PSYC 480 -- Dr. King OPTIONAL: How To Do Factorial ANOVA By Hand We will use the oddmonkeys data as an example. It's an easy example because the groups are small and everything comes out to whole numbers. Be prepared for more pain than this with most prblems. successes reward motivation 1 1 low weak 2 4 low weak 3 0 low weak 4 7 low weak 5 15 low strong 6 6 low strong 7 10 low strong 8 13 low strong 9 13 moderate weak 10 5 moderate weak 11 7 moderate weak 12 15 moderate weak 13 6 moderate strong 14 18 moderate strong 15 9 moderate strong 16 15 moderate strong 17 9 high weak 18 16 high weak 19 18 high weak 20 13 high weak 21 14 high strong 22 7 high strong 23 6 high strong 24 13 high strong Prepare an ANOVA summary table. This will help you keep things straight as you are doing the calculations. It does not matter which IV you put first in the summary table. Either way is the right way. Put the degrees of freedom in the summary table according to the usual rules: k - 1 for each factor, N - k for error, and N - 1 for total. For the interaction, recall that the dfs have to add to the total, so the interaction df can be obtained by subtracting all the other dfs from the total. Source df SS MS F reward 2 motivation 1 reward x motivation 2 error 18 total 23 Now prepare the design table with the data in each of the cells. Get the usual summary information in each of the cells. Once you get that information in each of the cells, add across the rows and down the columns to get sum, sumsq, and n in each row and each column margin. Then add across the column marginals to get grand totals. CHECK the grand totals by adding down the row marginals to see if you get the same result. reward low moderate high ----------------------------------------- | 1 13 9 | | 4 5 16 | weak | 0 7 18 | | 7 15 13 | | ----- ----- ----- | | sum 12 40 56 | sum 108 | sumsq 66 468 830 | sumsq 1364 | n 4 4 4 | n 12 | SS 30 68 46 | mean 9 motivation ----------------------------------------- | 15 6 14 | | 6 18 7 | strong | 10 9 6 | | 13 15 13 | | ----- ----- ----- | | sum 44 48 40 | sum 132 | sunsq 530 666 450 | sumsq 1646 | n 4 4 4 | n 12 | SS 46 90 50 | mean 11 ----------------------------------------------------- sum 56 88 96 | sum 240 sumsq 596 1134 1280 | sumsq 3010 n 8 8 8 | n 24 mean 7 11 12 | SS 610 | GM 10 Calculate the SS from the grand totals in the lower right corner of your table: SS=3010-240^2/24. This is SS.total. Put it in your ANOVA summary table. Source df SS MS F reward 2 motivation 1 reward x motivation 2 error 18 total 23 610 Add the SSes in each of the individual table cells. This is SS.error. Put it in your ANOVA summary table. Divide it by df.error to get MS.error, and put that in the summary table as well. Source df SS MS F reward 2 motivation 1 reward x motivation 2 error 18 330 18.333 total 23 610 The easy part is over! Now we need to get the main effects SSes. Calculate the marginal means for each row and each column and add those to the data table. Calculate a grand mean (GM) as well. The main effect SS = sum(n * (mean - GM)^2). I suggest you do that as follows. I'll do the column (reward) effect first. n mean GM 8 * ( 7 - 10)^2 = 72 8 * (11 - 10)^2 = 8 (add these to get SS.columns) 8 * (12 - 10)^2 = 32 ---- 112 This is the SS for the column main effect, or in this case SS.reward. Put it in the ANOVA table, divide it by df.reward to get MS.reward, and put that in the ANOVA table as well. Source df SS MS F reward 2 112 56.000 motivation 1 reward x motivation 2 error 18 330 18.333 total 23 610 Do the same to get the row (motivation) effect. n mean GM 12 * ( 9 - 10)^2 = 12 12 * (11 - 10)^2 = 12 ---- 24 Put it in the ANOVA table, divide it by df.motivation to get MS.motivation, and put that in the summary table as well. Source df SS MS F reward 2 112 56.000 motivation 1 24 24.000 reward x motivation 2 error 18 330 18.333 total 23 610 The interaction SS is obtained by subtraction. The effect and error SSes will add up to SS.total (in a balanced design only!), so we can get the SS.interaction as follows. SS.interaction = 610 - 330 - 24 - 112 = 144 Put that in the ANOVA table, then divide it by df.interaction to get MS.interaction and put that in the ANOVA table. Source df SS MS F reward 2 112 56.000 motivation 1 24 24.000 reward x motivation 2 144 72.000 error 18 330 18.333 total 23 610 To get the F values, divide each of the effect MSes by MS.error, as usual, except now you have to do it three times, once for each effect. Source df SS MS F reward 2 112 56.000 3.055 motivation 1 24 24.000 1.309 reward x motivation 2 144 72.000 3.927 error 18 330 18.333 total 23 610 Without software or some other form of electronic assistance, you won't be able to get the p-values, so you'll have to use an F table to get critical values. In this case, you need two of them, one for 1 and 18 degrees of freedom, and one for 2 and 18 degrees of freedom. Good luck!