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Introduction to Process Performance Qualification (PPQ) Sampling Plan R Package

Yalin Zhu (yalin.zhu@merck.com)

2020-10-07

This vignette summarizes the functions in the PPQplan package, and provides some examples to illustrates how to use the package.

Note: in order to better perform the dynamic plots, it is recommended to run the following code in RStudio.

Functions

This package provides several S3 functions listed as follows:

`rl_pp`: calculates probability of pass the specification test.

Example: Consider some sterile concentration assay as a CQA, the lower and upper specification limits are 95% and 105%, if the hypothetical mean and standard deviation are 98% and 1%, then the probability of passing the specification test will be calculated as follow.

rl_pp(Llim=95, Ulim=105, mu=98, sigma=1)

## [1] 0.9986501

The following four functions are mainly for the sampling plan using statistical intervals with general multiplier

PPQ_pp
PPQ_occurve
PPQ_ctplot
PPQ_ggplot

For the above example, assume the PPQ study reports a sample of 10 assay results per batch, test only one batch. Then a general multiplier for constructing 95% two-sided prediction interval can be calculated as \(k=2.373\).

`PPQ_pp`: calculates the probability of passing some critical quality attributes (CQA) PPQ test using a general constant multiplier `k`.

PPQ_pp(Llim=95, Ulim=105, mu=98, sigma=1, n=10, n.batch = 1, k = 2.373)

## [1] 0.8604419

Comparing different scenarios for hypothetical mean and standard deviation:

sigma <- seq(0.1, 4, 0.1)
 pp1 <- sapply(X=sigma, FUN =  PPQ_pp, mu=97, n=10, Llim=95, Ulim=105, k=2.373)
 pp2 <- sapply(X=sigma, FUN =  PPQ_pp, mu=98, n=10, Llim=95, Ulim=105, k=2.373)
 pp3 <- sapply(X=sigma, FUN =  PPQ_pp, mu=99, n=10, Llim=95, Ulim=105, k=2.373)
 pp4 <- sapply(X=sigma, FUN =  PPQ_pp, mu=100, n=10, Llim=95, Ulim=105, k=2.373)
 plot(sigma, pp1, xlab="Standard Deviation", main="LSL=95, USL=105, k=2.373, n=10",
 ylab="Probability of Passing", type="o", pch=1, col=1, lwd=1, ylim=c(0,1))
 lines(sigma, pp2, type="o", pch=2, col=2)
 lines(sigma, pp3, type="o", pch=3, col=3)
 lines(sigma, pp4, type="o", pch=4, col=4)
 legend("topright", legend=paste0(rep("mu=",4),c(97,98,99,100)), bg="white",
 col=c(1,2,3,4), pch=c(1,2,3,4), lty=1, cex=0.8)

 mu <- seq(95, 105, 0.1)
 pp5 <- sapply(X=mu, FUN =  PPQ_pp, sigma=0.5, n=10, Llim=95, Ulim=105, k=2.373)
 pp6 <- sapply(X=mu, FUN =  PPQ_pp, sigma=1, n=10, Llim=95, Ulim=105, k=2.373)
 pp7 <- sapply(X=mu, FUN =  PPQ_pp, sigma=1.5, n=10, Llim=95, Ulim=105, k=2.373)
 pp8 <- sapply(X=mu, FUN =  PPQ_pp, sigma=2, n=10, Llim=95, Ulim=105, k=2.373)
 pp9 <- sapply(X=mu, FUN =  PPQ_pp, sigma=2.5, n=10, Llim=95, Ulim=105, k=2.373)
 plot(mu, pp5, xlab="Mean Value", main="LSL=95, USL=105, k=2.373, n=10",
 ylab="Probability of Passing", type="o", pch=1, col=1, lwd=1, ylim=c(0,1))
 lines(mu, pp6, type="o", pch=2, col=2)
 lines(mu, pp7, type="o", pch=3, col=3)
 lines(mu, pp8, type="o", pch=4, col=4)
 lines(mu, pp9, type="o", pch=5, col=5)
 legend("topright", legend=paste0(rep("sigma=",5),seq(0.5,2.5,0.5)), bg="white",
 col=c(1,2,3,4,5), pch=c(1,2,3,4,5), lty=1, cex=0.8)

`PPQ_occurve`: plots OC curves for specification test and PPQ plan, with the options of customizing CQA name, unit, number of batch, optimizing the plans, etc.

PPQ_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC", Llim=95, Ulim=105, mu=98, sigma=seq(0.1, 10, 0.1), n=10, k=2.373)

The function can also optimize the baseline and high performance sampling plan¹ by using add.reference option.

PPQ_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC", Llim=95, Ulim=105, mu=98, sigma=seq(0.1, 10, 0.1), n=10, k=2.373, add.reference=TRUE)

We can also optimize and show the Baseline and High performance reference lines only:

PPQ_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC", Llim=95, Ulim=105, mu=98, sigma=seq(0.1, 10, 0.1), n=10, add.reference=TRUE)

Since \(k=2.373\) is between 1.798 (baseline) and 2.945 (high performance), the 95% confidence interval is suitable for this PPQ plan.

`PPQ_ctplot`: Heatmap (or Contour Plot) for PPQ assessment with parameter space.

mu <- seq(95,105,0.05)
sigma <- seq(0.1,1.75,0.05)
PPQ_ctplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma, k=2.373)

test <- data.frame(mu=c(97,98.3,102.5), sd=c(0.55, 1.5, 1.2))
PPQ_ctplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma, k=2.373, test.point=test)

`PPQ_ggplot`: Dynamic Heatmap (or Contour Plot) for PPQ assessment with parameter space.

 mu <- seq(95, 105, 0.05)
 sigma <- seq(0.1,1.75,0.05)
 PPQ_ggplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma, k=2.373, dynamic = FALSE)

 PPQ_ggplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma, k=2.373, test.point = test, dynamic = FALSE)

Plot a dynamic heat map. User can hover on the plot to interactively evaluate the plan with dynamic = TRUE option.

 PPQ_ggplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma, k=2.373, test.point = test, dynamic = TRUE)

The following three functions are used for sampling plan with prediction interval.

pi_pp
pi_occurve
pi_ctplot

`pi_pp`: calculates the probability of passing the PPQ test using prediction interval with confidence level \(100 \times 1-\alpha\).

Use the same example with alpha=0.05 option.

pi_pp(Llim=95, Ulim=105, mu=98, sigma=1, n=10, n.batch = 1, alpha=0.05)

## [1] 0.8606111

`pi_occurve`: plots OC curves for specification test and PPQ plan, with the options of customizing CQA name, unit, number of batch, optimizing the plans, etc.

 pi_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC",
 mu=97, sigma=seq(0.1, 10, 0.1), Llim=95, Ulim=105, n=10, add.reference=TRUE)

 pi_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC",
 mu=100, sigma=seq(0.1, 10, 0.1), Llim=95, Ulim=105, n=10, add.reference=TRUE)

 pi_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%LC",
 mu=seq(95,105,0.1), sigma=1, Llim=95, Ulim=105, n=10, add.reference=TRUE)

`pi_ctplot`: Heatmap (or Contour Plot) for PPQ assessment with parameter space.

mu <- seq(95, 105, 0.05)
sigma <- seq(0.1,1.75,0.05)
pi_ctplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma)

The following three functions are used for sampling plan with tolerance interval.

ti_pp
ti_occurve
ti_ctplot

`ti_pp`: calculates the probability of passing the PPQ test using one-sided or two-sided tolerance interval with confidence level \(100 \times 1-\alpha\).

Use the same example with alpha=0.05 option.

ti_pp(Llim=95, Ulim=105, mu=98, sigma=1, n=10, n.batch = 1, alpha=0.05, side=2)

## [1] 0.9942658

ti_pp(Llim = 100, Ulim = Inf, mu=102.5, sigma=2, alpha = 0.05, coverprob = 0.675, side=1)

## [1] 0.6185582

`ti_occurve`: plots OC curves for specification test and PPQ plan, with the options of customizing CQA name, unit, number of batch, optimizing the plans, etc.

 ti_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%",
 mu=97, sigma=seq(0.1, 10, 0.1), Llim=95, Ulim=105, n=10, add.reference=TRUE)

 ti_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%",
 mu=100, sigma=seq(0.1, 10, 0.1), Llim=95, Ulim=105, n=10, add.reference=TRUE)

 ti_occurve(attr.name = "Sterile Concentration Assay", attr.unit="%",
 mu=seq(95,105,0.1), sigma=1, Llim=95, Ulim=105, n=10, add.reference=TRUE)

Another example is test Extractable Volume using one-sided lower tolerance interval².

NV = 1mL, select 5 containers and pool them together

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=1mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.675, side=1, NV=1)

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=1mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.78, side=1, NV=1)

NV = 3mL, select 5 containers but not pool

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=3mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.97, side=1, NV=3)

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=3mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.992, side=1, NV=3)

NV = 6mL, select 3 containers but not pool

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=6mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.95, side=1, NV=6)

ti_occurve(attr.name = "Extractable Volume", attr.unit = "% of NV=6mL", Llim = 100, Ulim = Inf, mu=102.5, sigma=seq(0.2, 6 ,0.05), n=40, alpha = 0.05, coverprob = 0.987, side=1, NV=6)

`ti_ctplot`: Heatmap (or Contour Plot) for PPQ assessment with parameter space.

mu <- seq(95, 105, 0.05)
sigma <- seq(0.1,2.5,0.05)
ti_ctplot(attr.name = "Sterile Concentration Assay", attr.unit = "%LC", Llim=95, Ulim=105, mu = mu, sigma = sigma)

Also test the Extractable Volume using one-sided tolerance interval, for example, NV = 1mL with 95% / 67.5% one-sided lower tolerance interval.

ti_ctplot(attr.name = "Extractable Volume", attr.unit = "% of NV=1mL", Llim = 100, Ulim = Inf, mu=seq(100, 110, 0.5), sigma=seq(0.2, 15 ,0.5), n=40, alpha = 0.05, coverprob = 0.675, side=1)

The package also provides two functions for a general sampling plan based on lower and/or upper specification limits.

pp
heatmap_ly

`pp`: calculate the probability of passing general upper and/or lower specification limit.

ti_pp(Llim=-0.2, Ulim=0.2, mu=0.1, sigma=0.15, n=2)

## [1] 0.02568295

`heatmap_ly`: plot a plain or dynamic heatmap (or contour plot) for a general sampling plan with specification limit.

heatmap_ly(attr.name = "Thickness", attr.unit = "%",Llim = -0.2, Ulim = 0.2, mu = seq(-0.2, 0.2, 0.001), sigma = seq(0,0.2, 0.001), test.point=data.frame(c(0.1,-0.05),c(0.15,0.05)), n=2, dynamic = TRUE)

Burdick, R. K., LeBlond, D. J., Pfahler, L. B., Quiroz, J., Sidor, L., Vukovinsky, K., & Zhang, L. (2017). Statistical Applications for Chemistry, Manufacturing and Controls (CMC) in the Pharmaceutical Industry.↩
USP <1> https://www.usp.org/sites/default/files/usp/document/harmonization/gen-method/q08_pf_31_1_2005.pdf.↩

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.

Introduction to Process Performance Qualification (PPQ) Sampling Plan R Package

Yalin Zhu (yalin.zhu@merck.com)

2020-10-07

Functions

rl_pp: calculates probability of pass the specification test.

The following four functions are mainly for the sampling plan using statistical intervals with general multiplier

PPQ_pp: calculates the probability of passing some critical quality attributes (CQA) PPQ test using a general constant multiplier k.

PPQ_occurve: plots OC curves for specification test and PPQ plan, with the options of customizing CQA name, unit, number of batch, optimizing the plans, etc.

PPQ_ctplot: Heatmap (or Contour Plot) for PPQ assessment with parameter space.

PPQ_ggplot: Dynamic Heatmap (or Contour Plot) for PPQ assessment with parameter space.

The following three functions are used for sampling plan with prediction interval.

pi_pp: calculates the probability of passing the PPQ test using prediction interval with confidence level \(100 \times 1-\alpha\).

pi_occurve: plots OC curves for specification test and PPQ plan, with the options of customizing CQA name, unit, number of batch, optimizing the plans, etc.

pi_ctplot: Heatmap (or Contour Plot) for PPQ assessment with parameter space.