Presentation - Setting specifications, statistical ...

Setting Specifications

Statistical considerations

Enda Moran ? Senior Director, Development, Pfizer Melvyn Perry ? Manager, Statistics, Pfizer

Basic Statistics

Population distribution

1.5

(usually unknown).

Normal distribution

described by and .

1.0

0.5

True batch assay

Distribution of possible values

0.0

98

99

100

We infer the population from samples by calculating x and s.

1.5

Sample 1 1.0 Average

98.6

0.5

True batch assay

Distribution of possible values

1.5

Sample 2

True batch assay

1.0 Average 98.7

Distribution of possible values

0.5

1.5

Sample 3

True batch assay

Average 99.0

1.0

Distribution of possible values

0.5

0.0

0.0

0.0

98

99

100

98

99

100

98

99

100

2

Intervals

30 28 26 24 22 20 18 16

0

Population average

10

20

30

40

50

60

70

80

90

100

?100 samples of size 5 taken from a population with an average of 23.0 and a standard deviation of 2.0. ?The highlighted intervals do not include the population average (there are 6 of them). ?For a 95% confidence level expect 5 in 100 intervals to NOT include the population average. ?Usually we calculate just one interval and then act as if the population mean falls within this interval.

3

Intervals

Point Estimation The best estimate; eg MEAN

Interval Estimation A range which contains the true population parameter or a future observation to a certain degree of confidence.

Confidence Interval - The interval to estimate the true population parameter (e.g. the population

mean).

Prediction Interval - The interval containing the next single response.

Tolerance Interval - The interval which contains at least a given proportion of the population.

4

Formulae for Intervals

Intervals are defined as: x ? ks

Assuming a normal distribution

? Confidence (1-) interval

CI

=

x

?

1 n

0.5

t

1-

2

s

,n -1

? Prediction (1-) interval for m future observations

PI

=

x

?

1+

1 n

0.5

t

1-

,n 2m

s

-1

? Tolerance interval for confidence (1-) that proportion

(p) is covered

TI = x ?

( ) n

-

1

1+ 1 n 2

,n-1

z (21-2p )

s

5

Process Capability

Process capability is a measure of the risk of failing specification. The spread of the data are compared with the width of the specifications.

3s

3s

The distance from the mean to the nearest specification relative to half the process width (3s).

The index measures actual performance. Which may or may not be on target i.e., centred.

Ppk

= minUS3Ls-

x

,

x

- LSL

3s

LSL

x

USL

x - LSL

USL - x

6

Process Capability ? Ppk and Cpk

? Ppk should be used as this is the actual risk of failing specification. ? Cpk is the potential capability for the process when free of shifts and drifts.

Random data of mean 10 and SD 1, thus natural span 7 to 13. Added shifts to simulate trends around common average. With specs at 7 and 13 process capability should be unity.

Individual Value

15 14 13 12 11 10

9 8 7 6

14

I Chart of Shifted

1

7 10 13 16 19 22 25 28 Observation

UCL=13.589

_ X=10.092

LCL=6.595

P rocess D ata

LS L

7

T a rge t

*

USL

13

S ample M ean 10.0917

S ample N

30

S tD ev (Within) 1.16561

S tD ev (O v erall) 1.95585

Process Capability of Shifted

LSL

USL

W ithin Ov erall

P otential (Within) C apability C p 0.86 C P L 0.88 C P U 0.83 C pk 0.83

O v erall C apability

Pp PPL PPU P pk C pm

0.51 0.53 0.50 0.50

*

O bserv ed P erformance

P P M < LS L

0.00

P P M > U S L 100000.00

P P M Total 100000.00

6

8

E xp. Within P erformance P P M < LS L 3995.30 P P M > U S L 6296.59 P P M Total 10291.88

10

12

14

E xp. O v erall P erformance P P M < LS L 56966.34 P P M > U S L 68512.70 P P M Total 125479.03

When data is with trend Ppk less than Cpk due to method of calculation of std dev.

Ppk uses sample SD. Ppk less than 1 at 0.5.

Cpk uses average moving range SD (same as for control chart limits). Cpk is close to 1 at 0.83.

Individual Value

I Chart of Raw

13

UCL=13.038

12

11

_

10

X=10.092

9

8

7 14

7 10 13 16 19 22 25 28 Observation

LCL=7.145

Process Capability of Raw

LSL

P rocess Data

LS L

7

T a rge t

*

USL

13

S ample M ean 10.0917

S ample N

30

S tD ev (Within) 0.982187

S tD ev (O v erall) 1.14225

USL W ithin Ov erall

P otential (Within) C apability C p 1.02 C P L 1.05 C P U 0.99 C pk 0.99

O v erall C apability

Pp PPL PPU P pk C pm

0.88 0.90 0.85 0.85

*

7

8

9

10 11 12 13

O bserv ed P erformance P P M < LS L 0.00 P P M > U S L 0.00 P P M Total 0.00

E xp. Within P erformance P P M < LS L 822.51 P P M > U S L 1533.13 P P M Total 2355.65

E xp. O v erall P erformance P P M < LS L 3397.73 P P M > U S L 5446.84 P P M Total 8844.57

When data is without trend Ppk is same as Cpk. Only small differences are seen.

Cpk effectively 1 at 0.99.

Ppk close to 1 at 0.85.

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Measurement Uncertainty

Bad parts almost always rejected

LSL

Good

USL

parts almost

measurement

always

passed

total

Bad parts

almost

always

rejected

?3measurement

?3measurement

The grey areas highlighted represent those parts of the curve with the

potential for wrong decisions, or mis-classification.

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