Statistical control to monitor
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This document discusses statistical process control and monitoring validated processes. It describes determining what processes need to be monitored based on risk assessments and process validations. It also covers the basics of statistical process control including common and assignable cause variation, setting control limits, and using tools like X-bar and R charts to monitor processes. Western Electric rules for determining when a process is out of control are also summarized.
Statistical control to monitor
- 1. Statistical Control to Monitor and Control Validated Processes. Session 4 Steven S. Kuwahara, Ph.D. GXP BioTechnology LLC PMB 506, 1669-2 Hollenbeck Avenue Sunnyvale, CA 94087-5402 USA Tel. & FAX: (408) 530-9338 e-Mail: s.s.kuwahara@gmail.com Website: www.gxpbiotech.org GMPWkPHL1012S4 1
- 2. What to Monitor 1. • All functions mentioned in the GMP are GMP processes. – Not all GMP processes need to be monitored. – The processes that introduce variability into the product or can cause the production of unacceptable product need to be monitored. – This goes beyond manufacturing processes. – QC test methods need to be monitored. – The annual product review needs to be monitored to see if batch records are being properly executed. – Processes that generate critical business information need to be monitored. GMPWkPHL1012S4 2
- 3. What to Monitor 2. • Use information from the process validation and risk assessment to determine what needs to be monitored. (You need proof of your assertion.) – Even if the process validation shows that a step is very stable, if the risk assessment shows that that step can create a serious threat to the product or patient, it should be monitored. – Steps or processes can be monitored. If there are multiple small steps in a process, the outcome of the process can be monitored in place of each of the steps. – If the outcome of a process or step is immaterial to the quality of the product, it does not need to be monitored. GMPWkPHL1012S4 3
- 4. Data 1. • There can be two types of data. – Variable data forms a continuum and an individual result can fall anywhere within it. Most statistical procedures are designed to deal with variable data. – Attribute data comes in discrete units. Yes/no: red, green, blue, or yellow; pass/fail; high, medium, low. – One of the problems with attribute data is that it may be subjective or actually part of a continuum. Is a thing red, reddish, pink, or brown, brownish, rust; what is high or medium; passing? – Since attribute data must be discrete, a clear and firm definition is needed. GMPWkPHL1012S4 4
- 5. Data 2. • There are statistical methods for dealing with attribute data, especially since they may not follow the normal distribution. – One technique is to convert attribute data into variable data by using fractions or percentages, but this is good only with large numbers. With small numbers the discreteness creates large jumps. – Attribute data that is in binary (+/-, yes/no) form will follow a binomial distribution. – Attribute data is not as strong as variable data so you need more of it. • In what follows we will assume that you have variable data. GMPWkPHL1012S4 5
- 6. TOLERANCE vs. CONFIDENCE LIMITS With confidence limits you are trying to determine the value of the “true mean” (µ). The result gives an interval within which you expect to find µ with a certain probability (confidence). The results are means with the same n that was used to calculate the limits. With a tolerance interval you are asking for the values of the next n numbers of results. The result gives an interval within which you have x % confidence that it will contain y % of the results. It assumes that all of the results are from the same population. The result here is a single number. 6 IVTPHL1012S1
- 7. TOLERANCE LIMITS FOR A SINGLE RESULT 1. • DEFINITION: ± ks. Where k is a value to allow a statement that we have 100(1 - α) percent confidence that the limits will contain proportion p of the population. • Requires a normal distribution of the population. • Allows one to set limits for single determinations (not averages) with a limited number of replicates available. IVTPHL1012S1 7
- 8. TOLERANCE LIMITS 2. Selected Portions of a Table for Two-sided Limits • k 95% Confidence 99% Confidence • n p=95% p=99% p=95% p=99% • 2 37.67 48.43 188.5 242.3 • 3 9.916 12.86 22.40 29.06 • 4 6.370 8.299 11.15 14.53 • 5 5.079 6.634 7.855 10.26 • 10 3.379 4.433 4.265 5.594 • 20 2.752 3.615 3.168 4.161 • 50 2.379 3.126 2.576 3.385 • Table XIV, Applied Statistics and Probability for Engineers, D.C. Montgomery & G.C. Runger, John Wiley & Sons, 1994. IVTPHL1012S1 8
- 9. TOLERANCE LIMITS FOR A SINGLE RESULT 3. Example • A product is made at 2.25 mg/mL. n = 10 samples are taken from 5 early lots. The assays show 2.27, 2.25, 2.24, 2.22, 2.26, 2.23, 2.23, 2.24, 2.27, and 2.25 mg/mL. We calculate = 2.246, s = 0.017. From the table at n = 10, 99% confidence for 95% of results k = 4.265 so the Tolerance Interval is: • 2.246 ± (4.265)0.017 = 2.319 - 2.173. • We have 99% confidence that 95% of the results from this population will be in the interval. • The 95% confidence interval for the mean is ± 0.012, giving 2.258 to 2.234. IVTPHL1012S1 9
- 10. Statistical Process Control (SPC) Basics. 1. • At some level of discrimination, NO TWO THINGS ARE ALIKE. There is always some variation. • If things are not being hand made, the variation should be small enough to make each unit interchangeable with any other unit. – A 100 mg tablet may vary by ± 1 mg and be okay, but not a 1 mg tablet. • The allowable variation should not be larger than the variation that causes a unit to be “different.” GMPWkPHL1012S4 10
- 11. Statistical Process Control (SPC) Basics. 2. • In the old days, people made many units of product and the “production lot” underwent 100% inspection to segregate “good” units from the “bad.” “Bad” units were scrapped. Thus the name “Quality Control.” – It cost just as much to make scrap as to make a “good” unit. – 100% inspection is never 100% effective. – “Quality Control” was basically a sorting operation. Quality was really not controlled since the “bad” product had already been made. GMPWkPHL1012S4 11
- 12. Statistical Process Control (SPC) Basics. 3. • Walter Shewhart, in the late 1920’s, and, later, W. Edwards Deming basically proposed the idea that making scrap was not cost effective. • They proposed the idea that the elimination of scrap would produce the most cost effective manufacturing process. • Scrap was the result of excessive variation. • Variation comes in two forms. In the Shewhart/ Deming terminology there are common cause and assignable cause variations. GMPWkPHL1012S4 12
- 13. Statistical Process Control (SPC) Common Cause Variation. • Common (controlled) cause variation is the basic variation that is inherent in a process. It is the sum of all of the small, inherent, random errors associated with the process. Sources of error may be undetectable. • Since common cause variation is inherent in a process, it changes only when the process, or the way that the process is managed, changes. • Common cause variation, therefore, can only be reduced through the actions of management. GMPWkPHL1012S4 13
- 14. SPC: Assignable Cause Variation. • Assignable (special) cause variations result from unusual situations and are not built-in as a part of the process. • Because they arise from unusual events, assignable cause variations should be detectable by careful inspection of the system. • While management action may be needed, most assignable cause variations can be corrected by line workers themselves, if they have the required knowledge and experience. GMPWkPHL1012S4 14
- 15. X-bar Chart UCL x x x xx X bar x xx x x x x x x x xx LCL Time GMPWkPHL1012S4 15
- 16. R Chart • UCL o oo o Range oo o o o oo o o o o o o oo o o o o Time LCL GMPWkPHL1012S4 16
- 17. GMPWkPHL1012S4 17
- 18. Chart and Process Interpretation • Look at R chart first. x bar Chart is meaningless if R has changed significantly or is large. • If R has increased, identify any special causes that are responsible for the increased variability and change the process to stop them. • If R has decreased, identify special causes responsible and change process to incorporate them. GMPWkPHL1012S4 18
- 19. SETTING CONTROL LIMITS FOR YOUR SPC CHARTS. 1. • Control limits for the SPC charts should be incorporated as specifications, unless you are setting your release specifications to be wider than your control limits. • The problem here is that your SPC control limits should be based upon what your process is capable of delivering, if it is operating in a state of control. Therefore, if the limits are exceeded, it is an indication that the process is out of control, even if it is within the release limits. GMPWkPHL1012S4 19
- 20. SETTING CONTROL LIMITS FOR YOUR SPC CHARTS. 2. • First, calculate the average of the averages that you are using. This is: N ∑x1 x= N Now calculate the average range that comes from the ranges of the data that were used for the averages. This is: R GMPWkPHL1012S4 20
- 21. SETTING CONTROL LIMITS FOR YOUR SPC CHARTS. 3. • For “3-sigma limits” the Shewhart formulae are: UCLx = x + A2 R CL x = x LCL x = x − A2 R UCLR = D4 R CL R = R LCLR = D3 R if not negative. Note that D 3 = 0 for subgroup size < 7. GMPWkPHL1012S4 21
- 22. SETTING CONTROL LIMITS FOR YOUR SPC CHARTS. 4. • The constants, A2, D3, and D4 are combinations of other constants and are set for 3σ and change with the subgroup size. • The subgroup size (n) is the number of replicates for each average. (Also known as a “rational subgroup.”) • When possible, 20 to 30 subgroups should be used to establish the grand average and the average range. Fewer subgroups may be used in the beginning, but the numbers should evolve until 30 is reached. • The idea is to reach the point where you are dealing with µ and σ, but recent computer studies have shown that instead of 30, the real number should be around 200. GMPWkPHL1012S4 22
- 23. Western Electric Rules 1. Zones +3σ +2σ +1σ -1σ -2σ -3σ GMPWkPHL1012S4 23
- 24. Western Electric Rules 2. • The process is out of control when any one of the following happens: • 1. One point plots outside the 3σ control limits. • 2. Two out of 3 consecutive points are beyond the 2σ limits. • 3. Four out of 5 consecutive points plot beyond the 1σ control limits. • 4. Eight consecutive points plot on one side of the center line (). • These are one-sided rules. They only apply to events happening on one side of the chart. – If a point is beyond +2σ and the next point plots beyond -2σ, they are not 2 points beyond 2σ. GMPWkPHL1012S4 24
- 25. Western Electric Rules 3. Considerations • Based on the work of Shewhart, Deming, and others when they were with Bell Telephone. • Because they are based on the standard deviation zones, these are sometimes known as “zone rules.” • These rules enhance the sensitivity of the control charts, but require the use of standard deviations, not ranges. • A “point” on a control chart usually represents an average that was calculated for a “rational group.” GMPWkPHL1012S4 25
- 26. Rational Subgroups • The subgroup that is chosen to represent a “point” should be chosen so that it properly represents the product unit. – If you are checking lots over time then the subgroup units should be randomly chosen from the lot. – If you are more interested in the different parts of a lot, you should define the parts and choose from within the “part.” – The number of units in the subgroup should be calculated as an “n” value based on your desired confidence interval and the level of risk that you are willing to accept. GMPWkPHL1012S4 26
- 27. CUSUM Charts I. • There are CUmulative SUM control charts. – They are based on the Cumulative Sum of the Deviations from a target value. • The CUSUM chart is a relatively new invention designed to overcome the lack of sensitivity of “Shewhart Charts” and the tendency of the charts to give false alarms, especially when the Western Electric Rules are employed. • CuSum chart: A quality control chart with memory 27 GMPWkPHL1012S4
- 28. CUSUM Charts II. • For deviations greater than ± 2.5σ the Shewhart chart is as good or may be better than the CUSUM chart. • CUSUM charts are more sensitive and will detect changes (especially trends) in the ± 0.5σ - ± 2.5σ range. 28 GMPWkPHL1012S4
- 29. Types of ‘Time-weighted charts • There are 3 types of time-weighted charts. • Moving Average • - Chart of un-weighted moving averages • Exponential Moving Average (EWMA) • - Chart of exponentially weighted moving averages • CUSUM • - Chart of the cumulative sum of the deviations from a nominal specification 29 GMPWkPHL1012S4
- 30. Equations for CUSUM j S j = ∑ (x j − µ ) for single point data, where x j i is the observation at j. j ( ) S j = ∑ x j − µ for averages, where x j i is the average at j. S j is the CUSUM point at j and µ is the target value. j is the time period over which you will sum the deviations. 30 GMPWkPHL1012S4
- 31. Starting a CUSUM I. • Specification: Target Value: 100 = µ • n xj Calculation CuSum value = Sj j = 8 • 1 99 = 99 – 100 = -1 • 2 101 = 99+101-2*100 = 0 • 3 99 = 99+101+99-3*100 = -1 • 4 100 =99+101+99+100-4*100 = -1 • 5 102 = 99+101+99+100+102-5*100 = 1 • 6 101 = 99+101+99+100+102+101-6*100 = 2 • 7 100 = 99+101+99+100+102+101+100-7*100 = 2 • 8 101 = 99+101+99+100+102+101+100+101-8*100 = 3 • Sj= (Σxj ) – jµ 31 GMPWkPHL1012S4
- 32. Starting a CUSUM II. • In theory, the sum of the deviations should be zero, if the process is in statistical control. • So the cumulative run length (j) should be large enough that you would expect it to be at zero, if the target value (µ) is really the center point. • The target value (µ) should be the specification for the measurement. • Therefore the center line in a CUSUM chart should be at zero. 32 GMPWkPHL1012S4
- 33. Shift then return to Normal Theory Reality 33 GMPWkPHL1012S4
- 34. The V-Mask I. • A statistical t-test to determine if the process is in a state of statistical control. The V-mask is placed at a distance d from the last point of measurement. The opening of the V-mask is drawn at an angle of ± θ. • First, obtain an estimate of the standard deviation or standard error (σx) for the value of Sj. This should be known from the specification setting process or from development work. 34 GMPWkPHL1012S4
- 35. The V-Mask II. • Decide on the smallest deviation you want to detect (D). Calculate: δ = D/σx. If D = σx then δ = 1 • Decide on the probability level (α) at which you wish to make decisions. – For the usual ± 3σ level, α = 0.00135. • Determine the scale factor (k) which is the value of the statistic to be plotted (vertical scale) per unit change in the horizontal scale (lot or sample number). It is suggested that k should be between 1σx and 2σx, preferably closer to 2σx. • Using the value of δ, obtain the value of the lead time (d) from the following table. 35 GMPWkPHL1012S4
- 36. The V-Mask III. • Obtain the angle (ϴ) from the same table by setting δ = D/k. (Table BB in Juran’s Book) • Construct the V-Mask from these data. • Truncated Table BB. δ ϴ d 0.2 5o 43’ 330.4 0.5 14o 00’ 52.9 1.0 26o 34’ 13.2 1.8 41o 59’ 4.1 2.0 45o 00’ 3.3 2.6 52o 26’ 2.0 3.0 56o 19’ 1.5 36 GMPWkPHL1012S4
- 37. V-Masked CUSUM d ϴ Action? Failure Time X-bar 37 GMPWkPHL1012S4
- 38. CUSUM Rules • The sample size can be the same as used in an chart. One recommendation is to use. • n = 2.25s2/D where s2 is an estimate of the process (lot to lot) variance. • The V-Mask is always placed at the last point measured. • If all points are within the V-mask, the process is under control. • Any point out of the V-Mask shows a lack of control. • The first point out of the V-mask shows when the shift started even if later points are within the masks. 38 GMPWkPHL1012S4
- 39. Moving V-Mask Failure 39 GMPWkPHL1012S4
- 40. References: ISO/TR 7871:1997: Cumulative sum charts -- Guidance on quality control and data analysis using CUSUM techniques • http://www.iso.org/iso/iso_catalogue/ catalogue_tc/catalogue_detail.htm? csnumber=14804 • British Standards No. BS 5700 ff SR, 05.03.2009, page 16. • Wadsworth, H.M.; “Statistical Process Control,” in Juran’s Quality Handbook 5th Ed.; Juran, J.M. and Godfrey, A.B. (eds.); McGraw- Hill, New York, NY,401999, Sect. 45, page 45-17. GMPWkPHL1012S4