short course on bayesian applications to quality-by-design and assay development john peterson,...

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Short Course onBayesian Applications to Quality-by-Design

and Assay Development

John Peterson, Ph.D.Director, Statistical Sciences GroupGlaxoSmithKline Pharmaceuticals

Collegeville, Pennsylvania, USA

Non-Clinical Statistics Conference, Brugge, Belgium, 8 October, 2014

12 [email protected]

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OUTLINE• Some Bayesian preliminaries

• What is QbD? Some Basic Concepts

• The Flaw of Averages vs. Predictive Distributions

• Process Optimization using Bayesian Predictive Distributions

• A Bayesian Approach to ICH Q8 Design Space & Scale-up

• Bayesian Monte Carlo Studies for USP Test Assessment

• Assay Development & Dissolution

• Some Computational Recommendations

• Acknowledgements

• References

3333

The Bayesian Paradigm

Posterior distribution ofunknown model parameters

prior distribution ofunknown model parameters

Bayes’ ruleweighting function

data |

data |

L

L d

x = | datap 1212

44444





data |

data |

L

L d

x = | datap 1212

data*L

555555





*

data |

data

L

L

x = | datap

1212

• A simulation procedure known as “Markov Chain Monte Carlo” (MCMC)can be used to sample from the posterior distribution of unknown model parameters given experimental data and prior information.

• Other computational procedures can sometimes be used such as : numerical integration,generalized direct sampling, Laplace approximation approaches (e.g. INLA).

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Three Practical Benefits of Bayesian Statistics

• Provides a scientific way to incorporate informative prior information. - Probability theory and risk assessment can be used to build informative priors.

• Priors can be used to provide a certain degree of “regularization”. - Priors can be used to induce necessary boundaries (e.g. positive variance components).

- Priors can be used to stabilize parameter estimates (e.g. ridge regression)

- Priors on model forms can be used to obtain “model averaging”, which improves predictions. (This has been successfully applied in data mining competition.)

• Bayesian methodology provides a direct method for getting a predictive distribution. - For complex models (often needed for pharmaceutical process modeling) the frequentist approach may prove difficult here (e.g. nonlinear mixed-effect model).

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0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

• Prior distributions can be used to incorporate prior informationabout a process (in a scientific way ….via Bayes Rule)

- This can be done by using a distribution that maps out the relative possibilities of parameter values.

- Or, a prior can simply designate parameter boundaries. (e.g. a gamma distribution prior for a parameter that must be between positive.)

Prior Distributions

q

s0

shape=0.1rate=0.1

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• Prior distributions can be weak or non-informative e.g. using a flat prior for the mean in a normal-theory model.

• Sometimes priors are used to induce a certain degree of “regularization”.

- For example we may choose a multivariate normal prior for a vector of regression coefficients (intercept omitted) that has a mean vector of zero.

- This will tend to “pull” the posterior a bit towards the zero vector and thereby prevent regression coefficient estimates from becoming too large in situations of multi-collinearity. (Ridge regression can be thought of a Bayesian procedure with respect to a zero-centered prior for the regression coefficients.)

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Prior Distributions

99

• In some cases, care must be used in choosing priors:

(i) For small sample sizes or weak information about a model parameter.

(ii) For informative priors

(iii) Even for non-informative priors in certain situations. For details see: Seaman, J. W. III, et al. (2012). “Hidden Dangers of Specifying Noninformative Priors”, The American Statistician, 66(2), 77-84.

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Prior Distributions

Seaman et al. recommend that if one is interested a particular function of the model parameters, then one should use the chosen prior to observe theinduced prior for that function (e.g. by Monte Carlo simulation).

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Posterior Predictive Distributions

• Predictive distributions is why I became a Bayesian statistician.

• Process optimization and QbD is why I desire predictivedistributions.

Why?

1111




Why?

Because predictive distributions are a very good way to quantify process capability!

121212




Why?

Because predictive distributions are a very good way to quantify process capability!

A process can be optimized, but that does not imply that it is “capable”, i.e. likely to meet specifications.

13131313


Posterior distributionfor model parameters

0

Y e

standard normal distribution

Posterior predictive distribution

1414141414


Posterior distributionfor model parameters

0

Y e

standard normal distribution

Posterior predictive distribution

• Sampling from the posterior predictive distribution, we can then compute

as a measure of process capability. (Here, S is the specification interval.)

Pr dataY S

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What is QbD?

Quality-by-Design (QbD): Quality by Design (QbD) is a concept put forth by Joseph Juran. (See Juran, J. M. (1992) Juran on Quality by Design)

• Juran believed that quality could be designed into products and processes. He called this idea “Quality by Design”

• QbD has been applied in many industries, most notably the automotive industry.

• Recently, the US Food and Drug Administration has adopted QbD principles for drugmanufacture.

• The FDA QbD imperative is outlined in its report “Pharmaceutical Quality for the 21st Century: A Risk-Based Approach.”

• Quote from Juran on Quality by Design: “Organizations that have adopted such methodsof quantification (process capability) have significantly outperformed those which havenot.”

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• Risk is an important concept for the ICH Q8-Q10 Guidances. ICH Q8 - The word ‘risk’ or ‘risk-based’ occurs 35 times. ICH Q9 - The word ‘risk’ or ‘risk-based’ occurs 290 times. ICH Q10 - The word ‘risk’ or ‘risk-based’ occurs 34 times.

The growing importance of process reliability and risk assessment for pharmaceutical manufacturing

• The FDA continues to push its initiative on “Pharmaceutical Quality for the 21st Century”

• The ICH Quality (Q) Guidances are supported by the FDA. - They outline (at a high level) what the pharmaceutical companies should do to improve the quality of the manufacturing of their products.

• Such improved quality strategies are referred to as “Quality by Design” or QbD.

• Currently, QbD is “optional” for Rx sponsors, but there appears to be increasing pressure to incorporate QbD as a strategy in Rx submissions.

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Some Basic Quality Concepts

Process capability – is a measure of the inherent variability of a process measure about its target.

Quality improvement – reduction in variation of a process measure about its target.

Quality improvement can therefore be thought of as an improvementin process capability.

Quality concepts for a single quality measure:

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Process capability – is a measure of the inherent variability of a process measure about its target.

Quality improvement – reduction in variation of a process measure about its target.


Quality concepts for a single quality measure:

• But what about products and processes involving multiple quality criteria?

• What do we do with multiple variance (and covariance) measures? - Add them up? Find some function of a variance-covariance matrix?

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Process capability – is the inherent (multivariate) distribution of the process measures about a (vector) target.

Quality improvement – shrinking of the (multivariate) distribution of the process measures about a (vector) target

Quality concepts for a multiple quality measures:


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Process capability – is the inherent (multivariate) distribution of the process measures about a (vector) target.

Quality improvement – shrinking of the (multivariate) distribution of the process measures about a (vector) target

Quality concepts for a multiple quality measures:


• Generalizing from variances to distributions helps us to think more clearlyabout dealing with single or multiple quality endpoints.

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• A natural process capability index for both univariate and multivariatequality measures is the proportion of the distribution of qualitymeasures that are within the quality specification limits.

• This process capability index can be used as a desirability function for process optimization.

• Mathematically, we can express this as:

where Y is a vector of quality measures, S is a specification region, and x is a vector of process conditions.

• p(x) is a function of all of the means and variance components associatedwith the process.

• The expression for p(x) involves the posterior predictive distribution for Y.


Pr datap S | , x Y x

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Some confusion about process capability• Process capability indices have been criticized as being incapable of capturing all aspects of a process’s capability of meeting specifications because they are a single number.

• It has been argued that “process capability” should be assessed using various aspects of the distribution of a process’s responses, instead of using a single metric (Pignatiello and Ramberg, 1993)

• I suspect that this confusion arises from not recognizing the difference between “exploratory analysis” and “confirmatory analysis”.

• I believe that one should explore the nature of a process’s response distribution, but in the end one must choose some optimization criterion.

• If we are to optimize a process, we should optimize it to be as capable aspossible. Hence, the use of a process capability index makes sense for process optimization (Plante (2001), Jeang (2010) ). But care is required.

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Some Points about Process Capability Assessment• Process Capability Assessment is important. - Just because your process mean is on target does not necessarily imply that your process will meet specifications a high proportion of the time.

• Process Capability Assessment is statistically more difficult than other quality improvement methods, particularly for multiple responses. (This is why statisticians are important here!)

• It is important to have the process in control as much as possible forprocess capability assessment.

• We need to assess process capability in both exploratory and confirmatoryways. Exploratory: - Explore the process means relative to specifications. - Identify and measure process variance components and correlations. - Assess response distribution at various operating conditions. Confirmatory: - Compute process capability index (after process is in control) . - Validate process capability dynamically during SPC phase.

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Common Cause vs. Special Cause Variation Some Basic Quality Concepts

Common Cause Variation: - Common cause variation is the process variation due to “common causes”.

- Common causes are associated with the usual historically known, quantifiable variation of a process. - Typically, it is variation that can be modelled.

Special Cause Variation: - Special cause variation is the process variation due to “special causes” (sometimes referred to as “assignable causes”). - Special causes are typically come in the form of surprises or are outside the historical experience base.

- Some examples of special causes may be: operator error, machine malfunction, a surprisingly poor batch of raw material, or a change in a process condition that was not previously recognized as influential.

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This process capability index as a desirability function Pr datap S | , x Y x

• We assume that Y can be modeled as a (stochastic) function of process factors, x.

• For example:

• Note that

• We can optimize a process (under the above model) by maximizingp(x) with respect to the factors in x.

• p(x) forms a desirability function for process optimization that is easy tointerpret and avoids flaws associated with mean response surface optimization.

where , , 0Y Z x e e ~ N

dataPr data Pr, |p S | , E S | , , x Y x Y x

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Why is a multivariate reliability approach needed? (Accounting for correlation among the responses…a simple example)

• Suppose we have a process with four key responses, Y1, Y2, Y3, Y4

• For simplicity, let’s assume that

• Let

• Consider If S = I , then But if then

and if

, , , ~ ,Y Y Y Y Y N 01 2 3 4

Pr .A 050Y Pr AY

, , , ,A 1 1 1 1

. . .

. . .

. . .

. . .

1 09 09 09

09 1 09 09

09 09 1 09

09 09 09 1

Pr .A Y 075

-0.318812 -0.289382 -0.199902

-0.318812 -0.392855 -0.356863 then Pr .-0.289382 -0.392855 -0.29417

-0.199902 -0.356863 -0.29417

Y A

1

1042

1

1

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The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty by Dr. Sam Savage

The Flaw of Averages vs. Predictive Distributions

Average depth of river is 3 feet.

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• Inferences about population means are rarely sufficient toquantify good quality.

• Recall: Quality improvement – reduction in variation of a process measure about its target.

Or more generally…. Quality improvement – shrinking of the (multivariate) distribution of the process measures about a (vector) target

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• All classical textbooks on response surface methodology state a response surface model for a process in the basic form:

; ,y f x

where: x is a vector of process factors (e.g. temperature, pressure, etc.) and e is a “statistical error” or “random error” term, where E(e)=0.

• In these textbooks it is stated that is a “mechanistic model” with a vector of unknown model parameters b ).

• In cases where the form of is unknown, it is recommended that one approximate by a (second-order) Taylor series.

;f x

;f x ;f x

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What is the truth?: mean models vs. distribution models

• Given that the basic response surface model is represented in classical textbooks as

it is not too surprising that practitioners think (in the back of their minds) that a response surface model is:

observed process response = truth + measured error.

or possibly

observed process response = approx. truth + measured error.

• Since I have seen some authors refer to the“true mean model” with regard to process optimization.

; ,y f x

; ,E y f x

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Consider the following thought experiment…• Suppose you want to optimize a complex manufacturing process.

• Suppose further that you are able to acquire instruments to measure theprocess response that are extremely accurate (producing truly negligible measurement error).

• If you were to keep the process factors constant, but stop and started theprocess several times using different batches of raw materials each time, would you expect to measure the exact same process response values?

NO!

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Consider the following real experiment…Ar

NH

NH2

NH

RAr

NH

NH

R

NO2Substrate Product

0 02exp( / )[ ] / ( ) ,/t t Ha h tT Cat Py t tk E R K

where yt is the measured amount of substrate remaining after time t.

The parameters in red are unknown model parameters and the factors in blue are process conditions of temperature, amount of catalyst, etc.

20 22 24 26 28

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Time (t)

Log

Sub

stra

te• We want to be able to run the experiment long enough to removemost all of the substrate (less than 0.1% say)

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Process modeling: Scientific modeling is the backbone… but is there more?

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0 2 0log[Sub] log[Sub] exp( / )[ ] / ( )t t a H hk E RT Cat P K t t

20 22 24 26 28

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Time (t)

Log

Sub

stra

te

Chemistry equation:

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Process modeling: Scientific modeling is the backbone… but is there more?

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Chemistry equation:

20 22 24 26 28

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Time (t)

Log

Sub

stra

te

Three replicate batches run under the same experimental conditions:

Same chemistry but different profiles!

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0 1 2 2 0log[Sub] log[Sub] exp log exp( / )[ ] / ( )tj t a H h jj j tj jk E RT Cat P K t tb b e

Traditional mechanistic model equation for prediction

Stochastic-mechanistic model equation that models batch-to-batch variationMeasurementrandom error

Batch-to-batchrandom effects

Predictive distribution forrisk assessment

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-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Time (t)

Log

Sub

stra

tePossible drivers:• Reactor is charged with slightly different amounts of raw materials.• Proportion of catalyst is slightly different.• Changes in environmentalconditions.• Subtle operator differences

…

3636363636


0 1 2 2 0log[Sub] log[Sub] exp log exp( / )[ ] / ( )tj t a H h jj j tj jk E RT Cat P K t tb b e

Traditional mechanistic model equation for prediction

Stochastic-mechanistic model equation that models batch-to-batch variationMeasurementrandom error

Batch-to-batchrandom effects

Predictive distribution forrisk assessment

20 22 24 26 28

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Time (t)

Log

Sub

stra

te

Why is variation important to quality improvement ?

Because quality improvementcan be defined as “reduction in variation about a target”

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Y = f(x, z, e | )

Variation due to estimating unknownmodel parameters.

Multivariate predictivedistribution of qualityresponses

Variation due to noisy‘control’ variables, inputraw materials, etc.y1=Percent dissolved

(at 30 min.)

This ‘quality’ responsedistribution is froma poor x-pointin the process.

y2=

fria

bilit

y

99%

85%

60%

80% 100%0%

3%

99%

mean

Multivariate predictivemodel

A process with lowreliability

Probability of meetingboth specifications isabout 0.65

Note that themean is withinspecifications,but this is notgood enough!

Process control values

1,..., rY YYfor r response types.

1,... kx xx

1,..., hz zz

1,..., p


1,... kx xx

1,..., hz zz

1,..., p

Variation due tocommon-causeerror variability

Predictive Distributions and Multiple Response Process Optimization

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Y = f(x, z, e | )




Variation due to noisycontrol variables, inputraw materials, etc.

This quality responsedistribution is froma much better x-pointin the process.


A more reliable process

y2=

fria

bilit

y

99%

85%

60%

80% 100%0%

99%

mean

Probability of meeting bothspecifications is about 0.90


y1=Percent dissolved (at 30 min.)


1,... kx xx

1,..., hz zz

1,..., p


1,... kx xx

1,..., hz zz

1,..., p

3%


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Y = f(x, z, e | q )




Variation due to noisycontrol variables, inputraw materials, etc.

This quality responsedistribution is froma much better x-pointin the process.


A very reliable process

y2=

fria

bilit

y

85%

60%

80% 100%0%

99%

mean

Probability of meeting bothspecifications is well above 0.99 !


y1=Percent dissolved (at 30 min.)

h

3%

1,..., rY YY

1,..., kx xx 1,..., hz zz

1,..., p

Multiple response surface optimization:Beware the “sweet spot” for ICH Q8 Design Space

• The classical textbooks in response surface methodology (RSM)refer to “overlapping mean” response surfaces as a way to optimize processes with multiple response types.

• Popular point & click packages such as Design Expert and SAS/JMP will produce such a “sweet spot” graph with a few clicks of your mouse.

Box & Draper Myers, Montgomery, Anderson-Cook Box, Hunter & Hunter

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ICH Q8 Regulatory Guidance: The “Design Space” Concept

• ICH (International Congress on Harmonization) is an international body of representatives from the US, Europe, and Japan for the purposes of harmonizationof technical requirements for registration of pharmaceuticals for human use.

• The ICH has a set of quality guidelines, all with the prefix Q. ICH Q8 addresses“pharmaceutical development” (i.e. development of the actual pill, liquid, injectible drug product)

• ICH Q8 contains a key concept called “design space”. (This is not a statistical term, but refers to the set of all manufacturing recipes that should produce acceptable product quality results.)

• ICH Q8 defines the “design space” as: The multidimensional combination and interaction of input variables (e.g., materialattributes) and process parameters that have been demonstrated to provide assurance of quality.

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• A “design space” example from the ICH Q8 regulatory guidance document:

&

Contour plot of “dissolution” Contour plot of “friability” Design space (white region)

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• A “design space” example from the ICH Q8 regulatory guidance document:

Specification Region

Dissolution Spec. Limits

Friability spec. limits

75% out of spec.! 50% out of spec.

Process distribution



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Overlapping Mean Design Space Situation with Three Response Types

y1 = tablet disintegration time,y2 = friability, and y3 = hardness

y1y2

y3

Three-dimensional distribution of responses at a point on the boundary of an overlapping means design space.Red points are “out of spec”.

From Peterson, J. and Lief, K. (2010) “The ICH Q8 Definition of Design Space: A Comparison of the Overlapping Means and the Bayesian Predictive Approaches”, Statistics in Biopharmaceutical Research, 2, 249-259.

85% Out of Spec

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For this process, the overlapping meansDesign Space from the Design Expert package harbored process configurations with a low probability of meeting all three quality specifications.The worst probability was about 15%.

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ExampleNumber of factorsfor the DS

Number of responses for the DS

Minimum posterior probability of acceptance

Maximum posterior probability of acceptance

1 4 3 0.15 0.842 4 4 0.23 0.723 3 3 0.34 14 3* 2 0.26 0.745 4 3 0.11 0.336 3 4 0.11 0.84

* Mixture experiment. Factors sum to 1.From Peterson, J. and Lief, K. (2010) “The ICH Q8 Definition of Design Space: A Comparison of the Overlapping Means and the Bayesian Predictive Approaches”, Statistics in Biopharmaceutical Research, 2, 249-259.

Examples from Six Different (Real Data) Experiments

Results Corresponding to Overlapping Mean Design Spaces

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A Distribution Approach to Process Optimization1. Build your process model taking into account all key sources of variation

(e.g. batch-to-batch, measurement error)

2. Design an experiment to gather data that will enable you to compute aposterior distribution for the unknown model parameters.

3. Using the posterior distribution, compute a posterior predictive distribution for your process.

4. Compute the probability of meeting specifications (or some other utility measure) as a function of process factors.For example: Optimize

5. Optimize the process using the utility measure in 4. above.

Pr | ,data , where is a specification region.p S S x Y x

(Here, Y is a vector of response types and x is a vector of process factors)

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Example: An intermediate stage of a multi-stage route of manufacture for anActive Pharmaceutical Ingredient (API).

Measurements:• Four controllable quality factors (x’s) were used in a designed experiment. (x1=‘catalyst’, x2= ‘temperature’, x3=‘pressure’, x4=‘run time’.)

• A (face centered) Central Composite Design (CCD) was employed. (It was a Full Factorial + axial points + 6 center points [30 runs])

• Four quality-related response variables, Y ’s, were measured. (These were three side products and purity measure for the final API.)

Y1= ‘Starting material Isomer’, Y2=‘Product Isomer’, Y3=‘Impurity #1 Level’, Y4=‘Overall Purity measure’

• Quality Specification limits: Y1<=0.15%, Y2<=2%, Y3<=3.5%, Y4>=95%.

Multidimensional Acceptance region,

Overlapping Means vs. Bayesian Reliability Approach to Design Space:

An Example – due to Greg Stockdale, GSK.

[0,0.0015] [0,0.02] [0,0.035] [0.95,1]A

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Overlapping Means vs. Bayesian Reliability Approach to Design Space:

An Example – due to Greg Stockdale, GSK.

Prediction Models: Model Terms

Response x1 x2 x3 x4 x11 x22 x33 x44 x12 x13 x14 x23 x24 x34

SM Isomer D D D D D D D D

Prod Isomer D D D D D

Impurity D D D

Purity D D D

Temperature = x1 Pressure = x2 Catalyst Amount = x3 Reaction time = x4

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Design Space Table of Computed Reliabilities1

for the API (sorted by joint probability)

TempPressur

eCatalys

tRxntim

eJoint Prob

SM Isomer Prob

Prod Isomer Prob

Impurity

ProbPurity Prob

35 60 6 3 0.752 1 0.9985 0.8435 0.79

32.5 60 7 3 0.743 1 0.9995 0.7875 0.8295

37.5 60 6 3 0.7375 0.9995 0.9995 0.7855 0.8255

32.5 60 6.5 3 0.737 1 0.9975 0.821 0.7845

30 60 7.5 3 0.7335 1 0.9995 0.7775 0.8175

37.5 60 6.5 3 0.725 1 1 0.7485 0.845

35 60 6.5 3 0.7225 1 1 0.77 0.812

32.5 60 6 3 0.7195 1 0.9955 0.864 0.7415

30 60 7 3 0.717 1 0.999 0.8075 0.759

32.5 60 7.5 3 0.716 1 1 0.734 0.859

37.5 60 5.5 3 0.7145 1 0.993 0.8065 0.7565

35 60 7 3 0.712 1 1 0.731 0.8555

[1] This is only a small portion of the Monte Carlo output.

Optimal Reaction Conditions

Marginal Probabilities

Note that the largest probability of meeting specifications is only about 0.75

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Posterior Predicted Reliability with

Temp=20 to 70, Catalyst=2 to 12, Pressure=60, Rxntime=3.0

Catalyst

Tem

p

20

30

40

50

60

70

2 4 6 8 10 12

PressureRxntime

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Design-Expert® SoftwareOriginal ScaleOverlay Plot

ConversionSM IsomerProd IsomerImpurityPAR

Design Points

X1 = C: CatalystX2 = A: Temperature

Actual FactorsB: Pressure = 60.00D: Rxntime = 3.00

2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

55.00

60.00

65.00

70.00Overlay Plot

C: Catalyst

A: T

em

pera

ture

SM Isomer: 0.0015

Prod Isomer: 0.02

Impurity : 0.035

PAR: 0.95

Overlapping Mean Contours from analysis of each response individually.

such that Prob( is in | , ) 0.7Y A data x x

= Design SpaceContour plot of p(x) equal toProb (Y is in Agiven x & data).

The region inside thered ellipse is thedesign space. x1=

x2=

But this x-point (in the yellow “sweet spot”)has a probability of only 0.23 !

This x-point (in the yellow sweet spot)has only a probability of 0.75 .

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How was this model fit?

Prediction Models:

Model Terms

Response x1 x2 x3 x4 x11 x22 x33 x44 x12 x13 x14 x23 x24 x34

SM Isomer D D D D D D D D

Prod Isomer D D D D D

Impurity D D D

Purity D D D

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How was this model fit? … Using a SUR ModelSUR=Seemingly Unrelated Regressions

1 1 1 1 1 1 1 1 12 2 21 1 2 3 4 2 3 4 2 3 10 1 2 3 4 22 33 44 23

2 2 2 2 2 222 1 3 4 3 1 3 20 1 3 4 33 13

3 3 3 33 1 3 1 3 30 1 3 13

4 4 4 4 24 1 3 1 40 1 3 11

i i i i i i i i i i i

i i i i i i i i

i i i i i i

i i i i i

Y x x x x x x x x x e

Y x x x x x x e

Y x x x x e

Y x x x e

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Prediction Models:

Prediction Models in Matrix Form:

, Y X e

1i ,...,N

11

1

41

4

N

N

Y

Y

Y

Y

Y

9

4

1

4

N

N

0

0

X

X

X

1

4

11

1

41

4

N

N

e

e

e

e

e

~ N , 0e I

5353

Fitting the SUR Model

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Prediction Models in Matrix Form: , Y X e ~ N , 0e I

11 1ˆ

n n

X I X X I Y

1

1ˆ ˆ ˆˆ , whereN

j jj

e eN

The frequentist fit:

1ˆ ˆ ˆ,..., ,

ˆ ˆ , and 1,.., response types.

j j rj

ij ij ij i

e e e

e y i r

x

545454

Fitting the SUR Model

54

Prediction Models in Matrix Form: , Y X e ~ N , 0e I

11 1ˆ

n n

X I X X I Y

The frequentist fit:

These two formsstrongly suggest Gibbs samplingfor a Bayesian analysis!

1

1ˆ ˆ ˆˆ , whereN

j jj

e eN

1ˆ ˆ ˆ,..., ,

ˆ ˆ , and 1,.., response types.

j j rj

ij ij ij i

e e e

e y i r

x

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A Bayesian Analysis for the SUR Model

ˆ~ ,Inverse Wishart N N

Consider a non-informative prior approach:

1 /2, , where is the number of response types.r r

11ˆ~ , NN

X I X

Gibbs sampling:

Notes: See John Geweke’s book: Contemporary Bayesian Econometrics and Statisticsfor details.The rsurGibbs function in the bayesm R package uses Gibbs sampling for the SUR model. A similar analysis can also be done with BUGS with weak or informative priors.

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How do We Sample Posterior Predictive Valuesfrom the SUR Model?

11 ½ 0

r p pr

new new r, ~ N ,

Y X I

½where is the "square root" matrix of

also known as the Cholesky decomposition matrix.

Posterior distribution of

,

57

A Preposterior Analysis

• Suppose that is not large enough.

• The spread of the posterior predictive distribution depends upon both the process variation and the model parameter uncertainty.

• If the sample size for the experiment was small, it may be that the model parameter uncertainty is playing a significant factor in the size of

• By “simulating” additional data from the model, one can assess how muchdata is likely to be needed to substantially reduce the model parameteruncertainty and thereby increase

max Pr ,dataS

x

Y xE

Pr ,data .SY x

max Pr ,data .S

x

Y xE

Experimental region

5858

A Preposterior Analysis for the SUR Model

ˆ~ ,Inverse Wishart N N

11ˆ~ , NN

X I X

Gibbs sampling: Increase the rows of X to correspond to additional experimental runs.

Increase N to correspond to the additional rowsof X.

Note: In some cases, parametric bootstrap simulations will be needed to generatethe additional data. Simulating from the posterior predictive distribution may notwork! For details see: See Peterson, J. J., Miró-Quesada, G., and del Castillo, E. (2009), “A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models”, Quality Technology and Quality Management, 6(4), 353-369.

59

Design Space Scale-upfor the Maeda et al. (2012) Study

Process: Lubricant Blending process for theophylline tablets.

Optimization factors: x1=Froude number, x2=blending time

Responses: Y1=“compression rate of powder mixture” Y2=“tablet hardness (N)” Y3=“dissolution percentage at 30 min.”

Specification limits: Y1<=0.32, Y2>=30, Y3>=75

60


Small-scale design: 32 factorial.

Large-scale design: three replications at x1=0.36, x2=21 plus experimental runs at: x1=0.20, x2=2 x1=0.40, x2=58

61




Small-scale Model: Seemingly Unrelated Regressions (SUR) model.

1 1 1 11 1 2 1 2 10 1 2 12

2 2 2 2 22 1 2 2 20 1 2 22

3 3 3 3 3 32 23 1 2 1 2 1 2 30 1 2 12 11 22

Y x x x x e

Y x x x e

Y x x x x x x e

1

2

3

~ ,

e

e N

e

0 S

62


62



Small-scale Model: Seemingly Unrelated Regressions (SUR) model.

1 1 1 11 1 1 1 1 2 1 2 10 1 2 12

2 2 2 2 22 2 2 2 1 2 2 20 1 2 22

3 3 3 3 3 32 23 3 3 3 1 2 1 2 1 2 30 1 2 12 11 22

;

;

;

Y e x x x x e

Y e x x x e

Y e x x x x x x e

x

x

x

1

2

3

~ ,

e

e N

e

0 S

63


63



Large-scale Model: A SUR-like model

1

2

3

~ ,

e

e N

e

0 S

1 1 11 1 10 1

2 2 22 2 20 1

3 3 33 3 30 1

;

;

;

Y e

Y e

Y e

x

x

x

0

j is a additive adjustment coefficient.

1

j is a multiplicative adjustment coefficient.

j = 1,2,3

6464


64



Large-scale Model: A Reduced SUR-like model

1

2

3

~ ,

e

e N

e

0 S

1 11 1 11

2 22 2 21

3 3 33 3 30 1

;

;

;

Y e

Y e

Y e

x

x

x

0

j is a additive adjustment coefficient.

1

j is a multiplicative adjustment coefficient.

j = 1,2,3

656565


65

1 2 3, ,Y Y YY

; , , Pr , , ,p x Y S | x S is the specification region.

1

1Pr | ,data Pr | , ,

N i i i

iS S

N Y x Y x

simulated from the posterior

N = 1,000 simulations

Design Space : Pr | ,dataS R x Y x

for some reliability level R.

66666666


66

Data from the Small-scale & Large-scale Experiments:

X1 X2 Response TypesFroude no. Blending Time Y1 Y2 Y3 0.10 2 0.340 71.7 90.2 0.10 30 0.320 51.6 85.0 0.10 58 0.289 37.8 81.1 0.25 2 0.345 67.3 89.1 0.25 30 0.286 39.4 81.1 0.25 58 0.242 24.2 75.6 0.40 2 0.339 60.3 86.3 0.40 30 0.255 31.0 77.4 0.40 58 0.240 21.9 67.8 0.36 21 0.286 25.1 75.3 0.36 21 0.277 29.7 75.8 0.36 21 0.270 27.5 77.2 0.10 2 0.340 53.8 85.0 0.40 58 0.236 18.7 67.8

Large-scale data

Small-scale data

Specification limits: Y1<=0.32 Y2>=30 Y3>=75

6767676767


67

Data from the Small-scale & Large-scale Experiments:

X1 X2 Response TypesFroude no. Blending Time Y1 Y2 Y3 0.10 2 0.340 71.7 90.2 0.10 30 0.320 51.6 85.0 0.10 58 0.289 37.8 81.1 0.25 2 0.345 67.3 89.1 0.25 30 0.286 39.4 81.1 0.25 58 0.242 24.2 75.6 0.40 2 0.339 60.3 86.3 0.40 30 0.255 31.0 77.4 0.40 58 0.240 21.9 67.8 0.36 21 0.286 25.1 75.3 0.36 21 0.277 29.7 75.8 0.36 21 0.270 27.5 77.2 0.10 2 0.340 53.8 85.0 0.40 58 0.236 18.7 67.8

Large-scale data

Small-scale data


0.10 2 0.340 71.7 90.2 0.10 2 0.340 53.8 85.0 0.40 58 0.240 21.9 67.8 0.40 58 0.236 18.7 67.8

Small-scale dataLarge-scale data

Small-scale dataLarge-scale data

68


Model fits for small-scale experiments:

21 1 2 1 2

2 22 1 2 2

2 2 23 1 2 1 2 1 2

0 3 0 02 0 04 0 01 , 92 =0.81

40 7 7 98 19 2 + 6 53 , 98 =0.29

81 6 4 13 6 85 0 63 + 0 52 2 35 99 8 =0.99

y . . x . x . x x R %, p

y . - . x - . x . x R %, p

y . - . x - . x - . x x . x - . x , R . %, p

( -values are for the Wilks-Shapiro normality test on residuals.) p

Multivariate Wilks-Shapiro normality test on SUR residuals: p=0.014

1 2 3 4 5

12

34

5

Beta quantiles

Ma

ha

lan

ob

is D

2

Multivariate normal scores plot

eq1

-3 -1 1

-0.0

200.

005

-3-1

1

eq2

-0.020 0.000 -0.4 0.2 0.6

-0.4

0.2

eq3

Scatter plot matrix of SUR residuals

No outliers detected.

6969


Model fits for small-scale to large-scale analysis:

21 1

22 2

23 3

0 96 89

0 70 92

12 0 0 80 96

SS

SS

SS

ˆ ˆy . y , R %

ˆ ˆy . y , R %

ˆ ˆy . . y , R %

No outliers detected.

SS=“small-scale”

Large-scaleprediction

Large-scaleDesign Points X1 X2 Y1 (Y1-pred.) Y2 (Y2-pred.) Y3 (Y3-pred.) 0.36 21 0.270 (0.285) 27.5 (29.8) 77.2 (76.8) 0.10 2 0.340 (0.335) 53.8 (51.4) 85.0 (84.2) 0.40 58 0.236 (0.217) 18.7 (13.5) 67.8 (66.6)


70707070


70

1 2 3, ,Y Y YY

; , , Pr , , ,p x Y S | x S is the specification region.

1

1Pr | ,data Pr | , ,

N i i i

iS S

N Y x Y x

simulated from the posterior

N = 1,000 simulations

Design Space : Pr | ,dataS R x Y x

for some reliability level R.

71

Small Scale Design Space Contours

Froude number

Ble

nd

ing

Tim

e (

min

)

10

20

30

40

50

0.15 0.20 0.25 0.30 0.35

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

Large Scale Design Space Contours

Froude number

Ble

nd

ing

Tim

e (

min

)10

20

30

40

50

0.15 0.20 0.25 0.30 0.35

0.10.2

0.3

0.3

0.4

0.4

0.5

0.5

0.60.60.7

0.7

0.8

= 3 points= small-scale point = large-scale point

Contour Plots of Pr | ,dataSY x

Both small & large scale information used

0.92 probability

0.80 probability

72

The Posterior Predictive Approach Easily Solves the Multivariate Robust Parameter Design Problem

7272

.

and use p(x1,…,xk) to optimize the process.

See Miró-Quesada, del Castillo, Peterson (2004), Journal of Applied Statistics

for details.

1

1 ,..., 1 1Let ,..., Pr | ,..., , ,..., ,

k nk X X k k n

p x x E A x x X X

Y x = y

• Sometimes we have process factors that we can control in a lab setting but are noisy in a manufacturing plant.

• If such noise factors interact with other completely controllable factors, we may be able to dampen the effects of the noise factors to improve the probability of meeting specifications.

• Suppose where x1,…, xk are control factors and Xk+1,…, Xn are (random) noise factors.

1 1,..., , ,...,

k k nx x X X

x

7373

Advantages of the Multivariate Bayesian Predictive Approachto Process Optimization and Design Space

1. It provides a quantifiable assessment of process capability for a single operating target or a region of process operating conditions.

2. It considers the uncertainty of all of the model parameters.

3. It models the correlations among the response types.

4. It is easy to add noise variables.

5. It allows for a “preposterior” analysis to see how much further data wouldreduce the model parameter uncertainty.

6. It can be easily adapted to special desirability or cost functions.

7. It has Bayesian “flexibility” - Informative prior information can be used if desired. - Bayesian regularization can be used (e.g. to induce model stability or parameter constraints) - Predictive distributions for complex (e.g. nonlinear mixed effects) models can obtained.

74

Some Cautions on the Multivariate Bayesian Predictive Approach to Process Optimization and Design Space

1. Process capability surface is statistically more difficult to estimate than a mean response surface. - By its definition, process capability is sensitive to distributional assumptions. - Of course, this is not a reason to avoid inference for process capability.

2. In most all cases, process capability cannot be quantified to depend additionally on special cause variation. - For example, the “probability of meeting specification” in most cases cannotbe easily modeled to take into account issues related to machine failure,unexpected contamination in raw materials, mistakes by workers, etc.

- As such, the overall probability of being out of spec. will be somewhat largerthan that based upon the common cause variation of the process.

- Nonetheless, actions can be taken to lessen the extent of special cause variation (better machine maintenance, better raw material acquisition & screening, better staff training).

75

How Can We Deal with these Cautions?

1. Use good experimental design.

2. Use sensitivity analyses for regression models, not just priors. - If you can, fit more than one model form and compare the results.

3. Make sure clients and managers understand the difference between common cause and special cause variation….particularly if the

estimated process capability is large.

4. If the estimated process capability is too small, first consider a “preposterior” analysis, particularly for experiments with smallsample sizes. - It may be that by simulating additional data, one can show that the process capability can be increased to acceptable levels by additional experiments.

- If additional data still does not increase process capability far enough, further process variation reduction and/or mean improvement may be needed.

76

Some Future Challenges for Process Optimization using a Predictive Distribution Approach

• Processes with many response types and associatedspecification limits. - In biopharmaceutical manufacturing it is common to have about a dozen or more different response types, each with their own specification limits that need to be met. - Often, too little experimental runs will be available with which to check distributional assumptions in a clear way. - Current research underway involving use of a special desirability function to reduce the dimensionality of the problem.

• Processes that have many latent variables. - Modeling of raw material properties can help with building more realistic predictive models for pharmaceutical manufacturing.

- However, it is still “early days” for Bayesian predictive modeling using these types of models.

7777

Some Future Challenges for Process Optimization using a Predictive Distribution Approach

• Functional responses - Some manufacturing processes produces a functional trace or profile as a o measured response.

del Castillo, E., Colosimo, B. M., and Alshraided, H., (2012), “Bayesian Modeling and Optimization of Functional Responses Affected by Noise Factors”, Journal of Quality Technology 44, 117-135.

7878

Summary• Classical response surface methodology (based on means) has had a profound influence on process optimization…but more needs to be done!

• The “more” involves thinking about processes as entities which producedistributions of results over time.

• In other words, process optimization needs to be a distribution oriented endeavor.

• “Quality improvement” has been defined in a nutshell as “the reduction in variation about a target”. This can be generalized to “the shrinking of a multivariate distribution about a vector of quality target values”.

• The Bayesian posterior predictive distribution can be an effective tool for quality improvement, particularly for processes that involve complex modeling.

12

12

79

Some Further Applications

80

Bayesian Monte Carlo Studies for USP Test Assessment• Many USP tests involve multi-stage algorithms. - Here it is not obvious what the probability of acceptance or rejection would be for processes with assumed means and variance components (within and between batch variances).

8181

Bayesian Monte Carlo Studies for USP Test Assessment• Many USP tests involve multi-stage algorithms. - Here it is not obvious what the probability of acceptance or rejection would be for processes with assumed means and variance components (within and between batch variances).

• Suppose we have b batches of data

- Let A(X) = 1 if the batch is accepted, 0 if rejected., where A is a USP data algorithm.

- Assumed model:

- Consider the reliability measure:

- Based upon data, X, from several batches we can compute a posterior distribution

for to make process qualification decisions about a process.

- For example we could compute the posterior probability

111 1 1 bn b n bX ,...,X ,..., X ,...,X . X

1b e b er , , Pr A , , X

b er , ,

0 95b ePr r , , . x

2 20 0ij j ij j ij j b ij eX e b e , b ~ N , , e ~ N ,

batch 1,……...…..…, batch b

8282

Bayesian Monte Carlo Studies for USP Test Assessment

For further details see: LeBlond , D. and Mockus, L. (2014) “The Posterior Probability of Passing a CompendialStandard, Part 1: Uniformity of Dosage Units”, Statistics in Biopharmaceutical Research,DOI: 10.1080/19466315.2014.928231

• Using Monte Carlo simulations to generate data from given numbers of batches, we can determine the number of batches (of a given size) needed to have a specified degree of certainty for process qualification. - This determination would make use of for simulated values of x.

• A less conservative approach could involve the use of instead of

Note:

• Of course, prior information about the product quality endpoint (content uniformity, dissolution, etc.) can be used to possibly reduce the number of batches needed for process qualification.

0 95b ePr r , , . x

1Pr A X x

0 95b ePr r , , . x 1b e, , | b ePr A E r , , xX x

8383

Assay Development

• Assessing Assay Parallelism over a Pre-specified Interval , [xL, xU] - The reference standard and test sample are similar if they contain the same effective constituent.

- Under such condition, the test sample behaves as a dilution (or concentration) of the reference standard. - Graphically, this line is a copy of shifted by a fixed amount on the log-concentration axis.

- In other words, there exists a number such that for all x.

- This calibration constant is commonly known as log-relative potency of the test sample.

StandardTest

log10 Concentration

Ass

ay

Res

po

nse

xL xU

848484

Assay Development

• Assessing Assay Parallelism over a Pre-specified Interval , [xL, xU]

-If two lines are ”close” to parallel over a finite interval, then thereshould exist a shift that will bring the lines close together over that interval.

- As such, the criterion below could be used to quantify assayparallelism in a practical way, overa finite interval.

1 2; ;

L Ux x ,xmin max f x f x

85858585

Assay Development


1 2 1 2; ; 2

L Ux x ,xU Lmin max f x f x D D x x

1 1 1 2 2 2- Note: If and , then f x; C D x f x; C D x

- Therefore, one can assess assay parallelism (for linear assay profiles) by computing

- For sigmoidal assay profiles, one can assess parallelism by computing

1 2Pr 2 dataU LD D / x x

1 2Pr ; ; data

L Ux x ,xmin max f x f x

For details see: Novick, S. J., Yang, H., and Peterson, J. J., (2012) “A Bayesian Approach to Parallelism in Testing in Bioassay”. Statistics in Biopharmaceutical Research, 4(4) 357-374.

868686

Assay Development


- Note, for sigmoidal assay profiles, some people might prefer to assess differences after a logistic transformation has been done so that comparisons will be between linear forms.

10

Consider 1 10

This implies: = log

Dx Cy A B A / .

y* B y y A Dx C

1 2So do we compute Pr 2 transformed data ??U LD D / x x

87878787

Assay Development



10

Consider 1 10

This implies: = log

Dx Cy A B A / .

y* B y y A Dx C

But the transformed data depends upon the (unknown) parameters A and B !


8888888888

Assay Development



10

Consider 1 10

This implies: = log

Dx Cy A B A / .

y* B y y A Dx C

But the transformed data depends upon the (unknown) parameters A and B !

This does not stop the Bayesian approach!


898989898989

Assay Development



10

Consider 1 10

This implies: = log

Dx Cy A B A / .

y* B y y A Dx C

1 1 2 2 - 1 2 1 1 2 2Pr 2 *-dataA ,B ,A ,B |y data U LE D D / x x y A ,B ,A ,B

Compute instead:

9090

Assay Development

• Assay ruggedness

- Bayesian design space modeling ideas can be used here as well.

- Suppose one had several factors, say x1, x2,…, x8 , in a “screening design”over a small (robustness) factor region, X.

- Suppose also that one had one (or more) process response types:

- One could compute Pr dataR min S

x

YX

X

1 rY ,...,YY

For further details see: Peterson, J. J. and Yahyah, M., (2009) "A Bayesian Design Space Approach to Robustness and System Suitability for Pharmaceutical Assays and Other Processes", Statistics in Biopharmaceutical Research 1(4), 441-449.

Specificationregion

If is sufficiently large we can expect a rugged process over R .X X

9191

Assay Validation

• The “closeness” of a result X to the unknown true value of the sample, T, is simultaneously linked to both the size of the bias and precision of the method.

The “total error concept” in analytical method validation: A Bayesian perspective.

• Suppose we have an assay response, X, and a true (gold standard) value, T.

• Suppose further that

• Note:

• Consider

• This is equivalent to

2X ~ N ,

data, |E Pr X T , R

dataPr X T R

For further details see: Boulanger, B. et al. (2007), “Risk management for analytical methods based on the total error concept: Conciliating the objectives of the pre-study and in-study validation phases”,Chemometrics and Intelligent Laboratory Systems 86, 198–207.

and does not gaurantee that T Var X X -T ,

92

• The f2 statistic is commonly used to make decisions about equivalence of referenceand test batches.

Dissolution

222 10

2 1

100 1f 50 , where =

1

p

j R. j T . jj

log w Y Yp

D D

• Consider the dissolution data for a test and referencebatch:

11 1 1 11 1 1 and R Rn R k Rnp T Tn T k TnpY ,...,Y ,..., Y ,...,Y Y ,...,Y ,..., Y ,...,Y

th reference% dissolved for tablet at time point RijY i j

th test% dissolved for tablet at time point TijY i j

Reference Reference Test Test

9393

Dissolution

• The f2 statistic is commonly used to make decisions about equivalence of reference and test batches.

• F2 has been proposed as a population-based analogue of f2.

222 10

2 1

100 1f 50 , where =

1

p

j R. j T . jj

log w Y Yp

D D

222 10

2 1

100 1F 50 , where =

1

p

i R i Tii

log wp

949494

Dissolution

Guidance for Industry: SUPAC-MR: Modified Release Solid Oral Dosage Forms. Scale-Up and Postapproval Changes: Chemistry, Manufacturing and Controls; In Vitro Dissolution Testing and In Vivo Bioequivalence Documentation.

“An f2 value between 50 and 100 suggests the two dissolution profiles are similar. Also, theaverage difference at any dissolution sampling time point should not be greater than 15% betweenthe changed drug product and the biobatch or marketed batch (unchanged drug product)dissolution profiles.” 15i ,R i,T

imax Y Y %

f2=52.3

16%

So we might want to consider:

2 102

22

1

100F 50 15

1

1where =

Ri Tii

p

i R i Tii

log max % ,a

p

nd

w

95959595

Dissolution

f2=52.3

16%

2 102

100F 50 15

1Ri Ti

ilo andg max %

D

However, frequentist inference related to

could be difficult to implement.

But, if we have the posterior for the model parameters, the computation of

is straightforward (at least from a Monte Carlo approach).

2Pr 50 15 dataRi Tii

andF max % D

22

1

1Recall: =

p

i R i Tii

wp

96

Some Computational Recommendations

• Generally, the most difficult part of a Bayesian analysisis computing the posterior distribution of model parameters.

• Posterior predictive distributions and associated risk probabilities are more straightforward to compute.

• As a statistical consultant, it is advisable to have one (ormore) backup strategies for statistical computing.

• Sooner or later, WinBUGS or PROC MCMC will fail to produce adequate results!

9797


• “Access” to other Bayesian applications or specialty Rpackages can be useful.

• In addition to the software, “access” can mean - Working knowledge of the software. or - Access to someone who is knowledgeable and willing to help to help you get up to speed quickly.

• But it can be useful to know, and often use, 2 or 3 different Bayesian applications. - MCMC algorithms can be delicate and it may be prudent tocheck any important results using a different algorithm.

9898


Some alternative Bayesian applications to WinBUGS, jags, orPROC MCMC in SAS:

• STAN - Uses the ‘No-U-Turn Sampler’ algorithm. - Generally faster convergence for analysis situations where BUGS will have problems. - Can be invoked in R using the RStan package.

• Generalized Direct Sampling (GDS) - Does not use MCMC, but rather a sophisticated rejection algorithm. - GDS produces independent samples from the posterior! - Requires finding the posterior mode. - There is a R package for this: bayesGDS

999999


Some alternative Bayesian applications to WinBUGS, jags, or PROC MCMC in SAS:

• The R package, MCMCglmm - glmm stands for “generalized linear mixed-effect models”

- It can also sample from the posterior for multivariate normal mixed-effect models.

- It can sample from the posterior for binary, multinomial, and Poisson models.

100

Acknowledgements

• Steven Novick• Harry Yang• Stan Altan• David LeBlond• Yan Shen• Bruno Boulanger• Mohammad Yahyah • Kevin Lief

101

Some Useful Bayesian Books Christensen, R., Johnson, W., Branscum, A., and Hanson, T. E. (2011). Bayesian Ideas and Data Analysis: An Introduction for Scientists and Statisticians, Chapman & Hall, CRC, Boca Raton, FL. del Castillo, E. (2007), Process Optimization - A Statistical Approach, Springer, New York, NY. (Chapters 11 & 12 discuss Bayesian analysis for process optimization)

Krusche, J. K. (2010), Doing Bayesian Data Analysis: A Tutorial with R and BUGS, Academic Press, Oxford, UK.

Lunn, D., Jackson, C., Best, N., Thomas, A., Spiegelhalter, D., (2013), The BUGS Book – A Practical Introduction to Bayesian Analysis, CRC Press, Boca Raton, FL.

Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS, John Wiley and Sons, Inc. Hoboken, NJ. Gellman, A. and Hill, J, (2006), Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press, NY, NY.

102

Concerned about becoming a Bayesian?

You might consider reading the recent book:

The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy by Sharon Bertsch McGrayne.

103

Bernardo, J. M. and Irony, T. Z. (1996), "A General Multivariate Bayesian Process Capability Index", The Statistician, 45, 487-502.

del Castillo, E. (2007), Process Optimization - A Statistical Approach, Springer, New York, NY.(This is a good book for Bayesian process optimization)

del Castillo, E., Colosimo, B. M., and Alshraided, H., (2012), “Bayesian Modeling and Optimization of Functional Responses Affected by Noise Factors”, Journal of Quality Technology 44, 117-135.

Duncan, A. J. (1986). Quality Control and Industrial Statistics. Irwin, Homewood, IL.

Flaig, J. J. (1999), “Process Capability Sensitivity Analysis”, Quality Engineering, 11(4),587 - 592

Fu, Z., Leighton, J., Cheng, A., Appelbaum, E., and Aon, J. C. (2012) “Optimization of a Saccharomyces cerevisiae fermentation process for production of a therapeutic recombinant protein using a multivariate Bayesian approach”, Biotechnology Progress (to appear).

Gelman, A. and Hill, J., (2007) Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press, New York, NY. (This is a good book for Bayesian mixed effect modeling.)

References

104104

References (continued)Jeang, A. (2010), “Optimal Process Capability Analysis”, International Journal of Production Research, 48(4), 957-989.

LeBlond , D. and Mockus, L. (2014) “The Posterior Probability of Passing a CompendialStandard, Part 1: Uniformity of Dosage Units”, Statistics in Biopharmaceutical Research,DOI: 10.1080/19466315.2014.928231

Maeda, J., Suzuki, T., and Takayamab, K. (2012) “Novel Method for Constructing a Large-ScaleDesign Space in Lubrication Process by Using Bayesian Estimation Based on the Reliability of a Scale-Up Rule”, Chem. Pharm. Bull. 60(9) 1155–1163.

Miró-Quesada, G., del Castillo, E., and Peterson, J.J., (2004) "A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise variables", Journal of Applied Statistics, 31, 251-270

Montgomery, D. C. (2009), Introduction to Statistical Quality Control (6th ed), John Wiley & Sons,Inc., Hoboken, NJ.

Ng, S. H. (2010), “A Bayesian Model-Averaging Approach for Multiple-Response Optimization”, Journal of Quality Technology, 42, 52-68.

Peterson, J. J., (2004) "A Posterior Predictive Approach to Multiple Response Surface Optimization", Journal of Quality Technology, 36, 139-153.

105

References (continued)

Peterson, J. J. (2006), "A Review of Bayesian Reliability Approaches to Multiple Response Surface Optimization", Chapter 12 of Bayesian Statistics for Process Monitoring, Control, and Optimization , pp 269-290, (eds. Colosimo, B. M and del Castillo, E.) Chapman and Hall/CRCPress Inc.

Peterson, J. J. (2007), “A Review of Bayesian Reliability Approaches to Multiple Response Surface Optimization”, in Bayesian Process Monitorng , Control & Optimization, Chapman & Hall/CRC.

Peterson, J. J. (2008), “A Bayesian Approach to the ICH Q8 Definition of Design Space”, Journal of Biopharmaceutical Statistics, 18, 959-975.

Peterson, J. J., Miró-Quesada, G., and del Castillo, E. (2009), “A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models”, Quality Technology and Quality Management, 6(4), 353-369.

Peterson, J. J. (2009) “What Your ICH Q8 Design Space Needs: A Multivariate Predictive Distribution”, Pharmaceutical Manufacturing, 8(10) 23-28.

Peterson, J. J. and Lief, K. (2010) "The ICH Q8 Definition of Design Space: A Comparison of theOverlapping Means and the Bayesian Predictive Approaches", Statistics in Biopharmaceutical Research, 2,249-259.

106

References (continued)Plante, R. D. (2001), “Process Capability: A Criterion for Optimizing Multiple Response Product and Process Design”, IIE Transactions, 33, 497-509.

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Rajagopal, R., del Castillo, E., Peterson, J. J. (2005), "Model and Distribution-Robust Process Optimization with Noise Factors", Journal of Quality Technology 37, 210-222.

Rajagopal, R. and del Castillo, E. (2006), “A Bayesian approach for multiple criteria decision making with applications in Design for Six Sigma”, Journal of the Operational Research Society, 1–12.

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Stockdale, G. and Cheng, Aili (2009), “Finding Design Space and a Reliable OperatingRegion using a Multivariate Bayesian Approach with Experimental Design”, Quality Technology and Quantitative Management 6(4),391-408

References (continued)

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