Embed Size (px)
DESCRIPTION
Citation preview
2
“Designing an efficient process with an effec;ve process control approach is dependent on the process knowledge and understanding obtained. Design of Experiment (DOE) studies can help develop process knowledge by revealing rela;onships, including mul;-‐factorial interac;ons, between the variable inputs … and the resul;ng outputs. Risk analysis tools can be used to screen poten;al variables for DOE studies to minimize the total number of experiments conducted while maximizing knowledge gained. The results of DOE studies can provide jus;fica;on for establishing ranges of incoming component quality, equipment parameters, and in process material quality aKributes.”
3
What is it? The ability to accurately predict/control process responses.
How do we acquire it? Scien;fic experimenta;on and modeling.
How do we communicate it? Tell a compelling scien;fic story. Give the prior knowledge, theory, assump;ons. Show the model. Quan;fy the risks, and uncertain;es. Outline the boundaries of the model. Use pictures. Demonstrate predictability.
4
Screening Designs • 2 level factorial/ frac;onal factorial designs • Weed out the less important factors • Skeleton for a follow-‐up RSM design
Response Surface Designs • 3+ level designs • Find design space • Explore limits of experimental region
Confirmatory Designs
• Confirm Findings • Characterize Variability
5 Cau;on: EVERYTHING depends on gecng this right !!!
Key Factors Key
Responses
6
Make ACE
Tablets
Disint (A or B)
Drug% (5-‐15%)
Disint% (1-‐4%)
DrugPS (10-‐40%)
Lub% (1-‐2%)
Dissolu;on% (>90%) WeightRSD%(<2%)
Fixed Factors Responses
Day
Random Factors
7
Trial DrugPS Lub%
Disso%
1 25 1 85 2 25 2 95 3 10 1.5 90 4 40 1.5 70
DrugPS
Lubricant%
85
95
70 90
10 40 1
2
8
=
+ ×
− ×
+ε
Disso% 86.66710 Lub%0.667 DrugPS
DrugPS
Lubricant%
85
95
70 90
10 40 1
2
9
� Previous example had only 2 factors. Ø Factor space is 2D. We can visualize on paper.
� With 3 factors we need 3D paper. Ø Corners even further away
� Most new processes have >3 factors � OFAT can only accommodate addi;ve models � We need a more efficient approach
10
True response • Goal: Maximize response
• Fix Factor 2 at A. • Op;mize Factor 1 to B.
• Fix Factor 1 at B. • Op;mize Factor 2 to C.
• Done? True op;mum is Factor 1 = D and Factor 2 = E.
• We need to accommodate curvature and interac/ons
A
Factor 1
Factor 2
B
C
D
E 80 60 40
11
Response
Factor level A B C D
• A to B may give poor signal to noise • A to C gives beKer signal to noise and rela;onship is s;ll nearly linear
• A to D may give poor signal to noise and completely miss curvature
• Rule of thumb: Be bold (but not too bold)
12
Trial DrugPS Lub%
Disso%
1 10 1 75 2 10 2 100 3 40 1 75 4 40 2 80
DrugPS
Lubricant%
75
80
75
100
10 40 1
2
13
DrugPS
Lubricant%
75
80
75
100
10 40 1
2
=
+ ×
+ ×
− × ×
+ε
Disso% 43.330.667 DrugPS31.667 Lub%0.667 DrugPS Lub%
14
� Model non-‐addiKve behavior
› interacKons, curvature
� Efficiently explore the factor space
� Take advantage of hidden replicaKon
15
Planar: no interac;on
1 2Y a b X c X= + ⋅ + ⋅
Non-‐planar: interac;on
1 2 1 2Y a b X c X d X X= + ⋅ + ⋅ + ⋅ ⋅
16
17
18
19
DrugPS
Lub%
C
B
D
A
10 40 1
2
DrugPS
Lub%
DrugPS Lub%
B D A CMainEffect2 2
A B C DMainEffect2 2
C B A DInteractionEffect2 2×
+ += −
+ += −
+ += −
C B D
A
C B D
A C
B D
A
Trial DrugPS Lub% Disso% 1 10 1 C 2 10 2 A 3 40 1 D 4 40 2 B
20
Trial DrugPS Lub% 1 10 1 2 10 2 3 40 1 4 40 2
Trial DrugPS Lub% 1 -‐1 -‐1 2 -‐1 +1 3 +1 -‐1 4 +1 +1
Uncoded Units Coded Units
• Coding helps us evaluate design proper;es • Some sta;s;cal tests use coded factor units for analysis
(automa;cally handled by sotware) • Easy to convert between coded (C) and uncoded (U) factor levels
midmax mid mid
max mid
U UC U C(U U ) U
U U−
= ⇔ = − +−
21
Disso ab Lub%c DrugPSd Lub% DrugPS
=
+ ×
+ ×
+ × ×
+ε
DrugPS
Lub%
DrugPS Lub%
a ( A B C D) / 4b ME /2 ( A B C D) / 4
c ME /2 ( A B C D) / 4d IE /2 ( A B C D) / 4×
= + + + +
= = − + − +
= = + + − −
= = − + + −
DrugPS
Lub%
C
B
D
A
-‐1 +1 -‐1
+1 Trial DrugPS Lub%
DrugPS*Lub%
Disso%
1 -‐1 -‐1 +1 C 2 -‐1 +1 -‐1 A 3 +1 -‐1 -‐1 D 4 +1 +1 +1 B
22
Disso a b Lub c DrugPS d Lub DrugPS= + × + × + × × + ε
� It is obtained through the “magic” of regression.
� b measures the “main effect” of Lub
� c measures the “main effect” of DrugPS
� d measures the “interac;on effect” between Lub and DrugPS
Ø if d = 0, effects of Lub and DrugPS are addi;ve
Ø if d ≠ 0, effects of Lub and DrugPS are non-‐addi;ve
� ε represents trial to trial random noise
23
Trial DrugPS Lub% 1 -‐1 -‐1 2 -‐1 +1 3 +1 -‐1 4 +1 +1
DrugPS
Lub%
-‐1 +1 -‐1
+1
Trial DrugPS Lub% 1 -‐1 -‐1 2 -‐1 -‐1 3 +1 +1 4 +1 +1
DrugPS
Lub%
-‐1 +1 -‐1
+1
Inner product: +1-‐1-‐1+1=0 +1+0+0+1=2 +1+1+1+1=4
Trial DrugPS Lub% 1 -‐1 -‐1 2 -‐1 0 3 +1 0 4 +1 +1
DrugPS
Lub%
-‐1 +1 -‐1
+1
24
25
10 40
DrugPS
Dissolu;
on (%
LC)
2% Lubricant
1% Lubricant
90
26
Number of Factors (k)
Number of Trials (df =
2k) 0 1 1 2 2 4 3 8 4 16 5 32 6 64
• Average • Main Effects • 2-‐way interac;ons • Higher order
interac;ons (or es;mates of noise)
y a bA cB dC eAB fAC gBC hABC= + + + + + + + + ε
27
Trial I A B C D=AB E=AC F=BC ABC 1 + -‐ -‐ -‐ + + + -‐ 2 + + -‐ -‐ -‐ -‐ + + 3 + -‐ + -‐ -‐ + -‐ + 4 + + + -‐ + -‐ -‐ -‐ 5 + -‐ -‐ + + -‐ -‐ + 6 + + -‐ + -‐ + -‐ -‐ 7 + -‐ + + -‐ -‐ + -‐ 8 + + + + + + + +
y a bA cB dC eD fE gF= + + + + + + + ε
Main Effects
• Can include addi;onal variables in our experiment by aliasing with interac;on columns.
• Leave some columns to es;mate residual error for sta;s;cal tests
28
A
B
C
-1 +1 -1
+1
+1
-1
y a bA cB dC= + + +
Trial I A B C AB AC BC ABC 1 + -‐ -‐ -‐ + + + -‐ 2 + + -‐ -‐ -‐ -‐ + + 3 + -‐ + -‐ -‐ + -‐ + 4 + + + -‐ + -‐ -‐ -‐ 5 + -‐ -‐ + + -‐ -‐ + 6 + + -‐ + -‐ + -‐ -‐ 7 + -‐ + + -‐ -‐ + -‐ 8 + + + + + + + +
• Create a half frac;on by running only the ABC = +1 trials • Note confounding between main effects and interac;ons • Compromise: must assume interac;ons are negligible • In this case (not always) design is “saturated” (no df for sta;s;cal
tests).
29
• “I=ABC” for this 23-‐1 half frac;on is called the “Defining Rela;on” • Note that “I=ABC” implies that “A=BC”, “B=AC”, and “C=AB”.
• 3-‐way interac;ons are confounded with the intercept • Main effects are confounded with 2-‐way interac;ons • The number of factors in a defining rela;on is called the “Resolu;on”
• This 23-‐1 half frac;on has resolu;on III • We denote this frac;onal factorial design as 2III3-‐1
30
We like our screening designs to be at least resolu;on IV (I=ABCD)
• I=ABCD for this 24-‐1 half frac;on is called the Defining Rela;on • Note that I=ABCD implies
• A=BCD, B=ACD, C=ABD, and D=ABC. • AB=CD, AC=BD, AD=BC
• Main effects are confounded with 3-‐way interac;ons • Some 2-‐way interac;ons are confounded with others.
31
Number of Factors
2 3 4 5 6 7 8 9 10 11 12 13 14 15
Num
ber o
f Design Po
ints
4 Full III
6 IV
8 Full IV III III III
12 V IV IV III III III III III
16 Full V IV IV IV III III III III III III III
20 III III III III III
24 IV IV IV IV III III III
32 Full VI IV IV IV IV IV IV IV IV IV
48 V V
64 Full VII V IV IV IV IV IV IV IV
96 V V V
128 Full VIII VI V V IV IV IV IV
32
Trial DrugPS Lub%
Disso%
1 10 1 76 2 10 2 98 3 40 1 73 4 40 2 82 5 10 1 84 6 10 2 102 7 40 1 77 8 40 2 88
DrugPS
Lub%
76,84
88,82
73,77
98,102
10 40 1
2
FiKed model is based on averages individual
averageSD
SDnumber of replicates
=
33
Repeated measurement 1 batch
3 measurements per batch
ReplicaKng batch producKon
3 batches 1 measurement
per batch
34
ReplicaKon 1. Every operaKon that
contributes to variaKon is redone with each trial.
2. Measurements are independent.
3. Individual responses are analyzed.
RepeKKon 1. Some operaKons that
contribute variaKon are not redone.
2. Measurements are correlated. 3. The averages of the repeats
should be analyzed (usually).
Trial DrugPS Lub%
Disso%
1 10 1 76 2 10 2 98 3 40 1 73 4 40 2 82 5 10 1 84 6 10 2 102 7 40 1 77 8 40 2 88
Trial DrugPS Lub%
Disso%
1 10 1 76, 84 2 10 2 98, 102 3 40 1 73, 77 4 40 2 82, 88
35
� Frac;onal factorial designs are generally used for “screening”
� Sta;s;cal tests (e.g., t-‐test) are used to “detect” an effect.
� The power of a sta;s;cal test to detect an effect depends on the total number of replicates = (trials/design) x (replicates/trial)
� If our experiment is under powered, we will miss important effects.
� If our experiment is over-‐powered, we will waste resources.
� Prior to experimen;ng, we need to assess the need for replica;on.
36
( )22
11 2N (#points in design)(replicates/point) 4 z zα −β−
σ⎛ ⎞= ≅ + ⎜ ⎟δ⎝ ⎠
• While not exact, this ROT is easy to apply and useful.
• Commercial sotware will have more accurate formulas.
α z1-‐α/2 0.01 2.58 0.05 1.96 0.10 1.65
β z1-‐β 0.1 1.28 0.2 0.85 0.5 0.00
σ = replicate SD δ = size of effect (high – low) to be detected. α = probability of false detec;on β = probability of failure to detect an effect of size δ
2
N 16 σ⎛ ⎞≅ ⎜ ⎟δ⎝ ⎠
37
Disso% WtRSD Replicate SD σ 1.3 0.1
Difference to detect δ 2.0 0.2 False detecKon probability α 0.05 0.05
z1-‐α/2 1.96 1.96 DetecKon failure probability β 0.2 0.2
z1-‐β 0.85 0.85 Required number of trials N 13.3 8
( )22
11 2N (#points in design)(replicates/point) 4 z zα −β−
σ⎛ ⎞= ≅ + ⎜ ⎟δ⎝ ⎠
38
Run A B C D E 1 - - - - + 2 + - - - - 3 - + - - - 4 + + - - + 5 - - + - - 6 + - + - + 7 - + + - + 8 + + + - - 9 - - - + - 10 + - - + + 11 - + - + + 12 + + - + - 13 - - + + + 14 + - + + - 15 - + + + - 16 + + + + +
Confounding Table I = ABCDE A = BCDE B = ACDE C = ABDE D = ABCE E = ABCD AB = CDE AC = BDE AD = BCE AE = BCD BC = ADE BD = ACE BE = ACD CD = ABE CE = ABD DE = ABC
39
� Sta;s;cal test for presence of curvature (lack of fit) � Addi;onal degrees of freedom for sta;s;cal tests
� May be process “target” secngs
� Used as “controls” in sequen;al experiments.
� Spaced out in run order as a check for drit.
40
Complete RandomizaKon: • Is the cornerstone of sta;s;cal analysis • Insures observa;ons are independent • Protects against “lurking variables” • Requires a process (e.g., draw from a hat) • May be costly/ imprac;cal
Restricted RandomizaKon: • “Difficult to change factors (e.g., bath temperature) are “batched” • Analysis requires special approaches (split plot analysis)
Blocking: • Include uncontrollable random variable (e.g., day) in design. • Assume no interac;on between block variable and other factors • Excellent way to reduce varia;on. • Rule of thumb: “Block when you can. Randomize when you can’t block”.
41
42
Confounding Table I = ABCDE Blk = AB = CDE A = BCDE B = ACDE C = ABDE D = ABCE E = ABCD AC = BDE AD = BCE AE = BCD BC = ADE BD = ACE BE = ACD CD = ABE CE = ABD DE = ABC
43
StdOrder RunOrder CenterPt Blocks Disint Drug% Disint% DrugPS Lub% 11 1 1 2 A 5 1.0 10 2.0 13 2 1 2 A 5 4.0 10 1.0 19 3 0 2 A 10 2.5 25 1.5 15 4 1 2 A 5 1.0 40 1.0 18 5 1 2 B 15 4.0 40 2.0 14 6 1 2 B 15 4.0 10 1.0 20 7 0 2 B 10 2.5 25 1.5 16 8 1 2 B 15 1.0 40 1.0 17 9 1 2 A 5 4.0 40 2.0 12 10 1 2 B 15 1.0 10 2.0 9 11 0 1 A 10 2.5 25 1.5 7 12 1 1 B 5 4.0 40 1.0 1 13 1 1 B 5 1.0 10 1.0 2 14 1 1 A 15 1.0 10 1.0 4 15 1 1 A 15 4.0 10 2.0 3 16 1 1 B 5 4.0 10 2.0 10 17 0 1 B 10 2.5 25 1.5 5 18 1 1 B 5 1.0 40 2.0 8 19 1 1 A 15 4.0 40 1.0 6 20 1 1 A 15 1.0 40 2.0
44
RunOrder CenterPt Blocks Disint Drug% Disint% DrugPS Lub% Disso% WtRSD 1 1 2 A 5 1.0 10 2.0 100.4 1.6 2 1 2 A 5 4.0 10 1.0 103.0 2.1 3 0 2 A 10 2.5 25 1.5 88.8 1.6 4 1 2 A 5 1.0 40 1.0 94.3 2.3 5 1 2 B 15 4.0 40 2.0 78.9 1.6 6 1 2 B 15 4.0 10 1.0 102.9 2.0 7 0 2 B 10 2.5 25 1.5 90.9 1.4 8 1 2 B 15 1.0 40 1.0 91.8 2.2 9 1 2 A 5 4.0 40 2.0 76.3 1.4 10 1 2 B 15 1.0 10 2.0 103.4 1.6 11 0 1 A 10 2.5 25 1.5 89.9 1.8 12 1 1 B 5 4.0 40 1.0 91.8 2.2 13 1 1 B 5 1.0 10 1.0 101.2 2.2 14 1 1 A 15 1.0 10 1.0 101.8 2.6 15 1 1 A 15 4.0 10 2.0 102.5 1.4 16 1 1 B 5 4.0 10 2.0 100.3 1.5 17 0 1 B 10 2.5 25 1.5 91.2 1.6 18 1 1 B 5 1.0 40 2.0 76.3 1.3 19 1 1 A 15 4.0 40 1.0 92.4 2.1 20 1 1 A 15 1.0 40 2.0 76.8 1.6
45
46
47
48
49
50
Source DF Adj MS F P Blocks 1 2.21 0.11 0.745 Disint 1 0.30 0.01 0.905 Drug% 1 2.94 0.15 0.707 Disint% 1 0.30 0.01 0.905 DrugPS 1 1174.45 58.93 0.000 Lub% 1 258.61 12.98 0.004 Curvature 1 32.68 1.64 0.225 Res Error 12 19.93
2.179 is the 1-‐α/2 th quan;le of the t-‐distribu;on having 12 df.
51
Source DF Adj MS F P Blocks 1 0.01090 0.51 0.487 Disint 1 0.03751 1.77 0.208 Drug% 1 0.00847 0.40 0.539 Disint% 1 0.08282 3.91 0.071 DrugPS 1 0.00189 0.09 0.770 Lub% 1 2.10586 99.46 0.000 Curvature 1 0.21198 10.01 0.008 Res Error 12 0.02117
52
Disso% • Only DrugPS and Lub% show significant main effects • Plot of Disso% residuals vs predicted Disso% shows systema;c paKern.
• The residual SD (4.5) is considerably larger than expected (1.3) WtRSD • Only Lub% shows a sta;s;cally significant main effect • Curvature is significant for WtRSD Therefore • Only DrugPS and Lub% need to be considered further • The other 3 factors can fixed at nominal levels. • The predic;on model is inadequate. Addi;onal experimenta;on is needed.
53
Disso a b Lub% c DrugPS d Lub% DrugPS= + × + × + × × + ε
DrugPS
Lub%
C
B
D
A
10 40 1
2
Trial DrugPS Lub% Disso% 1 10 1 C 2 10 2 A 3 40 1 D 4 40 2 B
E
F
H G 5 25 1 E 6 25 2 F 7 10 1.5 G 8 40 1.5 H
2 2Disso a b Lub% c DrugPS d Lub% DrugPS e Lub% f DrugPS= + × + × + × × + × + × + ε
I
9 25 1.5 I
54
Respon
se
Factor
55
Factorial or fractional factorial screening design
Response surface design
56
57
• “Cube Oriented” • 3 or 5 levels for each factor In 3 factors
Factorial or FracKonal Factorial
Central Composite Design
+ +
=
Axial Points Center Points
58
59
60
61
Std Run Center Block Disint Drug% Disint% DrugPS Lub% Disso% WtRSD Order Order Point 11 1 1 2 A 5 1.0 10 2.0 100.4 1.6 13 2 1 2 A 5 4.0 10 1.0 103.0 2.1 19 3 0 2 A 10 2.5 25 1.5 88.8 1.6 15 4 1 2 A 5 1.0 40 1.0 94.3 2.3 18 5 1 2 B 15 4.0 40 2.0 78.9 1.6 … 10 17 0 1 B 10 2.5 25 1.5 91.2 1.6 5 18 1 1 B 5 1.0 40 2.0 76.3 1.3 8 19 1 1 A 15 4.0 40 1.0 92.4 2.1 6 20 1 1 A 15 1.0 40 2.0 76.8 1.6 21 21 -‐1 3 A 10 2.5 10 1.5 22 22 -‐1 3 A 10 2.5 40 1.5 23 23 -‐1 3 A 10 2.5 25 1.0 24 24 -‐1 3 A 10 2.5 25 2.0 25 25 0 3 A 10 2.5 25 1.5 26 26 0 3 A 10 2.5 25 1.5
62
Std Run Center Block Disint Drug% Disint% DrugPS Lub% Disso% WtRSD Order Order Point 11 1 1 2 A 5 1.0 10 2.0 100.4 1.6 13 2 1 2 A 5 4.0 10 1.0 103.0 2.1 19 3 0 2 A 10 2.5 25 1.5 88.8 1.6 15 4 1 2 A 5 1.0 40 1.0 94.3 2.3 18 5 1 2 B 15 4.0 40 2.0 78.9 1.6 … 10 17 0 1 B 10 2.5 25 1.5 91.2 1.6 5 18 1 1 B 5 1.0 40 2.0 76.3 1.3 8 19 1 1 A 15 4.0 40 1.0 92.4 2.1 6 20 1 1 A 15 1.0 40 2.0 76.8 1.6 21 21 -‐1 3 A 10 2.5 10 1.5 101.8 1.7 22 22 -‐1 3 A 10 2.5 40 1.5 84.0 1.7 23 23 -‐1 3 A 10 2.5 25 1.0 96.7 2.1 24 24 -‐1 3 A 10 2.5 25 2.0 82.8 1.4 25 25 0 3 A 10 2.5 25 1.5 92.3 1.5 26 26 0 3 A 10 2.5 25 1.5 91.9 1.2
63
64
2 2Y a b DrugPS c Lub% d DrugPS e Lub% f Drug PSLub%= + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ε
65
66
67
68
Source DF Adj SS Adj MS F P Blocks 2 2.27 1.13 0.48 0.625 Regression Linear DrugPS 1 1331.87 1331.87 567.73 0.000 Lub% 1 340.61 340.61 145.19 0.000 Square DrugPS*DrugPS 1 27.39 27.39 11.68 0.003 Lub%*Lub% 1 0.14 0.14 0.06 0.811 Interaction DrugPS*Lub% 1 222.98 222.98 95.05 0.000 Residual Error 18 42.23 2.35 Lack-of-Fit 7 25.15 3.59 2.32 0.103 Pure Error 11 17.07 1.55
69
Source DF Adj SS Adj MS F P Blocks 2 0.02341 0.01171 0.41 0.671 Regression Linear DrugPS 1 0.00118 0.00118 0.04 0.842 Lub% 1 2.31351 2.31351 80.72 0.000 Square DrugPS*DrugPS 1 0.04980 0.04980 1.74 0.204 Lub%*Lub% 1 0.09743 0.09743 3.40 0.082 Interaction DrugPS*Lub% 1 0.00234 0.00234 0.08 0.778 Residual Error 18 0.51589 0.02866 Lack-of-Fit 7 0.28587 0.04084 1.95 0.154 Pure Error 11 0.23003 0.02091
70
StaKsKcal Significance? Model Term Disso% WtRSD
DrugPS P P Lub% P P
DrugPS2 P P Lub%2 ?
DrugPS × Lub% P P Lack of Fit
2 2Y a b DrugPS c Lub% d DrugPS e Lub% f Drug PSLub%= + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ε?
71
• The simplest model that explains the data is best (Occam’s razor, rule of parsimony)
• Eliminate “least significant” terms one at a ;me followed by re-‐analysis
• Always eliminate highest order terms first
• Don’t eliminate lower order terms which are contained in significant higher order terms
• Any exis;ng theory or prior knowledge trumps these rules.
2 2Y a b DrugPS c Lub% d DrugPS e Lub% f Drug PSLub%= + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ε?
72
Estimated Regression Coefficients for Disso% using data in uncoded units Term Coef Constant 105.321 DrugPS -0.478970 Lub% 6.62343 DrugPS*DrugPS 0.0130426 Lub%*Lub% -0.959956 DrugPS*Lub% -0.497745
S = 1.49153 PRESS = 83.4051 R-Sq = 97.76% R-Sq(pred) = 95.79% R-Sq(adj) = 97.20%
73
Estimated Regression Coefficients for WtRSD using data in uncoded units Term Coef Constant 4.66698 DrugPS -0.0293187 Lub% -2.96608 DrugPS*DrugPS 0.000623945 Lub%*Lub% 0.763118 DrugPS*Lub% -0.00161165
S = 0.164211 PRESS = 0.850996 R-Sq = 83.93% R-Sq(pred) = 74.65% R-Sq(adj) = 79.92%
74
Acceptable performance more likely
• Difficult to do with > 2 factors • Does not take into account
• es;ma;on uncertainty • correla;on among responses • variability in control of factor levels
• variability in the underlying true model over ;me
75
76
77
Global Solution DrugPS = 11.2121 Lub% = 1.93939 Predicted Responses Disso% = 100.002 , desirability = 1.000 WtRSD = 1.500 , desirability = 0.117927 Composite Desirability = 0.343404
78
Predicted Response for New Design Points Using Model for Disso% Point Fit SE Fit 95% CI 95% PI 1 100.002 0.621070 (98.7063, 101.297) (96.6316, 103.372) Predicted Response for New Design Points Using Model for WtRSD Point Fit SE Fit 95% CI 95% PI 1 1.49952 0.0683772 (1.35689, 1.64216) (1.12848, 1.87057)
79
1. Number of trials ≥ Number of model coefficients
2. Each coded column adds to 0 (balance)
3. Inner product of any 2 coded columns = 0 (orthogonality)
4. Use resolu;on V (or at least IV) for screening designs 5. Factor ranges are bold (but not too bold) 6. Incorporate process knowledge & sequen;al strategies 7. Assure adequate sample size (power)
8. Randomize processing order
9. Block when you cannot randomize
10. Incorporate tests for model adequacy (e.g., center points)
11. Avoid PARC (Planning Ater Research is Complete)
80
1. Use graphics (picture = 1,000 words) 2. Always verify model assump;ons (normality, independence,
variance homogeneity)
3. In model reduc;on, follow rules of hierarchy tempered by prior process knowledge
4. Use coded factor levels in judging sta;s;cal significance of model coefficients.
5. Consider predic;on uncertainty when iden;fying op;mal factor secngs
6. Take advantage of curvature & interac;ons when choosing op;mal factor secngs
7. Always perform independent trials to confirm predic;ons.
81
Minitab • General purpose stat package • User friendly • Good learning tool JMP • General purpose stat package • Excellent for DOE & SPC • Very advanced features
• Monte-‐Carlo simula;on of DOE models • Good D-‐op;mal design features
• May need sta;s;cal support for some features Design Expert • Exclusive focus on DOE (may want addnl tools) • I have not used but my impression is very good
5
5
10
15
Hard%RSD
MixTime(min)5 7 9 11 13 15MixTime(min)5 7 9
15
20
2.015 17
3.02.5 Water(L)
2.0
3.0
Water(L)
Surface Plot of Hard%RSD
6 11 16
2.0
2.5
3.0
MixTime(min)
Wat
er(L
)
Overlaid Contour Plot of Hardness...Hard%RSD
Hardness
Hard%RSD
19.520.5
07
Lower BoundUpper Bound
White area: feasible region
82
Contour Profiling and overlay for design space idenKficaKon
Monte-‐Carlo SimulaKon to determine effect of poor factor control on future batch failure rate
67
83
• Robust design & Taguchi designs • Mixture (e.g.,gasoline blend) and constrained designs
• D-‐op;mal designs and custom augmenta;on
• Bayesian approaches • Probability of mee;ng specifica;ons • mul;ple correlated responses • incorpora;on of prior knowledge
• Variance component analysis & Gage R&R
• Split-‐plot experiments
84
1. Box, G. E. P.; Hunter, W. G., and Hunter, J. S. (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley and Sons.
2. Montgomery D (2005) Design and analysis of experiments, 6th edition, Wiley.
3. Myers R, Montgomery D, and Anderson-Cook C (2009) Response surface methodology, Wiley.
4. Diamond W (1981) Practical Experiment Designs, Wadsworth, Belmont CA
5. Altan S, et al (2010) Statistical Considerations in Design Space Development (Parts I-III) PharmTech Nov 2, 2010. Available on line at http://www.pharmtech.com/pharmtech/author/authorInfo.jsp?id=53118
6. Conformia CMC-IM Working Group (2008) Pharmaceutical Development case study: “ACE Tablets”. Available from the following web site: http://www.pharmaqbd.com/files/articles/QBD_ACE_Case_History.pdf
7. ICH Expert Working Group (2008) GUIDELINE on PHARMACEUTICAL DEVELOPMENT Q8(R1) Step 4 version dated 13 November 2008
8. ICH Expert Working Group (2005) Guideline on QUALITY RISK MANAGEMENT Q9 Step 4 version dated 9 November 2005
9. FDA CDER/CBER/CVM (November 2008) Draft Guidance for Industry Process Validation: General Principles and Practices (CGMP)
Thank You!!
