1. Shiny Tools for Sample Size Calculation in Process Performance Qualification of Large Molecules Qianqiu (Jenny) Li, Bill Pikounis May 24, 2017

2. Tools & Methods: RiskBinom Attribute Sampling Plans
For binary variables (e.g., pass/fail)
Inputs:
AQL & Producer’s Risk Rate
RQL(or LTPD) & Consumer’s Risk Rate
Others (e.g., AOQL - not used by RiskBinom yet)
Key Components: sample size and acceptance number (rejection number is needed also for multiple sampling plans)
Assumptions (e.g., “random” samples; samples are independent and come from the same distribution)

3. Tools & Methods: RiskBinom Attribute Sampling Plans
The lot is good
The lot is bad
The lot is accepted based on the sample
Correct Decision
Confidence=1−𝛼
Incorrect Decision
Consumer’s Risk
(Type II Error 𝛽)
The lot is rejected based on the sample
Producer’s Risk
(Type I Error 𝛼)
𝑃𝑜𝑤𝑒𝑟=1−𝛽
Acceptance Quality Limit (AQL): from Z1.4, “The AQL is the quality level that is the worst tolerable process average when a continuing series of lots is submitted for acceptance sampling.”
Rejection Quality Limit (RQL): or LTPD (Lot Tolerable Percent Defective); the smallest unacceptable lot defect rate.
Accept
Reject
Defect Rate
100% 0%
AQL
RQL

4. Tools & Methods: RiskBinom Why to Develop RiskBinom
Available attribute sampling plan standards, R resources (acc.samp, Planesmuestra, etc) are not flexible:
No RQL plans or the RQL plan requires lot size (in Planesmuestra)
require other inputs such as inspection level, etc
RiskBinom is useful when dichotomizing continuous or ordinal values to binary: =0 if within Acceptance range; =1 otherwise
MINITAB requires both AQL and RQL

5. Tools & Methods: RiskBinom Capability of RiskBinom
Sampling Plan Creation:
Single sampling plans (to-do: double/multiple)
AQL, RQL or AQL/RQL sampling plans
Adjusting sampling plans based on historical data
flexible inputs & constraint on maximum # samples

6. Tools & Methods: RiskBinom Capability of RiskBinom
Sampling Plan Evaluation:
Single/double/multiple sampling plans
AQL, RQL or AQL/RQL sampling plans
Allowing that different batches have different # of samples and different acceptance numbers

7. Tools & Methods: RiskBinom
Producer’s and Consumer’s Risk Rates in Single Sample Plans
Name
Type
Definition
Producer’s
Risk
Probability
Without Historical Data
𝑃𝑟 𝑋>𝐴 | 𝑝≤𝐴𝑄𝐿
With Historical Data: Frequentist
𝑃𝑟 𝑋>𝐴 | 𝑝≤𝑚𝑖𝑛 𝑈 𝑝 , 𝐴𝑄𝐿
e.g., 𝑈 𝑝 is the 95% upper confidence limit
With Historical Data: Bayesian
𝑃𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟 𝑚𝑒𝑎𝑛 𝑜𝑓
Consumer’s
𝑃𝑟 𝑋≤𝐴 | 𝑝≥𝑅𝑄𝐿
𝑃𝑟 𝑋≤𝐴 | 𝑝≥𝑚𝑎𝑥 𝐿 𝑝 , 𝑅𝑄𝐿
e.g., 𝐿 𝑝 is the 95% lower confidence limit
𝑃𝑟 𝑋>𝐴 | 𝑝≤𝑚𝑖𝑛 𝑈 𝑝 , 𝐴𝑄𝐿
1-pbinom(1,10,0.01) # [1]
1-pbinom(1,10,0.005) # [1]
𝑃𝑟 𝑋≤𝐴 | 𝑝≥𝑚𝑎𝑥 𝐿 𝑝 , 𝑅𝑄𝐿
pbinom(0,20,0.1) # [1]
pbinom(0,20,0.2) # [1]

8. Tools & Methods: RiskBinom
Prior & Posterior Distributions of Failure Probability
𝑋 𝑖 : number of Failed samples in i-th batch with 𝑛 𝑖 samples: 𝑋 𝑖 ~𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑛 𝑖 ,𝑝 , 𝑥 𝑖 ′ 𝑠 𝑎𝑟𝑒 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖=1,⋯,𝐵
Likelihood Function: 𝑖=1 𝐵 𝑝 𝑥 𝑖 1−𝑝 𝑛 𝑖 − 𝑥 𝑖 = 𝑝 𝑖=1 𝐵 𝑥 𝑖 1−𝑝 𝑖=1 𝐵 𝑛 𝑖 − 𝑥 𝑖
Prior 𝑝~𝐵𝑒𝑡𝑎 𝛼,𝛽 : PDF is 𝑢 𝛼−1 1−𝑢 𝛽−1 𝑑𝑢 𝑝 𝛼−1 1−𝑝 𝛽−1
Posterior: ∝ 𝑝 𝛼−1 1−𝑝 𝛽−1 𝑝 𝑖=1 𝐵 𝑥 𝑖 1−𝑝 𝑖=1 𝐵 𝑛 𝑖 − 𝑥 𝑖
Beta 𝛼+ 𝑖=1 𝐵 𝑥 𝑖 ,𝛽+ 𝑖=1 𝐵 𝑛 𝑖 − 𝑥 𝑖

9. Tools & Methods: RiskBinom Priors of Failure Probability
Jeffrey’s Prior: 𝜶=𝜷=𝟎.𝟓
Empirical Prior: 𝛼 & 𝛽 satisfy the equations below
Scenario
Equations for 𝛼 & 𝛽
With one historical batch
𝑥 𝑛 = 𝐸 𝑝 ≈ 𝛼 𝛼+𝛽
𝑉𝑎𝑟 𝑝 = 𝛼𝛽 𝛼+𝛽 2 𝛼+𝛽+1 ⇒ 𝑉𝑎𝑟 𝑝 ≈ 𝐸 𝑝 1− 𝐸 𝑝 𝛼+𝛽+1
With two or more historical batches
𝐸 𝑝 = 1 𝐵 𝑖=1 𝐵 𝑥 𝑖 𝑛 𝑖 ≈ 𝛼 𝛼+𝛽 𝑠 2 ≈ 1 𝐵 𝑖=1 𝐵 𝑉𝑎𝑟 𝑋 𝑖 𝑛 𝑖
Where 𝑠 2 =Var 𝑥 1 𝑛 1 ,⋯, 𝑥 𝐵 𝑛 𝐵 & 𝑉𝑎𝑟 𝑋 𝑖 𝑛 𝑖 is a function of 𝐸 𝑝 & 𝑛 𝑖
If the above equations give negative 𝛼 or 𝛽, then use 𝑉𝑎𝑟 𝑋 𝑖 𝑛 𝑖

10. Q/A Thanks!

11. Acknowledgements Ken Hinds & other key contributors of DS-VAL-68021
MTASS colleagues
Sirukumab & Stelara PPQ teams

12. References
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W. G. Howe (1969). Two-Sided Tolerance Limits for Normal Populations, Some Improvements, Journal of the American Statistical Association, 64,
K. Krishnamoorthy, X. Lian (2011). Closed-Form Approximation Tolerance Intervals for Some General Linear Models and Comparison Studies, Journal of Statistical Computation and Simulation, iFirst, 1-17
K. Krishnamoorthy, T. Mathew (2004). One-Sided Tolerance Limits in Balanced and Unbalanced One-Way Random Models Based on Generalized Confidence Intervals, Technometrics, 46(1), 44-52
K. Krishnamoorthy, T. Mathew (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons
R. W. Mee (1984). Beta-expectation and Beta-content tolerance limits for balanced one-way ANOVA random model. Technometrics, 26,

13. References
R. W. Mee, D. B. Owen (1983). Improved Factors for One-Sided Tolerance Limits for Balanced One-Way ANOVA Random Model, Journal of the American Statistical Association, 78:384,
M. G. Natrella (1963). Experimental Statistics, NBS Handbook 91, US Department of Commerce
DS-VAL-68021: Position Paper to Address Statistically Based Sampling Plan for Process Validation Stage 2 (Process Performance Qualification)
W. M. Bolstad, J. M. Curran (2016). Introduction to Bayesian Statistics. 
B. Puza (2015). Bayesian Methods for Statistical Analysis, ANU eView. 
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