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.
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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.
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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.
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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