Setting Specifications Statistical considerations Enda Moran - PowerPoint PPT Presentation

Setting Specifications Statistical considerations Enda Moran – Senior Director, Development, Pfizer Melvyn Perry – Manager, Statistics, Pfizer

B Basic Statistics i St ti ti Population distribution p 1 5 1.5 (usually unknown). True batch as s ay Normal distribution 1.0 described by μ and σ . Dis tribution of pos s ible values 0.5 0.0 98 99 100 We infer the population from samples by calculating and s. x 1.5 1.5 1.5 True batch as s ay y True batch as s ay y True batch as s ay y Sample 3 Sample 3 Sample 1 Sample 2 Average 99.0 1.0 1.0 Average 1.0 Average 98.7 98.6 Dis tribution of Dis tribution of Dis tribution of pos s ible values pos s ible values pos s ible values 0.5 0.5 0.5 2 0.0 0.0 0.0 98 99 100 98 99 100 98 99 100

I t Intervals l 30 30 28 26 24 Population average 22 20 18 16 0 10 20 30 40 50 60 70 80 90 100 •100 samples of size 5 taken from a population with an average of 23.0 and a standard deviation of 2.0. •The highlighted intervals do not include the population average (there are 6 of them). •For a 95% confidence level expect 5 in 100 intervals to NOT include the population average For a 95% confidence level expect 5 in 100 intervals to NOT include the population average. •Usually we calculate just one interval and then act as if the population mean falls within this interval. 3

I t Intervals l Point Estimation Point Estimation The best estimate; eg MEAN Interval Estimation A range which contains the true population parameter or a future observation to a certain degree of confidence. Confidence Interval - The interval to estimate the true population parameter (e.g. the population mean) mean). Prediction Interval Prediction Interval - The interval containing the next single response. Tolerance Interval - The interval which contains at least a given proportion of the population. 4

F Formulae for Intervals l f I t l Intervals are defined as: x ± ks A Assuming a normal distribution i l di t ib ti • Confidence (1- α ) interval 0 0 . 5 5 ⎛ ⎛ ⎞ ⎞ 1 = ± ⎜ ⎟ CI x t s α ⎝ ⎠ n − − 1 , n 1 2 • Prediction (1- α ) interval for m future observations Prediction (1 α ) interval for m future observations 0 . 5 ⎛ + 1 ⎞ = ± ⎜ ⎟ 1 PI x t s α ⎝ ⎝ ⎠ ⎠ n 1 − , − 1 n 2 2 m m • Tolerance interval for confidence (1- α ) that proportion (p) is covered ( ) i d ⎛ + ⎞ 1 ( ) − ⎜ ⎟ 2 1 1 n z − ( 1 ) p ⎝ ⎝ ⎠ ⎠ n n = ± 2 TI x s χ 2 5 α − , 1 n

P Process Capability C bilit s s 3 3 3 3 Process capability is a measure of the risk of failing specification. The spread of the data are compared with the width of the specifications. th ifi ti The distance from the mean to the The distance from the mean to the nearest specification relative to half the process width (3s). The index measures actual performance. Which may or may not be on target i.e., centred. x LSL USL − − x LSL USL x − − ⎧ ⎧ ⎫ ⎫ USL USL x x x x LSL LSL = ⎨ ⎬ P pk min , ⎩ ⎭ 3 s 3 s 6

P Process Capability – Ppk and Cpk C bilit P k d C k • Ppk should be used as this is the actual risk of failing specification Ppk should be used as this is the actual risk of failing specification. • Cpk is the potential capability for the process when free of shifts and drifts. Random data of mean 10 and SD 1, thus natural span 7 to 13. Added shifts to simulate trends around common average. Added shifts to simulate trends around common average With specs at 7 and 13 process capability should be unity. When data is with trend Ppk less than Cpk due to method of Process Capability of Shifted I Chart of Shifted calculation of std dev. calculation of std dev. 15 LSL USL 1 1 Process Data Within 14 LSL 7 Overall UCL= 13.589 Target * Ppk uses sample SD. Ppk less Potential (Within) Capability USL 13 13 Cp 0.86 Sample Mean 10.0917 CPL 0.88 Sample N 30 12 dividual Value CPU 0.83 than 1 at 0.5. StDev (Within) 1.16561 Cpk 0.83 StDev (Ov erall) 1.95585 11 O v erall Capability _ Pp 0.51 10 X= 10.092 PPL 0.53 C k Cpk uses average moving i PPU 0.50 Ind Ppk 0.50 9 Cpm * range SD (same as for control 8 7 6 8 10 12 14 chart limits). LCL= 6.595 6 Observ ed Performance Exp. Within Performance Exp. O v erall Performance PPM < LSL 0.00 PPM < LSL 3995.30 PPM < LSL 56966.34 1 4 7 10 13 16 19 22 25 28 Cpk is close to 1 at 0.83. p PPM > USL 100000.00 PPM > USL 6296.59 PPM > USL 68512.70 Observation PPM Total 100000.00 PPM Total 10291.88 PPM Total 125479.03 Process Capability of Raw I Chart of Raw When data is without trend When data is without trend LSL USL UCL= 13.038 13 Process Data Within LSL 7 Overall Ppk is same as Cpk. Only Target * 12 Potential (Within) Capability USL 13 Cp 1.02 Sample Mean 10.0917 CPL 1.05 Sample N 30 small differences are seen. Individual Value CPU 0.99 11 StDev (Within) 0.982187 Cpk 0.99 StDev (Ov erall) 1.14225 O v erall Capability _ X= 10.092 Pp 0.88 10 PPL 0.90 Cpk effectively 1 at 0.99. Cpk effectively 1 at 0.99. PPU 0.85 Ppk 0.85 9 9 Cpm C * * I 8 Ppk close to 1 at 0.85. 7 7 8 9 10 11 12 13 LCL= 7.145 7 O bserv ed Performance Exp. Within Performance Exp. O v erall Performance PPM < LSL 0.00 PPM < LSL 822.51 PPM < LSL 3397.73 1 4 7 10 13 16 19 22 25 28 PPM > USL 0.00 PPM > USL 1533.13 PPM > USL 5446.84 Observation PPM Total 0.00 PPM Total 2355.65 PPM Total 8844.57

M Measurement Uncertainty t U t i t Good USL LSL σ measurement parts t almost always passed p σ total Bad parts Bad parts almost almost always always always rejected j t d rejected ± 3 σ measurement ± 3 σ measurement The grey areas highlighted represent those parts of the curve with the potential for wrong decisions, or mis-classification. 8

Misclassification with less variable process i bl σ σ total USL LSL σ measurement measurement ± 3 σ measurement 3 σ measurement ± 3 σ measurement measurement 9

M Measurement Uncertainty t U t i t × σ Precision to Tolerance Ratio: 6 = MS P T / Tolerance How much of the tolerance is taken up by = − Tolerance USL LSL measurement error. = USL This estimate may be appropriate for evaluating This estimate may be appropriate for evaluating Upper pp Spec p Limit = how well the measurement system can perform LSL Lower Spec Limit with respect to specifications. = σ Std. Dev. of Measuremen t Sys. MS σ = × % & 100 MS R R Gage R&R (or GRR%) σ Total Total What percent of the total variation is taken up by measurement error (as SD and thus not additive). Use Measurement Systems Analysis to assess if the assay method is fit for purpose. It is unwise to have a method where the specification interval is consumed by the measurement variation alone. 10

Specification example – T l Tolerance Interval I t l Data from three sites used to set specifications. Data from three sites used to set specifications 16.5 16.5 16.0 Tolerance interval found from pooled data of 253 15.5 batches. 15.0 14.5 Tolerance interval chosen as 95% probability that 14.0 mean ± 3 standard deviations are contained. 13.5 13.0 Sample size: Mean ± 3s Tolerance 1 2 3 n=253 interval Code Mean = 14.77 ( (95% / % / N k multiplier Table of values for 95% 99.7%) 5 6.60 probability of interval s=0.58 10 4.44 containing 99% of 15 15 3 89 3.89 population values population values. R Range 13.03 - 16.51 13 03 16 51 12 89 12.89 - 16.65 16 65 30 3.35 ∞ 2.58 Note at ∞ value is 2.58 which is the z value for 99% coverage of a normal If sample size was smaller, difference between population these calculations increases. As the sample size approaches infinity the TI 11 approaches mean ± 3s.

Specification Example - St bilit Stability Batch history Total SD (process and (p measurement) Stability batch with SD (measurement) Shelf life set from 95% CI on slope from three clinical batches. Need to find release criteria for high probability of production batches meeting shelf life based on individual results being less than meeting shelf life based on individual results being less than specification. R Review of batch history will lead to a process capability statement i f b t h hi t ill l d t bilit t t t 12 against release limit.