Practical (Real-life)Implementation of Definitive Screening … · 2019-03-29 · Full factorials...
Transcript of Practical (Real-life)Implementation of Definitive Screening … · 2019-03-29 · Full factorials...
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Practical (Real-life) Implementation of Definitive Screening Designs Made Easy(ish)
Dr Paul Nelson
Prism Training & Consultancy Ltd
2018
The Plight of the Flight of the Phoenix
2© Prism Training & Consultancy Ltd
“Essentially, all models are wrong, but some are useful”
George BoxBrad Jones Chris NachtsheimDesign-Ease & then Design-Expert
the early years
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3© Prism Training & Consultancy Ltd
in the game
Design-Ease & then Design-Expert the early years
Yates, Plackett& Burman 1940s – 50s
What are the practicalities of…
• Implementing DSDs vs Classical sequential (SCO) strategy in real-life
• Managing their Analysis – handling multiple (likely correlated) responses, when the design possesses a desirable combinatorial structure
• Managing their supplementation/augmentation – be (pro)activefor new DSDs, or (re)active for existing DSDs, when it comes to potentially active factors…
• Managing their augmentation – to block or not to block, centre points & randomisation
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Standing on the shoulders of Giants
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I’m not saying I’ve (re)solved these practicalities in the context of DSDs and their supple/augmentation, but hopefully raised some interesting research ideas and challenges
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Choose a Design or Strategy
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Characterization
including Verification
Screening
Optimisation
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Customer Requirements / Criteria
Identify the vital few factors (culprits), from 6 suspects, causing an unexpected 30% drop in yield and propose alternative ranges likely to provide a robust optimum capable of routinely meeting customer specifications or QTPP
Solvent, Reagent, AcidCurrent process
Purity after isolation 99.9%
Early Stage Development
NC
O CO2H
NC
O CO2Me
Solution yield (by HPLC) 97a/a%
Example 1: Esterification of an Acid to Robustly Maximise Quality & Productivity
Screening: part of a sequential SCO strategy
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2k-p Fractional Factorial Screening Design:
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Fractionation & Resolution
Full factorials (white – no aliases). All possible combinations of factor levels are run. Provides information on all effects.
Resolution III (Red Warning) Designs: Main Linear Effects (MLEs) aliased with 2-factor interactions (2-FIs).
Resolution IV (Amber Caution) Designs: MLEs aliased with 3-FIs & 2-FIs are aliased with other 2-FIs.
Resolution V (Green Safe) Designs: MLEs are aliased with 4-FIs & 2-FIs with 3-FIs.
Alternative Minimum-Run Designs
Caution: partial aliasing – more on this later
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26-0 Full Factorial Design:
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Full Factorial
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26-1 Fractional Factorial Design:
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Half Fraction: Resolution V
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26-2 Fractional Factorial Design:
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Quarter Fraction: Resolution IV
Aliased Terms[A] = A + BCE + DEF [B] = B + ACE + CDF[C] = C + ABE + BDF [D] = D + AEF + BCF[E] = E + ABC + ADF [F] = F + ADE + BCD[AB] = AB + CE [AC] = AC + BE[AD] = AD + EF [AE] = AE + BC + DF[AF] = AF + DE [BD] = BD + CF[BF] = BF + CD
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Design-Expert® Software
Product
Error estimates
Shapiro-Wilk test
W-value = 0.989
p-value = 0.950
A: Solvent
B: Water Spike
C: Reagent
D: Acid
E: Temperature
F: Time
Positive Effects
Negative Effects
0.00 2.95 5.90 8.86 11.81
0
10
20
30
50
70
80
90
95
Half-Normal Plot
|Standardized Effect|
Hal
f-N
orm
al %
Pro
bab
ility
D-Acid
E-Temperature
F-Time
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26-3 Fractional Factorial Design:
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Eighth Fraction: Resolution III
Aliased Terms[A] = A + BD + CE + BEF + CDF[B] = B + AD + CF + AEF + CDE[C] = C + AE + BF + ADF + BDE[D] = D + AB + EF + ACF + BCE[E] = E + AC + DF + ABF + BCD[F] = F + BC + DE + ABE + ACD[AF] = AF + BE + CD + ABC + ADE
+ BDF + CEF
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Sequential Assembly of Fractions – Characterization
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Resolving ambiguities with a 2nd augmented foldover fraction
Fold over Res III designs to decouple ME’s from 2FI’s – fold over pairs
Check Alias List: Res IV design
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Sequential Assembly of Fractions – Characterization
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Resolving ambiguities with a 2nd augmented Semi-fold fraction
Semi fold over Res IV designs to decouple 2FI’s
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Sequential Assembly of Fractions – Characterization
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Resolving ambiguities with a 2nd optimal augmentation
Augment, build or repair using Alphabetic-optimal design – save resources
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Testing, Recognising & Handling Curvature
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Screening & Characterisation studies identified key effects to be D, F and DF interaction…
but with strong evidence of curvature – suggests design is in the region of an optimum:
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Optimization & RSM Designs
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Optimisation
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Added Bonus of Propagation of Error
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Risk Assess: part of a sequential SCO strategy
Risk Assessed Design Space
Knowledge Space
Control Space
Control Space
“Parameter movements that occur within the design space are not subjected to regulatory notification. However, movement out of the design space is considered to be a change and would normally initiate a regulatory post approval change process.”
Optimisation
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19© Prism Training & Consultancy Ltd
Resources: part of a sequential SCO strategy
36 Rxs!
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Alternative Minimum-Run Characterization Designs
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Resolution V for characterizing the vital few factors in more depth
Test for curvature but not estimable
Main effects not independent
but small correlation (±0.091)
2FI correlations
negligible (±0.183)
(±0.084& 0.099)
1 intercept + 6 MLEs + 15 2FIs = 22 runs + 2 CPs = 24 runs
Compared to 26-1 or 32 runsVI
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Alternative Minimum-Run Screening Designs
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Resolution IV for screening the factors to identify the vital few
Allowing for botched runs
0.1 – 0.55 .73 & .75
-0.1
–-0
.55
-.7
3&
-.7
5
2FI correlations
not all negligible
Main effects not independent
but orthogonal to 2FIs
Test for curvature but not estimable
(2x6)+2
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What more do you want from Screening?
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Small/minimum number of runs (2k… +1, for the intercept, at a minimum)
Main effects orthogonal & independent of potentially important 2nd order effects, such as 2FI’s & Quadratic terms. But, it’s a screening design!
Partial aliasing of 2nd order terms (not completely aliased) –small, ideally negligible – better than Res IV… don’t forget, it is supposed to be a screening design!
Drop/ignore unimportant factors – DSD retains properties, but greater power to detect effects of the retained factors (synonymous with fractional factorial design projective properties) – later… adding Fake Factors
Project on to any 3 active factors to enable fitting of a full second order RSM model, since 3 settings per factor… the definitive in Definitive Screening Design!
If Only...
Design
≥18 factors, a full RSM model is estimable for any 4 factors ≥24 factors, a full RSM model is estimable for any 5 factors
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Definitive Screening Design Structure
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6 foldoverpairs
& a centre point run
Model-oriented Designs “Design for the experiment, don't experiment for the design”
Jones & Nachtsheim Quality Technology, 43(1), 1-15, 2011
Near-minimal run (2K+1) DSD with 3 (not 2) levels
Design
0
0
±0.25
±0.5
±0.4655
+0.133
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Definitive Screening Design Structure
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Building a general minimum run DSD
A minimum run DSD is constructed using a kxk conference matrix C, such that
A C6 6x6 conference matrix
k foldover runs
k runs
centre point
k even
= 2k+1 runs
… k odd, C (k+1) matrix used with last column deleted
K
KK
k odd, 2*(k+1) + 1 or 2k+3 runs
Categorical factors, 2 centre runs: k even 2k+2, K odd 2k+4 runs
CTC=(k−1)Ik×k
To construct a DSD with more than the minimal number of runs, use a conference matrix with c > k columns and do not assign the last c – k (fake factor) columns to factors… later
Design
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What more do you want from Screening?
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Full 2nd order Response Surface Model (RSM) consists of:An intercept (1 of those)All main effects (=k for k factors)All main quadratic effects (=k)All 2-FIs (=k(k-1)/2)
#terms in a full RSM 1 + 2k + k(k-1)/2 or (k+1)(k+2)
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Ability to fit a full RSM model involving any 3 active factors
Minimum design is saturated by ML & Q effects. If you include the 2FI’s… supersaturated. Only estimate at most 13 model terms in the case of k=6 minimum run DSD and we need an estimate of σ. Is a simple analysis possible?
k=6 full (RSM) model consists of:1 intercept + 6 MLE’s + 6 QE’s + 15 2FI’s= 28 model terms
Simpler analysis when: Effects are >> σ & hereditableNo active 2FIs (eek! unlikely), so strong 2nd order terms don’t inflate σ or bias other important 2nd order effects… Sparsity of effects – # active factors → # effects ≤ ½ # runs > n/2, automated selection procedures tend to struggle
Definitive Screening
Analysis – aren’t we are asking a lot
Useful Effects Model Main Linear Effects Model
MLEs orthogonal/unbiased by 2nd order terms, but σ for a MLEs only model inflated by strong 2nd order terms. Separating the vital few difficult and the reliability of the coefficient SEs questionable
Can be Challenging. DSDs are supersaturated: # model terms > # runs e.g., k=6, # terms 28 > # runs 13: estimate at most 13 model terms & need to estimate σ.
Insufficient df to estimate full RSM model for 4 or more factors as well as σ. DSDs rely on sparsity, size of effects relative to noise, hierarchy & heredity.
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Definitive Screening Design Analysis
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(Multi-)Univariate or BORAT Analysis
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Definitive Screening Design Analysis
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# active effects < n/2 – just a few 2nd order effects – otherwise automated model selection procedures tend to struggle. We cannot fit a majority of the 28 terms
Power low to detect moderate 2nd order effects. Effects must be large. If many terms appear, or are likely to appear active… augment, or supplement
Anticipates the region might be ‘curvy’ so a RS model should be investigated
Previously Recommended Analysis Procedure: specify a RS model and use forward variable selection with stopping based on the AICc criterion to find out which effects are active and which are not
(Multi-)Univariate Analysis
Definitive Screening
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Definitive Screening Design Analysis
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Not designed to be a 1-step RSM
BUT can happen if ‘true’ set of active effects (6 here) <1/2 #Runs (13), so has gained in popularity
DSDs with more than 5 factors project onto any 3 factors to allow fitting the full quadratic model
(Multi-)Univariate Analysis
A problem of partial aliasing
Can fit full RSM model for any 3 active factors
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Definitive Screening Design Analysis
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New Recommended Analysis Procedure: use DSD foldover structure & correlation properties – odd/even 1st / 2nd order effects are orthogonal
Design-oriented Models Jones & Nachtsheim, Technolmetrics, 59(3), 319-329, 2017
Design Oriented (Multi-)Univariate Analysis
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 Response 1 Response 2 Response 3
A:Solvent B:Water SpikeC:Reagent D:Acid E:TemperatureF:Time Product Y_ME Y_2nd
vol %w/w eq eq deg C hour %
0 -1 -1 -1 -1 -1 76.23 -9.78 86.01
0 1 1 1 1 1 95.78 9.78 86.01
-1 0 -1 -1 1 1 95.26 -0.56 95.83
1 0 1 1 -1 -1 96.39 0.56 95.83
-1 -1 0 1 1 -1 97.68 0.91 96.78
1 1 0 -1 -1 1 95.87 -0.91 96.78
-1 -1 1 0 -1 1 94.99 6.25 88.75
1 1 -1 0 1 -1 82.5 -6.25 88.75
-1 1 1 -1 0 -1 73.45 -11.59 85.04
1 -1 -1 1 0 1 96.63 11.59 85.04
-1 1 -1 1 -1 0 98.12 3.27 94.86
1 -1 1 -1 1 0 91.59 -3.27 94.86
0 0 0 0 0 0 95.82 0.00 95.82
6 foldover pairs &
centre point run
Strong heredityWeak heredity
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 Response 1 Response 2 Response 3
A:Solvent B:Water SpikeC:Reagent D:Acid E:TemperatureF:Time Product Y_ME Y_2nd
vol %w/w eq eq deg C hour %
0 -1 -1 -1 -1 -1 76.23 -9.78 86.01
0 1 1 1 1 1 95.78 9.78 86.01
-1 0 -1 -1 1 1 95.26 -0.56 95.83
1 0 1 1 -1 -1 96.39 0.56 95.83
-1 -1 0 1 1 -1 97.68 0.91 96.78
1 1 0 -1 -1 1 95.87 -0.91 96.78
-1 -1 1 0 -1 1 94.99 6.25 88.75
1 1 -1 0 1 -1 82.5 -6.25 88.75
-1 1 1 -1 0 -1 73.45 -11.59 85.04
1 -1 -1 1 0 1 96.63 11.59 85.04
-1 1 -1 1 -1 0 98.12 3.27 94.86
1 -1 1 -1 1 0 91.59 -3.27 94.86
0 0 0 0 0 0 95.82 0.00 95.82
Combined ANOVA
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Definitive Screening Design Analysis
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New Recommended Analysis Procedure: use DSD fold-over structure & correlation properties – odd & even 1st & 2nd order effects are orthogonal
Fit All MLEs excl. Intercept to Y data. Then split into two orthogonal columns…
Predicted Y (YME) and Residual Y or Y – YME (Y2nd) are orthogonal (correlation = 0), so sum = Y
Use Forward Selection & p-value (0.1, 0.05, 0.01) to identify MLEs from YME
Use All Hierarchical & Adj R-Squared on Y2nd to identify 2nd order effects – start by including only 2nd order terms associated with the active MLEs to maintain hierarchy / heredity
Select combined YME & Y2nd model terms for Y to fit the design-oriented RSM model
Note: you can use this 2 response decomposition analysis for any fold-over design
Design Oriented (Multi-)Univariate Analysis
Additional Backward Elimination Step to tidy up
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Multiple Responses & Multivariate Analysis
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Exploiting DSD foldover structure, centring, scaling & correlation of MV data, along with PCA/PCR/PLS orthogonal projection properties:
PCA score & loadings Bi-plot displays correlation between inputs & outputs – correlation between Y’s and of Acid & Time with Y’s clearly evident from 1st PC alone
Model relationship between orthogonal Y PCs & X inputs as previously described, or…
Only 1st PC strong
Correlated responses benefitting from the DSD structure
Fit PLS MLEs models to correlated MV Y data to identify the vital few MLEs
Fit PLS models including All 1st & 2nd order terms associated with active MLEs; to maintain heredity & identify active 2FI & Quadratic effects – PLS orthogonal projection method
DSDs provide a definitive approach to screening in that main effects are not biased by any second-order effect and all quadratic effects are estimable. As screening designs they have many desirable characteristics. In the presence of sparsity in the number of active factors, our designs project to highly efficient response surface designs. Their most appropriate use is in the earliest stages of experimentation when there are a large number of potentially important factors that may affect a response of interest and when the goal is to identify what is generally a much smaller number of highly influential factors. DSDs work best when most of the factors are continuous. That is because each continuous factor has three levels, allowing an investigator to fit a curve rather than a straight line for each continuous factor. However, I want to make it clear that using a DSD is not a panacea. In other words, a DSD is not the solution to every experimental design problem.
DSD Farside
DSDs blah blah a definitive approach to screening blah blah blah blah blah blah blah blah blah blah blah blah blah all quadratic effects are estimable. Blah Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah highly efficient response surface designs. Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blahl blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah allowing an investigator to fit a curve rather than a straight line for each continuous factor. Blah blah blah blah blah blah blah blah blah using a DSD is blah a panacea. Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah.
What Brad Jones says about DSDsWhat is sometimes heard…
32© Prism Training & Consultancy Ltd
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What more do you want from Screening?
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What if >3 factors are active? Return to Design
Be Proactive add Fake Factor columns to improve estimate of σ2
Adds 4 extra runs & 2 error df (2 fold-over pairs).
If many terms are likely to be active / appear active, then Proactively /Reactively… consider DSDs with > the minimal # runs, or augmenting, to estimate 2nd order terms. Useful if you can retain the special DSD foldover pair structure with desirable properties
Analyse YME:
Analyse Y2nd:
Runs Error DF Power Potential MLEs
When in doubt, build it stout
MSq = 0.39/3 = 0.13SD=√(0.39/3) = 0.361
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DSD: Proactive Fake Factor Supplementation
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MSq = 0.39/3 = 0.13SD=√(0.39/3) = 0.361
Analyse Y2nd: 17 rows, but only 9 df (independent values) as each fold-over pair are the same. Intercept accounts for 1 df, so 8 dfleft to estimate 2nd order effectsUse heredity assumption, All Hierarchical & Adj R-Squared on Y2nd
to identify 2nd order effects
Fake factors are orthogonal to MLEs & 2FIs involving the real factors. When deleted, the resulting design is still a DSD, but fake factor df help create an unbiased estimate of σ2 (RMSE) → greater power to detect the effects of the real factors.
Analyse YME: 17 rows, but only 8 df (independent values) as centre point & each fold-over pair sum = 0. Six ‘real’ MLEs, 5 significant, so 8 – 5 = 3 df to estimate σ2
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DSD: Proactive Fake Factor Supplementation
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Term Estimate Forced correct model using DSD data
Intercept 93.32
B Water Spike %w/w(0.2,2) -1.25 Note: MLEs orthogonal to 2nd order terms
D Acid eq(0.1,0.4) 4.82 so are not partially aliased with BF & DF
F Time hour(1,4) 6.04 i.e., there alias coeffs = 0
BF Water Spike %w/w*Time hour 1.59
DF Acid eq*Time hour -4.77
FF Time hour*Time hour -2.46
Term Estimate Fake Factor DSD combined model with increased sensitivity
Intercept 91.297 includes additional MLEs & consequently 2nd order terms
A Solvent vol 0.2686 Aliased Matrix Coefficient BF FF
B Water Spike %w/w -1.251 Solvent vol*Temperature deg C 0.00 0.81
D Acid eq 4.8186 Water Spike %w/w*Acid eq -0.14 -0.81
E Temperature deg C -0.436 Acid eq*Temperature deg C 0.66 -0.81
F Time hour 6.0407
AE Solvent vol*Temperature deg C -1.732 0.00 -1.40 2nd order terms in red have been magnified
BD Water Spike %w/w*Acid eq -1.464 0.21 1.19 due to the partial aliasing, which is the hidden
DE Acid eq*Temperature deg C 2.2918 1.51 -1.86 cost of screening experiments in general
DF Acid eq*Time hour -4.411
SumProduct 1.72 -2.07
BF FF
“Essentially, all models are wrong, but some are useful”
George Box
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DSD: Proactive Fake Factor Supplementation
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Design Oriented Multivariate Modelling & Analysis
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DSD: Proactive Active Factor Supplementation
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Supplement – suspect any 4 or more subsets of active factors & 2nd order effects
Intelligent search for additional fold-over pairs (6 = 12 runs) to help fit a full RSM model involving potentially any subset of 4 active factors for each response. No need to unrealistically expand to ≥18 factors (37+ runs) for a full RSM model is estimable for any 4 factors
Additional fold-over pairs preserve fold-over structure, enabling estimation of the required model using previous methods
OPh
H
+
I
Ph
O
H
Pd(OAc)2 (0.03 eq)TBA-Cl (1.0 eq)NaOAc (1.7 eq)
DMF/60 oC/ 3.5 h
59% conversion56% yield
1.5 equiv
Cacchi et. al. Tetrahedron 1989, 813
Example 2: Improve Conversion of a Heck Reaction
SCO
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DSD: Proactive Active Factor Supplementation
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Simulation Model & Fit DS Model & Analysis
Forced Model Coefficients
RSM Model Coefficients & Plot Default 17 run DSD with Fake Factors
Reasonable Model Agreement, Fit, Makes Practical Sense?
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DSD: Multiple Responses & MVAnalysis
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Round 1: PLS fitting orthogonal MLEs to correlated Ys to identify the vital few MLEs
Orthogonal MLEs & Components – 6 factors. Only 4 VIPs!
Large # correlated responses, but also benefits from the DSD structure
Round 2: Fit PLS models incl. all 1st & 2nd order terms associated with active MLEs Validation possible given FFs & removal of non-significant terms
If only we had the luxury of knowing the model in real-life!
Block Augmented DSD* – assure each block is composed of fold-over pairs &…
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DSD: Proactive Active Factor Supplementation
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Augmented DSD any subset of 4 factors Un-blocked (Completely Randomised)
Main effects model
MLEs & Quad model
Design: when you don’t have luxury of knowing model at outset
Add centre run to second block
*Run original DSD & REACT if any subset of 4 potentially active factors appear after analysis of first 13 runs. This can only happen in the case of Multiple Responses
Additional 6 foldover pairs
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DSD: Proactive Active Factor Supplementation
© Prism Training & Consultancy Ltd
Simulation Model & Fit DS Model & Analysis25 run Augmented DSD any subset of 4 factors
26 Run Block Augmented DSD – assuring each block is composed of fold-over pairs (same data) except additional CP Block 2
Valid randomisation of a block design: randomise the order of the blocks and the units within each block when proactivelysupplementing a DSD.
If reactively augmenting, then original DSD comes first and…
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DSD: Reactive Active Factor Sequential Augmentation
© Prism Training & Consultancy Ltd
Augment for a specific set of 4 factors – you must’ve run the first DSD
Sequential reactive addition of 4 foldover pairs to preserve fold-over structure while enabling estimation of the required model
Intelligent search to identify an additional 4 fold-over pairs (8 runs) to fit a full RSM model for 4 identified active factors
The likelihood is the 2nd block of 8 augmented runs will be performed on a separate occasion, may be with different operators, equipment etc. The two blocks of runs should ideally include a blocking factor.
Another benefit of maintaining the foldover pair structure is that the two blocks are entirely composed of foldover pairs. Assuming fixed blocks, add a centre run to the augmented block to retain the ability to fit all quadratic effects.
Return to Design: we know the specific subset of 4 active factors
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DSD: Reactive Active Factor Sequential Augmentation
© Prism Training & Consultancy Ltd
Simulation Model & Fit DS Model & Analysis21 run Augmented DSD specific subset of 4 factors
22 Run Block Augmented DSD – assuring each block is composed of fold-over pairs (same data) except additional CP Block 2
Straight to Round 2: Fit PLS models incl. all 1st & 2nd order terms associated with the subset of 4 active factors
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DSD: Multiple Responses & MVAnalysis
© Prism Training & Consultancy Ltd
Large # correlated responses, but also benefits from the DSD structure
![Page 23: Practical (Real-life)Implementation of Definitive Screening … · 2019-03-29 · Full factorials (white –no aliases). All possible combinations of factor levels are run. Provides](https://reader033.fdocuments.in/reader033/viewer/2022041600/5e309d3e88d684039741e60d/html5/thumbnails/23.jpg)
Brad Jones
45© Prism Training & Consultancy Ltd
– Renaissance Men– Jimmy Stewart– Frank Towns, Lew & Heinrich
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Where to Go – Blogs and Articles
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