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.
Completely Randomized Factorial Design With Two Factors a1b1a1b1 a1b2a1b2 a1b3a1b3 a2b1a2b1 a2b2a2b2 a2b3a2b3 a3b1a3b1 a3b2a3b2 a3b3a3b
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?
Completely Randomized Factorial Design With Two Factors Hypotheses:
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 a1a ( )/3 25 a2a ( )/3 27 a3a3 42( )/3 32 Grand means ( )/3 23 ( )/3 23 ( )/
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 a1a ( )/ a2a ( )/3 27 a3a3 2042( )/ Grand means ( )/3 23 ( )/3 23 ( )/
Interactions
Linear Model Completely Randomized Factorial Design With Two Factors
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
Completely Randomized Factorial Design With Two Factors Hypotheses:
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)+( ) =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) (23-28) (38-28)
Estimated means from the data: A\Bb1b1 b2b2 b3b3 Grand Means a1a a2a a3a Grand means Interactions
Estimated means from the data: A\Bb1b1 b2b2 b3b3 Grand Means a1a = = = a2a = = = a3a = = = Grand means Interactions
Simple means vs. A-levels.
Interactions Simple means vs. B-levels.