also shown below SS DF MS F Source F 05 Driver 1 700 7 Car 2 400 5 Season 3 300 3 Interaction 12 50 Interaction 13 60 Interaction 23 70 Interaction 123 Within 100 Total 1700 1919 Since the sums of squares must add up to get the missing sum of squares subtract the sum of the numbers above the line 1680 from the total sum of squares 1700 and get 20 To get the degrees of freedom for interactions multiply the degrees of freedom for the factors For example DF12 DF1 DF 2 7 5 35 This is shown below SS DF MS F Source F 05 Driver 1 Car 2 Season 3 Interaction 12 Interaction 13 Interaction 23 Interaction 123 Within Total 700 400 300 50 60 70 20 100 1700 7 5 3 35 21 15 105 1919 Since the degrees of freedom must add up add the degrees of freedom above the line to get 191 and subtract from the total degrees of freedom to get 1919 191 1728 To get the mean squared column divide the items in the SS column by the corresponding items in the MS column Source Driver 1 Car 2 Season 3 Interaction 12 Interaction 13 Interaction 23 Interaction 123 Within Total SS 700 400 300 50 60 70 20 100 1700 DF 7 5 3 35 21 15 105 1728 1919 MS F F 05 100 00 80 00 100 00 1 4286 2 8571 4 6667 0 1905 0 05787 To get the items in the column for computed Fs divide the mean squares by the within mean square 1 252anovaex4 1 24 07 Open this document in page layout view Source SS DF MS F F 05 Driver 1 700 7 100 00 1728 Car 2 400 5 80 00 1382 Season 3 300 3 100 00 1728 Interaction 12 50 35 1 4286 24 68 Interaction 13 60 21 2 8571 49 37 Interaction 23 70 15 4 6667 80 64 Interaction 123 20 105 0 1905 3 29 Within 100 1728 0 05787 Total 1700 1919 Now look up values of F 05 on the F table The degrees of freedom for the numerator are given in the DF column The degrees of freedom for the denominator are 1728 SS DF MS F Source F 05 Driver 1 700 7 100 00 1728 F 7 1728 05 Car 2 400 5 80 00 1382 F 5 1728 Season 3 300 3 100 00 1728 F 3 1728 05 05 Interaction 12 50 35 1 4286 24 68 F 35 1728 Interaction 13 60 21 2 8571 49 37 F 21 1728 Interaction 23 70 15 4 6667 80 64 F 15 1728 Interaction 123 20 105 0 1905 3 29 F 105 1728 05 05 05 05 Within 100 1728 0 05787 Total 1700 1919 Look up the table values of F I have used the parts of the table where the denominator degrees of freedom are 1000 and have come as close as possible for the numerator We have tested seven null hypotheses The first three of these are that there are no significant differences between driver car and season means and the remaining four are that there is no interaction of the type described In every case the null hypothesis was rejected because the computed F exceeded the table F so the computed F was labeled with an s Source SS Driver 1 700 7 100 00 1728s Car 2 400 5 80 00 1382s Season 3 300 3 100 00 1728s Interaction 12 50 35 1 4286 24 68s Interaction 13 60 21 2 8571 49 37s Interaction 23 70 15 4 6667 80 64s Interaction 123 20 105 0 1905 3 29s 100 1700 1728 1919 0 05787 Within Total DF MS F F 05 F 7 1728 05 2 02 F 5 1728 05 2 22 F 3 1728 05 2 61 F 35 1728 05 1 4 4 F 21 1728 05 1 58 F 15 1728 05 1 70 F 105 1728 05 1 26 2 252anovaex4 1 24 07 Open this document in page layout view 3
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