1. Assumptions: In addition to the assumptions that we already talked about this design assumes: 1)Two or more factors, each factor having two or more levels. 2)All levels of each factor are investigated in combination with all levels of every other factor. If there are a (= 3) levels of factor A and b (= 3) levels of factor B then the experiment contains a x b (= 3 x 3 = 9) combinations. (the treatment levels are completely crossed). 3)Random assignment of experimental units to treatment combinations. Each experimental unit must be assigned to only one combination. Completely Randomized Factorial Design With Two Factors

2. Assignment of Experimental Units: Assume we have 3 factors. Factor A has three levels a 1, a 2 and a 3 and factor B has three levels b 1, b 2, and b 3 then the layout of the completely randomized design is as follows: a1b1a1b1 a1b2a1b2 a1b3a1b3 a2b1a2b1 a2b2a2b2 a2b3a2b3 a3b1a3b1 a3b2a3b2 a3b3a3b3 y 111 y 112 y 113 … y 11n y 121 y 122 y 123 … y 12n y 131 y 132 y 133 … y 13n y 211 y 212 y 213 … y 21n y 221 y 222 y 223 … y 22n y 231 y 232 y 233 … y 23n y 311 y 312 y 313 … y 31n y 321 y 322 y 323 … y 32n y 331 y 332 y 333 … y 33n Total sample is nab = n(3)(3) randomly assigned to the different combinations, with a minimum n = 1 (in this case we have to assume no interaction between the different factor levels). Completely Randomized Factorial Design With Two Factors

3. Linear Model Completely Randomized Factorial Design With Two Factors

4. Completely Randomized Factorial Design With Two Factors y ijk Response of the k th experimental unit in the ij factor combination.  The grand mean of all factor combinations’ population-means. ii Factor effect for population i, and should obey the condition: jj  ij Joint effect of factor levels i and j, and should obey both:  ijk The error effect associated with Y ijk and is equal to:

5. 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  Means

6. Completely Randomized Factorial Design With Two Factors Hypotheses:

7. Completely Randomized Factorial Design With Two Factors A\Bb1b1 b2b2 b3b3 Grand Means a1a1 a2a2 a3a3 Grand means Means

8. 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  23 =  +  2  +  1 + (  ) 21  23 =  +  2  +  2 + (  ) 22  23 =  +  2  +  3 + (  ) 23  2. =  +  2 a3a3  33 =  +  3  +  1 + (  ) 31  33 =  +  3  +  2 + (  ) 32  33 =  +  3  +  3 + (  ) 33  3. =  +  3 Grand means .1 =  +  1 .2 =  +  2 .3 +  3 

9. Completely Randomized Factorial Design With Two Factors Hypotheses:

10. Completely Randomized Factorial Design With Two Factors A\Bb1b1 b2b2 b3b3 Grand Means a1a1 a2a2 a3a3 Grand means Means Where

11. Completely Randomized Factorial Design With Two Factors

12. Completely Randomized Factorial Design With Two Factors (Fixed Effects)

13. Completely Randomized Factorial Design With Two Factors (Fixed Effects)