Statistical Methods for Biotechnology Products－Part I: Biopharmaceutical Product Statistical Quality Control

(1)

110/07/16 Copyright by Jen-pei Liu, PhD, Prod ucts are just for illustrations onl y

1

Statistical Methods for

Biotechnology Products

Part I: Biopharmaceutical Product

Statistical Quality Control

by

Professor, Jen-pei Liu, PhD, Professor

Division of Biometry, Department of Agronomy

National Taiwan University, and

Division of Biostatistics and Bioinformatics

National Health Research Institutes

(2)

2

Statistical Quality Control



1. Introduction



2. Concept



3. Simple Graphical Techniques



4. Control Charts



5. and R Charts



6. Process Capability



7. P Charts



8. C Charts



9. Six Sigma Concept



10. Summary

(3)

3

1 Introduction



Products or service of the highest quality



Avoid defective products



Avoid customer complaints

Total Quality Control (TQC)

(4)

4

2 Concepts



Single Process

(5)

5

2 Concepts



Breakdown of a product or service into

(6)

6

(7)

(8)

8

2 Concepts

Statistical Process Control (SPC)

The use of statistical quality control

techniques is called statistical process

control

(9)

9

2 Concept

(10)

10

2 Concept

(11)

11

2 Concept



Costs of Inspection vs Cost of Undetected

(12)

12

2 Concepts



The control chart is based on sample information,

measurable or qualitative from the process at

different point in time



Control charts for variables(measurable quantity)



Control charts for attributes(attribute data)

(13)

13

2 concepts



On-line quality control technique

- control charts



Off-line quality control technique

- Experimental design

- Taguchi methods for optimizing the

process to set level key process

variables for yielding the highest

possible quality

(14)

14

3 Simple Graphical Techniques



Specification Limits : A variable should be if it

is to meet function and quality standards



The largest allowable value of a variable is

called the upper specification limit(USL)

and the lowest allowable value is the lower

(15)

15

3 Simple Graphical Techniques



Example: Copper thickness of 50 printed

circuit board

Specification:

0.001~0.003 in.

26/50(52%) not meet

specification

(16)

16



A Pareto chart is a bar chart where each bar is associated with a

particular area of concern and the bars are drawn, from left to right, in

order of decreasing height

PC Rejection DATA

3 Simple Graphical Techniques

Type of Defect Number of Rejected Boards

Poor electroless coverage

35 Lamination problems 10

Low copper plating 112

Plating separation 8

Etching problems 5

Miscellaneous 12

(17)

17



Pareto Chart

(18)

18



The fishbone diagram is a graphical

way of displaying the possible reasons,

or causes, of a particular problem

3 Simple Graphical Techniques

The fishbone diagram is called the

cause-and-effect diagram

(19)

19



Construction of a cause-and-effect diagram

(20)

20



A cause-and-effect diagram

(21)

21

4 Control Charts



To differentiate controlled

variable-assignable causes from uncontrolled

variables-chance variation

(22)

22

4 Control Charts



Example of assignable causes: types of

raw materials used, differences in

workers used, slow wearing down of the

machinery, changes in temperature or

humidity



Statistical Control:

When all the assignable causes have

been found and eliminated,a process is

then said to be statistical control

(23)

23

4 Control Charts

(24)

(25)

25



Centerline:grand average of all the sample

statistics



Lower and upper Control Limits:

Determined by the sampling distribution and

are positioned 3 standard deviations above

and below the centerline



Out of Control: Points outside the control limits



In control: Points within the control limits

(26)

26



Classification of Control Charts

(27)

27



Difference between Control limits and specification limits

(28)

28



The Chart monitors the means of small

samples taken from a process



The R Chart monitors the range(or variability)

of small samples taken from a process



The Control Limits of the Chart are

computed by using the centerline of the R

chart



The Chart should not be used without

constructing the corresponding R Chart

5 and R Charts

_X

x

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110/07/16 Copyright by Jen-pei Liu, PhD, Prod ucts are just for illustrations onl y 29

5 and R Charts



Data

Sample Number

1 2 ………. K

Sample Size

n n ……… n

Sample mean ……

Sample Range

R

1 R

2 ……. R

k

k=20 to 25 , n=3 , 4 or 5

X

1

x

₂

x

_k

(30)

30

5 and R Charts



The R Chart

Centerline:

Upper control limit UCL=D

2 Lower control limit LCL=D

1 D

1 and D

2 depend on sample size n and

can be found in table

If n  6; D

1 =0 LCL=0

X



k i i=1

1 R =

R

k

R

(31)

31

5. and R Charts



Product: Precision ball bearings



Characteristic: bearing diameter



Target value: 0.500 in.



Specification Limits:0.490-0.510 in.



Data: Hourly measurements of 5

learning diameters (x1000) from the

target value

(32)

32

(33)

5 and R Charts



N=5 K=25 R=6.56 in.

D

2 =2.115

D

1 =0

UCL=D

2 =(2.115)(6.56)=13.874

LCL=D

1 =(0)(6.56)=0

X

R

(34)

34

5 and R Charts



R Charts for ball bearing diameters

(35)

35

5 and R Charts



Charts



Centerline:



Upper Control limit: UCL=



Lower Control limit: LCL=



A: depends upon the sample size and can be found

in Table

X



k i i=1

1 x=

x

k

2 x



A R

2 x



A R

(36)

5 and R Charts



Example: K=25 n=5

=0.048

=6.56

A

1

=0.577

UCL=0.048+(0.577)(6.56)=3.83

LCL=0.048 -(0.577)(6.56)=-3.74

X

x

R

(37)

37

5 and R Charts



Chart for ball bearing diameters

X

(38)

(39)

(40)

40

1. Most sample means are scattered at

random around the central line

2. A few sample means spread out and are

close to either the lower or upper control

limit.

3. No sample mean is outside the control

limits.

4. No recognized pattern of the distribution of

sample means is observed.

(41)

41

(42)

42

1. A sample mean is outside the control limits 3

The probability of this event under normality

assumption is about 0.0027. In general, from

Chebyshev's inequality, there is a probability of 0.11

that a sample mean is outside the three standard

deviation control limits (or action limits).

2. Under normality assumption, the probability of two

out of three consecutive means outside two

standard deviation warning limits is

(43)

43

3. Eight consecutive sample means are on the

same side of the central line. The probability

of this event for any distribution symmetrical

about the population mean is

4. Sometimes, it is also worthwhile noticing that

four of five successive sample means are

outside the one standard deviation limits

because that probability of this event under

the normality assumption is given by

8

2(0.5)



0.00078.

4

5 (0.3174) (0.6826) 0.0346.

4  

_

 

 

(44)

44

5. In many cases, samples collected from two or

more underlying distributions are combined

together for construction of control chart. An

unnatural pattern of all sample means

scattering around the central line with

unnaturally small fluctuations will occur. This

type of control charts is invalid because

samples from different distributions have

been combined. This phenomenon is referred

to as stratification

.

(45)

45

6. If half the samples come from one

distribution and the other half come from

another distribution, the sample means will

scatter within the control limits rather than

concentrate on the central line. This

(46)

46

7. The existence of a trend for a manufacturing

process can be identified by the following

unnatural patterns:

(a) One sample mean outside the control

limits on one side is followed by the next

consecutive sample mean, which is outside

the control limits on the other side.

(b) There are at least six consecutive sample

means, one of which is greater (or lower)

(47)

47

8. The presence of a systematic variable in the

process is suggested if at least eight

consecutive sample means alternate large,

small, large,and small without interruption.

This pattern may occur when samples are

selected alternately from different operators

or machines.

(48)

(49)

(50)

50

6 Process Capability



Process capability refers to the ability of

a process to stay within its specification

limits



Actual process spread: Under normal

assumption,all measurements

generated by the process should be

within a range of 6



Allowable process spread: The distance

between the specification limits

( 3 )

(51)

51

6 Process Capability



The process capability index is defined by

ˆ

p

USL-LSL

C =

6σ

Where is the SD of measurements from the

process

ˆ



A c

_p

1.67 Continue to maintain

B 1.33 c

_p

<1.67 Improve to A

C 1.00 c

_p

<1.33 Improve at once

D 0.67 c

_p

<1.00 Consider stopping product

E c

_p

<0.67 Stop production at once





(52)

52

(53)

53

(54)

6 Process Capability

The C

pk

Index

,

ˆ







pk

USL- x

C =min

3σ

Min=minimum

K-factor :









USL+LSL 2-x

k=

USL-LSL 2

ˆ







x-LSL

3σ

0≤k≤1

The relationship between c

_p

and c

_pk

C

_pk

=C

_p

(1-k)

(55)

55

6 Process Capability



Data:Ball bearing diameters

Target value: 500 USL=10 LSL=-10

6.56 R





 

ˆ

6.56

2.820

2.326 R 



10 ( 10)

1.18 ˆ

6 6(2.820)

p

USL LSL

C





 



ˆ







pk

USL - x

C =min

,

3σ

ˆ







x-LSL

3σ

10 0.048 0.048 ( 10)

min

,

min[1.176,1.188] 1.176

(3)(2.820)





 





_

_







(56)













USL+LSL 2-x

_{10-(-10) 2-0.048}

k=

=

=0.048

USL-LSL 2

10-(-10) 2

C

_pk

=C

_p

(1-k)=(1.182)(1-0.0048)=1.176

(57)

57

7 The P-Chart



A control chart for controlling the

percent of defectives in a sample is

called a P chart

Sample Number 1 2 … k

Sample Size n

1 n

2 … n

k

#

of defectives x

1 x

2 … x

k

% of defective

ˆp

1 ˆp

2 ...

p

ˆ

_k

(58)

58



Centerline :



Upper control limits UCL=



Lower control limits LCL=

7 The P-Chart

1

1 ˆ

k

i

p

k

_





(1- )

i

p

n



(1- )

i

p

n



(59)

59



# of rejected circuit boards

(60)

60



Since =0.054 and =305(given in table

16.11),the average sample size method gives

control limits of

7 The P-Chart

p(1-p)

(.054)(.946)

UCL =p+3

=0.54+3

=0.093

305 n

p(1-p)

(.054)(.946)

LCL =p-3

=0.54-3

=0.015

305 n

p

_n



p(1-p)

p 3

n

or

(.054)(.946)

0.054-3

286 or

.054 0.40



(61)

61

7 The P-Chart

(62)

62

7 The P-Chart

(63)

63

8 The C-Chart



A C-chart is used when the objective is to control

the number of defects per unit



A defective item may have more than one defect



A defect is just a flaw or nonconformity



To monitor defects requires determination of the

inspection unit



Example:



Accounting records: # of error per 10 records

(64)

64

8 The C-Chart (Poisson Distribution)



Inspection unit 1 2 …K



#of defects C

₁

C

₂

…C

_k



Centerline:



Upper Control limit UCL=



Lower Control limit LCL=

3 c



c

3 c



c

1

1 k

i

c

k

_





(65)

65

8 The C-Chart

(66)

8 The C-Chart

50

1

12.54

50 _i

i

c







3 12.54 3 12.54 23.16

UCL c

 

c







3 12.54 3 12.54 1.92

LCL c

 

c







(67)

67



Solder Defects

(68)

68

9 Six Sigma Concept



A management method to pursue

excellence in quality



Improvement in manufacturing capability



Eliminate waste in manufacturing process



Improve manufacturing quality

(69)

69



The usual concept

9 Six Sigma Concept

3 



(70)

70



The concept

1.5 off the target value

9 Six Sigma Concept

6 



(71)

71

9 Six Sigma Concept

PPM : Part per Million; Source: Pan and Lee (2003)

(72)

72



Motorola(1987) started six sigma program



Won the Baldridge award in 2 years



A five-fold growth between 1987 and 1997



A 20% profit increase per year



A reduction of 14 billion USD in cost



Allied Signal(1991)



GE(1995)



Application to all different business

(73)

73

9 Six Sigma Concept

(74)

10 Summary



Graphical Techniques



Charts



R Charts



P Charts



C Charts



Process Capability



Six Sigma

X