Experimental design
1. Mapping the experimental domain
pH
%γcyclodextrin
2
4
2.5
•
•
5
•
•
7.5
•
•
Knowledge based system for capillary
zone electrophoresis of chiral substances
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
1
Experimental design
2. Finding important variables
Factors and factor levels studied in connection
with a spheronization process
Factor
level
low (-)
high (+)
Water content (ml)
(A)
250
325
Extruder speed (rpm)
(B)
39
59
Screen size (mm)
(C)
0.8
1.5
Spheronizer (speed rpm)
(D)
700
1010
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
2
4.4
5.4
4.2
4.8
6.4
8.3
6.1
6.5
A
+
−
B
D
+
−
−
−
+
C
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
3
Conclusions :
B has no effect
A is most important (effect ∼ 2.2)
C and D have an effect that is much
less important (effect ∼ 0.8)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
4
3. Making a model and finding the optimal
combination
Cabernet
Syrah
Grenache
C
S
G
y = 2.81 x1 + 3.87 x2 + 5.87 x3 + 13.37 x1x2 +
49.37 x1x3 + 35.46 x2x3 - 106.41 x1x2x3
(y = quality, x1 = % Cabernet wine, x2 = % Grenache,
x3 = % Syrah)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
5
Why not use one-variable-at-a time
approach
x2
10
20
•
•
30
•
•
•
•
•B •
•
0
•
•
x1
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
6
x2
•
•
C
• • •
• • •B
•
10
20
30
•D
• •
•
•
0
•
•
x1
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
7
x2
10
B
20
•
•
A•
30
E
•D
•C
x1
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
8
SIMPLEX
HCHO-conc.
1.10
0.90
0.70
0.50
0.30
6
•
4
•
•
7
•
3
•
5
1•
•2
HCl-conc.
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
9
CHAPTER 1
TWO-LEVEL FULL
FACTORIAL
DESIGN
(with/without central point)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
10
• What is the effect on tablet hardness
(response) of the amount of water added, the
extrusion rate, the mesh size of a screen and
the spheronizer rate (variables or factors)?
• What is the optimal combination of the four
factors?
• How can I obtain an answer to this question
with a minimal number of experiments?
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
11
A spheronization process
Factor
level
low (-) high (+)
Water content (ml)
Extruder speed (rpm)
Screen size (mm)
Spheronizer (speed rpm)
(A)
(B)
(C)
(D)
250
39
0.8
700
325
59
1.5
1010
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
12
water
325
y4 •
• y1
250
y2 •
• y3
39
59
rpm
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
13
A
y2
• y1
•
B
• y3
y4 •
y6
•
y8 •
• y5
• y7
C
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
14
Two-level factorial designs
2k factorial design for k variables
2k experiments (all possible
combinations of the two levels for each
variable) = runs, treatments
levels :
+
−
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
15
A spheronization process
AB-
A+
B+
B-
B+
D-
6.1
(1)
6.4
b
4.2
a
4.4
ab
D+
4.7
d
6.3
bd
3.9
ad
3.4
abd
D-
6.5
c
8.3
bc
4.8
ac
5.4
abc
D+
6.7
cd
6.6
bcd
3.7
acd
3.7
abcd
C-
C+
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
16
Naming experiments
A +, B +, C −, D +
→ abd
A −, B −, C −, D −
→ (1)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
17
The spheronization example
Experiment
a
ac
(1)
acd
b
d
abcd
ab
c
bcd
abd
ad
cd
abc
bc
bd
A
+
+
+
+
+
+
+
+
-
B
+
+
+
+
+
+
+
+
C
+
+
+
+
+
+
+
+
-
D
+
+
+
+
+
+
+
+
Response
4.2
4.8
6.1
3.7
6.4
4.7
3.7
4.4
6.5
6.6
3.4
3.9
6.7
5.4
8.3
6.3
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
18
Standard orders
(1)
a
b
ab
c
ac
bc
abc
d
ad
(1)
a
b
ab
(1)
a
x
x
x
x
x
x
c
c
c
c
d
d
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
19
Design matrix for 23 factorial experiments
Run
A
B
C
Response
1
2
3
4
5
6
7
8
+
+
+
+
-
+
+
+
+
-
+
+
+
+
-
y1
y2
y3
y4
y5
y6
y7
y8
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
20
Run 1 : A + B + C +
Response : y1
Run 5 : A - B + C +
Response : y5
y1 - y5 = estimate effect of A
Any other estimates of A?
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
21
Other estimates : y2 - y6 , y3 - y7 , y4 - y8
Four estimates : y1 - y5 , y2 - y6 , y3 - y7 ,
y4 - y8
Effect A = (y1 + y2 + y3 + y4 - y5 y6 - y7 - y8) / 4
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
22
General :
(Σ positive level runs - Σ negative level runs) / 4
(for 3 factors)
Speroniser example :
Effect A = [(4.2 + 4.4 + 3.9 + 3.4 + 4.8 +5.4 +
3.7 + 3.7) -(6.1 + 6.4 + 4.7 + 6.3 + 6.5
+8.3 + 6.7 + 6.6)] / 8
= - 2.26
When going from 250 to 325 ml water the
estimated effect on hardness = - 2.26
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
23
Interaction
• Is the effect of the extruder rpm influenced
by the water content (or vice versa)
• Is the effect of the pH influenced by
concentration of the reagent (or vice-versa)?
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
24
Interaction between A and B
y1 - y5 = estimated effect of A
(at B + C +)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
25
Design matrix for 23 factorial experiments
Run
A
B
C
Response
1
2
3
4
5
6
7
8
+
+
+
+
-
+
+
+
+
-
+
+
+
+
-
y1
y2
y3
y4
y5
y6
y7
y8
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
26
Interaction between A and B
y1 - y5 = estimated effect of A at B + C +
y3 - y7 = estimated effect of A at B - C +
[(y1 - y5) - (y3 - y7)] / 2 =
[(y1 + y7) - (y3 + y5)] / 2 =
Estimate of how differences in B affect the
effect of A
Interaction of B on A (estimate)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
27
Interaction between A and B
[(y1 + y7) - (y3 + y5)] / 2
At C - :
[(y2 + y8) - (y4 + y6)] / 2
A x B = (y1 + y2 + y7 + y8 - y3 -y4 y5 - y6) / 4
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
28
Computation of interaction levels for 3 variables.
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
AB
+
+
+
+
AC
+
+
+
+
Run
1
2
3
4
5
6
7
8
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
29
Rules for interaction columns
sign AxB = sign A x sign B
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
30
Computation of interaction levels for 4 variables
A
+
+
+
+
+
+
+
+
-
B
+
+
+
+
+
+
+
+
C
+
+
+
+
+
+
+
+
-
D
+
+
+
+
+
+
+
+
AB
+
+
+
+
+
+
+
+
-
ABC ABCD
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
31
run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Effect ABC
= [(y1 + y5 + y7 + y9 + y12 + y13 + y14 + y16) (y2 + y3 + y4 + y6 + y8 + y10 + y11 + y15)] / 8
= [4.2 + 6.4 + 3.7 + 6.5 + 3.9 + 6.7 + 5.4 + 6.3) (4.8 + 6.1 + 3.7 + 4.7 + 4.4 + 6.6 + 3.4 + 8.3)] /8
= 0.1375
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
32
G.T. Wernimont
Use of statistics to develop and evaluate analytical
methods
AOAC 1985
Acetone in cellulose acetate
A:
- : acid
+ : alkali
B:
- : H2O
+ : methanol
C:
- : 3'
+ : 6'
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
33
The determination of acetone
Run
y1
y2
yT
y
(1)
a
b
ab
c
ac
bc
abc
4.04
7.02
4.16
5.68
4.08
7.23
4.26
5.72
4.06
6.82
4.12
5.80
4.04
7.20
4.20
5.86
8.10
13.84
8.28
11.48
8.12
14.43
8.46
11.58
4.05
6.92
4.14
5.74
4.06
7.21
4.23
5.79
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
34
A
+
B
+
−
−
−
+
C
5.74
5.79
6.92
7.21
4.14
4.23
4.05
4.06
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
35
4.4
4.2
5.4
4.8
A
+
6.4
6.1
−
8.3
6.5
3.4
3.9
B
D
+
−
−
+
C
3.7
A
3.7
+
−
6.3
4.7
−
6.6
B
D+
+
−
−
+
C
6.7
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
36
Computation of the effects of
the acetone data
Effects
Factor
T
5.27
A
2.30
B
- 0.59
AB
- 0.72
C
0.11
AC
0.06
BC
- 0.04
ABC
- 0.08
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
37
Computation of the effects of the
spheronization experiment
Factor
T
A
B
AB
C
AC
BC
ABC
D
AD
BD
ABD
CD
ACD
BCD
ABCD
Effects
5.32
- 2.26
0.49
- 0.41
0.79
- 0.36
0.09
0.14
- 0.89
- 0.14
- 0.24
- 0.09
- 0.19
- 0.19
- 0.39
0.41
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
38
• When no factor has an effect, they should
be distributed randomly around 0
• Can be verified by a normal plot
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
39
Rank
7
•A
•C
BC •
ABC •
4
• AC
•B
1
AB •
0
1
Effect
2
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
40
cumulative %
99.9
99
95
80
50
20
5
•D
•A
•
•
•••
•• •
•
•
•
•
•
C
1
- 2.0
- 1.0
0
1.0
Effect
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
41
ANOVA for the analytical data
Effect
A
B
AxB
C
AxC
BxC
AxBxC
Residual
F(1,8) = 5.32
df
1
1
1
1
1
1
1
8
SS
21.091
1.375
2.052
0.050
0.015
0.007
0.026
0.041
MS
21.091
1.375
2.052
0.050
0.015
0.007
0.026
0.0051
F
4136
269.6
402.3
9.8
2.94
1.37
5.18
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
42
ANOVA for the spheronization data
Factor
A
B
C
D
AB
AC
BC
AD
BD
CD
ABC
ABD
ACD
BCD
ABCD
Triple + Quadruple
interactions
SS
df
MS
F
20.48
0.95
2.48
3.15
0.68
0.53
0.031
0.075
0.225
0.14
0.076
0.031
0.14
0.60
0.68
1.527
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
5
20.48
0.95
2.48
3.15
0.68
0.53
0.031
0.075
0.225
0.14
0.076
0.031
0.14
0.60
0.68
0.305
67.1
3.11
8.13
10.33
2.23
1.74
-
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
43
CHAPTER 2
SCREENING DESIGNS
TWO-LEVEL FRACTIONAL
FACTORIAL DESIGNS
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
44
Expansion
chamber
Product
container
Top spray method used in
conventional granulation coaters.
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
45
Fluidised bed granulation
• Air flow rate?
• Air humidity?
• Temperature?
• Type of nozzle
• Flow rate of binder solution?
etc.
to optimise
• Particle diameter of granules
etc.
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
46
Full factorial designs
2k = 128
k=7
7
21
35
35
21
7
1
main effects
binary interactions
ternary
quaternary
5-factor
6-factor
7-factor
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
47
pH
CBE
CEM
•
•
•
•
•
•
•
•
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
48
pH
CBE
CEM
•1
•
3•
•
•
•
2
•
•4
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
49
A 23-1 experiment for determining effects on
resolution in capillary zone electrophoresis
Experiment
1
2
3
4
pH
+
+
-
CBE
+
+
-
CEM
+
+
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
50
Half-fraction
design
23 → 23-1
replica
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
51
half-fraction
design
23 → 23-1
replica
quarter-fraction
design
25-2
replica
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
52
Half replica of the spheronization design
ABD-
A+
B+
B-
6.1 (1)
B+
4.4 ab
CD+
6.3 bd
3.9 ad
D-
8.3 bc
4.8 ac
C+
D+
6.7 cd
3.7 abcd
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
53
24-1 design
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
D
+
+
+
+
-
Resp.
y1
y2
y3
y4
y5
y6
y7
y8
+
+
+
+
-
+
+
+
+
-
+
+
+
+
-
+
+
+
+
y9
y10
y11
y12
y13
y14
y15
y16
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
54
24-1 design
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
D
+
+
+
+
-
Resp.
y1
y2
y3
y4
y5
y6
y7
y8
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
55
Confounding
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
D
+
+
+
+
-
BC
+
+
+
+
AD
+
+
+
+
BCD
+
+
+
+
-
Resp.
y1
y2
y3
y4
y5
y6
y7
y8
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
56
Effect A = 1/4 (y1 + y2 + y3 + y4 - y5 - y6
- y 7 - y8 )
= Effect BCD!
= Effect (A + BCD)
A and BCD are confounded
AD and BC are confounded
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
57
Interpretation - half replica spheronization
design
Factor
Effect
Full factorial
y + ABCD
A+BCD
B+ACD
AB+CD
C+ABD
AC+BD
BC+AD
D+ABC
5.52
- 2.65
0.3
- 0.6
0.7
- 0.6
- 0.05
- 0.75
5.32
- 2.26
0.49
- 0.41
0.79
- 0.36
0.09
- 0.89
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
58
A 25-2 design
Experiments
Effects
2
+
+
-
3
+
+
-
4
+
+
+
5
+
+
+
6
+
+
-
7
8
A + BCD + ABCE + DE
B + ACD + CE + ABDE
C + ABD + BE + ACDE
D + ABC + BCDE + AE
E + ABCDE + BC + AD
1
+
+
+
+
+
+
+
-
+
AB + CD + ACE + BDE
AC + BD + ABE + CDE
y + ABCD + BCE + ADE
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
59
Successive strategy
• Highly fractional factorial : screening
• Two-level factorial : important variables
and interactions
• Centre point : curvature?
• More-level factorial : response surface
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
60
Robustness
Criteria for method validation
(International Conference on
Harmonisation - ICH)
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
61
• pH buffer = 4.0
• What happens when it is 3.9?
• - level : 3.9
+ level : 4.1
• Do this for all factors in the procedure
with the smallest design possible
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
62
Saturated fractional factorial design
for 7 factors : 27-4
Experiment
1
2
3
4
5
6
7
8
A
B
C
Factor
D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
E
F
G
+
+
+
+
+
+
+
+
+
+
+
+
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
63
Plackett-Burman design for eleven factors.
The response, y, is the capacity factor of
tetracycline.
Experiment
1
2
3
4
5
6
7
8
9
10
11
12
Factor
A
d1
B
d2
C
d3
D
d4
E
d5
F
y
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
+
+
-
1.84
1.16
1.45
1.64
1.44
1.21
1.15
1.27
1.46
1.13
1.24
1.53
D.L. Massart
massart@fabi.vub.ac.be
KD/crimi1
64