Validation of the Measurement Process

Validation of the Measurement Process James R. DeVoe, EDITOR Institute for Materials Research, National Bureau of Standa...

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Validation of the Measurement Process James R. DeVoe, EDITOR Institute for Materials Research, National Bureau of Standards

A symposium sponsored by the Division of Analytical Chemistry at the

171st

Meeting of the American Chemical Society, New York, N Y , April 5-6,

1976.

ACS SYMPOSIUM SERIES

AMERICAN CHEMICAL SOCIETY WASHINGTON, D. C. 1977

In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

63

Library of Congress

Data

Validation of the measurement process. (ACS symposium series; 63 ISSN 0097-6156) Includes bibliographies and index. 1. Chemistry, Analytic—Statistical methods—Congresses. I. DeVoe, James R. II. American Chemical Society. Division of Analytical Chemistry. III. Series: American Chemical Society. ACS symposium series; 63. QD75.4.S8V34 ISBN 0-8412-0396-2

543'.01'82 77-15555 ACSMC8 63 1-207 1977

Copyright © 1977 American Chemical Society A l l Rights Reserved. N o part of this book may be reproduced or transmitted in any form or by any means—graphic, electronic, including photocopying, recording, taping, or information storage and retrieval systems—without written permission from the American Chemical Society. PRINTED IN T H E UNITED STATES O F AMERICA


ACS Symposium Series Robert F. Gould,

Editor

Advisory D o n a l d G. Crosby Jeremiah P . Freeman E. Desmond Goddard Robert A. Hofstader J o h n L . Margrave N i n a I. M c C l e l l a n d J o h n B . Pfeiffer Joseph V. Rodricks Alan

C . Sartorelli

R a y m o n d B . Seymour Roy L. Whistler Aaron W o l d


FOREWORD The A C S S Y M P O S I U a medium for publishin format of the SERIES parallels that of the continuing A D V A N C E S I N C H E M I S T R Y SERIES except that i n order to save time the papers are not typeset but are reproduced as they are submitted by the authors i n camera-ready form. As a further means of saving time, the papers are not edited or reviewed except by the symposium chairman, who becomes editor of the book. Papers published i n the A C S S Y M P O S I U M SERIES are original contributions not published elsewhere i n whole or major part and include reports of research as well as reviews since symposia may embrace both types of presentation.


PREFACE The

existence of integrated electronic circuits has changed radically our thinking with respect to performing chemical analyses. L o w cost microprocessors are now integral parts of commercial analytical instrumentation. Minicomputers have the ability to control experiments, to collect data, and to perform calculations with ever increasing facility. Thus, there is considerable interest on the part of the chemical analyst to use computational technique t validat th t Chapters 1 and 2 describ control of the measurement process and emphasize the use of graphical techniques which can be implemented conveniently on digital computers. After control of the measurement process has been established, it is necessary to evaluate systematic errors; Chapters 3 and 4 are devoted to this subject. Chapter 5 describes an innovative procedure which uses a laboratory minicomputer to optimize the variables i n a chemical analysis. Chapter 6 outlines some examples for evaluating statistical control i n testing laboratories. I would like to thank the authors for their diligent effort and to express appreciation to Carol Shipley and the text editing staffs of the Analytical Chemistry Division and the Institute for Materials Research, N B S , for helping with the manuscripts. Institute for Materials Research, N B S

JAMES

R.

Washington, DC 20234 August 12, 1977

vii


DEVOE

1 Statistical Control of Measurement Processes GRANT WERNIMONT Department of Chemistry, Purdue University, Lafayette, I N 47905

Valid measurements are necessary whenever we make chemical test proper action can b the material. Measurements are not valid until we evaluate the performance characteristics of the process which produced the measurements and i t is essential that the statements about the future behavior of these characteristics be correct. Statistical control is concerned with removing the assignable causes of variation in a measurement process (or correcting for their effects) so that we can associate approximate levels of confidence with these statements. It is unfortunate, I think, that most academic courses involving measurement do not seem to make the student aware of how important i t is to achieve a state of s t a t i s t i c a l control when we set up and run a measurement process. I was able to find only one current text on the theory and practice of quantitative analysis which addressed i t s e l f to this most important performance characteristic. In contrast, applied analytical chemists have been involved in statistical control activities for more than 40 years. Some of the United States Government regulatory agencies are now becoming concerned about this important aspect of measurement operations. For example, the Nuclear Regulatory Commission requires (1): "The licensee shall establish and maintain a statistical control system including control charts and formal statistical 1 In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2

VALIDATION

OF T H E

M E A S U R E M E N T PROCESS

procedures, designed to monitor the q u a l i t y o f each type o f program measurement. C o n t r o l c h a r t l i m i t s s h a l l be e s t a b l i s h e d t o be e q u i v a l e n t t o l e v e l s o f s i g nificance o f 0.05 and 0.001. When e v e r c o n t r o l d a t a e x c e e d t h e 0.05 c o n t r o l limits, the licensee s h a l l i n v e s t i g a t e the c o n d i t i o n and t a k e c o r r e c t i v e a c t i o n i n a timely manner. The results of these investigations and actions shall be recorded. When e v e r the c o n t r o l data exceed t h e 0.001 control limits, the measurement system which generated the d a t a s h a l l n o t be u s e d f o r c o n t r o l l i m i t s the measuremen data shall no purposes until the deficiency has been b r o u g h t i n t o c o n t r o l a t t h e 0.05 l e v e l . " In t h i s c h a p t e r t h e meaning o f s t a t i s t i c a l cont r o l i s e x p l a i n e d , and t h e p r o c e d u r e s which we can use t o h e l p s e t up a n d r u n a m e a s u r e m e n t p r o c e s s a r e r e v i e w e d so t h a t i t i s i n a s t a t e o f s t a t i s t i c a l control .

WHAT I S MEASUREMENT Measurement h a s b e e n d e f i n e d as " t h e o p e r a t i o n o f a s s i g n i n g numbers t o r e p r e s e n t properties using arbitrary rules b a s e d on s c i e n t i f i c p r i n c i p l e s . O f c o u r s e t h i s i s an o v e r - s i m p l i f i c a t i o n ; a much b r o a d e r interpretation o f measurement f o r m u l a t e s a h i e r a r c h y of measurement s c a l e s : Nominal, Ordinal, Interval, and Ratio (_2) . The mathematical transformations p e r m i t t e d on e a c h s c a l e determine what statistical methodology c a n be a p p l i e d t o t h e m e a s u r e m e n t s . I n general, t h e more u n r e s t r i c t e d the permissable transformations, t h e more r e s t r i c t e d t h e s t a t i s t i c s ; n e a r l y a l l m e t h o d o l o g i e s c a n be applied to ratioscale measurements, but only a few serve f o r measurements on a n o m i n a l s c a l e . The most penetrating a n a l y s i s , by f a r , o f t h e basis f o r m a k i n g m e a s u r e m e n t s was formulated by Churchill Eisenhart ( 3 ) ; a n d i t s h o u l d be c a r e f u l l y s t u d i e d b y a l l p e o p l e who d e v i s e m e a s u r e m e n t methods and perform measurement operations as w e l l as b y t h o s e who u s e m e a s u r e m e n t r e s u l t s t o make decisions. E i s e n h a r t s t a t e s (_3, p . 1 6 3 ) :


1.

WERNIMONT

Statistical Control of Measurement Processes

"Measurement i s t h e a s s i g n m e n t o f numbers to m a t e r i a l t h i n g s t o r e p r e s e n t t h e r e l a tions e x i s t i n g among t h e m w i t h r e s p e c t t o particular properties. T h e number a s signed t o some p a r t i c u l a r p r o p e r t y s e r v e s to r e p r e s e n t t h e r e l a t i v e amounts o f t h i s property a s s o c i a t e d w i t h t h e o b j e c t concerned. Measurement always p e r t a i n s t o p r o p e r t i e s of things not t o t h e things themselves. Thus we c a n n o t measure a meter b a r , b u t c a n , a n d u s u a l l y do m e a s u r e i t s length; and we c o u l d a l s o measure i t s mass i t s d e n s i t y , and p e r h a p s The object o f measurement i s two f o l d : f i r s t , symbolic r e p r e s e n t a t i o n o f properties o f t h i n g s as a b a s i s f o r c o n c e p t u a l analysis; and second, to effect the representation i n a form amenable t o t h e powerful t o o l s o f mathematical analysis. The decisive feature i s symbolic repres e n t a t i o n o f p r o p e r t i e s , f o r which end numerals are n o t t h e usable symbols." There i s a form o f d i r e c t measurement w h i c h i s independent o f t h e p r i o r knowledge o f any o t h e r property; b u t t h e number s y s t e m u s e d t o e x p r e s s m a g n i t u d e s must behave l i k e t h e p r o p e r t y being measured. A s i m p l e example o f d i r e c t measurement i s t h e u s e o f JOHANSON b l o c k s t o c a l i b r a t e a m i c r o m e t e r . In this case i t i s e v i d e n t t h a t t h e p r o p e r t y we c a l l l e n g t h does behave l i k e numbers i n t h e f o l l o w i n g two ways: 1.

An e x p e r i m e n t a l p r o c e d u r e c a n be d e v i s e d w h i c h w i l l p r o d u c e an o r d e r e d sequence o f t h e b l o c k s .

2.

A n o t h e r e x p e r i m e n t a l p r o c e d u r e c a n be d e v i s e d t o combine (wring) t h e b l o c k s a d d i t i v e l y .

A more c o m p l e x e x a m p l e i s t h e p r o p e r t y we c a l l a b s o r b a n c e (A = - l o g T r a n s m i t t a n c e ) which behaves acc o r d i n g t o t h e r u l e s o f m a t r i x a l g e b r a (Z_>§_) not p r i m a r i l y concerned w i t h measurement processes, t h e y do p r e s e n t i d e a which b applied t them The 1939 book give statistical control, presentatio r e s u l t s , and t h e s p e c i f i c a t i o n o f p r e c i s i o n and a c c u racy. a

r

e

Eisenhart presents a section of the requirement o f s t a t i s t i c a l c o n t r o l (.3, p . 1 6 6 ) w h i c h summarizes Shewhart s ideas a n d d e m o n s t r a t e s how t h e y a p p l y t o measurement p r o c e s s e s ; I e x t r a c t some o f t h e s e i d e a s : 1

"The p o i n t t h a t S h e w h a r t makes f o r c e f u l l y , and s t r e s s e s r e p e a t e d l y , i s t h a t t h e f i r s t η measurements o f a q u a n t i t y g e n e r a t e d by a measurement p r o c e s s provide a logical basis f o r p r e d i c t i n g the behavior of fur t h e r m e a s u r e m e n t s o f t h e same q u a n t i t y by the same m e a s u r e m e n t p r o c e s s , i f a n d o n l y i f , t h e s e η m e a s u r e m e n t s may be regarded rancTom s a m p l e f r o m a p o p u l a t i o n o r universe of a l l conceivable measure m e n t s ... c h a r a c t e r i z e d by a p r o b a b i l i t y distribution...nothing i s said about the mathematical form of the d i s t r i b u t i o n . The important thing i s that there be one... a

s

a

1

Shewhart was w e l l a w a r e t h a t , f r o m a s e t of η measurements i n hand, i t i s n o t pos sible t o decide, w i t h c e r t a i n t y , whether t h e y do o r do n o t c o n s t i t u t e a random s a m p l e f r o m some d e f i n i t e s t a t i s t i c a l pop ulation characterized by a p r o b a b i l i t y distribution. He therefor p r o p o s e d (Z) t h a t i n a n y p a r t i c u l a r i n s t a n c e one s h o u l d decide t o a c t f o r t h e p r e s e n t as i f t h e measurements i n hand (and t h e i r immediate f

T


1.

WERNiMONT


successors)...meet the requirements of the s m a l l sample v e r s i o n o f C r i t e r i o n I o f h i s previous book (6J) a n d . . . show no e v i d e n c e of lack of statistical control when analyzed f o r randomness i n the order i n which t h e y were t a k e n by t h e c o n t r o l c h a r t techniques, f o r averages and s t a n d a r d d e v i a t i o n s t h a t he h a d f o u n d so v a l u a b l e in industrial process control, and by certain additional tests f o r randomness based on 'runs above and below average and r u n s up a n d d o w n . . . T

Experience shows t h a t i n t h e c a s e o f mea surement p r o c e s s e s statistical contro scribes, i s usually very d i f f i c u l t to attain, j u s t as i n t h e case o f i n d u s t r i a l production processes..." E i s e n h a r t a l s o quotes f r o m a p a p e r b y D r . R. B. Murphy, a n o t h e r B e l l Telephone e n g i n e e r , on t h e v a l i d i t y o f p r e c i s i o n and accuracy statements (9): " . . . a t e s t m e t h o d o u g h t n o t t o b e known a s a measurement p r o c e s s u n l e s s i t i s c a p a b l e of statistical c o n t r o l . . . ( w h i c h ) means t h a t e i t h e r t h e measurements a r e t h e pro duct o f an i d e n t i f i a b l e s t a t i s t i c a l u n i v e r s e , o r i f n o t , t h e p h y s i c a l causes pre v e n t i n g s u c h i d e n t i f i c a t i o n may t h e m s e l v e s be i d e n t i f i e d a n d , i f desired, isolated and suppressed. Incapability of control i m p l i e s t h a t t h e r e s u l t s o f t h e measure ment p r o c e s s a r e n o t t o be t r u s t e d as i n d i c a t i o n s o f t h e p r o p e r t y a t hand - i n short, we a r e n o t i n a n y v e r i f i a b l e s e n s e measuring anything...without this limi t a t i o n on t h e n o t i o n o f a measurement p r o c e s s , one i s u n a b l e t o go o n t o g i v e meaning t o those statistical measures which a r e t h e b a s i s f o r any d i s c u s s i o n o f p r e c i s i o n and a c c u r a c y . " I b e l i e v e we c a n now f o r m u l a t e t h e i d e a o f s t a t i s t i c a l c o n t r o l as f o l l o w s : A measurement process may be s a i d t o be i n a s t a t e o f s t a t i s t i c a l c o n t r o l i f the significant a s s i g n a b l e causes of variation have been removed o r c o r r e c t e d f o r , so t h a t a f i n i t e s e t o f η measurements f r o m t h e p r o c e s s c a n be u s e d t o


8

VALIDATION OF

THE


(a) predict limits o f v a r i a t i o n f o r the η measure m e n t s a n d (b) assign a level of confidence that f u t u r e measurements w i l l l i e w i t h i n these l i m i t s .

CONTROL CHART A N A L Y S I S The o p e r a t i o n a l p r o c e d u r e f o r d e m o n s t r a t i n g t h a t a process i s i n a s t a t e of statistical control is quite simple i n concept but r a t h e r complex i n prac tice. I t c o n s i s t s of arranging to gather η measure ments, i n some k i n d o f o r d e r , and i n t h e f o r m o f s o c a l l e d " r a t i o n a l subgroups", w i t h i n which the varia tions may be c o n s i d e r e d th basi f knowledg of the p r o c e s s , to b which, the v a r i a t i o n y suspecte assign able causes. To i l l u s t r a t e how we make a c o n t r o l - c h a r t - a n a l y s i s o f m e a s u r e m e n t s , l e t us e x a m i n e t h e r e s u l t s o f a s i m p l e experiment which Shewhart c a r r i e d out to simu late a "controlled" production process. He placed 998 circular c h i p s i n a l a r g e b o w l ; numbers between n e g a t i v e 3.0 and p o s i t i v e 3.0, a t 0.1 i n t e r v a l s , w e r e recorded on the c h i p s w h i c h w e r e one c o l o r f o r t h e n e g a t i v e n u m b e r s and a n o t h e r f o r t h e positive. The magnitudes o f t h e numbers were d i s t r i b u t e d a c c o r d i n g to a "normal" d i s t r i b u t i o n w i t h average = 0.0 and standard deviation = 1.007. The c h i p s w e r e d r a w n f r o m t h e b o w l one a t a t i m e , w i t h r e p l a c e m e n t , until 4000 v a l u e s w e r e o b t a i n e d and r e c o r d e d i n o r d e r . For f u r t h e r d e t a i l s , see ( 6 , pp. 164-165 and Appendix II). Shewhart observed that i n t h i s experiment we h a v e as n e a r an a p p r o a c h as i s l i k e l y f e a s i b l e t o t h e conditions i n w h i c h t h e law o f l a r g e numbers a p p l i e s s i n c e , t o t h e b e s t o f o u r k n o w l e d g e , t h e same essen tial c o n d i t i o n s were maintained. H o w e v e r , he o n c e t o l d me t h a t t h i s s i m p l e d r a w i n g o p e r a t i o n is prone t o show l a c k o f s t a t i s t i c a l c o n t r o l u n l e s s g r e a t c a r e i s t a k e n t o m i x up t h e bowl of chips between the d r a w i n g s and k e e p t h e b o o k k e e p i n g m i s t a k e - f r e e . I have plotted the results of d r a w i n g s as a c o n t r o l c h a r t i n F i g u r e 1, tional subgroup of four consecutive a v e r a g e s and s t a n d a r d d e v i a t i o n s o f t h e were c a l c u l a t e d a s ,

the f i r s t 200 using a ra values. The 50 subgroups



WERNiMONT

A SIMULATED MEASUREMENT PROCESS

J

2

I

I

I

I

10

I

ι

ι

ι

ι

20

ι

ι

ι

ι

ι

30

I

ι

i

ι

I

40

I

I

I

I

L

—I

50 991

1

1

1

1-

1000

RATIONAL SUBGROUP NUMBER Figure 1.

Consecutive drawings from Shewharfs bowl of chips


10

VALIDATION O F

X = E X / 4 , and i

THE


S = A ( X j - X ) V (4-1).

The g r a n d a v e r a g e , o£ a l l 200 v a l u e s i s -0.08 a n d t h e average o f the group s t a n d a r d d e v i a t i o n s is 0.912. T h r e e - s i g m a c o n t r o l l i m i t s f o r t h e 50 s u b g r o u p s a r e , Limits Upper Lower

Standard D e v i a t i o n (2.266 χ 0.912) = 2.07 (0. x 0.912) - 0.

Average -0.08 + (1.628 χ 0.912) = -0.08 - (1.628 χ 0.912) =

The factors, B = 0 , B = 2.266, and A t a b l e d i n v a r i o u s r e f e r e n c e s (1_0, 1_1, 12_, 3

o

4

3

1.40 -1.40

= 1.628 a r e 1_3, 14) .

To evaluate thes result fo statistical t r o l , we f i r s t e x a m i n standard deviation , , g r e a t e r t h a n the 3-sigma l i m i t . This indicates that no a s s i g n a b l e causes were a f f e c t i n g the o p e r a t i o n o f c o n s e c u t i v e l y d r a w i n g and r e p l a c i n g f o u r c h i p s . Lack of control f o r s t a n d a r d d e v i a t i o n w o u l d l e a d us t o look f o r l o c a l assignable causes i n the way each group o f f o u r c h i p s was r e m o v e d f r o m t h e b o w l . Per h a p s someone i s s u r r e p t i t i o u s l y e x c h a n g i n g the bowl with one w h i c h h a s a s t a n d a r d d e v i a t i o n g r e a t e r t h a n 1.007. Next, we examine the upper graph f o r subgroup a v e r a g e s , w h i c h a l s o shows n o n e o u t s i d e 3-sigma l i mits. T h i s i n d i c a t e s t h a t no a s s i g n a b l e c a u s e s w e r e a f f e c t i n g the drawing o p e r a t i o n through out the en tire s e q u e n c y o f t h e f i r s t 200 v a l u e s . Lack o f con t r o l would suggest that some n o n l o c a l assignable cause affected some s u b g r o u p s d i f f e r e n t l y t h a n o t h ers. Perhaps the s u r r e p t i t i o u s exchange involved a bowl with a d i s t r i b u t i o n w h i c h a v e r a g e s two r a t h e r than zero. Shewhart s u g g e s t e d t h a t c r i t e r i a f o r randomness should also i n c l u d e the behavior of urns f o r consec utive groups w i t h i n the 3-sigma l i m i t s . Duncan ex p l a i n s ( j ^ , p. 386) a r u n as "a s u c c e s s i o n o f items of t h e same c l a s s " s u c h as a s e r i e s o f i n c r e a s i n g o r d e c r e a s i n g v a l u e , or a s e r i e s of consecutive values above or below the average. We f i n d no r u n s , up o r down, g r e a t e r t h a n f i v e ; b u t two r u n s , o f s e v e n b e l o w the a v e r a g e , o c c u r r e d ( b e g i n n i n g w i t h subgroups 6 and 15). S t a t i s t i c a l t h e o r y and p r a c t i c a l e x p e r i e n c e i n d i c a t e t h a t a s s i g n a b l e c a u s e s c a n u s u a l l y be f o u n d t o e x p l a i n r u n s o f s e v e n o r more; o f c o u r s e i t is now i m p o s s i b l e t o l o o k f o r them.


1.

WERNiMONT


11

No other t y p e s o f s y s t e m a t i c v a r i a t i o n such as c y c l e s o r t r e n d s , a p p e a r t o be p r e s e n t f o r e i t h e r t h e standard d e v i a t i o n s or the averages. C a n we c o n c l u d e t h a t t h i s p r o c e s s was i n a s t a t e o f s t a t i s t i c a l control? Well, we h a v e t w o c h o i c e s : (a) t h e p r o c e s s was n o t i n c o n t r o l , o r ( b ) t h e p r o c e s s was i n c o n t r o l but two i m p r o b a b l e runs o c c u r r e d . This i sexactly t h e s i t u a t i o n we m e e t a l m o s t e v e r y time we examine results from a measurement p r o c e s s . No m a t t e r w h i c h c h o i c e we make, t h e r e i s some chance that i ti s wrong. I would conclude that t h e evidence f o r l a c k of c o n t r o l i sn o t c o n v i n c i n g based on knowledge o f the p r o c e s s , and p r e d i c t t h a t t h e 3-sigma l i m i t s , e s t i m a t e d f r o m t h e f i r s t 200 d r a w i n g s should als i n clude p r a c t i c a l l y a l see t h a t t h e l a s t 4 d r a w i n g are w e l l w i t h i n these l i m i t s . D u n c a n h a s g i v e n ( 1 3 , p . 3 9 2 ) t h e f o l l o w i n g summary o f c r i t e r i a f o r lacTf o f s t a t i s t i c a l c o n t r o l : 1. 2.

3. 4. 5. 6.

One o r more p o i n t s o u t s i d e 3 - s i g m a l i m i t s , One o r more p o i n t s i n t h e v i c i n i t y o f a " w a r n i n g l i m i t " suggesting that additional observations be t a k e n , A r u n o f s e v e n o r more p o i n t s , Cycles, trends, or other nonrandom patterns w i t h i n 3-sigma l i m i t s , A r u n o f t w o o r more p o i n t s outside o f 2-sigma limits, A r u n o f f o u r o r more p o i n t s o u t s i d e 1-sigma l i mits .

Of course we a r e a l w a y s f a c e d w i t h t h e r i s k o f b e i n g w r o n g when we d e c i d e w h e t h e r , o r n o t , a p r o c e s s is i n a state of s t a t i s t i c a l control. We f i x t h i s r i s k by a r b i t r a r i l y c h o o s i n g c r i t i c a l 3-sigma l i m i t s . Using wider limits, we increase the risk of erroneously concluding that the process i s i n s t a t i s tical control and decrease t h e chances o f d e t e c t i n g s i g n i f i c a n t a s s i g n a b l e causes. The u s e o f n a r r o w e r limits will have the opposite effects. Experience h a s shown t h a t t h e r i s k s a r e q u i t e t o l e r a b l e , i n m o s t cases, when a c t i o n l i m i t s a r e s e t b e t w e e n 2- a n d 3sigma f o r subgroup s t a n d a r d d e v i a t i o n s and averages.


12

VALIDATION OF T H E M E A S U R E M E N T

PROCESS

RATIONAL SUBGROUPS The key to s u c c e s s when we u s e c o n t r o l c h a r t a n a l y s i s t o examine r e s u l t s from a measurement p r o cess, l i e s i n t h e s t r a t e g y we u s e t o s e t up " r a t i o n al subgroups. The i d e a o f a r r a n g i n g t o g a t h e r the measurements i n s u b g r o u p s makes r e a l s e n s e , b e c a u s e i t i s my o b s e r v a t i o n t h a t a s s i g n a b l e c a u s e s a f f e c t i n g a measurement process f a 11 r a t h e r c l e a r l y i n t o two classes. n

The first class i s under the l o c a l c o n t r o l of t h e p e r s o n who o p e r a t e s t h e p r o c e s s ; i t i n c l u d e s s u c h o p e r a t i o n s as m a n i p u l a t i n equipment dispensin gents, c a l i b r a t i n g instruments points, and otherwise f o l l o w i n g procedural instruc t i o n s i n l o c a l time and space. O p e r a t o r s c a n be h e l d responsible f o r maintaining r i g i d c o n t r o l of these l o c a l o p e r a t i o n s , and good o p e r a t o r s soon learn how t o do i t . L a c k o f s t a t i s t i c a l c o n t r o l o f t h e s e l o c a l operations i s observed, o c c a s i o n a l l y , but only because o f b a s i c s h o r t c o m i n g s i n t h e method o r equipment w h i c h t h e o p e r a t o r i s u n a b l e t o p e r c e i v e o r c o p e with. The s e c o n d c l a s s o f a s s i g n a b l e c a u s e s i s n o t u n der t h e l o c a l c o n t r o l o f the o p e r a t o r ; i t includes such t h i n g s as l o n g - r a n g e m a i n t a i n a n c e o f l a b o r a t o r y c o n d i t i o n s and e q u i p m e n t , types and/or methods of calibration, d e t e r i o r a t i o n o f r e a g e n t s and i n s t r u ments, the nature o f i n t e r f e r e n c e s i n the material b e i n g t e s t e d , and numerous o t h e r t y p e s o f n o n l o c a l o r r e g i o n a l assignable causes. The l a b o r a t o r y supervis o r m u s t assume r e s p o n s i b i l i t y f o r f i n d i n g a n d r e m o v ing a s s i g n a b l e causes a f f e c t i n g these o p e r a t i o n s . I think i t i s obvious that c o n t r o l chart analys i s f o r v a r i a t i o n w i t h i n r at i o n a l subgroups ( s t a n d a r d information deviation or range) g i v es us i m p o r t a n t while the chart about the l o c a l a s s i g n a b l e causes, for averages r e v e a l s i n f ormation about t h e r e g i o n a l assignable causes. Two p o s s i b l e m i s t a k e s a r e e a s y t o make when we s e t up a s y s t e m o f r a t i o n a l s u b g r o u p s : (a) t h e r e p l i c a t i o n s a r e so c l o s e t o g e t h e r i n t i m e a n d / o r s p a c e t h a t t h e y do n o t i n c l u d e a l l the l o c a l assignable causes. F o r i n s t a n c e , we w o u l d n e v e r w a n t t o r e c o r d d u p l i c a t e r e a d i n g s o f an i n s t r u m e n t s c a l e b e c a u s e , as W. J . Youden o f t e n p o i n t e d o u t , t h i s i s m e r e l y "du-


1.

WERNiMONT


13

plicity". The subgroup s h o u l d i n c l u d e a l l t h e l o c a l random c a u s e s b e c a u s e a measurement p r o c e s s c a n n e v e r be b r o u g h t i n t o a s t a t e o f s t a t i s t i c a l c o n t r o l i f t h e r a t i o n a l subgroups a r e t o o r e s t r i c t e d , (b) t h e r e p l i cations a r e so f a r a p a r t i n time and/or space t h a t t h e y i n c l u d e some o f t h e r e g i o n a l a s s i g n a b l e causes. This l e a d s t o wide c o n t r o l l i m i t s which l a c k t h e power t o d e t e c t a s s i g n a b l e c a u s e s , l o c a l o r r e g i o n a l . I have a d e t a i l e d d i s c u s s i o n o f t h e concept o f r a t i o n a l s u b g r o u p s i n my p a p e r , "The U s e o f C o n t r o l Charts i n theA n a l y t i c a l Laboratory" (15). Specific i n s t r u c t i o n s cannot be f o r m u l a t e d t o d e v i s e rational subgroups which w i l l a p p l y t o a l l k i n d s o f measure ment p r o c e s s e s . I limited so t h a t variation e s s e n t i a l l y random a n d t h e y s h o u l d be sufficiently extended t o r e v e a l a s s i g n a b l e causes which t h e operator i sunable t o c o n t r o l .

how

Let u s now l o o k a t some r e a l w o r l d e x a m p l e s o f we c a n u s e c o n t r o l c h a r t a n a l y s i s .

A PROCESS WITH NO A S S I G N A B L E CAUSES Figure 2 shows a c o n t r o l c h a r t f o r a p r o c e s s t o d e t e r m i n e t h e w a t e r - e q u i v a l e n t o f a P a r r - t y p e bomb combustion calorimeter. Once e a c h m o n t h , t h e o p e r a t o r made f o u r i n d e p e n d e n t calibration runs on t h e same a f t e r n o o n b y w e i g h i n g a p p r o p r i a t e a m o u n t s o f NBS S t a n d a r d B e n z o i c A c i d and b u r n i n g i t i n t h e oxygenc h a r g e d bomb u n d e r e s s e n t i a l l y t h e same c o n d i t i o n s a s were used t o d e t e r m i n e h e a t s o f combustion of fuel. The material was i g n i t e d b y h e a t i n g e l e c t r i c a l l y a small p i e c e o f pure i r o n w i r e . The c a l o r i m e t e r cons t a n t was c o m p u t e d f r o m t h e o b s e r v e d t e m p e r a t u r e r i s e o f t h e w a t e r s u r r o u n d i n g t h e bomb, t h e w e i g h t o f b e n zoic acid, a n d t h e NBS c e r t i f i e d v a l u e f o r t h e h e a t of combustion o f the a c i d . A small correction f o r t h e h e a t g e n e r a t e d b y t h e w i r e was a p p l i e d . The d a t a f o r t h i s c h a r t was t a k e n f r o m h i s t o r i c a l r e c o r d s and you can see that d u r i n g t h e p r e v i o u s 11-month p e r i o d , no s i g n i f i c a n t a s s i g n a b l e causes w e r e a f f e c t i n g t h e s t a n d a r d d e v i a t i o n s s o we c a n c o n c l u d e t h a t t h e o p e r a t o r was c o n t r o l l i n g a l l t h e l o c a l operations. T h e c h a r t f o r a v e r a g e s a l s o shows s a t i s f a c t o r y c o n t r o l w h i c h means t h a t no r e g i o n a l a s s i g n a -


14 ble over

VALIDATION

OF

THE

MEASUREMENT

causes were a f f e c t i n g the c a l i b r a t i o n an e x t e n d e d p e r i o d o f t i m e .

PROCESS

operations

I t i s i n t e r e s t i n g t o n o t e t h a t p r i o r t o t h i s analysis of calibration mea s u r e m e n t s , the l a b o r a t o r y supervisor had been r e vi s i n g the water-equivalent each month. He now d e c i d e d t o a d o p t t h e l o n g range a v e r a g e o f 29030 b u t c o n t i nue c h e c k i n g i t e v e r y m o n t h as b e f o r e . T h i s was s o u n d strategy because a few months l a t e r t h e c a l i b r a t i o n a v e r a g e was o b s e r v e d t o be j u s t o u t o f c o n t r o l on t h e l o w s i d e . Investigat i o n r e v e a l e d t h a t a new s u p p l y o f i r o n w i r e h a d b e e n a c q u i r e d b u t t h e s u p e r v i s or n e g l e c t e d t o g i v e a rev i s e d c o r r e c t i o n f a c t o r t o th operator

A PROCESS WITH LOCAL A S S I G N A B L E

CAUSES

I have a l r e a d y i n d i c a t e d t h a t l a c k o f c o n t r o l o f l o c a l a s s i g n a b l e c a u s e s i s n o t commonly o b s e r v e d ; a n d I am a w a r e o f no s i m p l e t e c h n i q u e s , o t h e r t h a n c o n t r o l c h a r t a n a l y s i s , t o d e t e c t i t . T h i s example i n volved t h e u s e o f an i n s t r u m e n t t o m e a s u r e t h e t e a r i n g s t r e n g t h o f p l a s t i c s h e e t i n g u s e d t o s u p p o r t photographic emulsions. The i n s t r u m e n t (Thwing-Albert), d e s i g n e d t o measure t h e t e a r i n g s t r e n g t h of paper, consisted of a f a i r l y m a s s i v e pendulum a r r a n g e d so that i t c o u l d absorb the energy used to t e a r a small specimen o f m a t e r i a l , thus d e c r e a s i n g the amplitude of the pendulum. The i n s t r u m e n t h a d b e e n m o d i f i e d t o make i t more s e n s i t i v e t o t h e s m a l l e r s t r e n g t h s o f f i l m s u p p o r t by attaching a counterbalance t o the pendulum, thus r a i s i n g i t scenter of gravity. The m o d i f i e d instrument was m o n i t o r e d b y means o f a r e s e r v o i r o f " r e f e r ence" f i l m support p i c k e d from a uniform production lot, c u t i n t o t e s t s p e c i m e n s , and t h o r o u g h l y randomized. The s p e c i m e n s were conditioned and t e a r i n g strengths were measured once each day u s i n g r a t i o n a l subgroups o f f i v e s t r i p s from the r e s e r v o i r . Control chart analysis showed no e v i d e n c e f o r lack of s t a t i s t i c a l c o n t r o l for both standard deviat i o n a n d a v e r a g e d u r i n g t h e f i r s t 14 w e e k s as y o u c a n s e e i n F i g u r e 3. D u r i n g we ek 2 0 , l a c k o f c o n t r o l was i n d i c a t e d f o r one s u b g r o u p s t a n d a r d d e v i a t i o n a n d one a v e r a g e ; a n d b y w e e k 2 4 , i t became e v i d e n t t h a t both standard d e v i a t i o n and aver a g e were o u t o f s t a t i s t i cal control. The o p e r a t o r c o u l d f i n d no reasons to



WERNIMONT

A PROCESS WITH NO ASSIGNABLE 4θ|-

η=4

CAUSES

χ = observed value - 24,000

X sof20 10

S

5 0

MONTH NUMBER Figure 2.

Determination of the water-equivalent of a bomb calorimeter

A PROCESS WITH LOCAL ASSIGNABLE

CAUSES

η= 5 56.4 X

5

6

,

2

56.Oh

.

Λ /

S

0.2 0.0

ν

V

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I

I I 14

I

V / V

\ /'Χ

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25

WEEK NUMBER Figure 3.

Determination of the force to tear plastic film support


16

V A L I D A T I O N OF

THE


explain this and the m a t e r i a l behaved a l r i g h t t e s t e d on o t h e r i n s t r u m e n t s c u r r e n t l y i n u s e .

when

The i n s t r u m e n t was r e t u r n e d t o t h e m a c h i n e s h o p where t h e c o u n t e r b a l a n c e had been i n s t a l l e d , and i t was found that t h e b e a r i n g s , on w h i c h t h e p e n d u l u m was s u p p o r t e d , w e r e b e g i n n i n g t o d i s i n t i g r a t e b e c a u s e of the i n c r e a s e d l o a d of the c o u n t e r b a l a n c e . Larger b e a r i n g s w e r e i n s t a l l e d a n d , as y o u c a n s e e , t h e c o n trol chart f o r b o t h s t a n d a r d d e v i a t i o n and a v e r a g e returned to normal. New b e a r i n g s h a d t o be i n s t a l l e d on a l l t h e o t h e r i n s t r u m e n t s .

A PROCESS WIT When c o n t r o l c h a r t a n a l y s i s shows s a t i s f a c t o r y c o n t r o l f o r the v a r i a t i o n w i t h i n r a t i o n a l subgroups but l a c k o f c o n t r o l among s u b g r o u p a v e r a g e s , we m u s t look f o r r e g i o n a l a s s i g n a b l e causes. Most i n t e r l a b o ratory studies o f m e a s u r e m e n t p r o c e s s e s show l i t t l e o r no e v i d e n c e f o r l a c k o f c o n t r o l w i t h i n t h e laboratories over a s h o r t p e r i o d of time; but i t i s v e r y d i f f i c u l t to achieve statistical control among a g r o u p o f l a b o r a t o r i e s a l l u s i n g t h e same t e s t m e t h o d . F i g u r e 4 shows r e s u l t s o f a s t u d y of the Eberstadt method f o r determining the a c e t y l - c o n t e n t of c e l l u lose acetate. Samples o f a r e f e r e n c e material were a n a l y z e d i n e i g h t d i f f e r e n t l a b o r a t o r i e s w i t h two i n dependent o p e r a t o r s i n each l a b o r a t o r y making duplicate t e s t s on e a c h o f two d i f f e r e n t d a y s . The lower c h a r t f o r o p e r a t o r r a n g e s shows t h a t a s t a t e o f stat i s t i c a l c o n t r o l e x i s t e d f o r the v a r i a t i o n w i t h i n the l a b o r a t o r i e s , but i t i s obvious t h a t l a b o r a t o r y avera g e s v a r y more t h a n c a n be e x p l a i n e d by t h e v a r i a t i o n within laboratories. It is difficult to find the reasons for this because they are o f t e n d i f f e r e n t f r o m one l a b o r a t o r y t o a n o t h e r . In t h i s case i t was found that some o f t h e l a b o r a t o r i e s w e r e n o t r i g o r o u s l y f o l l o w i n g the t e s t method p r o c e d u r e s .

THE

PROBLEM OF

DUPLICITY

Let us r e t u r n t o t h e c r i t i c a l p r o b l e m o f d e v i s ing r a t i o n a l subgroups. I n F i g u r e 5, we s e e results f o r t h e d e t e r m i n a t i o n o f c o p p e r , made d u r i n g t h e p r o d u c t i o n o f b r o n z e c a s t i n g s . Two i n d e p e n d e n t samples were drilled f r o m e a c h c a s t i n g and a n a l y z e d , i n dup l i c a t e , u s i n g a p r e c i s e method of electrolytically



WERNIMONT

A PROCESS WITH REGIONAL ASSIGNABLE CAUSES η =2 39.4

Χ" 39.0 38.6 0.4

R

0.2 0.0 I

2

LABORATORY NUMBER Figure 4.

Determination of acetyl in cellulose acetate

A PROCESS WITH LIMITS BASED ON TEST

VARIATION

η= 2 85.8

X"

85.4 85.0

R

0.2 0.1 o.oh

-

-*r.

2

3

4

5

6

7

s

8

~

9

\

10 II

CASTING NUMBER Figure 5.

Determination of copper in bronze castings


18

VALIDATION OF T H E M E A S U R E M E N T PROCESS

depositing the copper a n d w e i g h i n g i t . The l o w e r chart f o r ranges i s i n c o n t r o l , but the chart f o r subgroup averages shows t h a t t h e d u p l i c a t e samples are e x c e e d i n g l y v a r i a b l e compared to the duplicate determinations. When c o n t r o l l i m i t s a r e b a s e d o n t h e v a r i a t i o n o f sample a v e r a g e s , w i t h i n c a s t i n g s , there is some r e a s o n t o b e l i e v e t h a t t h e m a n u f a c t u r i n g a n d t e s t i n g operations are both i n a state of s t a t i s t i c a l control, although a c y c l i c e f f e c t c a n n o t be r u l e d out, a s y o u c a n s e e i n F i g u r e 6.

S I M P L E AND COMPLEX

CONTROL

In a l l o f t h e a m a t h e m a t i c a l model w h i c h Eisenhart called SIMPLE statistical c o n t r o l (^3, p . 1 7 4 ) , t h a t i s , t h e v a r i a t i o n o f measurements within rational subgroups i s random and s e r v e s as a v a l i d e s t i m a t e o f t h e random v a r i a t i o n o f t h e s u b g r o u p a v e r a g e s . H o w e v e r , we o f ten f i n d p r o c e s s e s f o r w h i c h t h i s model i s i n a d e q u a t e because r e g i o n a l a s s i g n a b l e causes exist which we cannot identify and/or remove; i n such c a s e s , i t i s d e s i r a b l e t o determine whether the process is in a state o f COMPLEX, o r m u l t i s t a g e , s t a t i s t i c a l c o n t r o l (3, p. 1 7 8 ) . We do t h i s b y s e t t i n g up a c o n t r o l c h a r t f o r t h e v a r i a t i o n (standard d e v i a t i o n o r range) of measurements w i t h i n t h e r a t i o n a l s u b g r o u p s , j u s t as b e f o r e . H o w e v e r , we e s t i m a t e c o n t r o l l i m i t s f o r t h e subgroup averages by t r e a t i n g them as " i n d i v i d u a l " measurements and t h e n u s e t h e "moving range" method which calculates a l l t h e c o n s e c u t i v e d i f f e r e n c e s between the subgroup a v e r a g e s , t h u s p a r t i a l l y e l i m i n a t i n g t h e effects of the regional a s s i g n a b l e c a u s e s (1_3, p . 451) . Figure 7 shows r e s u l t s f o r t h e measurement o f the w a t e r c o n t e n t o f a s e r i e s o f p r o d u c t i o n lots of an organic solvent using t h e K a r l F i s c h e r method. The l o w e r c h a r t f o r standard deviations indicates t h a t t h e m e a s u r e m e n t p r o c e s s i s i n c o n t r o l when t h r e e r e p l i c a t e d e t e r m i n a t i o n s a r e made o n a s i n g l e sample of m a t e r i a l from each l o t . The u p p e r g r a p h shows t h e averages; t h e narrow l i m i t s a r e based on replicate measurement variation, w h i l e t h e wide l i m i t s c o r r e spond t o t h e moving range o f c o n s e c u t i v e l o t averages. O f c o u r s e , we w o u l d n o t e x p e c t t h e d i s t i l l a t i o n o f a n o r g a n i c m a t e r i a l t o be i n s i m p l e s t a t i s t i -


1. WERNIMONT


A PROCESS WITH LIMITS BASED ON MATERIAL VARIABILITY 86.2

_n=2

85.8

X

V

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CASTING NUMBER Figure 6.

Determination of copper in bronze castings

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LOT NUMBER Figure 7. Determination of water in an organic solvent


19


20 cal c o n t r o l i n a case o v e r a l l operation of d even i n the state The a s s i g n a b l e c a u s e s ment of the water l i k e l y t o be f o u n d i n

THE


l i k e t h i s ; b u t we s e e t h a t t h e i s t i l l i n g and m e a s u r i n g i s not of complex s t a t i s t i c a l c o n t r o l . f o r t h i s may be i n t h e m e a s u r e c o n t e n t , b u t t h e y a r e much m o r e the d i s t i l l a t i o n process.

UNORDERED DATA A N A L Y S I S C o n t r o l c h a r t a n a l y s i s was o r i g i n a l l y a p p l i e d t o measurements taken i n s e q u e n t i a l o r d e r from a c o n t i n uous p r o c e s s , b u t i t c a n a l s o be u s e d t o c o m p a r e r e s u l t s from d i f f e r e n t sources where l o g i c a l o r d e r can not be a s s i g n e d . A l a b o r a t o r y study o is necessary t o g i v e v e r y s e r i o u s t h o u g h t o f how t o arrange f o r subgroups w i t h i n the l a b o r a t o r i e s . Some people have defined a s u b g r o u p as t h e m e a s u r e m e n t s made b y a s i n g l e o p e r a t o r , using a single set of e q u i p m e n t , as c l o s e l y t o g e t h e r as p o s s i b l e . T h i s c a n be c o n s i d e r e d t o be d u p l i c i t y . A more useful subgroup includes the local a s s i g n a b l e causes over a more r e a s o n a b l e p e r i o d o f t i m e , f o r e x a m p l e , a week or more. A l o g i c a l r e a s o n f o r t h i s more e x t e n s i v e r a t i o n a l s u b g r o u p i s t h e f a c t t h a t t h e p e o p l e who use measurement results, often r e q u i r e c o m p a r i s o n s between r e p e a t e d measurements t o help make decisions relating t o s a m p l e r e c h e c k s , p r o d u c t i o n c h a n g e s , mat e r i a l s o u r c e s , e t c . , made o v e r t h e i n t e r v a l o f this p e r i o d of time. Many of t h e s e c o n t r o l c h a r t methods were devel o p e d b y S h e w h a r t a n d s u c c e s s f u l l y u s e d b y many people for nearly f i f t y years. During the l a s t three d e c a d e s , more s o p h i s t i c a t e d c o n t r o l c h a r t s for such t h i n g s as c u m u l a t i v e s u m s , l o t a c c e p t a n c e , m u l t i v a r i a b l e r e s p o n s e s , e t c . , have been developed (18); and some o f t h e s e t e c h n i q u e s w i l l be f o u n d u s e f u l t o h e l p e v a l u a t e measurement p r o c e s s e s .

RELATED A S S I G N A B L E CAUSES Many m e a s u r e m e n t p r o c e s s e s show l a c k o f s t a t i s t i c a l c o n t r o l of a type which o f t e n appears baffling because the assignable c a u s e s a c t t o g e t h e r so t h a t t h e e f f e c t s o f one a r e n o t t h e same a t v a r i o u s l e v e l s of the other. F o r e x a m p l e , i t h a s l o n g b e e n known that the o x i d a t i o n of ferrous iron with potassium


1.

WERNIMONT


21

permanganate gives high r e s u l t s i nhydrochloric acid solutions; the deviations increase with acid concentration. Also, the deviations are r e l a t i v e l y smaller as t h e i r o n c o n c e n t r a t i o n i n c r e a s e s , a n d t h e r a t e o f titration decreases. I t i s m o s t i m p o r t a n t t h a t we f i n d and remove t h e e f f e c t s o f t h i s k i n d o f d i f f e r e n tial response while a measurement p r o c e s s i s b e i n g developed. The c l a s s i c a l e x p e r i m e n t a l p r o c e d u r e (sometimes c a l l e d t h e s c i e n t i f i c method) f o r o p t i m i z i n g t h e r e sponse o f measurement p r o c e s s i s i n a d e q u a t e t o d e t e c t this kind of related behavior between assignable causes. In t h e cas s t u d i e s e a c h , a t som shown i n Figure 8 on t h e l e f t ; b u t i t never d e t e r mines whether t h e e f f e c t s o f changing t h e l e v e l s o f the f a c t o r s a r e independent o f each-other. Different i a l response i s e a s i l y detected using a complete factorial d e s i g n a s i s shown o n t h e r i g h t , w h e r e t h e e f f e c t s o f a l l c o m b i n a t i o n s o f t h e f a c t o r s a r e measu r e d w i t h l i t t l e o r no e x t r a work. In t h i s case, the factors are acting independently i f the difference between the diagonal averages i snot s i g n i f i c a n t l y greater than zero. Differential response (usually called interact i o n , o r n o n a d d i t i v i t y by s t a t i s t i c i a n s ) c a n be o f three types: ( a ) among f a c t o r s w i t h i n t h e m e a s u r e ment p r o c e s s , ( b ) b e t w e e n process factors and t h e type o f m a t e r i a l b e i n g t e s t e d , and (c) between t e s t methods and t h e t y p e o f i n t e r f e r e n c e s i n t h e m a t e r i a l being tested. The e x a m p l e d e s c r i b e d a b o v e f a l l s i n t o t h e f i r s t type. F i g u r e 9 shows t h e p r o b l e m o f d i f f e r e n t i a l r e s p o n s e when s e v e r a l m a t e r i a l s a r e t e s t e d u s i n g a meas u r e m e n t p r o c e s s s e t up i n v a r i o u s l a b o r a t o r i e s . The l a b o r a t o r i e s do n o t r a n k t h e m a t e r i a l s i n e x a c t l y t h e same o r d e r . T h i s b e h a v i o r i s n o t s e r i o u s as l o n g as the v a r i a t i o n among t h e l a b o r a t o r i e s i s no g r e a t e r than the r e p l i c a t i o n e r r o r o f the process. However, when u n k n o w n interferences are present i nd i f f e r e n t t y p e s o f m a t e r i a l , w h i c h a f f e c t some laboratory r e sults b u t n o t o t h e r s , i t soon becomes i m p o s s i b l e t o p r e d i c t t h e r e s p o n s e o f t h e t e s t method on t y p e s o f m a t e r i a l , other than those used i n t h e i n t e r l a b o r a t o ry study.


V A L I D A T I O N O F T H E M E A S U R E M E N T PROCESS

EXPERIMENTAL

TWO FACTORS TOGETHER

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Testing Basic Assumptions

41

e l l i p t i c a l s t r u c t u r e o f the a u t o c o r r e l a t i o n p l o t which i s a l s o i n d i c a t i v e o f the untenableness o f the randomness assumption. In t h i s l a s t case the l a c k o f randomness was, as i t turned out, due to an underlying c y c l i c s t r u c t u r e i n the data ( i . e . , the t r u e model was Y-j = c + a* s i η (ôi + φ) + e i (where i i s time) r a t h e r than the assumed Y = c + e . The reader should note the two p o i n t s i n the upper r i g h t p o r t i o n o f the p l o t which are o f f the ellipse. This i s due t o a s i n g l e o u t l i e r i n the data and demonstrates the secondary sensitivity o f the lag-1 a u t o c o r r e l a t i o n p l o t to o u t l i e r s . RUNS TEST The runs t e s t i s a technique t h a t i s s p e c i f i c a l l y used f o r testing randomness application of t h i s i l l u s t r a t e the technique, consider the run sequence p l o t o f 50 spectrophotometry transmittance data p o i n t s i n f i g u r e 3. I t i s apparent from the p l o t t h a t the data are not random (note how observations 35 to 45 are not random but r a t h e r near-monotonic i n nature). To s c r u t i n i z e the c o r r e l a t i o n s t r u c t u r e i n t h i s data set, consider the runs a n a l y s i s given i n f i g u r e 4. A run up of length i means t h a t there are e x a c t l y (i+1) successive observations such t h a t each observation i s g r e a t e r than (or a t l e a s t equal t o ) the previous observation. The underlying theory behind the runs t e s t i s t h a t i f the data are random and i f the sample s i z e i s known ( i n t h i s case, n=50), the number of runs up of length 1, o f length 2, etc. , may be considered as random v a r i a b l e s whose expected values and standard d e v i a t i o n s can be c a l c u l a t e d from t h e o r e t i c a l c o n s i d e r a t i o n s (9) and these c a l c u l a t i o n s w i l l not depend on the (unknown) d i s t r i b u t i o n o f the data but only on i t s assumed randomness. Having computed such t h e o r e t i c a l v a l u e s , the f i n a l step i n the t e s t i s to compute from the data the observed number o f runs (up) o f length 1, o f length 2, e t c . , and then determine how many t h e o r e t i c a l standard deviations that t h i s observed statistic falls from the t h e o r e t i c a l l y expected value. This i s most e a s i l y done by formation of the standardized v a r i a b l e : N. - E(N.) SD(N.) where Nj i s the observed number o f runs (up) o f length i , E(N-j) i s the t h e o r e t i c a l expected number o f runs up o f length i and SD(N-j) i s the t h e o r e t i c a l standard d e v i a t i o n o f the number o f runs up o f length i . This standardized v a r i a t e i s given i n the right-most column of f i g u r e 4. For random data, one would expect values o f , say, ±1, ±2, ±3 i n t h i s column, i . e . , the observed number o f runs o f length i should be only a few ( a t most) standard d e v i a t i o n s away from the t h e o r e t i c a l expected value f o r the number o f runs o f length i . For nonrandom data, the



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VALIDATION OF

44

THE


d e v i a t i o n s from the expected values w i l l , of course, be much l a r g e r and t h i s i s the crux of the runs t e s t . Note t h a t i n the spectrophotometry data, the randomness assumption i s e n t i r e l y untenable as i n d i c a t e d by the e x c e s s i v e l y l a r g e values of the standardized s t a t i s t i c i n the l a s t column (e.g., the number of runs up of length 10 i n the data i s 1 and y e t f o r n=50 observations and f o r random data, we should have e s s e n t i a l l y no runs up of length 1 0 - - t h i s 1 run up of length 10 i s over 1000 standard d e v i a t i o n s from i t s expected value and so the randomness assumption must be r e j e c t e d ) . The above-described runs a n a l y s i s i s a v a l u a b l e a d d i t i o n a l t o o l f o r t e s t i n g the s p e c i f i c hypothesis of randomness. The net e f f e c t o the sample mean X of th t h i s experiment, the u n c e r t a i n t y statement a s s o c i a t e d ^ w i t h X ( f o r example s- = the estimated standard d e v i a t i o n of X) would c e r t a i n l y have το be based on many fewer degrees of freedom than n - i = 49. As i s evident from the data, there are not 50 independent observations of the t r a n s m i t t a n c e ; there would be c o n s i d e r a b l y fewer--and t h i s would r e s u l t (from a p r a c t i c a l p o i n t of view) i n a l a r g e r (and more r e a l i s t i c ) value f o r s-. Λ BAND PLOTS The assumed u n d e r l y i n g model f o r a p p l i c a t i o n of technique i s again as i n eq. ( 1 ) . To g r a p h i c a l l y t e s t model, however, an a l t e r n a t i v e model, v i z . , response Ϋ. = f i X ^ - )

+

e r r o r e..

this this

(2)

where f(Xi_-j ) i s some unknown f u n c t i o n of the v a r i a b l e X and where X i s a p o s s i b l e v a r i a b l e a f f e c t i n g the response. A band p l o t (4) i s a s p e c i a l l y - c o n s t r u c t e d p l o t of the response v a r i a b l e Y versus another v a r i a b l e X . A band p l o t considers a l l the data w i t h i n v a r i o u s c l a s s e s of the h o r i z o n t a l a x i s v a r i a b l e and then, r a t h e r than p l o t t i n g a l l such p o i n t s , summarizes each subset of data i n t o f i v e s t a t i s t i c s : the median, the lower and upper q u a r t i l e s , and the two extremes (minimum and maximum). A l i n e connecting the medians across the h o r i z o n t a l a x i s adds c o n t i n u i t y to the p l o t and gives a more robust i n d i c a t i o n of whether the response v a r i a b l e s h i f t s l o c a t i o n w i t h respect to the h o r i z o n t a l a x i s v a r i a b l e . Lines connecting the various lower q u a r t i l e s provide a lower p r a c t i c a l l i m i t to the "body" of the data whereas l i n e s connecting the upper q u a r t i l e s d e l i n e a t e an upper edge to the body of the data. The f l a t n e s s ( l a c k of trend) of the band between upper and lower q u a r t i l e s i s an i n d i c a t i o n of whether or not model 1 above (the f i x e d l o c a t i o n model) i s tenable. The width of the band between upper and lower q u a r t i l e s i s an i n d i c a t i o n of whether the f i x e d v a r i a t i o n assumption ( w i t h respect to the h o r i z o n t a l a x i s v a r i a b l e ) i s tenable. ls

1$


2.

45


FiLLiBEN

The example that w i l l be used f o r the band p l o t w i l l demonstrate how i t can ( i n c e r t a i n s p e c i a l circumstances) be used t o t e s t randomness. The data s e t c o n s i s t e d o f 400 percentage measurements taken from a near-complete surface i n s p e c t i o n o f a c i r c u l a r a u s t e n i t e standard reference material specimen. A given reading i s the percentage a u s t e n i t e value f o r t h a t p a r t i c u l a r small sub-area o f the specimen. To t e s t the hypothesis t h a t the specimen was homogeneous ( t h a t i s , t h a t 2dimensional randomness e x i s t e d ) , a band p l o t o f the percentage a u s t e n i t e readings versus angle (from some reference r a d i a l spoke o f the c i r c u l a r specimen) was constructed. Figure 5 i l l u s t r a t e s the r e s u l t i n g band p l o t s when t h i s angle f a c t o r was d i v i d e d i n t o 24 c l a s s e s with a c l a s s width o f 15 degrees was used. Thus, a l l data i n a given 15 degree wedge were assembled and then summarize q u a r t i l e , upper q u a r t i l e s t a t i s t i c s were then p l o t t e d t o represent t h a t specimen wedge r a t h e r than p l o t t i n g a l l the data i n the wedge. I f the sample were homogeneous (with respect to a n g l e ) , the band p l o t should be n e a r - f l a t over the e n t i r e 360° range. As the p l o t i l l u s t r a t e s , t h i s i s not the case f o r these d a t a — t h e percentage a u s t e n i t e measurements tend t o be low i n the v i c i n i t y o f 135°, tend t o be high near 280°, and tend again t o be low near 330°. The p l o t c l e a r l y shows the (homogeneity) randomness assumptions t o be suspect f o r t h i s specimen. 2-VARIABLE GRAPHICAL ANALYSIS OF VARIANCE This technique i s a p p l i c a b l e when a m u l t i - f a c t o r model o f the f o l l o w i n g type i s suspected: response Y. . = constant c + B^.X^. +

B

2 j

+

e r r o r e

i j

with an a l t e r n a t i v e general model of the f o l l o w i n g form: response Y. . = "Ρ(Χ ·> X p 2

ΐΊ

+

e r r o r

e n

*j

where f i s an unknown f u n c t i o n r e l a t i n g the nonrandom v a r i a b l e s X] and X t o the response v a r i a b l e Y, and where Xn" and Xo-iindicate d i f f e r e n t d i s c r e t e l i m i t s w i t h i n the v a r i a b l e s )( ana X , r e s p e c t i v e l y . Although the p l o t o f Y versus K w i l l c e r t a i n l y give the a n a l y s t some i n d i c a t i o n of the nature o f the f u n c t i o n f , the main p o i n t i s whether and how the response i s a d d i t i o n a l l y a f f e c t e d by the second v a r i a b l e (say v a r i a b l e X ) f o r some (but not a l l ) values o f the v a r i a b l e X . I f the v a r i a b l e X a f f e c t s the response f o r only some (but not a l l ) of the values o f the v a r i a b l e X i n s t a t i s t i c a l terms t h i s i s r e f e r r e d t o as an " i n t e r a c t i o n " e x i s t i n g between X and X --thus the e f f e c t o f X on the response i s dependent on the value of X . Rather than apply the usual 2-factor a n a l y s i s of variance 2

1

±

2

2

x

2

l 5

x

2


2

x

SPECIMAN HOMOGENERITY CALL PLOTBD(Y,X,N,XDELTA)


2.

FiLLiBEN


47

(ANOVA) t o data from t h i s model, we apply the g r a p h i c a l procedure i l l u s t r a t e d i n f i g u r e 6. This g r a p h i c a l a n a l y s i s o f variance (GANOVA) (10,11) i s simply a p l o t of the response v a r i a b l e versus one f a c t o r , w i t h d i f f e r e n t l e v e l s of the second f a c t o r i n d i c a t e d by d i f f e r e n t types of p l o t characters w i t h i n the p l o t . The GANOVA procedure i s very r e v e a l i n g i n t h a t i t communicates a l l o f the l a t e n t r e l e v a n t information i n t h i s 2-factor system. This technique i s i l l u s t r a t e d i n f i g u r e 6 which p l o t s the r e s i d u a l s from a f i t o f the response v a r i a b l e (days t o f a i l u r e ) from a s t r e s s f a t i g u e experiment versus l a b (9 l a b s ) with the value o f t h e p l o t c h a r a c t e r representing v a r i o u s l e v e l s (3 l e v e l s ) o f a second v a r i a b l e (experiment c o n f i g u r a t i o n ) which could (but h y p o t h e t i c a l l y should not) a f f e c t the response. A p l o t c h a r a c t e r value data p o i n t was generate The two independent v a r i a b l e s are: l a b o r a t o r y (9 l e v e l s — p l o t t e d h o r i z o n t a l l y ) , configuration (3 levels--denoted by d i f f e r e n t characters).

plot

Making reference t o f i g u r e 6, i t i s seen t h a t the assumption t h a t a l l l e v e l s o f the c o n f i g u r a t i o n f a c t o r a f f e c t s the response i n a uniform fashion i s untenable. I t i s c l e a r from the p l o t t h a t a lab-configuration interaction exists. For example, c o n f i g u r a t i o n s 2 and 3 y i e l d c o n s i s t e n t l y low values f o r l a b 4 w h i l e c o n f i g u r a t i o n 1 y i e l d s low and r a t h e r v a r i a b l e values f o r labs 5 and 7. A suspicious low observation a l s o i s seen t o e x i s t f o r lab 3, c o n f i g u r a t i o n 2. Such an augmented p l o t - - a s described above--is a useful technique f o r examining the assumption t h a t the response i s not dependent on some p a r t i c u l a r v a r i a b l e . I t i s t o be noted i n passing t h a t although the p l o t c h a r a c t e r i s the recommended procedure f o r conveying information about the second v a r i a b l e , one could a l s o j u s t as w e l l use the type o f l i n e f o r conveying the second-variable information. The former i s recommended when generating computer p r i n t e r p l o t s - which are by nature d i s c r e t e . The l a t t e r i s recommended when a continuous p r i n t i n g device ( i . e . , one capable o f drawing d i f f e r e n t types o f continuous l i n e s ) i s a v a i l a b l e . Figure 7 i l l u s t r a t e s the above l i n e - t y p e a l t e r n a t i v e with an example based on measured voltages from electrical connectors. The two independent v a r i a b l e s here are: time i n days ( p l o t t e d h o r i z o n t a l l y ) connector type (3 levels--denoted by d i f f e r e n t l i n e types) The m u l t i p l e replication.

l i n e s o f t h e same type a r e due t o experimental Although much can be s a i d about the p l o t and about

American Chemical Society In Validation of theLibrary Measurement Process; DeVoe, J.;

ACS Symposium Series; American Society: Washington, DC, 1977. 1155 16thChemical St., N.W.


48

CO

.ο SX

CUD

Ο

Ο

fi

·

2 — [0 Jj H

0)

+3 W

Ρ bû -P -H -H «M

Un


2.


FiLLiBEN

49

the r e l a t i v e e f f e c t s o f the two f a c t o r s on the response, we concentrate herein on v i o l a t i o n s o f b a s i c assumptions and note t h a t an apparent v i o l a t i o n i n the form of an o u t l i e r i s evident from the p l o t — n o t e how the f o u r t h data p o i n t of the bottom l i n e of the p l o t i s i n c o n s i s t e n t w i t h the other data l i n e s i n t h i s bottom group. This f o u r t h p o i n t i s c l e a r l y an o u t l i e r , and y e t i t s d e t e c t i o n may very e a s i l y have been l o s t i n the numerical mechanics o f a standard ANOVA. 3-VARIABLE GANOVA This 3 - v a r i a b l e GANOVA technique i s a p p l i c a b l e where a multifactor model i s a p p r o p r i a t e , i . e . , the u n d e r l y i n g hypothesized model i s o f the form (e.g., f o r three f a c t o r s ) : response Y.

j R

= constan + e r r o r e.

w i t h an a l t e r n a t i v e general model of the f o l l o w i n g : Y

= f

X

X

X

ijk ( li» 2j- 3k>

+

™

e

ijk

w i t h f unknown, where the doubly-subscripted B's r e f e r t o f a c t o r e f f e c t s and the doubly-subscripted X's r e f e r t o coded dummy l e v e l s o f each f a c t o r . Again, r a t h e r than apply the standard 3f a c t o r a n a l y s i s of variance (ANOVA) t o data from t h i s model, we apply the g r a p h i c a l procedure i l l u s t r a t e d i n f i g u r e 7. This graphical a n a l y s i s o f variance (GANOVA) (10,11) i s a g e n e r a l i z a t i o n o f the type of p l o t discussed i n s e c t i o n 6 and i s defined simply as a p l o t o f the response v a r i a b l e versus one f a c t o r , w i t h d i f f e r e n t l e v e l s of the second f a c t o r i n d i c a t e d by d i f f e r e n t types o f p l o t characters w i t h i n the p l o t , and w i t h the d i f f e r e n t l e v e l s of the t h i r d f a c t o r i n d i c a t e d by d i f f e r e n t types o f l i n e s w i t h i n the p l o t . The 3 - v a r i a b l e GANOVA conveniently communicates a t a glance a l l o f the r e l e v a n t information i n t h i s 3 - f a c t o r system. Figure 8 i l l u s t r a t e s the a p p l i c a t i o n of the 3 - v a r i a b l e GANOVA t o d r i l l t h r u s t f o r c e . These data are drawn from the e x c e l l e n t a r t i c l e by Hamaker (10) whose main p o i n t was e x a c t l y as we are emphasizing h e r e — v i z . , t h a t there e x i s t v a l u a b l e graphical alternatives t o the usual ANOVA. The three independent v a r i a b l e s are: d r i l l speed (5 l e v e l s — p l o t t e d h o r i z o n t a l l y ) m a t e r i a l (2 levels--denoted by d i f f e r e n t p l o t c h a r a c t e r s ) feed r a t e (3 l e v e l s — d e n o t e d by d i f f e r e n t l i n e types) R e f e r r i n g t o f i g u r e 8, one concludes t h a t v a r i a b l e s 2 ( m a t e r i a l ) and 3 (feed r a t e ) both a f f e c t the response i n a n o n - n e g l i g i b l e


VALIDATION

CALL GAN0V3(Y,F1,F2,F3,N)

O F T H E M E A S U R E M E N T PROCESS

ELECTRIC CONNECTORS

20-)—,—ι—,—ι—,—ι—,—ι—,—ι—,—ι ure 7.

0 20 40 60 80 100 120 UOLTAGE DROP UERSUS ELAPSED DAYS 6 / 2 3 / 7 5 JJF6.CLARK1 HOT/COLD COPPER OPEN PLUG 20 AMP BRASS » SOLID, STEEL = SHALL DASH , INNOUATIUE «= LARGE DASH

Three graphic analyses of variance for electrical connectors data

CALL GAN0V3(Y,F1,F2,F3,K)

1500 ι

HAMAKER ( 1 9 7 1 PIRT) DRILL THRUST FORCE

—

Review of the International Statistical Institute

Figure 8. Three graphic analyses of variance for Hamaker (10). Drill thrust force data; thrust force (v) vs. drill speed (x); plot character = material (2 levels); type of line = feed rate (3 levels).


2.

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FiLLiBEN

way. The apparent e x i s t e n c e o f an o u t l i e r i s a l s o evident from the p l o t . I t i s t o be noted t h a t due t o the necessary use o f l i n e types, the 3-variable GANOVA can be done only with the continuous p r i n t i n g devices. YOUDEN PLOT The Youden p l o t (12,13) i s a useful graphical technique most commonly a p p l i c a b l e t o i n t e r l a b o r a t o r y experiments when there e x i s t s e x a c t l y 2 runs ( o r 2 specimens, or 2 l e v e l s of some p a r t i c u l a r f a c t o r , e t c . ) t o be t e s t e d f o r a c e r t a i n property o f interest. Ideally, l a b o r a t o r i e s are " a l i k e , ( 1 ) ; whereas i f a l a b o r a t o r y e f f e c t e x i s t e d , an appropriate model might be: response Y. = constant c + L. + e r r o r e. (where 1_· represents an e f f e c t due t o laboratory i - - i = 1, 2,...,k) and a d d i t i o n a l l y , i f both a laboratory e f f e c t and a run e f f e c t e x i s t e d , an a p p r o p r i a t e model might be: Ί

response Y. = constant c + L. + R. + e r r o r e^. (where R^ represents a run e f f e c t f o r run j — j = 1 , 2 ) . To t e s t which model i s appropriate, a Youden p l o t i s a p p l i e d which i s defined as a p l o t o f the k (where k = the number of l a b o r a t o r i e s i n the experiment) coordinate p a i r s : Y , Y ) (Y2i> ^ 2 2 ) » · · · ( k l > k 2 ) > where Y-jj represents the measured values obtained from l a b o r a t o r y i ( i = 1, 2, ... , k) on run j ( j = 1, 2). 13L

Y

i 2

5

Y

To f a c i l i t a t e the graphical a n a l y s i s , the p l o t c h a r a c t e r i s again used t o "pack" i n e x t r a i n f o r m a t i o n — i n t h i s case, about the laboratory f a c t o r . Thus, e.g., a p l o t character o f 4 i n d i c a t e s t h a t the measurement i n question came from l a b o r a t o r y 4. The Youden p l o t i s i l l u s t r a t e d i n f i g u r e 9 as a p p l i e d t o data from an ASTM s t r e s s c o r r o s i o n experiment where 7 ( k ) l a b o r a t o r i e s were being t e s t e d . I f no l a b o r a t o r y o r run e f f e c t s e x i s t e d , the r e s u l t i n g Youden p l o t w i l l appear as a random 2-dimensional s c a t t e r o f points. A l t e r n a t i v e l y , i f l a b o r a t o r y and/or run e f f e c t s do e x i s t , much useful information about the nature o f such e f f e c t s can be gleaned from the r e s u l t i n g p l o t . The p l o t i n f i g u r e 9 a c t u a l l y i s based on 7 χ 5 = 35 p l o t points (not a l l o f which In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

52

VALIDATION O F T H E

MEASUREMENT


PROCESS

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53


appear due t o computer p r i n t e r o v e r s t r i k i n g ) . The m u l t i p l i c i t y of 5 i s due t o the e x i s t e n c e o f 5 r e p l i c a t i o n s per lab--such r e p l i c a t i o n s pose no problems i n u t i l i z i n g the Youden p l o t . With respect t o how t o i n t e r p r e t a Youden p l o t , several c h a r a c t e r i s t i c s are t o be noted. A displacement o f p o i n t s from the same l a b o r a t o r y along the 45° diagonal i s i n d i c a t i v e t h a t t h i s l a b o r a t o r y i s c o n s i s t e n t l y generating low ( o r high) readings r e l a t i v e t o the other l a b o r a t o r i e s ( t h e c l u s t e r o f l a b o r a t o r y 4 p o i n t s i n f i g u r e 9 i s i l l u s t r a t i v e of t h i s negative l a b o r a t o r y b i a s ) . On the other hand, a c l u s t e r o f p o i n t s from the same l a b o r a t o r y d i s p l a c e d o f f the diagonal represents i n c o n s i s t e n t readings by t h a t l a b o r a t o r y from one run t o the next. Figure 9 i n d i c a t e v a r i a b i l i t y problem are c o n s i s t e n t l y higher than those f o r run 2. The Youden p l o t i s a s i m p l e — y e t extremely method f o r a n a l y z i n g i n t e r l a b o r a t o r y data.

effective--

EXAMINING DISTRIBUTIONAL INFORMATION The d i s c u s s i o n has already touched on three (randomness, f i x e d l o c a t i o n , f i x e d v a r i a t i o n ) o f the four assumptions t y p i c a l l y made about a measurement process. The f o u r t h assumption ( f i x e d d i s t r i b u t i o n ) w i l l now be addressed. From a s t a t i s t i c a l p o i n t of view, there are f i v e reasons why d i s t r i b u t i o n a l i n f o r m a t i o n should be r o u t i n e l y checked: 1. optimal parameters ;

estimators

f o r location

and

variation

2. v a l i d i t y of c r i t i c a l values used i n s t a t i s t i c a l t e s t s o f significance; 3.

assessment of goodness o f f i t i n r e g r e s s i o n ;

4. e x i s t e n c e of o u t l i e r s ; 5. assessment o f whether the measurement process control.

is in

The l a s t o f these reasons (assessment o f whether a measurement process i s i n s t a t i s t i c a l c o n t r o l ) i s the main one w i t h respect to the o v e r a l l purpose o f t h i s paper. The f i r s t four reasons provide a d d i t i o n a l m o t i v a t i o n f o r checking distributional assumptions, and w i l l be i n d i v i d u a l l y touched on a t t h i s time.


54


The f i r s t (optimal estimators) p o i n t r e f e r s to the case where one i s i n t e r e s t e d i n e s t i m a t i n g from a given data s e t the l o c a t i o n parameter c and v a r i a t i o n ( d i s p e r s i o n or s c a l e ) parameter σ i n the model described i n eq. (1). ( I t i s assumed t h a t the e r r o r , e^ i s a random v a r i a b l e w i t h mean 0 and (unknown) standard d e v i a t i o n , σ. ) Various estimators of c would, f o r example, include the usual sample mean of η observations c = SY|/n, the sample median (c = the middle observation i n the ordered s e t of o b s e r v a t i o n s ) , or the sample midrange (c = the average of the s m a l l e s t and l a r g e s t o b s e r v a t i o n s ) . It is a statistical "fact-of-life" t h a t i n e s t i m a t i n g l o c a t i o n and v a r i a t i o n parameters, the goodness (accuracy) of a p a r t i c u l a r estimator and the choice of an optimal estimator are dependent on the underlying d i s t r i b u t i o n d i s t r i b u t i o n which generate (normal or Gaussian), the best estimator of c would be the sample mean. However, i f the underlying d i s t r i b u t i o n were uniform ( i . e . , i t had a f l a t - - r a t h e r than bell-shaped p r o b a b i l i t y f u n c t i o n ) , it can be t h e o r e t i c a l l y demonstrated that the sample midrange, c = ( s m a l l e s t + l a r g e s t ) / 2 i s a much more accurate estimator of c than the sample mean. A l t e r n a t i v e l y , i f the underlying d i s t r i b u t i o n f o r the data were, e.g., very "longt a i l e d " l i k e the Cauchy ( i ^ j e . , the p r o b a b i l i t y f u n c t i o n i s b e l l shaped but higher valued i n the t a i l s than the normal), then theory d i c t a t e s and p r a c t i c e confirms t h a t the sample median i s a much more accurate estimator of c than e i t h e r the sample mean or the simple midrange. Thus, i t i s seen t h a t f o r e s t i m a t i n g the constant c i n the s i m p l e s t p o s s i b l e response model (Y = c + e ) , a necessary p r e l i m i n a r y step i s t o "estimate" the underlying distribution. Although the c e n t r a l l i m i t theorem provides a t h e o r e t i c a l b a s i s f o r suggesting t h a t f o r many p h y s i c a l science experiments, the normal d i s t r i b u t i o n "should" be the underlying d i s t r i b u t i o n , such normality should never be a u t o m a t i c a l l y assumed. As w i l l be seen i n the remaining s e c t i o n s , s t a t i s t i c a l techniques do e x i s t which a l l o w the a n a l y s t t o e a s i l y and r o u t i n e l y check such d i s t r i b u t i o n a l models. The second reason why d i s t r i b u t i o n a l information should be checked deals w i t h the v a l i d i t y o f t e s t s t a t i s t i c s . In the m u l t i f a c t o r s t a t i s t i c a l techniques r e f e r r e d t o as r e g r e s s i o n and a n a l y s i s of v a r i a n c e , there are a v a r i e t y of t e s t s t a t i s t i c s (mostly t and F s t a t i s t i c s ) which are a p p l i e d t o t e s t the s i g n i f i c a n c e of various f a c t o r s i n the m u l t i - f a c t o r model. I t i s an important s t a t i s t i c a l f a c t t h a t the v a l i d i t y of these t e s t s t a t i s t i c s holds only i f the r e s i d u a l s ( d e v i a t i o n s ) a f t e r the f i t are normally d i s t r i b u t e d . That i s t o say, i t i s the d i s t r i b u t i o n a l c h a r a c t e r i s t i c of the r e s i d u a l s a f t e r the f i t t h a t d i c t a t e the v a l i d i t y of the t and F s t a t i s t i c s . I f the t r u e underlying d i s t r i b u t i o n of the r e s i d u a l s i s non-normal, t h i s w i l l a f f e c t the t r u e s i g n i f i c a n c e l e v e l s of the t e s t s t a t i s t i c s . The net r e s u l t i s t h a t u l t i m a t e l y the conclusions about the


2.

FiLLiBEN


55

s i g n i f i c a n c e o f various f a c t o r s i n r e g r e s s i o n and ANOVA may be incorrect. Again, as emphasized before, no b l i n d assumptions need be made about the d i s t r i b u t i o n o f such residuals. Techniques w i l l be demonstrated t o a l l o w the d i s t r i b u t i o n t o be r o u t i n e l y checked. The t h i r d reason f o r checking d i s t r i b u t i o n a l i n f o r m a t i o n i s r e l a t e d t o the aforementioned r e g r e s s i o n and ANOVA. The p o i n t t o be emphasized i s t h a t an a d d i t i o n a l important reason f o r examining the d i s t r i b u t i o n o f r e s i d u a l s a f t e r t h e f i t i s t o determine whether o r not one has a r r i v e d a t a reasonable d e t e r m i n i s t i c o r f u n c t i o n a l model f o r the data. I f the f i t t e d r e g r e s s i o n or ANOVA model i s c o r r e c t , the r e s i d u a l s a f t e r the f i t should i d e a l l y have the same four p r o p e r t i e s as has been p r e v i o u s l y discusse variable, viz. : random fixed location fixed variation fixed distribution In a large m a j o r i t y of cases, the r e s i d u a l s a f t e r the c o r r e c t f i t w i l l not only f o l l o w some f i x e d d i s t r i b u t i o n , but w i l l a l s o rather specifically follow a normal distribution. The i m p l i c a t i o n o f course i s t h a t i n order t o assess whether o r not one has a c o r r e c t f i t , one ought t o examine the d i s t r i b u t i o n o f the r e s i d u a l s t o check f o r such normality. Though not a s u f f i c i e n t c o n d i t i o n i n i t s e l f f o r adequate f i t , the normality of the r e s i d u a l s serves as a p r a c t i c a l necessary c o n d i t i o n which may p r o f i t a b l y be used i n determining model adequacy. From a pragmatic p o i n t o f view, t h i s t h i r d reason f o r examining d i s t r i b u t i o n a l information i s an extremely important one. The f o u r t h reason f o r checking d i s t r i b u t i o n a l information deals w i t h the o u t l i e r problem. How does one t e l l i f a suspicious-looking observation i s i n fact an o u t l i e r ? ( " O u t l i e r " as here used r e f e r s t o an observation t h a t was generated from a d i f f e r e n t model o r a d i f f e r e n t d i s t r i b u t i o n than was the main "body" o f the data.) Frequently, an o u t l i e r w i l l manifest i t s e l f i n one o r another o f the p l o t s already discussed i n previous s e c t i o n s . However, an a d d i t i o n a l and a t times more s e n s i t i v e check i s given by a d e t a i l e d examination o f the d i s t r i b u t i o n of the data. An observation which appears t o be a b o r d e r l i n e o u t l i e r i n some previous p l o t s f r e q u e n t l y turns out to be a w e l l - d e f i n e d o u t l i e r when examined r e l a t i v e t o the d i s t r i b u t i o n o f the r e s t o f the data. The same numerical observation may very well be a " t y p i c a l " extreme observation r e l a t i v e t o one d i s t r i b u t i o n but an o u t l i e r r e l a t i v e t o another d i s t r i b u t i o n . By examining the d i s t r i b u t i o n of the data (and/or the r e s i d u a l s a f t e r a f i t ) , the a n a l y s t gives himself a much more s e n s i t i v e t o o l f o r o u t l i e r d e t e c t i o n and i d e n t i f i c a t i o n . In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

56


The f i f t h and f i n a l p o i n t w i t h respect t o the importance of checking f o r d i s t r i b u t i o n a l i n f o r m a t i o n deals w i t h the main p o i n t of t h i s p a p e r — p r e d i c t a b i l i t y and the determination of whether a process i s " i n c o n t r o l . " P r e d i c t a b i l i t y means being able t o make p r o b a b i l i t y statements about f u t u r e output from the process. These p r o b a b i l i t y statements w i l l most commonly r e f e r to expected v a r i a t i o n (about some t y p i c a l value) of output from the process. The main p o i n t i s t h a t such p r o b a b i l i t y statements w i l l change depending on the t r u e u n d e r l y i n g d i s t r i b u t i o n of the process. A statement such as: "97-1/2% of the f u t u r e observations from t h i s measurement process should f a l l w i t h i n (approximately) 3 standard d e v i a t i o n

w i l l of course be t r u e i f the u n d e r l y i n g generating d i s t r i b u t i o n i s normal but on the other hand w i l l be f a l s e i f the u n d e r l y i n g d i s t r i b u t i o n i s ( f o r example) uniform, Cauchy or e x p o n e n t i a l . I t i s important f o r a n a l y s t s t o keep i n mind t h a t f o r non-normal d i s t r i b u t i o n s , a p r o b a b i l i t y statement about expected f u t u r e occurrences (e.g., w i t h i n two standard d e v i a t i o n s of the mean) w i l l change from d i s t r i b u t i o n t o d i s t r i b u t i o n . The exact proba b i l i t y value (= 97-1/2% f o r the normal) must be (and can be) determined once the u n d e r l y i n g d i s t r i b u t i o n i s determined. I t is a r e c u r r i n g requirement t o "estimate" the u n d e r l y i n g distribution. With these motivations and j u s t i f i c a t i o n s f o r examining d i s t r i b u t i o n a l i n f o r m a t i o n , the next two s e c t i o n s w i l l present various data a n a l y s i s techniques t o c a r r y out such examinations. PROBABILITY PLOTS A p r o b a b i l i t y p l o t (14,15,16,17,18,19,20,21) i s a g r a p h i c a l t o o l f o r a s s e s s i n g the goodness of f i t of some hypothesized d i s t r i b u t i o n (e.g., normal, uniform, Poisson, e t c . ) t o an observed data set. In d e s c r i b i n g a p r o b a b i l i t y p l o t , i t w i l l be assumed t h a t the model i s as i n d i c a t e d i n eq. ( 1 ) . However, i t i s t o be kept i n mind t h a t the p r o b a b i l i t y p l o t technique has much g r e a t e r g e n e r a l i t y inasmuch as i t can be a p p l i e d t o the r e s i d u a l s a f t e r any m u l t i f a c t o r f i t as w e l l as t o the raw observations from the simple Y. = c + e.. model. A p r o b a b i l i t y p l o t i s ( i n general) simply a p l o t of the observed ordered ( s m a l l e s t t o l a r g e s t ) observations Y j on the vertical axis versus the corresponding typical ordered observations based on whatever d i s t r i b u t i o n i s being hypothesized. Thus, f o r example, i f one were forming a normal p r o b a b i l i t y p l o t , the f o l l o w i n g η coordinate p l o t p o i n t s would


2.


FiLLiBEN

57

be formed: ( Y M J , Y , M ) , . . . ( Y , M ) where Y i s the observed s m a l l e s t data p o i n t , and M i s the t h e o r e t i c a l "expected" value o f the s m a l l e s t data p o i n t from a sample o f s i z e η normally d i s t r i b u t e d p o i n t s . S i m i l a r l y , Y would be the second s m a l l e s t observed value and M would be the "expected value" o f the second s m a l l e s t o b s e r v a t i o n i n a sample o f s i z e η normally d i s t r i b u t e d p o i n t s . This proceeds up t o Y which would be the l a r g e s t observed data value and M would be the "expected value" of the l a r g e s t observation i n a sample o f s i z e η from a normal d i s t r i b u t i o n . Thus, i n forming a normal p r o b a b i l i t y p l o t , the v e r t i c a l a x i s values depend only on the observed data, w h i l e the horizontal a x i s values a r e generated independently o f the observed data and depend only on the t h e o r e t i c a l d i s t r i b u t i o n being t e s t e d o r hypothesized ( n o r m a l i t y i n t h i s case) and a l s o the value o f the sampl s i m p l e s t terms a p l o "expected." i $

2

2

n

1

n

x

2

2

n

t

h

The crux o f the p r o b a b i l i t y p l o t i s t h a t the i ordered observation i n a sample o f s i z e η from some d i s t r i b u t i o n i s i t s e l f a random v a r i a b l e which has a d i s t r i b u t i o n unto i t s e l f . This d i s t r i b u t i o n o f the i^h ordered o b s e r v a t i o n can be t h e o r e t i c a l l y d e r i v e d and summarized ( i . e . , mapped i n t o a s i n g l e " t y p i c a l value") as can any other random v a r i a b l e . One can then pose the r e l e v a n t question as t o what s i n g l e number best t y p i f i e s the d i s t r i b u t i o n a s s o c i a t e d w i t h a given ordered o b s e r v a t i o n i n a sample o f s i z e , n. A computational disadvantage t o the use o f the mean i s t h a t d i f f e r e n t i n t e g r a t i o n techniques may be needed f o r d i f f e r e n t types o f d i s t r i b u t i o n . For some d i s t r i b u t i o n s the mathematical i n t e g r a t i o n does not e x i s t . These c o n s i d e r a t i o n s d i c t a t e t h a t the median i s s u p e r i o r t o the mean i n terms o f forming a t h e o r e t i c a l "expected" o r " t y p i c a l " value t o summarize the e n t i r e d i s t r i b u t i o n o f the l'th ordered o b s e r v a t i o n i n a sample o f s i z e η from the d i s t r i b u t i o n being t e s t e d . Thus, t o be p r e c i s e , the M-j on the h o r i z o n t a l a x i s of the p r o b a b i l i t y p l o t i s taken t o be the median o f the d i s t r i b u t i o n o f the i ™ ordered observation i n a sample o f s i z e η from whatever u n d e r l y i n g d i s t r i b u t i o n i s being t e s t e d . I t i s t o be noted t h a t the s e t o f Mj as a whole w i l l change from one hypothesized d i s t r i b u t i o n t o another--and t h e r e i n l i e s the distributional s e n s i t i v i t y o f the p r o b a b i l i t y plot technique. For example, i f the hypothesized d i s t r i b u t i o n i s uniform, then a uniform p r o b a b i l i t y p l o t would be formed and the w i l l be approximately equi-spaced t o r e f l e c t the f l a t nature of the uniform p r o b a b i l i t y d e n s i t y f u n c t i o n . On the other hand, i f the hypothesized d i s t r i b u t i o n i s normal, then the Mj w i l l have a r a t h e r sparse spacing f o r the f i r s t few ( M , M , M ,...) and l a s t few (..., M __ , M , M ) values but w i l l become more densely spaced as one proceeds xoward the middle of the s e t (... , ^-1/2» n/2> n + l / 2 behavior f o r the M-j i s of x

n

M

M

2

2

3

n - 1

S u c h


VALIDATION OF

58 course r e f l e c t i n g density function.

the

bell-shape

of

THE

the

MEASUREMENT

normal

PROCESS

probability

In summary, f o r a s p e c i f i c hypothesized d i s t r i b u t i o n , D Q the l'th value Mj i n the corresponding p r o b a b i l i t y p l o t i s a t h e o r e t i c a l (but computable) value c l o s e to what one t y p i c a l l y would "expect" f o r the value of the i order observation i f i n f a c t one had taken a random sample of s i z e η from the distribution D . ο How does one use and i n t e r p r e t p r o b a b i l i t y p l o t s ? In l i g h t of the above, i t i s seen t h a t i f i n f a c t the observed data do have a d i s t r i b u t i o n t h a t the a n a l y s t has hypothesized, then (except f o r an unimportant l o c a t i o n and scale f a c t o r which can be determined a f t e r the f o r a l l i , t h a t i s , ove of Y j versus Μ · w i l l be n e a r - l i n e a r . This l i n e a r i t y i s the dominant f e a t u r e to be checked f o r i n any p r o b a b i l i t y p l o t . A linear probability plot indicates t h a t the hypothesized d i s t r i b u t i o n , D gives a good d i s t r i b u t i o n a l f i t to the observed data set. This combination of s i m p l i c i t y of use along w i t h distributional s e n s i t i v i t y makes the probability plot an extremely powerful t o o l f o r data a n a l y s i s . η

Q

The next l o g i c a l question to be examined i s what w i l l the p r o b a b i l i t y p l o t look l i k e i f the hypothesized d i s t r i b u t i o n , D i s not c o r r e c t - - ! . e . , i f the u n d e r l y i n g d i s t r i b u t i o n t h a t generated the data i s not the same as the d i s t r i b u t i o n , D hypothesized by the a n a l y s t . In t h i s case, the Υ· and will not match over the e n t i r e set and so the r e s u l t i n g p r o b a b i l i t y p l o t w i l l be nonlinear. A very useful aspect of the p r o b a b i l i t y p l o t i s t h a t the type of n o n l i n e a r i t y e x h i b i t e d by a given p r o b a b i l i t y p l o t w i l l give the a n a l y s t useful i n f o r m a t i o n as to how the d i s t r i b u t i o n a l hypothesis, D should be adjusted so as to a r r i v e at a b e t t e r d i s t r i b u t i o n a l f i t to the data. This l a s t p o i n t i s an important asset of the p r o b a b i l i t y p l o t technique f o r t e s t i n g assumptions i n d i s t r i b u t i o n . For example, i f the a n a l y s t b e l i e v e s t h a t the true underlying d i s t r i b u t i o n i s i n general a symmetric d i s t r i b u t i o n ( i . e . , a d i s t r i b u t i o n which has a p r o b a b i l i t y f u n c t i o n as i l l u s t r a t e d i n f i g . 10) as opposed to a skewed d i s t r i b u t i o n (e.g., w i t h a p r o b a b i l i t y f u n c t i o n as i l l u s t r a t e d i n f i g . 11), then the p r o b a b i l i t y p l o t a n a l y s i s to be p r e s e n t l y d e s c r i b e d i s r a t h e r t y p i c a l . The f i r s t step i n such a n a l y s i s i s u s u a l l y to t e s t the normal d i s t r i b u t i o n hypothesis (the normal being the most commonly-employed symmetric d i s t r i b u t i o n ) by forming a normal p r o b a b i l i t y p l o t . In forming such a p l o t , l e t us c o n s i d e r the f o l l o w i n g f i v e types of most commonly-encountered appearances of the normal p r o b a b i l i t y p l o t : l i n e a r , S-shaped, N-shaped, nonsymmetric c r o s s - o v e r , and convex (see f i g . 12). 0

0

Ί

Q


FiLLiBEN




60

I f the normal p r o b a b i l i t y p l o t has the l i n e a r appearance of f i g u r e 12a, t h i s i n d i c a t e s t h a t the normal d i s t r i b u t i o n y i e l d s an acceptably good f i t t o the data; so no f u r t h e r p r o b a b i l i t y p l o t s need be formed and the d i s t r i b u t i o n a n a l y s i s i s completed. I f the normal p r o b a b i l i t y p l o t has the S-shaped appearance of f i g u r e 12b, t h i s i n d i c a t e s t h a t the D = normal hypothesis i s i n c o r r e c t , and t h a t the t r u e underlying d i s t r i b u t i o n f o r the data i s symmetric but i s s h o r t e r - t a i l e d than normal. Examples of such symmetric d i s t r i b u t i o n s , s h o r t e r - t a i l e d than normal, would be a U-shaped d i s t r i b u t i o n , a uniform d i s t r i b u t i o n , or a truncated bell-shaped d i s t r i b u t i o n . (These three d i s t r i b u t i o n s have p r o b a b i l i t y f u n c t i o n s as i l l u s t r a t e d i n f i g . 13.) In such a case, the second i t e r a t i o n by the a n a l y s t would be t o form an additional probabilit (e.g., from a unifor uniform p r o b a b i l i t y p l o t i s s t i l l S-shaped, the t h i r d i t e r a t i o n i s t o form a p r o b a b i l i t y p l o t f o r a d i s t r i b u t i o n t h a t i s even s h o r t e r - t a i l e d than uniform (e.g., some U-shaped d i s t r i b u t i o n ) . On the other hand, i f the uniform p r o b a b i l i t y p l o t has a form as i n f i g u r e 12c (and which w i l l be represented very crudely as an "N shape"), the t h i r d i t e r a t i o n would be t o form a p r o b a b i l i t y p l o t f o r some d i s t r i b u t i o n s h o r t e r - t a i l e d than normal but l o n g e r - t a i l e d than uniform. Such i t e r a t i o n i s continued u n t i l there i s convergence t o an acceptable l i n e a r p r o b a b i l i t y p l o t . In p r a c t i c e , the a n a l y s i s w i l l u s u a l l y converge t o an acceptable d i s t r i b u t i o n i n a r e l a t i v e l y small number of i t e r a t i o n s . 0

To consider another p o s s i b i l i t y , i f the o r i g i n a l normal p r o b a b i l i t y p l o t has the "N-shaped" appearance o f f i g u r e 12c, t h i s suggests t h a t the D = normal hypothesis i s i n c o r r e c t , and t h a t the true underlying d i s t r i b u t i o n f o r the data i s s t i l l symmetric but i s l o n g e r - t a i l e d than normal. An example would be the Cauchy ( a l s o known as the Lorentzian) d i s t r i b u t i o n which i s a bell-shaped d i s t r i b u t i o n whose " t a i l s " are "longer" or " f a t t e r " than the normal. Figure lOd i l l u s t r a t e s the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r the Cauchy d i s t r i b u t i o n . The t y p i c a l nature of l o n g - t a i l e d d i s t r i b u t i o n s l i k e the Cauchy i s t h a t i f the measurement process i s generating data from such a d i s t r i b u t i o n , i t i s more l i k e l y t o generate some observations which are c o n s i d e r a b l y removed from the "body" of the data than i n sampling from a more m o d e r a t e - t a i l e d d i s t r i b u t i o n (such as the normal). As b e f o r e , since the o r i g i n a l normal p r o b a b i l i t y p l o t was not l i n e a r , the a n a l y s t should perform the i t e r a t i v e a n a l y s i s t o produce a l o n g e r - t a i l e d probability plot ( l i k e a Cauchy p r o b a b i l i t y p l o t ) . I f t h i s second p l o t i s l i n e a r , t h i s i m p l i e s t h a t the Cauchy y i e l d s an acceptable d i s t r i b u t i o n . I f t h i s second p l o t i s not l i n e a r , other i t e r a t i o n s on D must be made based on the S-shaped or N-shaped appearance of the Cauchy p r o b a b i l i t y p l o t . A r o u t i n e computerized procedure t o c a r r y out such i t e r a t i o n s f o r the symmetric f a m i l y of d i s t r i b u t i o n s w i l l be presented i n s e c t i o n 11. 0

0


2.

FiLLiBEN


χ

61

χχχχχχχΧ

Figure 12. Typical shapes of probability plots, (a.) Linear; (b.) s-shaped; (d.) nonsymmetric crossover; (e.) convex.

Figure 13. Distributions shorter-tailed than normal, a. Tukey λ = 1.5 distribution (very short-tailed); b. uniform distribution (shorttailed); c. truncated normal distribution (moderate/short-tailed); d. normal distribution (moderate-tailed).


VALIDATION OF

62

THE


I f the o r i g i n a l normal p r o b a b i l i t y p l o t has the appearance of f i g u r e 12d where the diagonal l i n e d i v i d e d the data p o i n t s on e i t h e r side unequally or as i n 12e where the diagonal l i n e does not d i v i d e the data at a l l , t h i s i s i n d i c a t i v e t h a t not only may the s p e c i f i c hypothesis t h a t D = normal, be i n c o r r e c t , but a l s o t h a t the hypothesis of a symmetric d i s t r i b u t i o n may be i n c o r r e c t . In such a case, the t r u e u n d e r l y i n g d i s t r i b u t i o n f o r the data would then be some type of skewed d i s t r i b u t i o n (e.g. , of the types w i t h p r o b a b i l i t y d e n s i t y f u n c t i o n s as i l l u s t r a t e d i n f i g u r e 11). In forming a d d i t i o n a l p r o b a b i l i t y p l o t s to f i t the data, the a n a l y s t should consequently consider d i s t r i b u t i o n s which are skewed. To enumerate but a few of the skewed d i s t r i b u t i o n s t h a t might be considered i n subsequent i t e r a t i o n s , one would i n c l u d e the log-normal d i s t r i b u t i o n , the half-normal d i s t r i b u t i o n , the exponentia of d i s t r i b u t i o n s , th the Pareto f a m i l y of d i s t r i b u t i o n s . For an e x c e l l e n t general d e s c r i p t i o n of various d i s t r i b u t i o n s and d i s t r i b u t i o n a l f a m i l i e s (both skewed and symmetric) the reader i s r e f e r r e d to the comprehensive t e x t s by Johnson and Kotz (22,23). 0

One f i n a l p o i n t regarding o u t l i e r - d e t e c t i o n i s noteworthy. I f i n forming, f o r example, a normal p r o b a b i l i t y p l o t , the p l o t turns out to be l i n e a r w i t h the exception of one or two p o i n t s (see f i g . 14), how i s t h i s to be i n t e r p r e t e d ? This type of p l o t i s i n d i c a t i n g t h a t the normal f i t i s acceptable f o r most of the data but t h a t one or two p o i n t s are o u t l i e r s and do not seem to agree w i t h the normality assumption. The p r o b a b i l i t y p l o t i s thus seen to be usable f o r d e t e c t i n g o u t l i e r s . The next step i n the a n a l y s i s i s f o r the a n a l y s t to d e l e t e the one or two o f f e n d i n g p o i n t s and to form a p r o b a b i l i t y p l o t w i t h the remaining p o i n t s . I f t h i s second p l o t i s s t i l l s t r o n g l y l i n e a r , t h i s gives a d d i t i o n a l support to the hypothesis t h a t the data are n o r m a l l y - d i s t r i b u t e d and t h a t the one or two questionable p o i n t s are i n f a c t o u t l i e r s . The use of the p r o b a b i l i t y p l o t as a t o o l f o r o u t l i e r d e t e c t i o n i s g e n e r a l l y more s e n s i t i v e than any of the techniques discussed i n previous s e c t i o n s . The experimenter i s a l s o reminded t h a t although such o u t l i e r s may be deleted from f u r t h e r a n a l y s i s , these o u t l i e r s e x i s t " f o r a reason" and the experimenter ought to s a t i s f y himself t h a t he has determined what set of experimental circumstances had l e d to them. The examination of o u t l i e r s almost i n v a r i a b l y leads to improved design of the experiment and ultimately to an improved understanding of the experimental f a c t o r s which prevent a measurement process from being " i n c o n t r o l . " Having discussed what a p r o b a b i l i t y p l o t i s and how one i s to be interpreted, we now enumerate b r i e f l y some of the advantages of using a p r o b a b i l i t y p l o t as opposed to other methods of checking f o r d i s t r i b u t i o n a l information (e.g., histogram, χ s t a t i s t i c , f i t to p r o b a b i l i t y d e n s i t y f u n c t i o n ) . 2


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64


THE

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Although i t w i l l be shown t h a t the p r o b a b i l i t y p l o t technique i s to be highly recommended, the various techniques are complementary. An o u t l i n e of the advantages of the p r o b a b i l i t y p l o t approach i s as f o l l o w s : Graphical Technique The p r o b a b i l i t y p l o t i s a graphical technique and b e n e f i t s from a l l of the advantages of graphics as o u t l i n e d the end of s e c t i o n 1. Easy to

so at

Use

The dominant f e a t u r e to be checked i n a p r o b a b i l i t y p l o t i s l i n e a r i t y . This i s th is easily detectable p r o b a b i l i t y p l o t s i s no longer a problem. A p p l i c a b l e to a Wide Range of D i s t r i b u t i o n The p r o b a b i l i t y p l o t technique can be a p p l i e d to a wide range of d i s t r i b u t i o n s — c e r t a i n l y f o r a l l d i s t r i b u t i o n s commonly encountered i n p r a c t i c e . These d i s t r i b u t i o n s would cover those of both the continuous (e.g., normal) and the d i s c r e t e (e.g., Poisson) types. Such d i s t r i b u t i o n s would i n c l u d e the normal (Gaussian), uniform, v a r i o u s U-shaped d i s t r i b u t i o n s , Cauchy, L o g i s t i c , h a l f - n o r m a l , log-normal, e x p o n e n t i a l , gamma, beta, Wei b u l l , extreme v a l u e , Pareto, b i n o m i a l , Poisson, geometric, and negative binomial. For each such d i s t r i b u t i o n D, there nonetheless remains the same uniform approach i n i n t e r p r e t i n g the r e s u l t i n g p r o b a b i l i t y p l o t ; v i z . , to check f o r l i n e a r i t y and if nonlinear to make adjustments to the hypothesized d i s t r i b u t i o n s D a c c o r d i n g l y — b a s e d on the type of n o n l i n e a r i t y encountered. 0

No a p r i o r i Location and V a r i a t i o n Estimates Needed 2

One problem a s s o c i a t e d w i t h the χ goodness of f i t techniques and w i t h the e m p i r i c a l technique of superimposing a f i t t e d p r o b a b i l i t y d e n s i t y f u n c t i o n over a histogram of the data i s t h a t a p r i o r i values of the parameter ( u s u a l l y l o c a t i o n and v a r i a t i o n ) are needed before the technique can a c t u a l l y be a p p l i e d . This i s f r e q u e n t l y i m p r a c t i c a l f o r two reasons: 1. Such available.

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Since the p r o b a b i l i t y p l o t technique does not need a p r i o r i values t o be a p p l i e d , i t i s s u p e r i o r and d e f i n i t e l y f a r more p r a c t i c a l than the χ and f i t t e d p r o b a b i l i t y d e n s i t y f u n c t i o n methods f o r d i s t r i b u t i o n a l t e s t i n g . 2

Automatic Estimate of Location and V a r i a t i o n Obtained An a d d i t i o n a l advantage o f a p p l y i n g the technique i s t h a t estimates o f l o c a t i o n and s c a l e parameters a r e a u t o m a t i c a l l y produced as a secondary output. These l o c a t i o n and v a r i a t i o n estimates a r e d e r i v a b l e , r e s p e c t i v e l y , from the v e r t i c a l a x i s i n t e r c e p t and t h e slope o f the r e s u l t i n g p r o b a b i l i t y p l o t . Although the a n a l y s t i s reminded t h a t such l o c a t i o n and v a r i a t i o n estimates a r e not t o be considered as the optimal (minimum v a r i a n c e ) estimates practical indication should be. No Grouping o f Data Need be Done A problem a s s o c i a t e d w i t h the histogram technique (whereby the a n a l y s t simply forms a histogram o f the data and notes i t s general shape without applying or f i t t i n g a specific d i s t r i b u t i o n t o i t ) f o r gathering d i s t r i b u t i o n a l i n f o r m a t i o n i s t h a t o f choosing the grouping i n t e r v a l (the c l a s s width) f o r the histogram. The appearance o f the r e s u l t i n g histogram i s r a t h e r s t r o n g l y a f f e c t e d by the choice o f t h i s c l a s s width. A c l a s s width which i s "too narrow" w i l l r e s u l t i n a histogram i n which the true d i s t r i b u t i o n a l shape i s obscured by excessive v a r i a b i l i t y i n the height o f the bar a s s o c i a t e d w i t h each c l a s s , a c l a s s width which i s "top wide" w i l l r e s u l t i n a histogram i n which the t r u e d i s t r i b u t i o n a l shape i s obscured by "leakage" across neighboring c l a s s e s so t h a t the d i s t r i b u t i o n a l content f o r a given c l a s s w i l l be "smeared" out over several c l a s s e s . Although r u l e s o f thumb do e x i s t f o r choosing a reasonable c l a s s w i d t h , t h i s nevertheless c a l l s f o r an intermediate judgment t o be made by the a n a l y s t . The use o f the p r o b a b i l i t y p l o t technique e l i m i n a t e s the need f o r such a choice. Inasmuch as a p r o b a b i l i t y p l o t uses each observation i n d i v i d u a l l y and r e q u i r e s no grouping, t h i s f r e e s t h e a n a l y s t from making choices about c l a s s widths and e l i m i n a t e s ( i f the wrong c l a s s width happens t o have been chosen) a p o s s i b l e undesirable approach-dependency on the u l t i m a t e c o n c l u s i o n s . The net p o s i t i v e e f f e c t o f the p r o b a b i l i t y p l o t i s t h a t i t allows a d i s t r i b u t i o n a l a n a l y s i s t o be performed i n a completely d i r e c t and automatic f a s h i o n w i t h no intermediate d e c i s i o n s (such as c l a s s width) t o be made by the a n a l y s t . Thus, the conclusions from the d i s t r i b u t i o n a l a n a l y s i s w i l l r e f l e c t only the content o f the data and w i l l avoid p o s s i b l e biases introduced by the a n a l y s i s .



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For the Josephson J u n c t i o n v o l t a g e counts data (see f i g . 24), the MPPCC c r i t e r i o n i n d i c a t e d t h a t the normal d i s t r i b u t i o n was the best f i t which i s i n agreement w i t h t h e already-seen l i n e a r i t y of the ( f i g . 17) normal p r o b a b i l i t y p l o t . For the wind v e l o c i t y data (see f i g . 25), the MPPCC c r i t e r i o n i n d i c a t e d t h a t the normal d i s t r i b u t i o n (though not p r e c i s e l y optimal) was nevertheless near-optimal and i s i n agreement w i t h the nearl i n e a r i t y o f the ( f i g . 18) normal p r o b a b i l i t y p l o t . The MPPCC c r i t e r i a as a p p l i e d t o the beam d e f l e c t i o n data i s presented i n f i g u r e 26. As expected, the b e s t - f i t symmetric d i s t r i b u t i o n t o t h i s data s e t i s i n the U-shaped d i s t r i b u t i o n region. The a n a l y s i s o f the x-ray c r y s t a l l o g r a p h y r e s i d u a l s (see f i g . 27) a l s o confirms what was p r e v i o u s l y s u s p e c t e d — v i z . , t h a t the best d i s t r i b u t i o n i s i n the moderate to moderate l o n g - t a i l e d region. The f i n a l exampl MPPCC f o r a skewed d i s t r i b u t i o n a l f a m i l y - - i n t h i s case the extreme value f a m i l y . The data set considered i s annual maximum wind speeds a t Walla W a l l a , Washington. In a f a s h i o n s i m i l a r t o the symmetric f a m i l y a n a l y s i s , a r e p r e s e n t a t i v e s e t o f 46 members o f the extreme value d i s t r i b u t i o n a l f a m i l y were s e l e c t e d and the p r o b a b i l i t y p l o t c o r r e l a t i o n c o e f f i c i e n t was computed for each. The r e s u l t s o f the a n a l y s i s i n d i c a t e d t h a t an extreme value d i s t r i b u t i o n w i t h shape parameter γ = 7 y i e l d s the best fit. The use o f the MPPCC c r i t e r i a i s recommended not as a replacement f o r examination of i n d i v i d u a l p r o b a b i l i t y p l o t s , but r a t h e r as an important complement t o such analyses. The automated procedures presented above a l l o w the a n a l y s t t o q u i c k l y "converge" t o a neighborhood o f d i s t r i b u t i o n s which provide good f i t s to the data set under examination. 4-PLOT ANALYSIS The general question posed i n t h i s s e c t i o n i s as f o l l o w s : Given t h a t one would l i k e t o prepare a completely automated (computerized) f i r s t - p a s s a n a l y s i s t h a t would be a p p l i c a b l e to a wide v a r i e t y o f data sets and which would take no more than one computer page, what would one i n c l u d e on t h a t s i n g l e page? As has been s t r e s s e d throughout t h i s paper, the b a s i c assumptions which must be t e s t e d t o assure t h a t a measurement process i s i n c o n t r o l i n c l u d e randomness, f i x e d l o c a t i o n , f i x e d v a r i a t i o n , and f i x e d d i s t r i b u t i o n . The value o f the run sequence p l o t and the lag-1 a u t o c o r r e l a t i o n p l o t s have already been amply discussed w i t h respect to the f i r s t three p o i n t s . The p r o b a b i l i t y p l o t has been discussed a t length w i t h respect t o the l a s t p o i n t . And so the following 4-plot a n a l y s i s i s presented as an automated



BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN BETWEEN

THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE THE

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ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED CRDEREO ORDERED ORDERED ORDERED OROERED ORDERED ORDERED ORDERED ORDERED ORDEREO ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED ORDERED

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LAMBDA = 2 . 0 DI S T . LAMBDA = 1 .9 DIST. LAMBDA 1.8 D I S T . LAMBDA = I . 7 DI S T . LAMBDA = 1 .6 DIST. LAMBDA = 1.5 DIST. LAMBDA = 1 .4 DIST. LAMBDA 1 .3 DIST. LAMBDA = 1 .2 DIST . LAMBDA = 1 .1 DI S T . LAMBDA = 1.0 D I S T . LAMBDA = • 9 DIST. LAMBDA = • 8 DI S T . LAMBDA .7 DIST. LAMBOA = • 6 DIST. LAMBDA = . 5 DIST . • 4 DIST. LAMBDA = LAMBDA = . 3 DI S T . LAMBDA = • 2 DIST. NORMAL D I S T R I B U T ION LAMBDA = • 1 DI S T . LOGISTIC DIST. DIST. DOUBLE E X P . DIST. LAMBDA -.1 LAMBDA - . 2 DIST. LAMBDA = - . 3 DIST. LAMBDA = - . 4 DIST. LAMBDA = - . 5 DIST. LAMBOA = - . 6 DI S T . - .7 D I S T . LAMBDA = LAMBDA = - . 8 OIST. LAMBDA = - . 9 DIST. CAUCHY D I S T R I B U T I O N DIST. LAMBDA = - 1 . 0 DI S T . LAMBDA -1.1 LAMBDA = -1 . 2 D I S T . LAMBDA = - 1 . 3 D I S T . LAMBDA - 1 . 4 DI S T . DIST. LAMBDA = - 1 . 5 LAMBDA = -1 . 6 D I S T . LAMBDA = - 1 . 7 DI S T . LAMBDA -1 . 8 D I S T . L A M B D A = - 1 . 9 D I ST . L A M B D A = - 2 . 0 DI S T . I s IS IS IS

I s IS IS IS IS IS IS IS IS

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.95543 .95447 .95374 .95308 .95265 .95244 .95249 .95267 .95329 .95407 .95544 .95706 .95911 .96161 .96446 .96737 .97048 .97315 .97492 .97484 .97434 .97045 .95613 .96028 .94076 .90818 .85924 .79357 .71518 .63128 .55017 .47743 .42313 .41619 .36553 .32518 .29315 .26772 .24749 .23128 .21820 .20752 . 19871 .19139

Figure 24. Printout plot correlation coefficient analysis for Josephson Junction cryothermometry voltage counts

T H E CORR E L A T I C N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I O N T H E CORR E L A T I C N T H E CORR E L A T I C N T H E CORR E L A T I O N T H E CORR E L A T I C N

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CORRELATION CORR EL AT I C N CORRELATION CORRELATICN CORRELATION CORRELATION CORRELATION CORRELATICN CORR E L A T I O N CORRELATION CORRELATION CORRELATION CORRELATION CORRELATICN CORRELATION CORRELATICN CORRELATICN CORRELATION CORRELATION CORRELATICN CORRELATION CORRELATICN CORRELATION CORRELATION CORRELATICN CORRELATION CORRELATICN CORRELATICN CORRELATION CORRELATION CORRELATICN CORRELATION CORRELATICN CORR ELAT I C N CORRELATION CORRELATION CORRELATION CORRELATION CORRELATICN CORRELATICN CORRELATICN CORRELATICN CORRELATION C O R R E L A T I ON

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LAMBDA = 2.0 D I S T . LAMBDA = 1 .9 DI S T . LAMBDA = 1.8 D I S T . 1.7 D I S T . LAMBDA = LAMBDA = 1 .6 DI S T . LAMBDA = 1 .5 D I S T . 1 .4 D I S T . LAMBDA = LAMBDA 1 .3 D I S T . LAMBDA = 1 .2 D I S T . LAMBDA = 1.1 D I S T . 1 .0 D I S T . LAMBDA LAMBDA • 9 DIST. LAMBDA • 8 DIST. LAMBDA .7 D I S T . LAMBDA = • 6 DIST. LAMBDA .5 D I S T . .4 D I S T . LAMBDA = .3 D I S T . LAMBDA LAMBDA = . 2 DIST. NORMAL DI: ST RIBUT ION LAMBDA = • 1 DIST. L O G I S T I C DI ST. DOUBLE E X P . D I S T . LAMBOA - - . 1 DI S T . -.2 DI S T . LAMBDA = LAMBDA = -.3 DIST. LAMBDA = - .4 DI S T . LAMBDA = - .5 DI S T . LAMBDA = -.6 D I S T . - . 7 DI S T . LAMBDA = LAMBDA - .8 D I S T . LAMBDA = -.9 DIST. CAUCHY DI[ S T R I B U T I O N LAMBDA = - I .0 D I S T . LAMBDA - - 1 . 1 D I S T . LAMBDA = -1 .2 DI S T . LAMBDA = - 1 . 3 D I S T . LAMBDA -1 .4 D I S T . LAMBDA = -1 .5 DI S T . LAMBDA -1.6 DIST. LAMBDA = - 1 . 7 D I S T . LAMBDA = -1 .8 DI S T . LAMBDA = - 1 . 9 D I S T . LAMBDA -2.0 OIST.

Printout plot correlation coefficient analysis for beam deflection

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Figure 27. Printout plot correlation coefficient analysis for x-ray crystallography residuals

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normal p r o b a b i l i t y p l o t was chosen because the normality assumption i s most commonly employed. The histogram i s included as an a d d i t i o n a l g r a p h i c a l p o i n t o f reference i n case the normality assumption i of about a dozen usefu summarizing s t a t i s t i c s i s included on t h i s s i n g l e page. This p a r t i c u l a r 4-plot a n a l y s i s has proved to be i n v a r i a b l y informative i n terms o f a s s e s s i n g the v a l i d i t y of the underlying assumptions i n a measurement process. The a n a l y s i s can be a p p l i e d to both raw response data and to r e s i d u a l s a f t e r a m u l t i factor (e.g., r e g r e s s i o n , ANOVA) f i t . The technique i s recommended only as a f i r s t pass i n a data a n a l y s i s and should be' complemented by more d e t a i l e d a n a l y s i s . The a p p l i c a t i o n o f t h i s technique t o several examples i s now discussed. The f i r s t example ( f i g . 29) i s the 500 Rand (24) normal random numbers o f f i g u r e s 15 and 22. The run sequence plot indicates f i x e d l o c a t i o n and v a r i a t i o n . The lag-1 autocorrelation plot indicates randomness. The histogram i n d i c a t e s a bell-shaped symmetric d i s t r i b u t i o n . The normal p r o b a b i l i t y p l o t i n d i c a t e s normality. The second example ( f i g . 30) i s the 700 Josephson J u n c t i o n cryothermometry voltage counts o f f i g u r e s 17 and 24. The run sequence p l o t i n d i c a t e s f i x e d l o c a t i o n and v a r i a t i o n and a l s o the rather discrete nature o f the data. The lag-1 a u t o c o r r e l a t i o n p l o t i n d i c a t e s randomness and r e i n f o r c e s the d i s c r e t e aspect o f the data; the histogram i n d i c a t e s symmetry and a bell-shape; the normal p r o b a b i l i t y p l o t indicates normality. The t h i r d example ( f i g . 31) i s the 200 beam d e f l e c t i o n s data on f i g u r e s 19 and 26. The run sequence p l o t i n d i c a t e s f i x e d l o c a t i o n and v a r i a t i o n and perhaps a s i n g l e o u t l i e r ( h i g h ) . The lag-1 a u t o c o r r e l a t i o n p l o t i n d i c a t e s w e l l - d e f i n e d nonrandomness and a d d i t i o n a l evidence f o r an o u t l i e r . The histogram i n d i c a t e s symmetry and a U-shaped i n d i c a t e s t h a t a d i s t r i b u t i o n s h o r t e r t a i l e d than normal i s needed) and a d d i t i o n a l o u t l i e r evidence. A remodeling t o take i n t o account the dominant a u t o c o r r e l a t i o n s t r u c t u r e of the data i s c l e a r l y c a l l e d f o r i n t h i s case.


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pharmaceutical process validation an international

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