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1 Sampling Plans Part 2 of 3 Acceptance Sampling Plans for Inspection by Variables Peter Knepell, PhD Peak Quality Services pete@pcisys.net 1 Introducing the Presenter Peter Knepell, President of Peak Quality Services PhD, Cornell University, Operations Research Certified Quality Engineer (CQE) and Software Quality Engineer (CSQE) by the American Society for Quality Started assisting medical device and pharmaceutical manufacturers in Presented over 10 popular, AAMI-sponsored webinars. Since 1998, specialized in Lean Six Sigma & Design for Six Sigma implementation for a variety of industries & organizations AAMI faculty for: Statistics, Design of Experiments, Risk Management, and Process Validation workshops 2

2 Sampling Plans Webinar Series Overall Objectives At the end of this series, you will be able to: Create a risk-based sampling plan for the validation of a new production process Create acceptance sampling plans for inspection by variable Create acceptance sampling plans for inspection by attribute Explain the risks involved with decisions Balance cost of inspection with risks of making a wrong decision 3 Sampling Plans - Part 2 Objectives At the end of this webinar, you will be able to: Define the fundamental concepts for Acceptance Sampling Plans Create Acceptance Sampling Plans for inspection by variable Establish an appropriate sample size for inspection by variable Explain the risks involved with decisions 6/24/ Association for the Advancement of Medical Instrumentation 4

3 Webinar Outline Quick Review of Key Topics Q&A and Review from Part 1 FDA References Definitions Scenario from Part 1 Process Capability Measures for Variable Data Acceptance Sampling Plans for Variables 5 Q&A and Review from Part 1 Small Production Lots For certain manufacturing methods, first article or first-and-last article inspections are adequate. (eg, CNC manufacturing) For other manufacturing methods, 100% inspection may be appropriate. (eg, manual assembly) Acceptance Sampling Plans can reflect these strategies but will not depend on statistical techniques. Process monitoring should be considered. Use/Misuse of AQL vs. LTPD These terms will be completely covered in Part 3 If you MUST have an immediate answer, drop me an request for some definitions (pete@pcisys.net) 6

4 Q&A and Review from Part 1 Censored Data for Reliability Scenario: Want a device to last at least 50,000 cycles. Test a sample to 100,000 cycles and collect the times to failure. Attribute Data Example: One out of twenty tested failed in less than 50,000 cycles. Estimated Failure Rate = 1/20 = 5% + X%. This will be covered in a future webinar: How Much Is Enough? Variable Data Example: You want to establish a mean time to failure (MTTF) for the device. Beyond scope for these webinars. Consult a reliability expert. 7 Confidence in Statistical Conclusions Depend on Sampling Sampling plans, when used, shall be written and based on a valid statistical rationale. Each manufacturer shall establish and maintain procedures to ensure that sampling methods are adequate for their intended use and to ensure that when changes occur the sampling plans are reviewed. These activities shall be documented. 21 CFR (b) 6/24/ Association for the Advancement of Medical Instrumentation 8

5 Acceptance Activities Establish and maintain procedures for acceptance activities. Include inspections, tests or other verification activities. Receiving acceptance In-process Final acceptance 21 CFR (a) (d) 6/24/ Association for the Advancement of Medical Instrumentation 9 Definition of an Acceptance Sampling Plan Acceptance Sampling Plan A pathway for deciding on the disposition of a product based on the inspection of one or more samples. Goal To minimize the cost of inspection while understanding the risks of making a wrong decision. 6/24/ Association for the Advancement of Medical Instrumentation 10

6 High Level Acceptance Sampling Plan Receive Lot (N) Inspect Sample (n) Meets Criteria? Yes Accept Lot No Disposition Decision Some Alternatives: Inspect another sample Inspect 100% Fine your supplier Reject lot 6/24/ Association for the Advancement of Medical Instrumentation 11 Specific Example for Variable Data Receive Lot N = 1,000 Cpk > 1.25? Yes Inspect Sample n = 15* Inspect New Sample n = 24* Cpk > 1.5? No For all 39*, Cpk > 1.25? Yes Yes Accept the lot with 95% confidence the Cpk > 1.0 No No Inspect 100% * Based on Cpk Sample Size Table, Slide 40. 6/24/ Association for the Advancement of Medical Instrumentation 12

7 Noteworthy Observations Acceptance plans can be innovative Previous example is called a Double Sampling Plan Decisions Expressed in terms of confidence in a quality level To perform 100% inspection of the lot after failure to accept is an expensive disposition Need to be based on the cost of inspection versus the benefit to the customer 6/24/ Association for the Advancement of Medical Instrumentation 13 Key Assumptions for Acceptance Sampling Plans Results of inspections are valid (ie, measurement system is good and samples are randomly selected) Cost of 100% inspection exceeds the benefits Underlying probability distributions are appropriate Setting a performance goal is not permission to produce defectives it is guidance for disposition decisions 14

8 Scenario from Part 1 Scenario: A manufacturer is going to set up two parallel production lines for a new reagent. During Operational Qualification (OQ) they found that a key quality characteristic for the reagent is very sensitive to: ph of a raw material purity of water used in the process In January they will set up both production lines that will operate over two shifts a day, five days a week. Each day, each line will produce a batch of reagent. At the end of the second shift, the equipment will be cleaned. How should they evaluate PQ results for ph? 15 Analysis of Results Cpk for Each Batch s ph Goal: Cpk > 1.33 Each Sample Size = 16 Cpk Line 1 Batch 1 Batch 2 Batch 3 Batch Cpk Line 2 Batch 1 Batch 2 Batch 3 Batch Analysis is incomplete. No level of risk is assigned these performance capability measures. We ll revisit this table later. 16

9 Webinar Outline Quick Review of Key Topics Q&A and Review from Part 1 FDA & GHTF References Definitions Scenario from Part 1 Process Capability Measures for Variable Data FDA & GHTF References Definitions of Cpk, Ppk, level, & DPM Technical Details Acceptance Sampling Plans for Variables 17 Process Capability Requirement Where appropriate, each manufacturer shall establish and maintain procedures for identifying valid statistical techniques required for establishing, controlling, and verifying the acceptability of process capability and product characteristics. [emphasis added] 21 CFR (a) 18

10 Capability Studies Capability studies are performed to evaluate the ability of a process to consistently meet a specification. The most commonly used capability indices are C p and C pk. Capability studies are frequently used towards the end of the validation to demonstrate that the outputs consistently meet the specifications. Global Harmonization Task Force (GHTF), Process Validation Guidance, page Statistical Measures of Process Capability Name Symbol Estimate of: Process Capability Index Cpk Defect-free work Sigma Level level Defect-free work Defects per Million DPM or PPM Defect-free work Defects per Unit dpu Defect-free work First Pass Yield FPY Defect-free work Do you notice a trend here? 20

11 Cpk Process Capability If Metric is Variable AND Normally Distributed Process Capability Index LSL USL C pk = minimum { USL -, -LSL } 3 Common measure Normally distributed data that do not have to be centered on target Red Area represents proportion out of specification (aka, proportion defective) DPM = Red Area x 1,000, Sigma Level Six Sigma Quality Systems Metric is Variable AND Normally Distributed level = minimum { USL -, -LSL } Exactly 3 times Cpk Represents number of standard deviations between current mean of process & closest specification limit 22

12 Process Capability Example The packaging team established Seal Strength specs of 10 and 30 lbs. They took a random sample and found: an average of 25 lbs a standard deviation of 2.5 lbs. Compute Cpk, Cp and level. Cpk = Min {30-25, } / (3 x 2.5) = 5 / 7.5 = 0.67 level = min {30 25, 25-10} / 2.5 = 5 / 2.5 = 2.0 DPM = x 1,000,000 = 22,750 (from statistics software) 23 Equivalence of Measures Cpk level DPM* , , , DPM drops as Cpk increases Double Cpk by halving standard deviation Increasing Cpk dramatically decreases DPM * For non-centered processes (ie., Cp Cpk) 24

13 Technical Notes Cpk vs. Ppk Standard Deviation Short-Term Capability ST = R / d 2 LT = Long-Term Capability i1 n 2 yi y n 1 Process Capability Measure C pk = Min { USL -, -LSL } 3 ST P pk = Min { USL -, -LSL } 3 LT NOTE: In theory ST < LT, thus C pk > P pk (theoretically) P pk is often the preferred measure of capability 25 Technical Notes Non-Normal Data Most common approach is transform data to make it look more Normal. Original Time-to-Beep Data to <= to <= to <= to <= to <= 54.0 USL 54.0 to 60.0 to <= <= time log(time) # Observations Normal Distribution Mean = Std Dev = KS Test p-value =.3398 Transformed Data to <= to <= to <= to <= Class to <= to <= log(usl) = log(60) = USL to <=

14 Webinar Outline Quick Review of Key Topics Process Capability Measures for Variable Data FDA & GHTF References Definitions of Cpk, Ppk, level, & DPM Technical Details Acceptance Sampling Plans for Variables Key Assumptions Motivational Example (Scenario 2) Deriving Sample Size Acceptance Sampling Plan Pathway Final Examples (Scenario 1) 27 Key Assumptions for Acceptance Sampling Plans (repeated) Results of inspections are valid (ie, measurement system is good and samples are randomly selected) Cost of 100% inspection exceeds the benefits Underlying probability distributions are appropriate Setting a performance goal is not permission to produce defectives it is guidance for disposition decisions 28

15 Scenario 2 for Variable Data 35 You sampled 100 temperature sensors 30 from a lot of 1,000 and created a histogram. 25 The upper and lower 20 spec limits (USL & LSL) are shown. 15 Is there an simple way 10 to express how well or poorly your supplier of 5 temperature sensors is performing? 0 LSL USL Process Capability Measures for the Sample of 100 Sensors LSL USL N = 100 Mean = Std Dev = USL = 98.9 LSL = 98.3 Cpk = 1.73 Sigma Lvl = 5.19 DPM = We would like a Cpk > 1.5. Should we be happy with our supplier of sensors? LSL USL

16 Key Points to Consider About a Result from a Sample Sample mean and standard deviation are estimates of the true parameters of the lot. Using them to calculate Cpk also gives an estimate of the true value for Cpk. Could get over- or under-estimate of the true Cpk and DPM Intuition says that large sample sizes help reduce the risk of over or under estimating. That is correct but how much is enough? Need a way to quantify the risk in deciding that the lot is acceptable 31 Confidence Intervals for Cpk 1. Take a sample (n) 2. Compute estimate for Cpk Area = 3. Compute lower bound for confidence interval based on: Sample size (n) Estimated Cpk (Est Cpk) Lower Confidence Level (1- )% Bound Estimated Cpk Probability distribution for Cpk Lower Bound* = Est Cpk Z 2 1 estcpk 9n 2n 2 *Ref: Crossley, Mark, The Desk Reference of Statistical Quality Methods (2 nd Ed), ASQ Quality Press, 2007, pg

17 Computing Confidence Interval for Cpk Using SPC XL Cp and Cpk Confidence Interval (lower bound) User Defined Parameters Sample Size 100 Sample Mean Sample Standard Deviation Upper Spec Limit (USL) 98.9 Lower Spec Limit (LSL) 98.3 Confidence Level 95.00% Statistics and Confidence Intervals Cp 1.79 Lower Bound for Cp 1.58 Cpk 1.73 Lower Bound for Cpk 1.52 Based on our sample of 100 sensors we are 95% confident that the lot has a Cpk of at least Since our goal was Cpk > 1.5, we will accept the lot of 1,000 temperature sensors from our supplier. 33 Computing Confidence Interval for Cpk Using Minitab 34

18 Discussion of Results If Lower Bound Cpk > Desired Cpk, then we have met our goals for Desired Cpk with a certain level of confidence Notice for this scenario, our Estimated Cpk was 1.73, which is 0.23 higher than the Desired Cpk of Cp and Cpk Confidence Interval (lower bound) User Defined Parameters Sample Size 100 Sample Mean Sample Standard Deviation Upper Spec Limit (USL) 98.9 Lower Spec Limit (LSL) 98.3 Confidence Level 95.00% Statistics and Confidence Intervals Cp 1.79 Lower Bound for Cp 1.58 Cpk 1.73 Lower Bound for Cpk 1.52 For this scenario we can say: We are 95% confident that the lot has a Cpk of at least Illustration of Decision Logic for Sample Size 1) Decide on a minimum acceptable Cpk ( Desired Cpk or Spec Cpk ) 2) Take a sample & compute a Cpk for the sample ( Estimated Cpk ) 3) Compute a confidence interval to get a Lower Bound Cpk 4) If Lower Bound Cpk < Desired Cpk, then the risk is too great that the lot s Cpk is less than the Desired Cpk. 5) If we sample more, the lower bound should move up. So we can sample more to see if we can get Lower Bound Cpk > Desired Cpk. Desired Cpk Lower Bound Cpk Desired Cpk Estimated Cpk Ideal Situation After Sampling More Estimated Cpk Lower Bound Cpk 36

19 An Estimated Cpk must be greater than the Desired Cpk. But how much greater? Answer: The Lower Bound Cpk must be at or above the Desired Cpk Equivalent Statement: CI Width < Est Cpk Desired Cpk Introduction to Sample Size for Cpk Area = Limiting Case CI Width Lower Est Cpk Bound = Desired Cpk Confidence Interval (CI) = Width Z 2 1 estcpk 9n 2n 2 37 Determining Lower Bound Cpk Area = CI Width Confidence Interval (CI) = Width Z 2 1 estcpk 9n 2n 2 Lower Est Cpk Bound Cpk Lower Bound Cpk = Est Cpk - CI Width 38

20 95% Lower Bound Cpk Table Lower Bound Cpk = Est Cpk - CI Width Sample Estimated Cpk Size Determining Sample Size for Variable Data Sampling Plan Confidence Interval (CI) = Width Z 2 1 estcpk 9n 2n 2 Confidence Level (1- )% Estimated Cpk Sample Size (n) max CI Width = Est Cpk Desired Cpk 40

21 Cpk Sample Size Table Sample Size Needed for 95% Confidence That At Least the Desired Cpk is Achieved Est Cpk - Desired Cpk Desired Cpk Acceptance Sampling Plan for Inspection by Variables 1. Decide on a Desired Cpk and a Confidence Level, ( )%. 2. Determine an Estimated Cpk based on a small sample, say 10 items. 3. Is Est Cpk < Desired Cpk? YES: Then sampling more will probably not help. Decide on lot disposition. DONE NO: Compute the Lower Bound Cpk using Table. 4. Is Lower Bound Cpk > Desired Cpk? YES: Then accept the lot with at least ( )% confidence. DONE NO: Go to Cpk Sample Size Table (previous slide) and decide on how many additional samples to take. 5. Take a additional sample and compute Cpk based on all samples. 6. Compute the Lower Bound Cpk using Table. 7. Is Lower Bound Cpk > Desired Cpk? YES: Then accept the lot with at least ( )% confidence. DONE NO: Decide on lot disposition. DONE 42

22 Acceptance Sampling Plan Pathway Decide on Desired Cpk and Conf Lvl (1- ) Take a small sample (n>10) Est Cpk > Desired Cpk? Yes A No B A Get Lower Bound Cpk from table Lower Bnd Cpk > Desired Cpk? Yes Accept the Lot No Cpk based on small sample? Yes Get Cpk sample size from table Take additional sample & compute Est Cpk A No B Decide on Lot Disposition 43 Case 4 Analysis of Results Cpk for Each Batch s ph Goal: Cpk > 1.33 Each Sample Size = 16 Cpk Line 1 Batch 1 Batch 2 Batch 3 Batch Cpk Line 2 Batch 1 Batch 2 Batch 3 Batch Case 1 Case 3 Case 2 Analysis is incomplete. Need to assign a level of risk to these performance capability measures. 44

23 Case 1: Est Cpk = 2.07 Basic Information: Estimated Cpk = 2.07 for n=16 Want 95% Confidence Level that Cpk > 1.33 Pathway Result Est Cpk > Desired Cpk Lower Bound Cpk 1.36 (from Table) Lower Bound Cpk > Desired Cpk of 1.33 Accept the lot with 95% confidence that Cpk > Case 2: Est Cpk = 1.71 Basic Information: Estimated Cpk = 1.71 for n=16 Want 95% Confidence Level that Cpk > 1.33 Pathway Result Est Cpk > Desired Cpk Lower Bound Cpk 1.16 < Desired Cpk of 1.33 Sample Size from Table = 30 (using interpolation) Must sample additional: = 14 Compute Estimated Cpk based on all 30 samples Compute Lower Bound Cpk based on 30 samples If Lower Bound Cpk > 1.33, then accept lot otherwise decide on lot disposition 46

24 Case 3: Est Cpk = 1.56 Basic Information: Estimated Cpk = 1.56 for n=16 Want 95% Confidence Level that Cpk > 1.33 Pathway Result Est Cpk > Desired Cpk Lower Bound Cpk 1.06 < Desired Cpk of 1.33 Sample Size from Table = 68 (using interpolation) Must sample additional: = 52 Must decide if sampling this much is worth the investment. NOTE: For a Cpk at the lower bound (1.06) we will have fewer than 1,350 DPM or a defect rate lower than 0.135%. 47 Case 4: Est Cpk = 1.28 Basic Information: This is the first batch produced Estimated Cpk = 1.28 for n=16 Want 95% Confidence Level that Cpk > 1.33 Pathway Result Est Cpk < Desired Cpk so go directly to deciding on lot disposition Potential disposition decisions: Sample more to bring up lower bound Cpk. Then decide on disposition based on potential DPM or defect rate. Blend with Batch 3 that has Cpk = 2.07 Scrap this lot and caulk it up to learning curve cost 48

25 Key Takeaways Computing Cpk or Ppk alone is not enough Must establish level of risk with your decision Must match assumptions Can review results in terms of risk to the customer and DPM 49 Webinar Series on Statistical Methods & Tools Scheduled for 2014: Sampling Plans Part 3 July 16 Acceptance Sampling Plans for Attribute Data Future Webinars Based on Your Feedback: Sampling - How Much is Enough? Keeping Score - Cpk, FPY, and other process capability metrics Visualizing Your Data - Quick & Cheap Tricks Control Charts - Visually Monitoring Your Processes Design of Experiments A Three-Part Series 50

26 On-Site Workshops Available Statistical Methods & Tools for a Quality System 3-day Workshop Hands on with user-friendly, Excel-based statistical software Can integrate your challenges or data into workshop No prerequisite knowledge of statistics necessary Design of Experiments for a Quality System 3-day Workshop Hands on with user-friendly, Excel-based design of experiments software Can integrate your challenges or data into workshop No prerequisite knowledge of statistics necessary 51 Questions? Type your question in the Q&A box on the left side of your screen and press Enter Or press *1 on your telephone keypad 52

27 Closing Reminders Be sure to fill out the evaluation form at: Optional exam for this webinar is available through the elearning Portal at: AAMI is planning the following webinars that may be of interest to you: July 16: Sampling Plans Part 3 of 3: Attribute Acceptance Sampling Plans 53 Closing Reminders Announcing AAMI University - a better way to manage your professional development Online and live comprehensive education resources for medical technology professionals Access to AAMI s industry-leading curriculum and instructors Please visit AAMI U at Learn. Think. Implement. 54

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