1. Completely Randomized Factorial Design With Two Factors Example A police department in a big city want to assess their human relations course for new officers. The independent variables are the type of neighborhood the officers get to be assigned to during the period of the course, factor A, and the amount of time they spend in the course, Factor B. Factor A has three levels: a 1 =upper-class, a 2 =middle-class and a 3 =inner-city. Factor B also has three levels: b 1 =5 hours, b 2 =10 hours and b 3 =15 hours. The dependent (response) variable, y, is attitude towards minority groups following the course.

2. Completely Randomized Factorial Design With Two Factors a1b1a1b1 a1b2a1b2 a1b3a1b3 a2b1a2b1 a2b2a2b2 a2b3a2b3 a3b1a3b1 a3b2a3b2 a3b3a3b3 244438303526214142 333629214027183952 372528392736105053 292747263146313649 424348342245203464

3. Completely Randomized Factorial Design With Two Factors A\Bb1b1 b2b2 b3b3 Grand Means a1a1  11  12  13  1. a2a2  21  22  23  2. a3a3  31  32  33  3. Grand means .1 .2 .3  What do we want to compare?

4. Completely Randomized Factorial Design With Two Factors Hypotheses:

5. Interactions Assuming we have these two factors together in one experiment and that we know the following true means. Is there an interaction effect? BABA b1b1 b2b2 b3b3 Grand Means a1a1 20 35(20+20+35)/3 25 a2a2 22 37(22+22+37)/3 27 a3a3 42(27+27+42)/3 32 Grand means (20+22+27)/3 23 (20+22+27)/3 23 (35+37+42)/3 38 28

6. Interactions

8. Assuming we have these two factors together in one experiment and that we know the following true means. Is there an interaction effect? BABA b1b1 b2b2 b3b3 Grand Means a1a1 202735(20+20+35)/3 27.33 a2a2 22 37(22+22+37)/3 27 a3a3 2042(27+27+42)/3 29.66 Grand means (20+22+27)/3 23 (20+22+27)/3 23 (35+37+42)/3 38 28

9. Interactions

11. Linear Model Completely Randomized Factorial Design With Two Factors

12. Completely Randomized Factorial Design With Two Factors What are we comparing? A/Bb1b1 b2b2 b3b3 Grand Means a1a1  11 =  +  1  +  1 + (  ) 11  12 =  +  1  +  2 + (  ) 12  12 =  +  1  +  3 + (  ) 13  1. =  +  1 a2a2  21 =  +  2  +  1 + (  ) 21  22 =  +  2  +  2 + (  ) 22  23 =  +  2  +  3 + (  ) 23  2. =  +  2 a3a3  31 =  +  3  +  1 + (  ) 31  32 =  +  3  +  2 + (  ) 32  33 =  +  3  +  3 + (  ) 33  3. =  +  3 Grand means .1 =  +  1 .2 =  +  2 .3 =  +  3 

13. Completely Randomized Factorial Design With Two Factors Hypotheses:

14. Interactions If there is no interaction and we have these two factors together in one experiment we will have the following results (effects model) : A\Bb1b1 b2b2 b3b3 Grand Means a1a1 28+(25-28)+(20- 25-23+28) =28+(25-28)+0 28+(25-28) 30-3 a2a2 28+(27-28) 30-1 a3a3 28+(32-28) 30+4 Grand means 28+(23-28) 28-5 28+(23-28) 28-5 28+(38-28) 28+10 28

15. Estimated means from the data: A\Bb1b1 b2b2 b3b3 Grand Means a1a1 33353835.33 a2a2 30313632.33 a3a3 30405240.67 Grand means 3135.334236.11 Interactions

16. Estimated means from the data: A\Bb1b1 b2b2 b3b3 Grand Means a1a1 36.11-0.78-5.11 +2.78 = 33 36.11-0.78-0.78 +0.45 = 35 36.11-0.78+5.89 - 3.22 = 38 36.11-0.78 a2a2 36.11-3.78-5.11 +2.75 = 30 36.11-3.78-0.78 - -0.55 = 31 36.11-3.78+5.89 -2.22 = 36 36.11-3.78 a3a3 36.11+4.59-5.11 - 5.59 = 30 36.11+4.56-0.78 +0.11 = 40 36.11+4.56+5.89 +5.44 = 52 36.11+4.56 Grand means 36.11-5.1136.11-0.7836.11+5.8936.11 Interactions

17. Simple means vs. A-levels.

18. Interactions Simple means vs. B-levels.