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

1. Outline
Overview
Shiny Tools & Statistical Methods
Q/A

2. Overview: Process Validation (PV) What is process validation?
Process validation is not a one-time milestone event

3. Overview: PV Guidance 1978 CGMP: Validation
2011 FDA PV Guidance (obsolete the 1987 document)
“The sampling plan, including sampling points, number of samples, and the frequency of sampling for each unit operation and attribute. The number of samples should be adequate to provide sufficient statistical confidence of quality both within a batch and between batches.”
“Before any batch from the process is commercially distributed for use by consumers, a manufacturer should have gained a high degree of assurance in the performance of the manufacturing process such that it will consistently produce APIs and drug products meeting those attributes relating to identity, strength, quality, purity, and potency.”

4. Overview: PV Guidance
2014 EU PV Guidance: “sampling plan - where, when and how the samples are taken;”
DS-VAL-68021: Janssen R&D Position Paper to Address Statistically Based Sampling Plan for Process Validation Stage 2 (Process Performance Qualification)
Others:
PDA Tech Report 60
Canadian
WHO
ICH Q7, Q8, Q9, Q10, etc

5. Overview: PPQ Rationale of PPQ Designs/Sampling Plans
To fulfil regulatory expectations: e.g., the FDA PV guidance states “the level of sampling and testing in validation must provide statistical assurance that the process is reproducible and consistently delivers quality products”.
To evaluate process performance: via critical quality attributes (CQAs) using the samples from the intended manufacturing process; including process input parameters in the design/sampling plan;
To gain benefits: Quality, Consumers, Business

6. Overview: PPQ Number of PPQ Batches
Mainly influenced by product knowledge, process and risk understanding, control strategy, feasibility (e.g., production rate), etc.
Statistically supported by
process-specific tolerance intervals for CQAs with acceptance ranges
Batch failure rates via Beta-Binomial Bayesian approach
Others (e.g., process capability indices)
Number of Samples within Batches: Primary Objective of Shiny Tools

7. Overview: Sampling Plan References
ANSI/ASQ Z1.4 & ISO (MLD-STD-105E; AQL attribute sampling plans; single/double/multiple; lot size; inspection level)
ANSI/ASQ Z1.9 (MLD-STD-414; variable sampling plans; AQL: 0.04%~15%; also depend on lot size, etc)
ASTM E122-17:
𝑛= 3 𝜎 0 𝐸 or 𝑛= 3 𝑉 0 𝑒 2 : 𝐸=𝑀𝑎𝑥 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑏𝑙𝑒 𝜇− 𝜇 𝑉 0 = 𝜎 0 𝜇
𝑛= 3 𝜎 0 𝐸 𝑓 : 𝑓 is the DF for 𝜎 0
ASTM E2334 (Binary with 0% observed defect rate; sample size vs upper confidence limit of defect rate)
ASTM E2709 (Lot conformance rate w.r.t. USP dosage uniformity test; maximum tolerable RSD vs average given different sample sizes)
ASTM E2587 (control charts), ASTM E2281 (Cpk, Ppk)
Compendial (e.g., USP <905>, <1010> (𝑛≥ 2 𝜎 2 𝛿 2 ) )

8. Multi-factor historical Results for means & variances
Overview:
Shiny Tools
Data Type?
Continuous
Binary
Multi-factor historical
data?
Others
Consult a statistician
RiskBinom
Yes No
VarCompLM
SSNormTI
Results for means & variances

9. Overview: Shiny Applications
What is Shiny?
An R package for interactive web applications based on R analytics, without need of HTML CSS, or JavaScript knowledge
for updates:
Versions: 0.13~0.14(2016); 1.0(Jan 2017)
R-based Resource
End Users
Shiny App

10. Overview: Shiny Applications
Build up a Shiny App
Preparation:
- RStudio
- R packages (shiny, shinyFiles, ggplot2, xlsx,
readxl, tolerance, nlme, etc)
- Others (e.g., R functions, Packrat, data, images)
ui.R & server.R (Prior to Shiny 0.10; ui.R: the user interface defines the structuring part of the app; server.R: makes the app get its content)
or App.R (with Shiny ~; including ui.R & server.R; at last row: shinyApp(ui=ui,server=server))
Shiny app (on a Shiny server or Rstudio Connect; run from Rstudio, a URL, Gists on Github, in the cloud Shinyapps.io)

11. Overview: Shiny Applications

12. Overview: Shiny Applications
Example App.R
ui <- shinyUI(pageWithSidebar(
headerPanel("Hello Shiny!"),
sidebarPanel(
sliderInput("nobs", "Number of Data Points:",
min = 0, max = 1000, value = 500)),
mainPanel( plotOutput("distPlot")) # plot of simulation data ))
server <- shinyServer(function(input,output) {
output$distPlot <- renderPlot({
dist <- rnorm(input$nobs)
hist(dist) }) })
shinyApp(ui=ui, server=server)
Reference:

13. Overview: SSNormTI For sampling plans of continuous CQAs via normal tolerance intervals

14. Overview: SSNormTI Specified Not Specified
1-Sided Batch TIs: 𝑌 𝑜𝑟 − 𝐾 1 𝜎 𝐸
Exact, Natrella (1963)
2-Sided Batch TIs: 𝑌 .. ± 𝐾 2 𝜎 𝐸
Close to Exact for 𝐾 2 : using Eq (2.3.4) in Krishnamoorthy & Mathew (2009)
Howe (1969)
Not Specified
Batch TIs are used to calculate sample sizes via
DIR approach
FW approach
YZGO approach
1-Sided Process TIs
Mee & Owen (1983)
Hoffman (2010)
Krishnamoorthy & Mathew (2004)
2-Sided Process TIs
Mee (1984)
Hoffman & Kringle (2005)
Krishnamoorthy & Lian (2011)

15. Overview: VarCompLM For prior information in sampling plans of continuous CQAs

16. Overview: RISKBinom For sampling plans of binary CQAs via consumer’s & producer’s risk rates

17. Tools & Methods: SSNormTI
For a CQA with Acceptance Range: taking the Three-Step Procedure as recommended by DS-VAL-68021
criticality analysis report: to help define coverage probability and confidence level of the tolerance intervals
prior knowledge: to help specify distribution & parameter estimates
numbers of samples: e.g., minimum n to control normal tolerance intervals within the acceptance range
For a CQA without Acceptance Range: to use statistical criteria (e.g., estimation accuracy); via Faulkenberry & Weeks approach (1968); to consider other CQAs (e.g., sampled simultaneously)

18. Tools & Methods: SSNormTI EMS & EMS-Based Variances under One-Way Random Effects Models
𝒀 𝒊𝒌 is the k-th observation from the i-th level of A (e.g., batch)
𝜇 is the overall mean
With equal variances for 𝐴 𝑖 and 𝐸 𝑘 𝑖 , across all 𝑖 𝑎𝑛𝑑 𝑘,
𝑨 𝒊 ~𝑁 0, 𝜎 𝐴 𝑖=1,⋯, 𝐼 𝑬 𝒌 𝒊 ~𝑁 0, 𝜎 𝐸 𝑘=1,⋯, 𝐾 𝑖
𝒀 .. = 1 𝑁 𝑖=1 𝐼 𝑘=1 𝐾 𝑖 𝑌 𝑖𝑘 𝒀 𝒊. = 1 𝐾 𝑖 𝑘=1 𝐾 𝑖 𝑌 𝑖𝑘 𝒏 𝟎 = 𝑁 2 − 𝑖=1 𝐼 𝐾 𝑖 2 𝑁 𝐼−1 𝑵= 𝑖=1 𝐼 𝐾 𝑖

19. Tools & Methods: SSNormTI EMS & EMS-Based Variances under One-Way Random Effects Models
Source DF
Sum of Squares (SS)
Mean Squares
(MS)
EMS
Based on Balanced Data (Scenario for PPQ sample size calculation)
A (or Batch)
𝑓 𝐴 =𝐼−1
𝑆𝑆𝐴=𝐾 𝑖=1 𝐼 𝑌 𝑖. − 𝑌
𝑀𝑆𝐴= 𝑆𝑆𝐴 𝐼−1
𝜔 𝐴 2 =
𝜎 𝐸 2 +𝐾 𝜎 𝐴 2
Residual
𝑓 𝐸 =𝐼 𝐾−1
𝑆𝑆𝐸= 𝑖=1 𝐼 𝑘=1 𝐾 𝑌 𝑖𝑘 − 𝑌 𝑖. 2
𝑀𝑆𝐸= 𝑆𝑆𝐸 𝐼 𝐾−1
𝜔 𝐸 2 = 𝜎 𝐸 2
Corrected
Total
𝑓 𝑇 =𝐼𝐾−1
𝑆𝑆𝑇= 𝑖=1 𝐼 𝑘=1 𝐾 𝑌 𝑖𝑘 − 𝑌
Based on Unbalanced Data A
𝑆𝑆𝐴= 𝑖=1 𝐼 𝑘=1 𝐾 𝑖 𝑌 𝑖. − 𝑌
𝜎 𝐸 2 + 𝑛 0 𝜎 𝐴 2
𝑓 𝐸 =𝑁−𝐼
𝑆𝑆𝐸= 𝑖=1 𝐼 𝑘=1 𝐾 𝑖 𝑌 𝑖𝑘 − 𝑌 𝑖. 2
𝑀𝑆𝐸= 𝑆𝑆𝐸 𝑁−𝐼
𝑓 𝑇 =𝑁−1
𝑆𝑆𝑇= 𝑖=1 𝐼 𝑘=1 𝐾 𝑖 𝑌 𝑖𝑘 − 𝑌
𝝈 𝑨 𝟐 and 𝝈 𝑬 𝟐 are the EMS-based between- and within- batch variances; they can be estimated by letting
𝑴𝑺≈𝑬𝑴𝑺

20. Tools & Methods: SSNormTI Batch Tolerance Intervals
One-Sided Tolerance Intervals: 𝑌 𝑜𝑟 − 𝐾 1 𝜎 𝐸
Exact: 𝐾 1 = 𝑡 1−𝛼,𝐾−1 𝑍 𝑝 𝐾 𝐾
Natrella (1963): 𝐾 1 = 𝑍 𝑝 + 𝑍 𝑝 2 − 1− 𝑍 1−𝛼 2 2 𝐾−1 𝑍 𝑝 2 − 𝑍 1−𝛼 2 𝐾 1− 𝑍 1−𝛼 2 2 𝐾−1
Two-Sided Tolerance Intervals: 𝑌 .. ± 𝐾 2 𝜎 𝐸
Close to Exact: 𝐾 2 is the solution of integral equation (2.3.4) in Krishnamoorthy & Mathew (2009)
Howe (1969): 𝐾 2 = 𝐾−1 1+1/𝐾 𝑍 1+𝑝 𝜒 𝛼,𝐾−1 2

21. Tools & Methods: SSNormTI
Process Tolerance Intervals under One-Way Random Effects Models
One-Sided
Tolerance Intervals
Two-Sided
Mee & Owen (1983)
Mee (1984)
Hoffman (2010)
Hoffman & Kringle (2005)
Krishnamoorthy & Mathew (2004)
Krishnamoorthy & Lian (2011)
All of the above TIs depends on:
Number of batches (I)
Number of samples within batches (K)
confidence level (1−𝛼);
coverage probability (𝑝)
𝑀𝑆𝐴 & 𝑀𝑆𝐸 (as functions of EMS-based variances)
Only Hoffman TIs depends on: 𝝈 𝒀 .. 𝟐 & 𝝈 𝒀 𝟐 (as functions of 𝑀𝑆𝐴 & 𝑀𝑆𝐸)

22. Tools & Methods: SSNormTI Mee-Owen Process Tolerance Intervals
One-Sided Tolerance Interval for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Mee & Owen 1983)
𝑌 𝑜𝑟 − 𝐾 𝑀𝑂 𝑐 1 𝑀𝑆𝐴+ 𝑐 2 𝑀𝑆𝐸
Where 𝐾 𝑀𝑂 = 1 𝐼𝐾 𝑅 0 ∗ 𝑡 𝑓 ∗ ,1−𝛼 𝑧 𝑝 𝐼𝐾 𝑅 0 ∗ , 𝑅 0 ∗ = 𝑅 ∗ +1 𝐾 𝑅 ∗ +1 and
Two-Sided Tolerance Interval for 𝑵 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Mee 1984)
𝑌 .. ± 𝐾 𝑀 𝑐 1 𝑀𝑆𝐴+ 𝑐 2 𝑀𝑆𝐸
Where 𝐾 𝑀 = 𝑓 ∗ 𝜒 1,𝑝, 1 𝐼𝐾 𝑅 0 ∗ 𝜒 𝑓 ∗ ,𝛼 2
𝑓 ∗ = 𝑅 ∗ 𝑅 ∗ + 1 𝐾 2 𝐼−1 + 𝐾−1 𝐼 𝐾 2 and 𝑅 ∗ =𝑚𝑎𝑥 0, 1 𝐾 𝑀𝑆𝐴 𝑀𝑆𝐸 ∗𝐹 𝑓 𝐴 , 𝑓 𝐸 , 1−𝛼 −1
R=MSA/MSE ~ F(I-1, I*(K-1))

23. Tools & Methods: SSNormTI Hoffman Process Tolerance Intervals
One-Sided Tolerance Interval for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Hoffman 2010)
𝑌 𝑜𝑟 − 𝑧 𝑝 𝜎 𝑌 𝐻 1 𝑐 1 𝑀𝑆𝐴 𝐻 2 𝑐 2 𝑀𝑆𝐸 𝑧 1−𝛼 𝜎 𝑌
𝑧 𝑝 is the 100p% percentile of normal distribution; 𝝈 𝒀 .. 𝟐 = ℎ 1 𝑀𝑆𝐴+ ℎ 2 𝑀𝑆𝐸, ℎ 1 = 1 𝐼𝐾 , ℎ 2 =0, 𝐻 1 = 1 𝐹 𝛼, 𝑓 𝐴 ,∞ −1, 𝐻 2 = 1 𝐹 𝛼, 𝑓 𝐸 ,∞ −1, with 𝐹 𝛼,𝑓,∞ as the lower −𝛼 % lower percentile of F distribution with 𝑓 𝑎𝑛𝑑 ∞ degrees of freedom
Two-Sided Tolerance Interval for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Hoffman & Kringle 2005)
𝑌 .. ± 𝑧 1+𝑝 𝑁 𝑒 𝜎 𝑌 𝐻 1 𝑐 1 𝑀𝑆𝐴 𝐻 2 𝑐 2 𝑀𝑆𝐸 2
Where 𝑁 𝑒 = 𝑐 1 𝑀𝑆𝐴+ 𝑐 2 𝑀𝑆𝐸 ℎ 1 𝑀𝑆𝐴+ ℎ 2 𝑀𝑆𝐸 , 𝝈 𝒀 𝟐 = 𝑐 1 𝑀𝑆𝐴+ 𝑐 2 𝑀𝑆𝐸 𝑤𝑖𝑡ℎ 𝑐 1 = 1 𝐾 , 𝑐 2 =1− 1 𝐾

24. Tools & Methods: SSNormTI
Krishnamoorthy Process Tolerance Intervals
Via Modified Large-Sample Approximation
One-Sided Tolerance Interval for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Krishnamoorthy & Mathew 2004)
𝑌 𝑜𝑟 − 𝑡 𝑓 𝐴 𝛿 1 ,1−𝛼 𝑀𝑆𝐴 𝐼𝐾 with 𝛿 1 = 𝑧 𝑝 𝐼+ 𝑀𝑆𝐸 𝑀𝑆𝐴 𝐼 𝐾−1 𝐹 𝛼, 𝑓 𝐴 , 𝑓 𝐸
Two-Sided Tolerance Interval for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 (Krishnamoorthy & Lian 2011)
𝑌 .. ± 𝑧 1+𝑝 𝑈 1−𝛼
Where 𝑈 1−𝛼 = 𝑎 1 MSA+ 𝑎 2 MSE+ 𝑎 1 MSA 𝑓 𝐴 𝜒 𝑓 𝐴 ,𝛼 2 − 𝑎 2 MSE 𝑓 𝐸 𝜒 𝑓 𝐸 ,𝛼 2 −1 2
𝑎 1 = 𝑐 1 + ℎ 1 , 𝑎 2 = 𝑐 2 + ℎ 2

25. Tools & Methods: SSNormTI
Krishnamoorthy Process Tolerance Intervals
Via Generalized Pivotal Quantities
Assumptions: 𝑓 𝐴 𝜔 𝐴 2 𝜔 𝐴 2 ~ 𝜒 𝑓 𝐴 2 and 𝑓 𝐸 𝜔 𝐸 2 𝜔 𝐸 2 ~ 𝜒 𝑓 𝐸 2 are independent
Generalized pivotal quantities
for 𝜔 𝐴 2 : 𝑓 𝐴 𝜔 𝐴 2 𝜒 𝑓 𝐴 2 ; for 𝜔 𝐸 2 : 𝑓 𝐸 𝜔 𝐸 2 𝜒 𝑓 𝐸 2
for 𝑎 𝐴 𝜔 𝐴 2 + 𝑎 𝐸 𝜔 𝐸 2 𝑎 𝐴 & 𝑎 𝐸 𝑎𝑟𝑒 𝑘𝑛𝑜𝑤𝑛 𝑣𝑎𝑙𝑢𝑒𝑠 : 𝑎 𝐴 𝑓 𝐴 𝜔 𝐴 2 𝜒 𝑓 𝐴 𝑎 𝐸 𝑓 𝐸 𝜔 𝐸 2 𝜒 𝑓 𝐸 2
for 𝜇 : Equation (5) in Krishnamoorthy & Mathew (2004)
One-Sided TI for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 : Krishnamoorthy & Mathew (2004)
Two-Sided TI for 𝐍 𝝁, 𝝈 𝑨 𝟐 + 𝝈 𝑬 𝟐 : Krishnamoorthy & Lian (2011)

26. Tools & Methods: SSNormTI Precision-Based Calculation: using norm.ss()
Approach
Notes
DIR: Owen (1964)
A quick back-of-the-envelope calculation simply for planning purposes
Without ensure any specific bounds relative to the nominal coverage probability
Population mean (mu.0) and SD (sig2.0) are assumed known, thus the coverage probability for a 2-sided TI is the central proportion of the data population
FW:
Faulkenberry &
Weeks (1968)
No direct guidance on setting P.prime and 𝛿
No need to provide true or estimated values of population mean and SD
YGZO:
Young et al (2016)
With use of historical data to help reduce the subjectivity in choosing P.prime and 𝛿
Via frequentist and Bayesian approaches

27. Tools & Methods: SSNormTI Precision-Based Calculation : DIR Approach
Goal: to find the minimum sample size 𝑛 ∗ to control the tolerance interval within prespecified acceptance range
Without use of historical data
Inputs:
mean: mu.0, SD: 𝑠𝑖𝑔2.0, specification range: spec (e.g., spec=c(70,NA))
confidence level: 1−𝛼, coverage probability: P
Tolerance intervals (within specification range)
2-sided: mu.0±𝑠𝑖𝑔2.0∗ 𝐾 2 𝑛 ∗ , 𝛼,𝑃
1-sided: mu.0+𝑠𝑖𝑔2.0∗ 𝐾 1 𝑛 ∗ , 𝛼,𝑃 or mu.0+𝑠𝑖𝑔2.0∗ 𝐾 1 𝑛 ∗ , 𝛼,𝑃
K factors 𝐾 1 𝑛 ∗ , 𝛼,𝑃 and 𝐾 2 𝑛 ∗ , 𝛼,𝑃 are calculated using Owen approach (method=“OCT”)
K.factor(n=10,side=2,method=c("EXACT")) #[1]
K.factor(n=10,side=2,method=c("OCT")) #[1]

28. Tools & Methods: SSNormTI Precision-Based Calculation : FW Approach
Faulkenberry & Weeks (1968): to determine a sample size such that there is only a small probability that the TI covers too large a proportion of the data population:
𝑃𝑟 𝑇𝐼 𝐶𝑜𝑣𝑒𝑟𝑎𝑔𝑒 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦≥𝑃.𝑝𝑟𝑖𝑚𝑒 ≤𝛿 (𝑰𝑬𝟏)
Without use of historical data
Inputs:
𝑃.𝑝𝑟𝑖𝑚𝑒 > P; 𝛿 (e.g., 0.01, 0.05, 0.1)
confidence level: 1−𝛼, coverage probability: P
Tolerance interval (TI):
𝑃𝑟 𝑇𝐼 𝐶𝑜𝑣𝑒𝑟𝑎𝑔𝑒 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦≥𝑃 ≤1−𝛼 (𝑰𝑬𝟐)
Output 𝒏 ∗ : the smallest n such that two K factors below are equal: 𝐾 𝑗 𝑛, 𝑃 ′ ,𝛿 = 𝐾 𝑗 𝑛,𝑃,1−𝛼 𝑗=1 𝑜𝑟 2

29. Tools & Methods: SSNormTI Precision-Based Calculation: YGZO Approach
Young et al (2016): modified FW approach with 𝑃.𝑝𝑟𝑖𝑚𝑒 and 𝛿 determined by historical data (and hyper-parameters and acceptance limits if specified)
With use of historical data
Inputs: prior hyper-parameters ( 𝜇 0 , 𝜎 0 2 , 𝑚 0 >0, 𝑛 0 >0); used for calculation of Bayesian TI; if not specified, then frequentist TI based on historical data will be calculated.
Priors: 𝜇| 𝜎 2 ~ 𝑁 𝜇 0 , 𝜎 2 𝑛 0 and 𝜎 2 ~ 𝑆𝑐𝑎𝑙𝑒𝑑−𝑖𝑛𝑣− 𝜒 2 𝑚 0 , 𝜎 0 2
When a scaled inverse chi-squared random variable 𝜒 2 𝜐, 𝜏 2 is divided by 𝜐 𝜏 2 , it results in an inverse chi-squared random variable with 𝜐 degrees of freedom.
Joint posterior distribution:𝑃 𝜇, 𝜎 2 =𝑃 𝜇| 𝜎 2 𝑃 𝜎 2 , with 𝑥 and 𝑆 2 as the average and variance of the historical data, 𝑥 = 𝑛 0 𝜇 0 +𝑛 𝑥 𝑛 0 +𝑛 , 𝑞 2 = 𝑚 0 𝜎 𝑛−1 𝑆 2 + 𝑛 0 𝑛 𝑥 − 𝜇 𝑛 0 +𝑛 𝑚 0 +𝑛−1
𝜇| 𝜎 2 ~ 𝑁 𝑥 , 𝜎 2 𝑛 0 +𝑛 and 𝜎 2 ~ 𝑆𝑐𝑎𝑙𝑒𝑑−𝑖𝑛𝑣− 𝜒 2 𝑚 0 +𝑛−1, 𝑞 2

30. Tools & Methods: SSNormTI Precision-Based Calculation : YGZO Approach
Input: Relative Error 𝛿; if not specified, then 𝛿= 𝑃 𝑟 𝑁 𝜇 0 , 𝜎 0 2 ∈𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑇𝐼 −𝑃 𝑃
Calculated Tolerance Interval (TI):
2-sided Bayesian TI: 𝑥 ± 𝐾 2 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼 𝑞, where 𝐾 2 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼
satisfies 2 𝑛+ 𝑛 0 𝜋 0 ∞ 𝑃 𝜒 𝑚 0 +𝑛−1 2 > 𝑚 0 +𝑛−1 𝜒 1;𝑃 2 𝑧 𝐾 2 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼 𝑒 − 𝑛+ 𝑛 0 𝑧 𝑑𝑧=1−𝛼
1-sided Bayesian TI: 𝑥 − 𝐾 1 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼 𝑞 𝑜𝑟 𝑥 + 𝐾 1 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼 𝑞
where 𝐾 1 𝑛, 𝑛 0 , 𝑚 0 , 𝛼,𝑃,1−𝛼 = 1 𝑛+ 𝑛 0 𝑡 𝑚 0 +𝑛−1;1−𝛼 𝑛+ 𝑛 0 𝑍 𝑃
Frequentist TIs: using EXACT method via normtol.int()
Input: P.prime; if not specified, then it is given below, or 𝑃 ′ = 1+𝑃 2 if 𝑃 ′ ≤P
𝑃 ′ = 1+𝑃 2 𝑊𝑖𝑡ℎ𝑜𝑢𝑡 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝐿𝑖𝑚𝑖𝑡(𝑠) 𝑃 𝑟 𝑁 𝜇 0 , 𝜎 𝑤𝑖𝑡ℎ𝑖𝑛 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝐿𝑖𝑚𝑖𝑡(𝑠) 𝑊𝑖𝑡ℎ 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝐿𝑖𝑚𝑖𝑡(𝑠)
Output 𝒏 ∗ : the smallest n such that the K factors below are equal
𝐾 𝑗 𝑛, 𝑃 ′ ,𝛿 = 𝐾 𝑗 𝑛,𝑃,1−𝛼 𝑗=1 𝑜𝑟 2
via Howe(1969) & EXACT methods for 𝐾 2 & 𝐾 1 , respectively; with adjustment if 𝑛 ∗ <4 or the above equation has no solution

31. Tools & Methods: SSNormTI Illustration
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