1. 2k Experiments, Incomplete block designs for 2k experiments, fractional 2k experiments

2. Factorial Experiments

3. k Categorical independent variables A, B, C, … (the Factors) Let
Dependent variable y
k Categorical independent variables A, B, C, … (the Factors)
Let
a = the number of categories of A
b = the number of categories of B
c = the number of categories of C
etc.
t = abc... Treatment combinations

4. The Completely Randomized Design
We form the set of all treatment combinations – the set of all combinations of the k factors
Total number of treatment combinations
t = abc….
In the completely randomized design n experimental units (test animals , test plots, etc. are randomly assigned to each treatment combination.
Total number of experimental units N = nt=nabc..

5. The ANOVA Table three factor experiment

6. If the number of factors, k, is large then it may be appropriate to keep the number of levels of each factor low (2 or 3) to keep the number of treatment combinations, t, small.
t = 2k if a = b =c = ... =2 or
t = 3k if a = b =c = ... =3
The experimental designs are called 2k and 3k designs

7. The ANOVA Table 23 experiment
Source
Sum of Squares
d.f. A
SSA 1 B
SSB C
SSC AB
SSAB AC
SSAC BC
SSBC
ABC
SSABC
Error
SSError
23(n – 1)

8. Notation for treatment combinations for 2k experiments
There are several methods for indicating treatment combinations in a 2k experiment and 3k experiment.
A sequence of small letters representing the factors with subscripts (0,1 for 2k experiment and 0, 1, 2 for a 3k experiment)
A sequence of k digits (0,1 for 2k experiment and 0, 1, 2 for a 3k experiment.
A third way of representing treatment combinations for 2k experiment is by representing each treatment combination by a sequence of small letters. If a factor is at its high level, it’s letter is present. If a factor is at its low level, it’s letter is not present.

9. The 8 treatment combinations in a 23 experiment
(a0, b0, c0), (a1, b0, c0), (a0, b1, c0), (a0, b0, c1),
(a1, b1, c0), (a1, b0, c1), (a0, b1, c1), (a1, b1, c1)
000, 100, 010, 001, 110, 101, 011, 111
1, a, b, c, ab, ac, bc, abc
In the last way of representing the treatment combinations, a more natural ordering is:
1, a, b, ab, c, ac, bc, abc
Using this ordering the 16 treatment combinations in a 24 experiment
1, a, b, ab, c, ac, bc, abc, d, da, db, dab, dc, dac, dbc, dabc

10. Notation for Linear contrasts treatment combinations in a 2k experiments
The linear contrast for 1 d.f. representing the Main effect of A
LA = (1 + b + c + bc) – (a + ab +ac + abc)
= comparison of the treatment combinations when A is at its low level with treatment combinations when A is at its high level.
Note: LA = (1 - a) (1 + b) (1 + c)
also
LB = (1 + a) (1 - b) (1 + c)
= (1 + a + c + ac) – (b + ab +bc + abc)
LC = (1 + a) (1 + b) (1 - c)
= (1 + a + b + ab) – (c + ca +cb + abc)

11. The linear contrast for 1 d.f. representing the interaction AB
LAB = (1 - a) (1 - b) (1 + c)
= (1 + ab + c + abc) – (a + b +ac + bc)
= comparison of the treatment combinations where A and B are both at a high level or both at a low level with treatment combinations either A is at its high level and B is at a low level or B is at its high level and A is at a low level.
LAC = (1 - a) (1 + b) (1 - c)
= (1 + ac + b + abc) – (a + c +ab + bc)
LBC = (1 + a) (1 - b) (1 - c)
= (1 + bc + a + abc) – (b + c +ac + ab)

12. The linear contrast for 1 d.f. representing the interaction ABC
LABC = (1 - a) (1 - b) (1 - c)
= (1 + ab + ac + bc) – (a + b + c + abc)
In general Linear contrasts are of the form:
L = (1 ± a)(1 ± b)(1 ± c) etc
We use minus (-) if the factor is present in the effect and plus (+) if the factor is not present.

13. + × + = + - × + = - + × - = - - × - = +
The sign of coefficients of each treatment for each contrast (LA, LB, LAB, LC, LAC, LBC, LABC) is illustrated in the table below:
For the main effects (LA, LB, LC) the sign is negative (-) if the letter is present in the treatment, positive (+) if the letter is not present. The interactions are products of the main effects:
+ × + = + - × + = - + × - = - - × - = +

14. Yates Algorithm
This is a method for computing the Linear contrasts of the effects and their sum of squares (S.S.) The algorithm is illustrated with the Table on the next slide
The algorithm is as follows:
Treatments are listed in the standard order (1, a, b, ab, c, ac, bc, abc etc.) i.e. Starting with 1, then adding one letter at a time followed by all combinations with letters that have been previously added.

15. Table: Illustration of Yates Algorithm for a 23 factorial Design (# of replicates n = 4)
Treatment
Total yield I II
III
Effect SS 1
121
302
663
1362
Total a
181
361
699
408 A b
104
296
213
166 B
861.12 ab
257
403
195
188 AB c
123 60 59 36 C
40.50 ac
173
153
107
-18 AC
10.12
129 50 93 48
72.00
abc
274
145 95 2
ABC
0.12

16. Yates Algorithm (continued)
In the yield column enter the total yields for each treatment combination.
Fill in as many columns headed by Roman numerals as there are factors in the experiment in the following way.
Add successive pairs in the previous column. (1st +2nd), (3rd + 4th) etc
Subtract successive pairs in the previous column (2nd - 1st), (4th - 3rd) etc
To obtain entries in column II repeat steps 3a and 3b on the entries of column I.
To obtain entries in column III repeat steps 3a and 3b on the entries of column II
Continue in this way until as many columns have been filled as factors.

17. Yates Algorithm (continued) - Computation of SS’s
Square the effect total (entry in last column).
Divide the result by the number of observations n2k.

18. Strategy for a single replication (n = 1)
The ANOVA Table 23 experiment
Source
Sum of Squares
d.f. A
SSA 1 B
SSB C
SSC AB
SSAB AC
SSAC BC
SSBC
ABC
SSABC
Error
SSError
23(n – 1)
If n = 1 then there is 0 df for estimating error. In practice the higher order interactions are not usually present. One makes this assumption and pools together these degrees of freedom to estimate Error

19. In a 7 factor experiment (each at two levels) there are 27 =128 treatments.

20. ANOVA table: Pool together these degrees of freedom to estimate Error
Source
d.f.
Main Effects 7
2-factor interactions 21
3-factor interactions 35
4-factor interactions
5-factor interactions
6-factor interactions
7-factor interaction 1
Pool together these degrees of freedom to estimate Error

21. Randomized Block design for 2k experiments
Blocks
... n 1 2 3 4 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc
A Randomized Block Design for a 23 experiment

22. The ANOVA Table 23 experiment in RB design
Source
Sum of Squares
d.f.
Blocks
SSBlocks
n - 1 A
SSA 1 B
SSB C
SSC AB
SSAB AC
SSAC BC
SSBC
ABC
SSABC
Error
SSError
(23 – 1)(n – 1)

23. Incomplete Block designs for 2k experiments Confounding

24. ... A Randomized Block Design for a 23 experiment Blocks n 1 2 3 4 1 aab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc 1 a b ab c ac bc
abc

25. Incomplete Block designs for 2k experiments
A Randomized Block Design for a 2k experiment requires blocks of size 2k. The ability to detect treatment differences depends on the magnitude of the within block variability. This can be reduced by decreasing the block size.
Blocks 1 2
Example: a 23 experiment in blocks of size 4 (1 replication). The ABC interaction is confounded with blocks
abc 1 a bc b ac c ab

26. In this experiment the linear contrast 1 2
Blocks
In this experiment the linear contrast 1 2
LABC = (1 + ab + ac + bc) – (a + b + c + abc)
In addition to measuring the ABC interaction it is also subject to block to block differences.
abc 1 a bc b ac
The ABC interaction it is said to be confounded with block differences. c ab
The linear contrasts
LA = (1 + b + c + bc) – (a + ab +ac + abc)
LB = (1 + a + c + ac) – (b + ab +bc + abc)
LC = (1 + a + b + ab) – (c + ca +cb + abc
LAB = (1 + ab + c + abc) – (a + b +ac + bc)
LAC = (1 + ac + b + abc) – (a + c +ab + bc)
LBC = (1 + bc + a + abc) – (b + c +ac + ab)
are not subject to block to block differences

27. LABC = 1 + ab + ac + bc – a – b – c – abc
To confound an interaction (e. g. ABC) consider the linear contrast associated with the interaction:
LABC = 1 + ab + ac + bc – a – b – c – abc
Assign treatments associated with positive (+) coefficients to one block and treatments associated with negative (-) coefficients to the other block
Blocks 1 2
abc 1 a bc b ac c ab

28. The ANOVA Table 23 experiment in incomplete design with 2 blocks of size 4
Source
Sum of Squares
d.f.
Blocks
SSBlocks 1 A
SSA B
SSB C
SSC AB
SSAB AC
SSAC BC
SSBC
Total
SSTotal 7

29. Confounding more than one interaction to further reduce block size

30. Example: contrasts for 23 experiment
If I want to confound ABC, one places the treatments associated with the positive sign (+) in one block and the treatments associated with the negative sign (-) in the other block.
If I want to confound both BC and ABC, one chooses the blocks using the sign categories (+,+) (+,-) (-,+) (-,-)
Comment: There will also be a third contrast that will also be confounded

31. LABC = (1 + ab + ac + bc) – (a + b + c + abc) and
Example: a 23 experiment in blocks of size 2 (1 replicate). BC and ABC interaction is confounded in the four block.
Block 1
Block 2
Block 3
Block 4 1 a ab b bc
abc ac c
LABC = (1 + ab + ac + bc) – (a + b + c + abc) and
LBC = (1 + bc + a + abc) – (b + c +ac + ab)
are confounded with blocks
LA = (1 + b + c + bc) – (a + ab +ac + abc)
is also confounded with blocks
LB = (1 + a + c + ac) – (b + ab +bc + abc)
LC = (1 + a + b + ab) – (c + ca +cb + abc
LAB = (1 + ab + c + abc) – (a + b +ac + bc)
LAC = (1 + ac + b + abc) – (a + c +ab + bc)
are not subject to block to block differences

32. The ANOVA Table 23 experiment in incomplete design with 4 blocks of size 2 (ABC, BC and hence A confounded with blocks)
Source
Sum of Squares
d.f.
Blocks
SSBlocks 3 B
SSB 1 C
SSC AB
SSAB AC
SSAC
Total
SSTotal 7
There are no degrees of freedom for Error.
Solution: Assume either one or both of the two factor interactions are not present and use those degrees of freedom to estimate error

33. Rule: (for determining additional contrasts that are confounded with block)
“Multiply” the confounded interactions together.
If a factor is raised to the power 2, delete it
Example:
Suppose that ABC and BC is confounded, then so also is (ABC)(BC) = AB2C2 = A.
A better choice would be to confound AC and BC, then the third contrast that would be confounded would be (AC)(BC) = ABC2 = AB

34. If I want to confound both AC and BC, one chooses the blocks using the sign categories (+,+) (+,-) (-,+) (-,-). As noted this would also confound (AC)(BC) = ABC2 = AB.
Block 1
Block 2
Block 3
Block 4 1 b a ab
abc ac bc c

35. The ANOVA Table 23 experiment in incomplete design with 4 blocks of size 2 (AC, BC and hence AB confounded with blocks)
Source
Sum of Squares
d.f.
Blocks
SSBlocks 3 A
SSA 1 B
SSB C
SSC
ABC
SSABC
Total
SSTotal 7
There are no degrees of freedom for Error.
Solution: Assume that the three factor interaction is not present and use this degrees of freedom to estimate error

36. Partial confounding

37. Example: a 23 experiment in blocks of size 4 (3 replicates)
Example: a 23 experiment in blocks of size 4 (3 replicates). BC interaction is confounded in 1st replication. AC interaction is confounded in 2nd replication. AB interaction is confounded in 3rd replication.
Replicate 1
BC confounded
Replicate 2
AC confounded
Replicate 3
AB confounded
Block 1
Block 2
Block 3
Block 4
Block 5
Block 6 1 ab
abc bc 1 a bc c b a c b
abc ac 1 ab ab ac a b ac c
abc bc
The main effects (A, B and C) and the three factor interaction ABC can be estimated using all three replicates. The two factor interaction AB can be estimated using replicates 1 and 2, AC using replicates 1 and 3, BC using replicates 2 and 3,

38. The ANOVA Table Source Sum of Squares d.f. Reps SSBlocks 2
Blocks within Reps
SSBlocks(Reps) 3 A
SSA 1 B
SSB C
SSC AB
SSAB
Reps I,II AC
SSAC
Reps I,III BC
SSBC
Reps II,III
ABC
SSABC
Error
SSError 11
Total
SSTotal 23

39. Example: A chemist is interested in determining how purity (Y) of a chemical product, depends on agitation rate (A), base component concentration (B) and concnetration of reagent (C). He decides to use a 23 design. Only 4 runs can be done each day (Block) and he wanted to have 3 replications of the experiment.
Replicate 1
BC confounded
Replicate 2
AC confounded
Replicate 3
AB confounded
day 1
day 2
day 3
day 4
day 5
day 6 1 25 ab 43
abc 39 bc 38 26 a 34 c 30 b 29 37 32 42 ac 40 27 46 52 33 51 36

40. The ANOVA Table F0.05(1,11) = 4.84 and F0.01(1,11) = 9.65 Source
Sum of Squares
d.f.
Mean Square F
Reps
111.00 2
55.50
Blocks within Reps
108.00 3
36.00 A
600.00 1
40.6** B
253.50
17.2** C
54.00
3.7(ns)
AB (Reps I,II)
6.25
< 1
AC (Reps I,III)
1.00
BC (Reps II,III)
ABC
13.50
Error
162.50 11
14.77
Total 23
F0.05(1,11) = and F0.01(1,11) = 9.65

41. Fractional Factorials

42. In a 2k experiment the number of experimental units required may be quite large even for moderate values of k.
For k = 7, 27 = 128 and n27 = 256 if n = 2.
Solution:
Use only n = 1 replicate and use higher order interactions to estimate error. It is very rare thqt the higher order interactions are significant
An alternative solution is to use ½ a replicate, ¼ a replicate, 1/8 a replicate etc. (i.e. a fractional replicate)
2k – 1 = ½ 2k design, 2k – 2 = ¼ 2k design

43. In a fractional factorial design, some ot he effects (interactions or main effects) may not be estimable. However it may be assumed that these effects are not present (in particular the higher order interactions)

44. Example: 24 experiment, A, B, C, D - contrasts
To construct a ½ replicate of this design in which the four factor interaction, ABCD, select only the treatment combinations where the coefficient is positive (+) for ABCD

45. The treatments and contrasts of a ½ 24 = 24-1 experiment
Notice that some of the contrasts are equivalent
e.g.
A and BCD, B and ACD, etc
In this case the two contrasts are said to be aliased.
Note the defining contrast, ABCD is aliased with the constant term I. To determine aliased contrasts multiply the any effect by the effect of the defining contrast
e.g. (A)×(ABCD) = A2BCD = BCD

46. Aliased contrasts in a 24 -1 design with ABCD the defining contrast
A with BCD
B with ACD
C with ABD
D with ABC
AB with CD
AC with BD
AD with BC
If an effect is aliased with another effect you can either estimate one or the other but not both

47. The ANOVA for a 24 -1 design with ABCD the defining contrast
Source df A 1 B C D AB AC AD
Total 7

48. Example: ¼ 24 experiment
To construct a ¼ replicate of the 24 design. Choose two defining contrasts, AB and CD, say and select only the treatment combinations where the coefficient is positive (+) for both AB and CD

49. The treatments and contrasts of a ¼ 24 = 24-2 experiment
Aliased contrasts
I and AC and BD and ABCD
A and C and ABD and BCD
B and ABC and D and ACD
AB and BC and AD and CD

50. The ANOVA for a 24 -1 design with ABCD the defining contrast
Source df A 1 B AB
Total 3
There may be better choices for the defining contrasts
The smaller fraction of a 2k design becomes more appropriate as k increases.