The 2k Factorial Design 

Case 1:  Two Level: Two factor.


•Study the effects of 2 or more factors with factorial experiments

•Each factor and each combination of factors are studied

•     Example

–         Factor A has 2 levels (high, low)

–         Factor B has 2 levels (on, off)

–         Then the total number of experiments is 2x2=4, or

–         high-on, high-off, low-on, low-off

•Experiments measure the difference of the response from one level of the factor (high for A) and another level (low for A).
          

•Example

–Factor A- 2 levels- A1, A2

–Factor B- 2 levels- B1, B2

–Measured values are



– 

A= 30+40 - 10+20 = 20

           2            2

B= 20+40 - 30+10 = 10

           2            2


Conclusion

Changes in Factor A causes more of an effect than B. Factor A is more significant than Factor B

 

•In the 2k design:

·All factor effects will have 1 d.f

·If there are n replicates, SSE will have (n-1)2k d.f.

·Replicates are very important in testing for lack of fit

·If n=1, we have no estimate for error [Why?]

·Use higher order interactions to get an estimate.

-Plot the estimates on a normal probability paper. All effects that are insignificant will fall on a line.










Consider Only Factor A  (Factor B is collapsed and really looking at the difference between going from a LOW A setting to a HIGH A setting.



A effect  =   (Y4+Y2)  - (Y3+Y1)
                          2                2

A effect  = 1/2 ( Y4+Y2-Y3-Y1)

Therefore the generic formula can be stated as:  Effect = (Sum of Matrix Colum) / (2k-1  x n)

if AVERAGE is used n = 1,  if  TOTAL  is used n = # replicates.  NOTE:  THIS IS THE ONLY PLACE WHERE N WILL BE DIFFERENT IN YOUR CALCULATIONS!
                                                                                                   IN ALL OTHER FORMULAE FOR 2 LEVEL, n = NUMBER OF REPLICATES (TRAILS).

The A effect can also be written as follows:





             




Next, consider only Factor B (Factor A is collapsed, and now only the effect of going from a LOW B to a HIGH B is considered.









Finally, consider the Interaction effect.  Now the difference of a HIGH A, LOW  B  to  a  HIGH B, LOW 
is compared to the difference from  a LOW A, LOW B to a HIGH A, HIGH B










NEXT : EXAMPLE and using EXCEL to calculate EFFECTS, CONTRASTS, and ANOVA.