# Ten Deadly Statistical Traps in Pharmaceutical Quality Control by C2bZ8rAz

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```									Ten Deadly Statistical Traps in
Pharmaceutical Quality Control

Lynn Torbeck

Pharmaceutical Technology
29 March 2007

1

“In theory there is no
difference between
theory and practice, but
in practice there is.”
Yogi Berria
2

1. Graphs
2. Normal Distribution
3. Statistical Significance
4. Xbar 3S
5. %RSD

3

6. Control Charts
7. Setting Specifications
8. Cause and Effect
9. Variability
10. Sampling Plans

4
Graph? What &%\$# Graph?

 Q#1 “Have you graphed the data?”
 I have solved many statistical problems by
simply graphing the data.
 Always, always, always plot your data.
 No ink on the page that isn’t needed.
 Cause and effect on the same page.
 Make the answer appear obvious.
5
Anscombe’s Astounding Graphs
Average    9.0     7.5      7.5      7.5      9.0      7.5
Std Dev   3.32     2.03     2.03     2.03     3.32     2.03

X Axis Y Axis 1 Y Axis 2 Y Axis 3 X Axis 2 Y Axis 4
10.0    8.04     9.14     7.46          8 6.58
8.0    6.95     8.14     6.77          8 5.76
13.0    7.58     8.74    12.74          8 7.71
9.0    8.81     8.77     7.11          8 8.84
11.0    8.33     9.26     7.81          8 8.47
14.0    9.96     8.10     8.84          8 7.04
6.0    7.24     6.13     6.08          8 5.25
4.0    4.26     3.10     5.39         19 12.5
12.0   10.84     9.13     8.15          8 5.56
7.0    4.82     7.26     6.42          8 7.91
5.0    5.68     4.74     5.73          8 6.89       6
Anscombe’s Astounding Graphs

N=11
Average of X’s = 9.0
Average of the Y’s = 7.5
Regression Line Y=3+0.5X
R2 = 0.67
Std Error of the Slope = 0.118
Residual Sums of Squares = 13.75
7
12.00

10.00

8.00

6.00

4.00
y = 0.5001x + 3.0001

2.00

0.00
0.0   2.0   4.0   6.0   8.0      10.0    12.0    14.0   16.0
X                                      8
14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.00
0.0   2.0   4.0   6.0   8.0   10.0   12.0   14.0   16.0
X                                 9
14

12

10

8

6

4

2

0
0   5   10   15   20
X2             10
12.00

10.00

8.00

6.00

4.00

2.00

0.00
0.0   2.0   4.0   6.0   8.0   10.0   12.0   14.0   16.0
X                                 11
Prolonged Acting Pro-Stuff

An ulcer drug from the late 1960’s.
In 1980 a change in a raw material resulted
in more rejects.
In-process control using a UV assay
Composite of 5 tablets assayed

12
Prolonged Acting Pro-Stuff

Sample from the top of each can
Specs were 95% to 105%
If value in spec, accept the can
If value out of spec, reject the can
Accepting and rejecting specific cans
About 50% of the cans were rejected

13
Histogram of UV Assay
90        95             100           105         110
9

8

7

6
Frequency

5

4

3

2

1

0
90   93        96      99       102    105   108
UV Assay                          14
Histogram of UV Assays
90        95         100   105   110
14

12

10
Frequency

8

6

4

2

0
80   85       90        95         100   105   110
UV Assays                         15
Histogram of Retests
90          95              100     105         110
3.0

2.5

2.0
Frequency

1.5

1.0

0.5

0.0
88        92          96         100   104     108
Retests                         16
Prolonged Acting Pro-Stuff

No good cans or bad cans.
Some “good” cans when retested are now
out of specifications.
The cans accepted are just as bad or good
as the cans rejected.
45% of the values are OOS
The product was taken off the market.
A personal story
17
Shipping Decision
3

2.5
Number of Complaints

2

1.5

1

0.5

0

-0.5

-1
20   30   40      50        60       70   80   90
Outside Temperature              18
A Little Normal History

The concept of the Normal is basic.
Also called Gaussian or Bell Curve.
First published in November 12, 1733.
First set of tables in 1799 !
Used by the astronomer Laplace for errors.
First called the Normal in 1893 by the
statistician Karl Pearson.
19
They Were Blown Away

“I know of scare anything so
apt to impress the imagination
as the wonderful form of
cosmic order expressed by the
‘Law of Frequency of Error.’”
Francis Galton in Natural Inherence, 1888

20
Histogram of All Data
Normal
80   85   90       95      100   105   110   115
18                                                      Mean 95.98
StDev 4.787
16                                                      N        77

14

12
Frequency

10

8

6

4

2

0
80   85   90       95      100   105   110   115
All Data
21
Hunting the Elusive Normal

I have never met a real Normal
distribution. Gotten close a couple of
times.
There are no real Normal distributions
It’s a theoretical fiction that is useful part
of the time.
We must separate reality from theory.

22
“Normal Distribution”

-6                           +6
-3            +3
Mean

23
Normal Facts

In theory, the tails of the distribution
stretch from minus infinity to plus infinity,
but there are real physical limits.
It is unique in that it is fully described by
just its mean, mu, , and its standard
deviations, sigma, , which are almost
never actually known for certain.
Probabilities are represented by areas.
24
What’s Normally Normal?

Tablet and capsule weights
Most manufactured parts
Student test scores, the ‘bell curve’ again
Things that grow in nature:
–   Apples
–   Bird eggs
–   Flowers
–   Peoples heights
25
Ain’t Never Gonna be Normal

Particle sizes
LAL, EU/mL
Bioburden, cfu/mL
Failures of most anything
Telephone calls per unit of time
Church contributions
Floods
26
Watch Out!

The tails are the most volatile and unstable
But, that is often the area of most interest!
Difficult to tell if data are normally
distributed by looking at a small sample.
Crude rule is that we need at least 100
representative data values to determine if it
is even approximately normal.

27
Statistical Significance:
Who Cares ?
The role of statistical analysis is as an
additional tool to assist the scientist in
making scientific interpretations and
conclusions and not an end in itself.

28
Differences

A scientific analysis often takes the form
of looking for significant differences.
Is drug A different from drug B?
Is the increase in yield significantly better
with the new centrifuge?
A difference can be significant in two
ways, practical and statistical.

29
Practical Significance

Practical significance comes form
comparing a difference to an absolute
reference or absolute truth.
How big a difference can you accept for:
–   Number of seconds of tooth pain?
–   Number of phone rings before hanging up?
–   How long will you wait for a bus?
–   How big your next raise is?

30
Statistical Significance

Statistical significance testing is one of the
great tools of statistics and science.
Statistical significance comes from
comparing a difference, a signal, to a
relative reference of random variability or
the best estimate of noise in the data.

31
Practical vs.Statistical

Practical Significance always wins and
takes precedence over statistical
significance!
In most applications, statistical
significance should not be tested until
practical significance is found.

32
Are The Analysts Different?

 Sam                    Barb
 98.2                   100.2
 99.3                   100.5
 99.7                   100.8

 Xbar=99.1              Xbar=100.5

 Spec= 90.0 to 110.0    Two Sided t, P=0.04
33
Signal to Noise

All statistical significance testing is only a
comparison of the signal to the noise.
If the signal can be shown to be larger than
the noise, than we would expect by chance
variation alone, we say it is significant.
Bigger signal more significant.
Smaller noise more significant.

34
Significance?

Practical /           NO                    YES
Statistical
Nothing going           1.   May be due to
NO           on here it                   chance.
seems.                  2.   May need
more data.
1. Small noise          Great!
YES           2. Large sample size.
Everybody is
What does it mean?
happy.
35
Why Do It To It?

 The primary purpose of statistical tests of
significance is to prevent a us from accepting an
apparent result as real when it could be just due
to random chance.
 Statistical significance without practical
significance could in some circumstances be a
 What if the spec was changed to 98.0 to 102.0?
 We may want to find out why different

36
The Biggest Lie in Statistics?

Is Xbar±3S ever Correct?
For ever complex problem there is a
solution that is quick, simple,
understandable and absolutely wrong!
More grief has been perpetuated by this
formula than any in statistics.

37
The Biggest Lie in Statistics?

What is true is that     3  will bracket
99.73% of the area under the normal cures.
Note that this assumes we know the true
values for the mean mu, , and standard
deviation, sigma, , which we never do of
course. We have to estimate them with the
small samples we take.
Thus, there is uncertainty in the estimates.
38
Side Line

Did you hear about the statistician’s wife
who said her husband was just average?
She was being mean.

39
So, What Do I Do Now?

Don’t use Xbar±3S as generalized monkey
wrench and apply it to all of your statistical
questions. Use the right tool for the job.
Use Confidence Intervals to bracket the
unknown mean.
Use Tolerance Intervals to bracket a given
percentage of the individual data values.

40
%RSD: Friend or Foe?

S= SQRT[(X-Xbar)2/(n-1)]
%RSD = (100 * S) / Xbar
They are two different summary statistics
They measure two different concepts
They are not substitutes for each other
We need to report both.

41
Control Charts

Having just told you not to use Xbar±3S, I
now have to tell you that is how control
charts define the control limits.
This is an artifact of history.
Control charts were developed by Dr.
Walter Shewhart in 1924 while working at
Western Electric in Cicero Ill.

42
Control Chart
I and MR Chart for Yield %
103.5
limits to a line                                                                                UCL=103

Individual Value
102.5
101.5
plot.                                   100.5
Mean=100
99.5

 A chart for the                               98.5
97.5
LCL=97
response.                    Subgroup
96.5
0               50               100

 A chart for the                                      4
UCL=3.686

moving range to                       Moving Range
3

estimate                                             2

R=1.128
1
variability.                                         0                                          LCL=0

43
Do You Trust Your
Control Chart?
Control charts are crude tools and not exact
probability statements.
They don’t take into account the number of
samples in the data set for the limits.
They are intended as early warning devices
and not accept/reject decision tools.
Don’t use for large \$\$ decisions.

44
Oh Wow, I Don’t Believe It !

You did what to set the
specification criteria for
product?

45
Setting Specifications

A specification is a document that contains
methods and accept/reject criteria
Criteria can be determined several ways
–   Wishful thinking
–   Clinical results
–   Compendial standards
–   Historical data and statistics

46
Million \$\$ Decisions?

Regulatory Limits - External
Release: accept/reject - Internal
Action limits
– Warning limits
– Trend limits
– Validation limits

47
Idealized Specification Limits

Action

/
Accept Reject

Regulatory
48
Calculating Criteria

Don’t use Confidence Intervals, they
shrink toward zero with large sample sizes.
Don’t use X bar ± 3 S. They are too
narrow for small sample sizes
Use Tolerance Intervals, preferably
99%/99%. This will take into consideration
the sample size and uncertainty of the
average and the standard deviation.

49
Setting Specification Criteria

For action limits, expect the average to
vary and widen the Tolerance Limits
For accept/reject limits, add a further
allowance for stability.
Consider the clinical results when possible
as part of the justification for limits.

50
Drunken Teachers

Did you know that there is a positive
correlation between alcohol consumption
and High School teacher’s salaries?
That there is a negative correlation
between average student’s test scores for a
state and the distance of the state capital

51
Cow Magnets Cure Gout

What’s a cow magnet?
What is gout?
How do we test a cause and effect
relationship to see if this works?
Should we just ask people what they think?
“No causation without manipulation.”
Gold Standard is double blind clinical trial.
52
Variability is the Enemy

How many OOS values were documented
in the lab last year?
How many manufacturing deviations were
investigated last year?
How many lots were rejected last year?
How many of your quality problems would
go away if there were no variation?

53
Misconceptions of variability

We have variability because the equipment
needs to be replaced with new technology.
We do too many tests.
Variability exists because some idiot didn’t
do their job correctly.
Variability is an inherent fact of life and
there isn’t a darn thing we can do about it
except to live with it. It’s cost of business.
54
Variability is the Enemy

“Special Cause” variation is the result of a
single source. Use CAPA to solve it.
“Common Cause” variation is the result of
multiple small sources all contributing to
the sum total.
CAPA will not work for common cause
We need a culture change to address
common cause variation
55
Sources of Variation:

Common cause variation:
–   People
–   Materials
–   Methods
–   Measurement
–   Machines
–   Environment

56
Common vs. Special Causes

 A plot of the data                                   I Chart for Yield%
with X bar ± 3 S                         106
illustrates common                       105
1

cause variation.                         104

Individual Value
103                                    UCL=103
 A value that is                          102
larger than would                        101

be expected by                           100                                    Mean=100
99
chance alone is                           98
assumed to be due                         97                                    LCL=97
to a special cause.                       96
0          50              100
Observation Number
57
Deming’s Message

Dr. W. Edwards Deming was the very
famous statistician that taught statistical
quality control to the Japanese in the 50’s.

“If I had to reduce my message for
management to just a few words, I’d say it
all had to do with reducing variation.”

58
Deming’s Message

If you reduce variability, you will reduce
scrap, rejects and rework. You can then
make a better product at less cost. You will
capture a larger market share. Your people
will be employed and you will prosper.

• Paraphrase of Deming’s message

59
Confronting the Enemy

Operational Definitions
Achieve the Target
Flexible Consistency
Hold Constant Controllable Factors
Mistake Proofing
New Technology
Continuous and forever improvement
60
The Black Hole of Quality

Like a black hole with light, sampling
plans just suck the common sense right out
of people’s brains.
Normal, logical and rational people
suddenly become willfully and terminally
stupid.
what sampling plans can and can not do.

61
Black Hole Facts

A sample is only a small part of the whole
Each sample is going to be different
Some samples will have many defects
Some samples will have few defects
Bigger sample, better estimate.
On average, the defect percent can only be
estimated and not known perfectly.
62
Black Hole Facts

There is a small but real probability that a
good lot of product will be rejected.
Called the “Producer’s Risk, usually 5%.
There is a small but real probability that a
“Consumer’s Risk, usually 5% or 10%
Most common plan is ANSI/ASQ Z1.4.

63
Black Hole Facts

“The AQL is the quality level that is the
worst tolerable process average … .”
“The acceptance of a lot is not intended to
“The standard is not intended as a
procedure for estimating lot quality or for
segregating lots.”

64
Black Hole Facts

“The purpose of this standard is, through
the economic and psychological pressure
of lot non-acceptance, to induce a supplier
to maintain a process average at least as
good as the specified AQL while at the
same time providing an upper limit on the
consideration of the consumer’s risk of
accepting occasional poor lots.”
65
Misunderstandings

Double and multiple sampling plans are
not testing into compliance.
It is not possible to have an AQL=0.0
Accept on zero, reject on one is not always
the best plan for critical defects.
If the lot size is ten times or more than the
sample size, then the lot size doesn’t
matter.
66
Summary

“Statistical thinking will
one day be as necessary for
efficient citizenship as the
H. G. Wells

67
References

NIST online statistics textbook
– http://www.itl.nist.gov/div898/handbook/inde
x.htm
Edward Tufte’s website
– http://www.edwardtufte.com/tufte/
W. Edwards Deming’s book
– Out of the Crisis

68
References

Torbeck, Lynn.,Using Statistics to Measure
and Improve Quality, DHI Publishing
2004.
De Muth, James (1999). Basic Statistics
and Pharmaceutical Statistical
Applications, Marcel Dekker.

69
“That’s All Folks”

Thank you !
Questions ?

70

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