Statistical Techniques for Validation Sampling
Tie Risk to Sampling Data Type Confidence Level Reliability and Risk Typical Performance Levels One-sided or two-sided spec Distribution (variables)
Risk in Sampling Data Type Confidence Level Risk Defective Rate Reliability Variables 95% Attribute 90% or 95% FDA Mandated 0.% 99.90% High 0.30% 99.70% Low 5% 95% High 1% 99% Low 3% 97% Source: Taylor, W. A., Guide to Acceptance Sampling, Taylor Enterprises, 1992
Attribute Single & Double Sampling Plans
LTPD.05 = 3% Attribute Plans with 95% Confidence Type Parameters AQL LTPD 0.05 Single n=0, a=0 0.05% 3% Double n1=1, a1=0, r1=2, n2=120, a2=2 0.2% 3% Single n=2, a=2 0.39% 3%
Variables Sampling Plans LSL USL
P pk P pk is a measure of how close the process is to the nearest spec relative to the variation Variables sampling plans for 1-sided spec limits are based on P pk P pk = Distance from mean to nearest spec 3 s
P p Variables sampling plans for 2-sided spec limits are based on Ppk and Pp s is standard deviation (total) Compares width of process (6 s) to width of spec (USL - LSL) P p is similar to C p but uses total rather than within subgroup standard deviation P p = USL - 6 s LSL
2-Sided Variables Sampling Plans LTPD 0.05 = 1% 95% confidence Parameters AQL LTPD 0.05 n=15, P pk =1.17, P p =1.17 0.00016% (P pk =1.55) 1% (P pk =0.7) n=20, P pk =1.11, P p =1.13 0.001% (P pk =1.2) 1% (P pk =0.7) n=30, P pk =1.03, P p =1.07 0.007% (P pk =1.27) 1% (P pk =0.7) n=0, P pk =0.99, P p =1.0 0.01% (P pk =1.19) 1% (P pk =0.7)
Interactive Exercise: Process validation, 3 lots, 95% confidence level, 99.7% reliability based on high risk, continuous data, 1-sided spec: tensile force 2.5 lb/in 2
Choose a Sampling Plan 1-sided LTPD 0.05 = 0.3% 95% confidence Given: Ppk = 1. (historic data) Parameters AQL LTPD 0.05 n=15, P pk =1.7 =1.37 0.000003% (P pk =1.0) 0.3% (P pk =0.92) n=20, P pk =1.29 0.0000% (P pk =1.69) 0.3% (P pk =0.92) n=30, P pk =1.20 0.0005% (P pk =1.7) 0.3% (P pk =0.92) n=0, P pk =1.15 0.002% (P pk =1.3) 0.3% (P pk =0.92) n=15 has a 50% probability of acceptance n=30 has a 95% probability of acceptance
Collect the Data Collect data on 30 samples per lot Repeat for all 3 lots
Analyze the Data Normality Test Stability Fail (p<.05) Pass (p.05) *Transformation Capability Analysis
Lot 1: Normality Test
Sample Range Sample Mean Lot 1: Stability Xbar-R Chart of Lot1 16 12 _ UC L=11.56 X=. LC L=.63 0 1 2 3 5 6 Sample 7 9 16 12 0 UC L=3.70 _ R=1. LC L=0 1 2 3 5 6 Sample 7 9
Lot 1: Capability Analysis Process Capability of Lot1 Process Data LSL 2.5 Target * USL * Sample Mean.0952 Sample N 30 StDev (Within) 0.072 StDev (O v erall) 0.960366 LSL Within Overall Potential (Within) C apability C p * C PL 2.99 C PU * C pk 2.99 O v erall C apability Pp * PPL 2.6 PPU * Ppk 2.6 C pm * 3.0.5 6.0 7.5 9.0.5 12.0 O bserv ed Performance PPM < LSL 0.00 PPM > USL * PPM Total 0.00 Exp. Within Performance PPM < LSL 0.00 PPM > USL * PPM Total 0.00 Exp. O v erall Performance PPM < LSL 0.00 PPM > USL * PPM Total 0.00
Lot 2: Normality Test Summary for Lot2 A nderson-darling Normality Test A -Squared 2.30 P-V alue < 0.005 Mean.15 StDev 3.7526 V ariance 1.01 Skew ness 1.031 Kurtosis 1.2116 N 30 12 16 Minimum 3.66 1st Q uartile 6.050 Median 6.766 3rd Q uartile 9.27 Maximum 17.9937 95% C onfidence Interv al for Mean 6.7576 9.5601 95% C onfidence Interv al for Median 6.215 7.92 95% Confidence Intervals 95% C onfidence Interv al for StDev 2.96 5.06 Mean Median 6 7 9
Sample Range Sample Mean Lot 2: Stability Xbar-R Chart of Lot2 16 UC L=1.9 12 _ X=.16 0 1 2 3 5 6 Sample 7 9 LC L=1.3 16 UC L=17.16 12 _ R=6.67 0 LC L=0 1 2 3 5 6 Sample 7 9
StDev Lot 2: Data Transformation Box-Cox Plot of Lot2 30 25 20 Lower CL Upper CL Lambda (using 95.0% confidence) Estimate -1. Lower CL -2.26 Upper CL -0.73 Rounded Value -1.00 Don t forget to transform the specification! 15 5 Limit 0-5.0-2.5 0.0 Lambda 2.5 5.0
Transformed Normality Test
Lot 2: Capability Analysis Process Capability of Lot2 Using Box-Cox Transformation With Lambda = -1 Process Data LSL 2.5 Target * USL * Sample Mean.15 Sample N 30 StDev (Within) 3.9377 StDev (O v erall) 3.7506 A fter Transformation LSL* 0. Target* * USL* * Sample Mean* 0.1321 StDev (Within)* 0.050002 StDev (O v erall)* 0.0521765 transformed data LSL* Within O v erall Potential (Within) C apability C p * C PL 1.90 C PU * C pk 1.90 O v erall C apability Pp * PPL 1.6 PPU * Ppk 1.6 C pm * 0.05 0. 0.15 0.20 0.25 0.30 0.35 0.0 O bserv ed Performance PPM < LSL 0.00 PPM > USL * PPM Total 0.00 Exp. Within Performance PPM > LSL* 0.01 PPM < USL* * PPM Total 0.01 Exp. O v erall Performance PPM > LSL* 0.3 PPM < USL* * PPM Total 0.3
Lot 3: Normality Test Summary for Lot3 A nderson-darling Normality Test A -Squared 0.99 P-V alue 0.011 Mean 9.2630 StDev 1.6659 V ariance 2.775 Skew ness -1.990 Kurtosis 3.21573 N 30 6 12 Minimum 3.992 1st Q uartile.73 Median 9.312 3rd Q uartile.322 Maximum 11.503 95% C onfidence Interv al for Mean.609 9.51 95% C onfidence Interv al for Median 9.0309.131 9 5 % Confidence Intervals 95% C onfidence Interv al for StDev 1.326 2.2395 Mean Median.50.75 9.00 9.25 9.50 9.75.00
Sample Range Sample Mean Lot 3: Stability Xbar-R Chart of Lot3 16 12 UC L=12.20 _ X=9.26 LC L=6.33 0 1 2 3 5 6 Sample 7 9 16 12 UC L=7.3 _ R=2.7 0 LC L=0 1 2 3 5 6 Sample 7 9
Percent Percent Percent Percent Lot 3: Data Transformation 90 50 Probability Plot for Lot3 2-Parameter Exponential - 95% C I Weibull - 95% C I 90 50 Goodness of F it Test 2-Parameter Exponential A D = 7.207 P-V alue < 0.0 Weibull A D = 0.69 P-V alue = 0.239 1 0.01 90 0. 1.00.00 Lot3 - T hreshold 3-Parameter Weibull - 95% C I 0.00 1 90 5 Lot3 Smallest Extreme V alue - 95% C I 3-Parameter Weibull A D = 0.2 P-V alue > 0.500 Smallest Extreme V alue A D = 0.2 P-V alue > 0.250 50 50 1 160 165 Lot3 - T hreshold 170 1 3 6 Lot3 9 12
Lot 3: Capability Analysis Process Capability of Lot3 Calculations Based on Weibull Distribution Model Process Data LSL 2.5 Target * USL * Sample Mean 9.26301 Sample N 30 Shape 7.5165 Scale 9.757 O bserv ed Performance PPM < LSL 0.00 PPM > USL * PPM Total 0.00 LSL O v erall C apability Pp * PPL 1.31 PPU * Ppk 1.31 Exp. O v erall Performance PPM < LSL 29.96 PPM > USL * PPM Total 29.96 6 12
Conclusion All three lots met criteria to conclude that the validation passes. With 95% confidence, the process average across each lot produces at least 99% reliability, or With 95% confidence, the process average across each lot produces less than 1% defective. Note: all 3 lots combined are at a 99.99% confidence level.
Distribution Analysis Attribute sampling plans Normality established Data transformation (special cases) Distribution-free methods such as VP: Require unimodality Requires sufficient distance between mean and specification limit Can be used with very skewed distributions
References EN ISO 135:2012 - Medical Devices Quality Management Systems. 21 CFR 20, Quality System Regulation, Subparts C, G & O (design control, production and process controls, statistical techniques). Taylor, W., Guide to Acceptance Sampling, Taylor Enterprises, Inc., 1992. D. F. Vysochanskij, Y. I. Petunin (190). "Justification of the 3σ rule for unimodal distributions." Theory of Probability and Mathematical Statistics 21: 25 36.