Stephens Performance of Two New Algorithms

8/16/2019 Stephens Performance of Two New Algorithms

1/7

C L IN . C H E M . 3 4 /9 , 1 8 0 5 -1 8 1 1 (1 9 8 8 )

C L IN IC A L C H E M IS T R Y , V o l. 3 4 , N o . 9 , 1 98 8 1 8 0 5

P e rfo rm a n c e o f T w o N e w A lg o rith m s fo r E s tim a tin g W ith in - a n d B e tw e e n -M e th o d C a rry o v e r

E v alu ate d S ta tis tic ally

T h o m as W . S t e p h e n s

A c cu ra te a nd p re c ise a lg o r ith m s fo r es tim a tin g w ith in -m e th -

o d c a rryo v e r, b a s e d o n th e m in im iza tio n o f a u n iq u e “c a r-

ry o v e r s u m o f s q u a re s ,” a n d b e tw e e n -m e th o d c a rry o v e r ,

b a s e d o n a w e ig h te d D e m in g re g re s s io n o f firs t s a m p le

recovery v s ca rry o v e r-c o rre c te d “ tru e ” re c o v e ry , a re d e -

s c rib e d a n d c o mp a re d w ith tra d itio n a l m e th o d s b y u s e o f a

M on te C arlo s tu d y . In ad d itio n , I h a ve s tu d ie d th e e xp erim en -

ta l p a ra m e te rs th a t in flu e n c e th e a c c u ra c y a n d p re c is io n o f

c a rry o v e r e s tim atio n . T h e n e w a lg o r ith m fo r e s tim a tin g w ith -

in -m e th o d c a rry o v e r is u n b ia s e d u n d e r m o s t c o n d it io n s ,

w h e re a s th e tra d it io n a l a lg o rith m is b ia s e d lo w u n d e r m o s t

c o n d itio n s . T h e n e w a lg o rith m is a ls o m o re p re c is e , o w in g to

m ore -e ff ic ie n t u tiliz atio n o f in fo rm atio n c o n ta in e d in a n a n a -

ly t ica l ru n p e r fo rme d fo r c a rry o v e r e s tim a tio n . B e tw e e n -

me th o d c a rry o v e r in a ra n d o m -a c c e s s a n a ly ze r is e s tim ate d

q u a n tita tiv e ly b y th e s e c o n d p ro p o s e d a lg o r ith m a n d is fo u n d

to b e re a d ily a n d p re c is e ly d e te rm in a b le . U s e o f th e s e

m e th o d s in c om b in a tio n to e va lu a te a n a ly tica l in te ra c tio n

s h o u ld a llo w th e p re d ic tio n o f c a rry o v e r e rro r u n d e r m o s t

c urre nt a na ly tic al s itu atio ns .

A d di ti on al K e yp hr as es : “random -access” ana lys is . analytical

e rro r M o n te C ar lo s im u la tio n . w eigh ted D em ing regress io n

In te rac t ion

be tw e en a ssa ys

in

a u t o m a t e d

o r m an ua l a n a -

ly tica l s ys te m s, o p e ra tin g in e ith e r batch or ra n d o m-a c c e s s

m o d e s ,

h a s b e e n a

t rad i t iona l

p ro b le m in c lin ic a l c h e m is try .

In batch ana lyses , th is in te ra ctio n le a d s to inaccu ra te

est imat ion of t he c on ce nt ra tio n o f a n a n aly te in a s a mp le

that

fo l lows

a h ig h ly a typ ic a l s a m p le in a ru n se q u e n c e

w i th in -method c arr yo ve r). A lte rn ativ ely , in random-access

ana lyses , re a g e n ts th a t in te rfe re w ith o th e r

ana ly t ica l

m et h o d s

a l te r th e a c c u ra c y o f a na ly te d e te rm in a tio n s in

samples tha t ar e th e firs t to fo llo w a tra n s itio n b e tw e e n th e

m et h o d s (b e tw ee n -m eth o d c a rryo ve r).

Th e carryover fa c to r is a v a ria b le th a t w a s d e v e lo p e d to

es t ima te th e d e g re e o f in te ra c tio n in a b atc h ana ly t ica l

e n v i ro n me n t (1 ). Th is tra d it io n a l a lg o rith m h a s b e e n u ti-

l ized w ith little c h a n g e s in c e its in c e p tio n . D ixo n 2 der ived

th e c a rry o v e r fa c to r m o re r ig o ro u s ly a n d th e re b y a lte re d

s lig h tly th e e q u a tio n u s e d to e stim a te its magni tude . N ew

d e v e lo p m e n ts in in s tru m e n ta tio n a n d m e th o d o lo g y h a v e

reduced

th e carryover p ro b le m in s o m e s itu a tio n s (3 ) an d

in c re a s e d it o r n o t c h a n g e d it in others (4 ) . U n til n o w , th e

per fo rmance

o f th e tra d itio n a l a lg o rith m a n d th e in flu e n ce

o f s uc h e xp e rim e n ta l fa c to rs a s

ana ly t ica l

precision, n u mb e r

o f r e p l ic a t e s, an d concen tra t ions o f s tandards w as n ot deter -

m in e d .

T he t ra dit io na l a lgor i thm is n o t a p p lic a b le to b e t w e e n -

m eth o d c arry o v e r, fo u n d in ra n d o m -a c c e s s a n a ly s is , b e c a u s e

this s itu a tio n d o e s n o t fit th e b a s ic m od e l fo r w ith in -m eth o d

D e p a rtm en t o f Biochemistry , L illy R es ea rc h Labora to r ies , E li

L illy a n d C o., L illy

Corpo ra te

C e n te r ,

In d ia na po lis , IN 46 2 85 .

R ec e iv e d J an ua ry 2 6 , 1 9 8 8 ; a cce pte d M ay 12 , 1 9 8 8 .

c a rry o v e r; c u rre n tly , th e re is n o q u a n tita tiv e m e th o d fo r

es t ima t ing

c ar ry ov er b et we en -m e th od s (5 ). T h e i nt ro d uc ti on

o f “ra n d o m -a c c e s s ” a n a ly ze rs c re a te d a n e w p ro b le m o f

analytica l

in te rac t ion

th a t d id n o t p re vi ou s ly exist. T h e fa c t

th at ra nd om -a cc es s ana lyzers h a v e b e c o m e c o m m o n p lac e in

th e clinical la b o ra to ry h a s inc reased th e n e e d fo r a m eth o d

to e s tim a te th e

m a g n i t u d e

o f th is ty p e o f erro r .

C a rry o v e r e s tim a tio n is a necessa ry s tep in th e c o m ple te

e v a lu atio n o f a n a n a ly tic a l m e th o d o lo g y in th e c lin ic a l

chemis t ry

la b o ra to ry o f to d a y . S ta tis tic a l e v a lu a tio n s o f

ca r ryove r -es t ima t ion

a lg o rith m s a n d o f th e fa c to rs th a t

in f lu e n c e th is e s tim atio n a re

necessa ry to unders tand

th e

a c c u ra c y an d prec is ion o f t he se processes under th e var ious

ana ly t ica l e n v iro n m e n ts e x p e c te d . In th e a b s e n c e o f s u c h

s ta t is t ica l in fo rm a tio n , th e d e s ig n o f a n e x p erim e n t to

e s t ima te ca rry ov e r is le ft to th e in tu it iv e a p p ro a ch .

H e re I descr ibe a n e w a p p ro a c h to e stim atin g w ith in -

m eth od c arry ov er , based o n th e min imiza t ion of a un ique

“ca rry o ve r s u m of

squares . ”

I h a v e e v a lu a te d th e n e w an d

tra d itio n a l a lg o rith m s fo r p re c is io n a n d accu racy o v e r a

ra n g e of

ana ly t ica l

v a ria b le s (a n a ly tic a l p re c is io n , n u m be r

o f re p lic a te s p e r g ro u p , n u m b e r o f s a m p le s p e r ru n , ra tio o f

s a m p le m e a n s ) b y a M o n te C ar lo study , i.e ., b y u s in g

p s e u d o -ra n d o m n u mb e rs ge ne r a t e d to s im ula te a n a ly tic a l

req u ire m e n ts fo r a la rg e s e t o f d a ta samples a n d th e n

assess ing th e accuracy a n d p re c is io n o f th e e s tim a te s s o

o b ta in e d . A s I wil l s h o w , th e n e w a lg o rith m is both m o r e

n e a rly a c c u ra te u n d e r

a lm os t a ll

cond i t ions an d m o r e pre-

c ise . I a lso p res e n t a n a lg o rith m fo r es t ima t ing b e tw e e n -

m eth od c arry ov er,

b a s e d

o n w e ig h te d D e m in g re g re s s io n o f

recovered

f i rs t -samp le

re s u lts v s c a rry o v e r-c o rre c te d s e c o n d -

s a mp le

resu l ts

fo r e a c h se t o f s am p le s tha t fo llo w a m eth o d -

to -m e th o d tra n s itio n ; th e accu racy a nd p re cis io n o f th is

a lg o rith m a re e v a lu a te d o v e r a ra n g e o f

ana ly t ica l

varia -

b le s .

M aterIa ls a n d M eth o d s

P ro g ra m s w e re

wr i t ten

in th e “ C ”

language ,

c o mp i le d

w ith th e M ic ro s o ft “C ” c o m p iler a n d ru n o n a T a n d y 1 2 0 0

c o mp u te r

equ ipped

with a n 8 08 7 flo atin g-p oin t

cop rocesso r .

T h e s o u rc e

code

fo r w ith in -m e th o d c a rry o v e r e s tim a tio n b y

th e minimiza t ion of carryover su m o f s q u a re s ( M C S S )

m e t h o d is w ri t ten to a ssu m e a m in im u m o f library s u p p o r t

fo r m ax im um

portabil ity.’

F o r e a s e o f co m p a riso n , th e d a ta s t ruc tu re in a ru n w as

se t up

to b e co m pa tib le

w ith th e tra d it io n a l m e th o d o f

carryover e s tim atio n . T h e s e d a ta c o n s is t o f a s e rie s o f

responses fo r r re p lica te s pe r g ro u p , v c o n c e n tra tio n s o f

s a m p le (v =

2,

o r h ig h a n d lo w fo r

th is

s tudy), a n d a s e ts o f

groups p e r ru n . L imi ta t ion to th is data s t ruc tu re is n o t

necessa ry

fo r th e M C S S m e th o d o f c a rry o v e r e s tim a tio n ,

h o w e v e r , an d th e in v e stig a to r h a s g re a te r fle x ib il ity in

experimen tal d e s ig n w ith th e n ew m e th o d .

‘Available f rom th e a u th o r b y

re qu e s t in

compi led o r s ou rc e f or m

on 5 {188}-inchiskettes in P C D O S

fo rma t .


2/7

T h e s im p le s t lin e a r m o d e l fo r

w ith in -m eth od c a rryov e r

is :

Y 1 = (1 -k )X 1 + k Y 1 _ , + e j

4)

1 8 0 6

C L IN IC A L C H E M IS T R Y , V o l. 3 4 , N o . 9 ,

1 9 8 8

w h er e :

Y 1

=

m e a s u re d re s p o n se fo r a s a m p le in a s e rie s o f s a mp le s

X

= tru e re sp on se fo r th e sam p l e

k =

true

ca rryo ve r fa c to r an d 0 k

<

1

=

e rro r w ith a n e x p e c te d v a lu e o f 0 , v a ria n c e cons tant,

a n d a g a u s s ia n d is trib u tio n

i

=

1 ton , n

=

r

{1 4 9 }

If th e s e q u e n c e c o n s is ts o f a g ro u p o f lo w sa m p le s ( S l im)

fo llo w e d b y a g ro u p o f h ig h samples (S rep l ica ted r tim es

(i

=

1 to r) an d th e n repeated fo r s s e ts (m

=

1 to s ), th e n :

S e q u e n ce (S 11 ,,5 2 1 ,, . . ., S,,11 ,

S 1 2 1 ,S 2 2 1 ,

. .

.,

S r ,

S 1 1 2 ,5 2 1 2 , .

.

.,

S 1 S 2 2 2 , . .

.,

2)

a n d Y = re s p o n s e fo r sa m p le S

T he

ca rryo v e r fa c tor

is tra d itio n a lly d e te rm in e d a s :

kj m

= (Y 1 ,

n ext j ,m

- Y r,

ne xt j ,m)1 ( ’ , j ,m -

Yr _ i ,

n e xt j,m )

(3)

j

=

1 o r 2 , m

=

ito s ,j < 2 w h e n m

=

s

w h er e

“n e x t j,m ” re fers to th e n e x t se t in th e s e q u e n c e ,

w h e th e r

this

re fe rs to a n

inc rease

in th e

j

o r m in d e x .

Th is

unusual notation becomes

necessa ry because

o f th e o rd e r

d e p e n d e n c e fo r th is s e q u e n c e . U s u a lly in d ic e s a re o rd e re d b y

th e freq u e n c y a n d s e q u e n c e o f v a ria tio n ; h o w e v e r, in th is

c a s e , th e s e c o n d in d e x m u s t b e th e c o n c e n tra tio n s in c e it is

var ied second afte r re p lic a te s e v e n th o u g h co n c en tra tio n is

va r ied

les s th a n s e ts (in th is c a s e , w h e re v

<

s ). k , d e te r-

m in e d fo r e a c h tran s it io n b e tw ee n -g ro u p s (c o n c e n tra tio n s ),

is a v e ra g e d a s a n e s tim a te fo r th e e n tire ru n .

=

(> > .k jm) / (2 -

1 ) j< 2 w h e n m = s

T h e s e fo rm u la s a re a g e n e ra liz a tio n o f th e c a rry o v e r fac to r

d e riv e d fo r th e s p e c if ic c a s e o f th re e re p lic a te s p e r g ro u p b y

D i x o n

2 .

T h e re is n o re a s o n fo r

a s s u m in g

th a t assay ing

th ree re p lic ate s is o p tim a l fo r e s tim a tin g c a rry o v e r fa c to r

a n d , a s I w ill s h o w la te r, th e tru e o ptim um fo r th e tra ditio n-

a l a lg o rith m is tw o re p lic a te s . Th is m ay h a v e b ee n o bse rv ed

b y B ro u g h to n , w h o in a re c e n t p u b lic a tio n

5

g av e a

fo rm u la fo r c a rry o v e r fa c to r fo r th e tw o -re p lic a te s c a s e (a

c a s e w h e re th e D ix o n an d B ro u g h to n fo rm u la s a re a ls o in

agreement).

T w o te rm s n e e d d e fin itio n fo r th e p re s e n ta tio n o f th e

M C S S a lg o rith m : th e w ith in -c o n c e n tra tio n su m o f s q u a res

(S S , a n d th e “ca rry o ve r s u m o f sq u are s ” (S S .

55 w

=

- Y)2 ]

ss

= -

Y)2 ]

T h e m e a n s q u a re o f th e w ith in -c o n c e n tra tio n

su m

o f

s q u a re s is eq u iva le n t to th e ana ly t ica l va r ia tio n in th e

a b s e n c e o f c a rry o v e r o r w h e n th e d a ta a re co r rec ted fo r th e

in flu e n c e o f ca rry o v e r. T h e S S , c o m p o n e n t is in c re a s e d b y

th e v a ria n c e c a u s e d b y th e in flu e n c e o f c a rry o v e r re s u lt in g

in a la c k o f e q u iv a le n ce b e tw e e n p o s itio n s in th e s e q u e n c e

( s a m p le s im m e d ia te ly fo llo w in g a g ro u p o f a n o th e r c o n c e n -

tra tio n o f s a m ple ar e

exp e c ted to b e

e ith e r h ig h o r lo w ,

d e p e n d in g o n th e d iffe re n c e b e tw e e n s a m p le s). A n a ly tic a l

v ar ia tio n a ls o in flu en c es th is s ta tis t ic a n d , w h e n th e d ata

a re co rre cte d fo r ca r ryove r , th e m e a n s q u a re o f th e c a r-

ry o v e r

su m

o f s q u a re s is e q u iv a le n t to th e a n a ly tic a l var i -

a n c e .

T h e M C S S a lg o r ith m w o rk s b y m a k in g s u c c e s s iv e q u a -

d ra tic e stim ate s o f th e c a rry o v e r fa c to r th a t min im iz e s S S

T h e b es t es tim a te o f th e c arry o v e r fa c to r is th a t w h ic h

m in im ize s S S

E ith e r S S ,, o r 55 w c a n b e u s e d a s a n in dic ath i

o f th e c lo se n e s s to th e le as t-sq ua re s

ca rryo ve r fa c to r,

b u t S S ,

gives le ss b ia s e d e s tim a te s . T h e a lg o rith m p ro c e e d s b y

m a k in g a s e rie s o f

g u e s s e s

a t th e c a rry o v e r fa c to r, a n d S S i

d e te rm in e d fo r e a c h guess. A n e w e s tim a te o f th e c a rry o v e i

fac tor is m a d e b y fin d in g th e m in im u m o f a q u a d ra tk

func t ion fit to S S f rom th re e p r io r e s tim a te s b y a L a g ra n g e

p o ly n o m ia l. T h e n e w e s tim a te re p la c e s th e w o rs t o ld e s ti-

m a t e , a n d ite ra tio n s co n tin u e u n til n o s ig n ifica n t d iffe re n ce

ex ists in

5 5 ,, be tw e en the las t

th ree e s tim ate s (re la tive

d if fe ren c e < 0 .00 0 1 ). N u m e ric a lly , th is p ro c e d ure is

qu i te

s imi la r

to th e s u p e r m o d ifie d s im p le x a lg o rith m a p p lie d te

o n e d im en s io n (6 ). O th e r n u m er i ca l a lg or it hm s f or min imi-

zat ion m a y b e m o re o r les s e ffec tiv e th an th e o n e c h o s e a

h e re b u t w ere n o t e v a lu a te d in th is s tu d y (i.e ., F ib o n a c c i

search o r G old en S ectio n S ea rch ;

7 9 .

Simulated data samples w e re p ro d u c ed fo r th is M o n te

C a rlo s tu d y b y g e n e ra tin g a s e q u e n c e o f n u m b e rs w ith e

fix e d ra tio o f m e a n s b e tw e e n h ig h a n d lo w c o n c e n tra tio n s

C a rry o v e r w a s th e n in tro d u c e d b y u s in g e q u a tio n

1

above,

w ith th e f irst s a m p le in a d a ta s e t a d ju s te d a s if th e

s e q u e n ce w a s b e in g s ta rte d fro m a re a g e n t b la n k (z e rc

s a mp le ) . P s e u d o -ra n d o m v a ria tio n w a s th e n su p er im p o s e d

to h a v e a

cons tan t

v a ria tio n a n d a g au ss ia n distribution,

B o th a lg or ith m s g ive re su lts s im ila r to th o s e shown w h e r

v a ria tio n is a d d e d s u c h th a t it is p ro p o rtio n a l to th e m e a ii

(da ta n ot

sh o w n ) .

Carryover fa c to rs w e re d e te rm in e d fo i

e v e ry t ra ns it io n b et we en -c on ce nt ra ti on s (e x c e p t th e

transi

tio n a t th e

s ta r t

o f th e s e q u e n c e ) b y th e tra d it io n a l a lg o .

rith m a n d th e n a v e ra g e d fo r a c a rry o v e r fa c to r e s tim a te fo i

th e ru n . T h e ca rry o v e r fa c to r w as a ls o d e te rm in e d b y th e

M C S S m e th o d w ith th e first s u b s e t o f th e s e q u e n c e o m itte d ,

b e c a u s e th e tra n sit io n fro m re a g e n t b la n k to s a mp le is n o l

equ iva len t to su b s eq u en t tra n s itio n s . O n e th o u s a n d d a ta

s a mp le s w e re g e n e ra te d a n d th e m e a n a n d s ta n d a rd d e v i.

a tio n fo r th e carryover fa c to r e s tim a te w e re d e te rm in e d fo i

b o th th e tra d itio n a l an d M C S S m e th o d a t e a c h s e t o l

va ria ble p ara mete rs ( si mu l at ed c ar r yo ve r

= 1% ,

exp ressed

as

p e rc e n t o r

ca r ryove r

fa c to r m u ltip lie d b y 1 0 0 % ; c o e ffi.

c ie nt o f v aria tio n = 1 % a t a m e a n o f 1 0 0 ; ra tio o f m e a n s o

simulated

s tandards =

20 ; n u m b e r o f re p lic a te s p e r g ro u p

=

3 ; num be r o f d a ta p o in ts p e r ru n = 3 0 ; h ig h s tandard = 10 0

e x c e p t a s ind ica ted ) .

C a r r y o v e r

in a ra n d o m -a c c e s s e n v iro n m en t is es s e n tia ll)

d ifferent,

b e c a u s e

t he i nt er ac ti on

occu rs

o n ly b e tw ee n th

las t s a m p le in a s e q u e n c e o f o n e m e th o d (m e th o d A fo i

(6 ) e x a m p le ) a n d th e firs t sa m p le in a se q u e n ce of a n o th e i

m e th o d im m e d ia te ly fo llo w in g a tra n s itio n (m e th o d B ). T h i

in te rac tio n m ay n o t b e d u e to a d iffe re n ce in c on c e n tra tio n

o f a co m m o n a n a ly te a s in w i th in -me th o d ca r ryove r b ui

ins tead d u e to a d iffe re n c e in c h e m ic a l c o m p o s it io n o f th i

ana ly t ica l re a g e n ts . A dd itio n a l p o s s ib ilitie s e x is t, b e c a u s e

re a g e n t m ay p rod u ce a s im p le s h ift u p o r d o w n in th e firsi

re s u lt o f m e th o d B , o r it m a y e n h a n c e o r in h ib it th


3/7

1.5

1.0

0. 5

a

0

a

U

I

a

a

E

a

w

C

0

a

a

a

V

C

a

C o effic ien t o f V aria tio n

C L IN IC A L C H E M IS T R Y , V o l. 34 , N o . 9 , 1 9 8 8 1 8 0 7

ana ly t ica l re a ctio n o f m eth o d B , res u ltin g in a p ro p o rtio n a te

error, o r b o th . E s tim a tio n o f b o th w ith in -m e th o d a n d b e -

tw e e n -me th o d carryover is n e c e s s a ry fo r a ra n d o m-a c c e s s

ana lyzer an d m a y b e p e rfo rm e d b e st b y tw o se p a ra te

e xp e rim e n ts . W ith in -m e th o d ca rry o ve r c a n b e e s tim a te d b y

th e M C S S m e th o d b y re v er tin g to th e b a tc h m o d e o r

co n tin u a lly p ro gra m min g th e ra n d o m -a c c e s s a n a ly ze r to

ac cep t sa m ple s

in a sequence as above fo r o n e m e th o d . If th e

a ssu m p tio n is m a d e th a t th e in flu e n ce b e tw e e n th e f i rs t

a s s a y in m e th o d B (fo llo w in g a tra n s itio n fro m m e th o d A )

an d

th e

seco n d

is g o v e rn e d b y th e sa m e fo rce s a s m e a s u re d

b y th e w ith in -m eth od carryover, th e n a s e rie s o f d u p lic a te s

c a n b e ru n b y th e a lte rn a tiv e m e th o d s a n d th e difference

b etw e e n th e firs t-s a m ple re s u lt a n d th e s e c o n d , c o rre c te d fo r

w i th in -me th o d c arryo ve r, ca n b e u s e d to e s tim ate b etw ee n -

m e t h o d carryover. It m u s t b e re m e m b e re d th a t th e t rans i-

t ion fro m m eth o d A to B w ill n o t h a v e th e s a m e e ffe c t a s th e

tra n s itio n fro m m e th o d B to A , s o b o th types o f carryover

m u st b e e s tim a te d . If s u c h a n e xp erim en t is p erfo rm ed

w h e re th e concentra t ion o f ana ly te fo r m e th o d B is var ied

through th e ana ly t ica l ra n g e , th e d e te rm in e d resu lt ca n b e

r e gr e s s e d aga ins t th e es t imated t rue r e s u l t ( w i t h in - m e t ho d

car ryover-co r rec ted re su lt fo r th e se c o n d sa m p le ) a n d th e

ex is tence of co n s ta n t an d p ro po rtio n a l b ia s e s tim a ted fro m

th e

s ign i f icance

o f th e in te rc e p t an d th e d iffe re n c e o f th e

s lo p e f rom 1 .0 10 . W ith th is re g re s s io n e q u a tio n in h an d , a

tru e re su lt ca n b e e s tim a te d b y in ve rtin g th e e q u a tio n an d

s o lv in g fo r th a e xp e c te d tru e re su lt g ive n th e d e term in e d

o n e . L in e a r re g re s s io n u n d e r th e s e c i rcumstances v io la tes

c er ta in a s s u m p tio n s . T h e in d e p e n d e n t v a ria b le , fo r e x a m -

p le , is n o t a k n ow n fix e d v a ria b le b u t in s te a d h as a va ria nce .

T h e s itu a tio n w h ere b o th in d e p e n d e n t a n d d e p e n d e n t v a ria -

b le s h a v e a v a ria n c e is o n e w h e re th e D e m in g m e th o d o f

re g re s s io n a p plie s (11). In a d d itio n , th e var iance o f a n

a u to ma te d ana ly t ica l m eth o d o fte n v a rie s with th e lev e l o f

a n a ly te . T h e w ay in w h ic h v a ria n c e c h a n g e s w ith th e m e a n

o f th e in d e p e n d e n t v a ria b le (o r d e p e n d e n t v a ria b le in th is

c a s e )

inf luences

th e p re c is io n o f re g re s s io n e s tim a te s .

We ig h te d

reg res s io n c o rrec ts fo r va ria n ce ,

w h ich is de p e n -

d e n t o n th e m e a n o f th e in d e p e n d e n t v ar ia b le (10).

In th is s t ud y , p r op o rt i on a l an d cons tan t b ia s ( si m ul at in g

m e th o d -to -m e th o d in te ra c tio n ) is in tro d u ce d in th e first

s a mp le o f a p a ir a n d th e n th e s e co n d sa m ple is ad jus ted fo r

within-method carryover as descr ibed above. T his p ro ce ss is

re p e a te d u n til th e n u m b e r o f s a mp le s is e q u iv a len t to th e

to ta l number p e r ru n . P s e u d o -ra n d o m var ia t ion is th e n

in t roduced to b e e ith e r constan t , p ro p o rtio n a l to th e m e a n ,

or h alf c on sta nt an d ha l f p ro p o r tio n a l. T h e firs t s a m ple of a

pa ir in th is s im u la te d d a ta s e t is th e n re g re s s e d a g a in s t th e

w ith in -m e th o d c arryo ve r-c o rre c te d s e co n d sa m p le o f a p a ir

us i ng

leas t s qu ares , D e m ing ,

weighted, an d w e ig h ted D e m -

in g re g re ss io n a n aly s is . O n e th o u s a n d sa m p le s o f d a ta w e re

g e n e ra te d

an d

re g re s s io n e s tim ates fo r s lo p e a n d in te rc e p t

(ob ta ined b y th e fo u r m e th o d s ) w e re a v e ra g e d a n d th e

s ta n d a rd d e v ia tio n fo r th e e s tim a te s c a lc u la te d fo r e a c h

se t

o f v ar ia b le p a ra m e te rs (co e ffic ie n t o f va r ia tio n a t a m e a n o f

50

=

5 % , ra n g e c o v e re d b y s ta n d a rd s

=

5 0 , n u m b e r o f d a ta

po in ts p e r ru n = 22 , w it hi n- me th od c ar ry ov er = 2 % , c o n -

s tan t b e tw e e n -m e th o d c a rryo ve r

= 5 .0 ,

p ro p o r t io n a l erro r

=

0.9 , ii s tandards co n ce n tra tio n s ce n te re d a ro u n d 5 0 e xc ep t

as ind ica ted , va rian ce p ro p o rtio n a l to th e m ea n ). T h is w o u ld

b e e q u iv a le n t to a g lu c o s e assay , fo r e x a m ple , w ith analyti-

c al ra n g e o f 0 to 5 0 mmo l /L , an d s ta n d a rd s ru n i n d u pl ic at e

at c o n c e n tra tio n s o f 1 2 .5 , 1 5 .0 , 1 7 .5 , 2 0 .0 , 2 2 .5 , 2 5 .0 , 2 7 .5 ,

3 0 .0 , 32 .5 , 3 5 .0 , a n d 3 7 .5 mmo lJ L fo llo win g d up lic ate assays

of a second m e t h o d (ca lc ium, fo r e x a m p le ). T h e n , in

th is

e x a m p le ,

th e p rec is io n o f th e m e d ia n s ta n d a rd w o uld b e 2 5 .0

±

1 .25 m m o l/L (m ea n

±

s tandard d ev ia tio n) a n d p ro p ortio n -

a l to th e m e a n . T h e f irst sa m p le o f d up lic ate s tandards

fo llo w in g a m e t h o d t ra ns it io n w ou ld be sh ifte d u p w a rd s a

cons tant

2. 5

mmo l /L an d d o w n w a rd s by 1 0 % . T h e firs t a n d

la s t s ta n d a rd d u p lic a te s w o u ld b e expec ted to h a v e v a lu e s o f

13 .7 5 , 12 .5 25 , a n d 3 6.2 5 , 3 7 .4 7 5 , respect ive ly .

R e su lts a re

g ive n in th e te x t a s m e a n ± s tandard d e v i -

a tio n fo r th e e s tim a te fo r 1 0 0 0 s a m ple s o f d ata .

Resul ts

W ith in -M eth o d C a rry ov e r E s tim a tio n

When th e ra tio o f c o e ffic ie n t o f variat ion to p e rc e n t

carryover is le s s th a n 2 .0 , th e

accuracy

o f th e

MCSS

m e t h o d

is g re a te r th a n

99% (F ig u re 1 , o p e n

circles).T he t ra dit io na l

m e th o d g iv e s a n e s tim ate th a t is cons is ten t ly lo w b y -1 1 %

an d is le ss precise. A s th e ana ly t ica l precision is

decreased ,

b o th

m e th o d s

b e co m e b ias e d

lo w , w ith th e

tra d itio n a l m eth -

o d sh o w in g g re a te r s ta b ility a t v e ry lo w a n a ly tica l p re c i-

s io n . If ru n c o n d itio n s a re ch a n g e d to in c lu d e 4 0 d a ta p o in ts

pe r ru n o f d u p lic a te s w ith th e ra tio o f h ig h to lo w m e a n o f

1 0 0 , th e p re c is io n o f b o th est imates is im p ro v e d b y 3 0 % ,

b e co m in g v ir tu a lly in d is tin g u ish a b le , a n d th e a c c u ra c y of

th e tra d itio n a l m eth o d is im p ro v e d to -9 5 % (F ig u re 1 , s o lid

s y mb o ls ) . Th e MCSS m e th o d is n o w accura te o v e r a m u c h

w id e r ra n g e o f an a ly tic a l im pre c is io n , b u t th e t rad i t iona l

m e t h o d b e c o m e s b iased h ig h b y 1 0 0 % w h e n th e ra tio o f

c o e ffic ie n t o f v a ria tio n to p e rc e n t carryover is 1 0 .0 . C a r-

ry o v e r e s tim a tio n b y e ith e r a lgor i thm is v e ry s e n s it iv e to

ana ly t ica l

p re c is io n , a fa c t

tha t m u s t b e considered w h e n

o n e is attempt ing to m e a su re th is p a ra m ete r. A ratio of

s tandard m e a n s in th e ra n g e o f 2 0 to 1 0 0 m ay a p p e a r h ig h

b u t is u su ally a tta in ab le w ith s ta te -o f-th e -a rt m etho d o lo g y

F ig . 1 . E fF e c t o f a n al yt ic al prec is ion o n e stim atin g w ith in -m eth od

carryover

M e a n (A ) a n d s ta n da rd d e vi at io n ( f o r p e rc e nt c a rr yo v er e s ti m at ed with dm fe s

3 0 d a ta po in ts pe r run , th ree re p lica tes pe r g ro up , ra tio

o f

h ig h to

lo w s tandard =

20 , or square s 4 0 d a ta po in ts pe r sa m p le , tw o rep lica tes p e r g roup , ra t io o f h igh

to

lo w s tandard = 10 0 , by e ithe r th e M C S S m ethod

(0 ,0 ) o r th e tra d itio na l

m ethod C , {149} ) .JI o ther con dftio ns as g iven

in

M a te ria Ls a nd M et h o da


4/7

{ 149 } _

a

>

0

;

1.0

0.8

U

1

0.6

V

a

a

0.4

;

o

0.2

0.6

A

I I I

2

;

a

0.4

E

a

V

C

0.2

B

and ,

if

precision is fa ir ly g o o d , is n o t

necessa ry

fo r e ith er

p re c ise o r a cc u ra te e s tim a te s o f ca rry o ve r.

In a ccu rac y as so -

ciated w ith th e MCSS algorithm w h e n p re c is io n is e x c e p -

tion a lly p o o r is a ttr ib u ta b le to o v e r-co rre c tio n fo r carryover,

w h ic h te n d s to d e cre a se ra n d o m va ria tio n a n d h e n ce S S

Inaccuracy in th e tra d it io n a l m eth o d is a p p a re n tly inherent

in

th e assumptions m a d e in its d er iv atio n.

T he tru e c a rryo ve r ca n be e s tim a te d o v e r a w id e ra n g e o f

expec ted

v a lu e s with accu r acy a n d p re c is io n b y th e M C S S

a lgo rith m (F igu re

2 ). T h e M C S S

a lg o rithm re m a in s

a c c u -

ra te to bette r th a n 9 9 % u n d er a lm os t a ll c o n d it io n s , w h e re a s

th e a c c u ra c y o f th e tra d itio n a l a lg o r ith m is ro u g h ly 8 9 % .

In c re a s in g th e n u m b e r o f d a ta p o in ts p e r ru n im p ro v e s

th e p re c is io n o f th es e e s tim ate s with th e M CS S a lg o rith m

im p ro ve d s lig h tly m o re than th e tra d itio n a l a lg o rith m (F ig-

u re 3 ). In a d d itio n to im p ro v e m e n t o f prec is ion , th e a c c u ra c y

o f th e tra d itio n a l m e th o d is co n tin u o u s ly im p ro v e d a s th e

n u m b e r o f d a ta p o in ts p e r ru n is in c re a sed . T h is e ffec t m ay

b e d ue to th e in h e re n t a s su m p tio n in th e tra d itio n a l ca r-

ryo ver m od e l

o f a n in fin ite s e rie s o f s a mp le inte rac t ions ,

w h ic h is be tte r a p p ro x im a te d a s th e ru n length is

inc reased .

B o th m e tho d s a re in sen s itive

to th e ra tio o f m e a n s

b e tw ee n s ta n da rds

o r s a m p le s u s e d to

es tim ate ca rryo ve r

w h e n th is ra tio is g re a te r th a n 1 .1 (F ig u re 4 ). T h is ratio is

re a d ily e x c e ed e d w he n an

e x p er im e n t is

se t u p to d e te rm in e

carryover a n d th e re fo re s h o u ld n o t represent a s ign i f icant

factor in e x p e rim en ta l d e s ig n .

Increasing th e n u m b e r o f re p lic a te s in a g ro u p w h ile th e

to ta l n u m b er o f d a ta p o in ts p e r ru n is h e ld fixe d h a s a

d e le te r io u s effect o n th e p re cis io n o f e s t ima te s

g e n era ted b y

e ith e r th e M O S S o r t ra d it io n al a lg or it hm s ( Fi gu re 5 ). T h e

accu racy o f th e M C S S a lgor i thm is u n a ffe c te d , w h e re a s th e

a c c u ra c y o f th e tra d itio n al a lg o r ith m is

d ram atic a lly re -

d u c e d b y in c re a s in g th e n u m b e r o f re p lic a te s p e r g ro u p .

O ne o f th e b e n e fits o f co rre c tio n fo r ca rryo ve r is a n

15 25 35 45 55

D ata P oin ts pe r Sa m p l e

Fig . 3. Effec t

of th e n u m b e r o f d a ta

p o in ts p e r run o n e s ti m a ti ng with in -

m e th od c ar ry ov er

M e a n

(A )

a nd s ta nd ar d d ev ia tio n ( fo r p e rc en t c a rr yo v er est imated wt h al l o the r

c on ditio ns a s in di ca te d i n M ateria ls an d Methods . S y m b o l s fo r Figs . 3 -5 as in F ig .

2

im p ro v e d e s tim a te o f th e

tru e w ith in -g ro up variance

w h e n

data o f m o re th a n o n e le v e l ar e s u b mi t te d

to a n a ly s is o f

var iance fo r th e e v a lu a tio n o f th e p re c is io n o f a m e th o d o lo -

g y . F o r e xa m p le , w h en th e t rue c o e ffic ie n t o f v a ria tio n is 1 %

(me a n

=

1 0 0 .0 ) a n d c a rry o v e r is 1 0 % (ra tio o f m e a n s

=

20 ,

3 0 d a ta p o in ts p e r ru n a n d thre e re p e a ts p e r

group ) ,

th e

es tim ate d w ith in -g ro up

co e ffic ie n t o f va r ia tio n is 5 .3 1 %

b e fo re c a rryo ve r c o rre c tio n a nd 1 .0 8 % afte r M C S S carryover

a

0

a.

1.0

I

a

E

‘Ii

a

0.9

a

a

a

4IlIIIlIllI

0

a

U

I

‘0

a

a

E

a

UI

C

0

a

a

a

0

V

a

V

C

a

‘I,

1.0

0.8

0.6

0.4

0.2

5

4

3

2

---------

a

-,

A

- I I

I I I I I I I I

B

0

0 2 4 6 8 1 0

A ctu s i % Ca r r y o v e r

F ig . 2 . Accuracy a n d p rec is io n o f w ith in -m eth o d ca rryo ve r e s tim ation a s

t ru e w i th in -m e t ho d

ca r ryove r

is var ied

T h e m ean re la tive pe rce n t ca nyo ve r ( m ea n/ tr ue m e an ) (A ) a nd s ta nd ard

d ev ia tio n ( fo r percen t ca rryo ve r e s tim ate d from 1 00 00 d a ta sa m ple s (in th is

c as e) , w ith

al l

o ther

c on di tio ns a s i nd ic at ed in

M a te ri al s a nd M e th od s, b y

e i the r

the M CSS m ethod (0 ) or th e t ra d it io n a l m e t h od (C ) as the tru e c a rryo ve r is va ried

a s s ho wn

0 2 0 40 6 0 80 10 0

R atio o f S im p le M ea n s

F ig . 4 .

Effec t

o f th e ra t io o f s ta n d a rd m ea n s o n

estimating

with in -

m et ho d c ar ry ov er

M e a n

(A )

a nd s ta nd ard d ev ia tio n ( fo r p e rc e nt c a rr yo v er e st im a te d with a l l o t he r

cond i t ions

a s In dica te d In

Mater ia ls arid Me th o d s

1808 C L IN IC A L C H E M IS T R Y , V o l. 3 4 , N o . 9 , 1 9 8 8

0.48

C

0

a

; 0.46

0.44

o .ss


5/7

a

a

0

a.

U

o

I

V

a

U

E

a

UI

C

0

a

a

a

0

V

a

V

C

a

0

1.0

0.9

0.8

0. 7

0.6

0.4

0.2

B

0

UI

a

C

0

0

0.

0

a.

V

a

a

S

a

UI

C

0

a

a

a

0

V

a

V

C

a

0

0 a 10 15 2 0

C LIN IC A L C H EM IS TR Y , V ol. 3 4 , N o . 9 , 1 9 8 8 1 8 0 9

2 3 4 5 6

Repllca taa p.r G r o u p

Fig . 5. Effec t o f t he n um b er o f r ep lic at es p er g ro up o n e st im a tin g w it hin -

m e t h o d

ca r ryove r

M e a n (A ) a nd s ta nd ar d d ev ia ti on ( f or p e rc e nt c ar ry ov er e st im a te d w it h a ll o th er

c on ditio ns a s in dic ate d in M ate ria ls a nd M eth od s

correc t ion . L ittle e ffec t

is s ee n if

carryover is v e ry s m a ll a s

co m p ar ed w ith

ana ly t ica l

imprecision. F or e x a m pl e ,

if

th e

t ru e c oe ff ic ie nt o f v ar ia t io n is 1 % a n d carryover is 0 .1 % , th e

e stim a ted c oe ffic ien ts o f v ar ia tio n a re

1 .0 0 % a n d 0 .9 8 %

b e fo re a n d a fte r M C S S c a rryo ve r co rre c tio n .

T h e tra d it io n a l m e th o d fo r carryover fa c to r es tim a tio n is

n o n ite ra tive , w he re a s th e M C S S m eth o d is ite ra tiv e , le a d -

in g

to

concern fo r a p o te n tia l

inc rease

in p ro ce s s in g tim e.

D u rin g th is study, > 9 9 % o f th e d a ta s e ts

ana lyzed c o n -

v e rg e d with in 2 0 ite ra tio n s w ith a to ta l p roc e ss in g tim e le ss

tha n 2

s fo r th e c a s e o f 3 0 d a ta

po in ts

p e r ru n , th ree d a ta

p o in t p e r g ro u p . It is p o s s ib le th a t t his p er fo rm a nc e c o u ld b e

fu r the r im p ro v e d b y u s e o f o n e o f th e o th e r n u me r i c a l

minimiza t ion m et ho ds r ef er en ce d in Ma te ria ls and Me thod s .

B etw e en -M eth o d C a rryo v e r E s tim atio n

T h e s lo p e , de te rm in ed b y re g re s s in g th e f i rs t-sample

resu l t

o n th e w ith in -m etho d

car ryover-co r rec ted

s e c o n d -

sa m ple re su lt,

is a n e s tim a te o f

p ro p o rtio n a l e rro r caused by

b e tw e e n -m e th o d ca rry o ve r. W h e n th e s lo p e is s ign i f icant ly

le s s th a n o n e , th e n e g a tiv e error is in d ic a tiv e o f a n in hib i-

tio n o f t he a na ly tic al re a c tio n . T h e in te rc e p t is th e e s tim ate

of cons tan t error. B ecau se th e e s tim a te s o f s lo p e a n d in te r-

c e p t a re n e g a tiv e ly

co r re la ted ,

w h e n th e a c c u ra c y o f th e

s lo p e is p o o r th e n th e in te rc e p t is a ls o in a c c u rate ly e s tim at-

ed . Prec is ions of es t imat ing slope a nd in te rc ep t

are a lso

re la ted ,

an d w h e n th e s lo p e is p o o rly d e te rm in e d , s o a ls o w ill

b e th e in te rc ep t. T h u s th e re s u lts a re p re se nte d in te rm s o f

th e

accu racy

a n d p re c is io n o f th e s lo p e e s tim ate an d w ill b e

considered

to

a p p ly a s w e ll to th e e s tim a te o f th e intercept.

Fur the r in fo rm a tio n o n th e s ta tis tic s o f e s tim a tin g both

in te rce p t a n d

s lo p e a re av a ila b le u p on re q u es t.

W h e n v a ria tio n is co n s ta n t, th e D e m in g re g re s s io n e s ti-

m ate fo r th e s lop e is v ery a c cu ra te (m = s l ope =

0 .9 0 5 ±

0 .1 81 fo r a

tru e M

= 0 .9 ), w h ere as th e le a s t-s q u are s

es t ima te is b iased lo w (m = 0 .7 90 ±

0 .1 47 ), an d w eig h tin g

h as n o effe c t o n a cc u ra c y o r p rec is io n . W h e n va ria tio n is

C o effic ien t of va ria tio n

F ig . 6 . E ffe c t

o f a n al yt ic a l

prec is ion

on es tim atin g p ro po rtio na l e rro r

c au se d b y b etw ee n-m eth od carryover

M e a n (A )

a nd s ta nd ar d d ev ia tio n ( fo r th e s lo pe e stim ate d by

l e as t s q u a re s

( {149}) ,

w e ig ht ed l ea st s qu ar es

(U) ,

D e m ln g ( 0), o r w eig hte d D em ln g (0 ) re gre ss io n

ana lys is . Al o th e r c o nd it io n s a s g iv e n

In

M a te ria ls a nd M e th od s: C V ,

de te rmined

at

a

m e a n c o n ce n tr a ti o n

of

5 0 .0 ,1 8 va rie d a s sh ow n

p ro p o rtio n a l to th e m e a n o f th e in de pe nd en t variab le, h o w -

e v e r, th e

D e m in g

reg ression

es t ima te

fo r s lo p e is b iased

s lig h tly h ig h

(0 .916 ± 0 .2 01 ) an d is im pro ved b y

w e i gh t i ng

(0.905 ±

0 .1 8 3 ) . Th e

le as t-s qu a res e s tim a te b ec om es le ss

p re cis e (0 .7 88

±

0 .162) an d is im p ro v e d b y w e ig h tin g (0 .8 1 6

±

0 .152)

w h e n

var ia t ion is

p ro po rtio na l to th e m e a n .

When

variat ion is pa rtly p ro po rtio na l an d part ly cons tan t , th e

resu l ts

a re b e tw ee n tho se p re sen ted .

Inc reas ing th e

coeff icien t

o f v ar ia tion b eyo n d 5 .0 in -

c re a s e s th e lo w b ia s o f th e s lo p e

es t ima ted

b y lea s t

squares

a n d w e ig h te d le a s t s q u are s a nd in t roduces a h ig h b ias in th e

D e m in g an d w eig h ted D em in g s lo p e e s tim a tes (F ig u re 6A ).

T h e D e m in g an d w eig h ted D e m in g are

stil l

m o r e accu ra te

th a n lea s t s q u a re s an d w e ig h te d leas t squ a res u n til a

c o effic ie n t o f

va r ia t ion o f 20 .0 is rea ch e d , w h en th e D em in g

e s t ima te s b eco m e to ta lly u n re liab le . T h e p rec is io n o f th e

D e m in g a n d w eig h te d D em in g es tim ate s is le ss th a n th a t o f

leas t squares an d

w e ig h te d

leas t squares

(F ig u re 6 B ).

T h e an a ly tica l ran g e c o vered b y th e sa m p le s u se d fo r

e s tim atio n o f cons tan t

an d

propo r t iona l e rro r is an im po r-

tant de te rminan t o f th e a ccu rac y an d

prec is ion

o f th e

result ing e stim ate s (F ig u re 7A) .

A s th e

co ve re d a na ly tica l

range

d e cre a se s be lo w

20 (a ll o th er

c o nd itio n s a s s ta te d in

m e th o d s), th e lo w b ias o f th e le a s t sq ua re s an d

w e i gh t e d

leas t-squares e s tim a tes in c re ase s an d th e D e m in g a n d

w e ig h te d

D e m ing e s tim ate s

b e co m e b ia se d h ig h . T h e prec i-

s io n o f th e s e e s tim a tes b eco m e s les s a s th e covered ana ly t i -

c a l ra n g e

b e c o m e s

le ss than 60 (Figure

7B . T h e

c o v e re d

ana ly t ica l

ra n g e re q u ire d fo r accu rately

e s tim a tin g the

s lo pe is in ve rs ely re la ted to th e a n a ly tic a l p re c is io n , a n d , a s

th e p re c is io n im p ro v e s , th e lo w e r is th e range o f s a mp le s

requ i red fo r accura te a nd p rec ise e s tim ate s (a ccu ra te an d

precise e s t ima te s obtained w h e n e v e r c o v e re d ana ly t ica l

range/coeff ic ien t o f va ria tio n > 1 0 in th is s tu d y ) .

C o ns ta n t e r ro r ca n be a ccu ra te ly a n d p re cis ely e stim ate d


6/7

4

0

UI

a

C

0

0.

0

0.2

V

a

a

El

a

UI

40

C

0

;30

a

0

20

‘1o

2 0 4 0 60 80

R an g e In D .pen d .n t/In d .p .n d .n t v .r lab ls

0

UI

0 .90

0

0.

0

a.

0 .85

a

S

a

UI

C

0

0.1

a

a

0

V

a

0.1

a

0

10 20 30 4 0 50 60

N u m b e r

of

D ata P oin t. pe r Ru n

1 8 1 0 C L IN IC A L C H E M IS T R Y , V o l. 34 , N o . 9 ,

1 9 8 8

F ig . 7 .

Effect

o f

th e

ra n g e in d e p e n d e n t a n d

in d e pe n d e nt v ar iab le o n

es tim atin g p ro po rtIo na l e rro r

caused b y b e t w e e n - m e t h o d carryover

M e a n (A ) a nd s ta nd ard d ev ia tio n ( fo r

th e

s lop e . S ym bo ls a s in F ig . 6

th rou gh a w ide

ra n g e o f v a lu e s , w h e th e r th e var iance is

cons tan t o r p ro p o rtio n a l, a lth o u g h m o re p re c is e v a lu e s

a p p e a r to b e ob ta in e d w ith ne ga t i v e

c on s ta n t e rro r

-9 .35

±

5 .31 fo r -1 0 actua l , 9.31

±

7 .3 2 fo r +1 0 ac tua l by

D e m i n g

reg ression w ith pro portiona l varian ce). T h e re la tiv e e s ti-

m a te s o f s lo p e b y D e min g a n d w e ig h te d D e m in g re g re s s io n

in c re as e as th e t rue s lo p e inc reases w h e n th e ana ly t ica l

va r ia tio n is propo r t iona l to th e m e a n (0.481 ± 0 .0 8 0 fo r 0 .5

ac tua l ,

1 .5 8 8

±

0 .255 fo r 1 .5 a c tu a l b y D em in g re g re ss io n ;

an d

0 .489 ±

0 .0 7 2 fo r 0 .5

ac tua l ,

1 .5 3 8

±

0 .2 14 fo r 1 .5

ac tua l

b y w eig hte d

D e m in g

reg re ss io n ). T h is m a y b e d u e to th e

a ss um p tio n m a d e in th is s tu d y th a t th e va ria tio n in th e

in d e p e n d e n t a n d d e p e n d e n t va r ia b le s is eq u a l (because th e y

o b v io u s ly a re th e re su lt o f th e sa m e a n a ly tic a l m eth o d ).

W h e n th e va r ia n c e is p ro p o rtio n a l to th e m e a n , th e g rea te r

th e s lo p e th e g re a te r is th e dif fe rence in variances in th e

in d e pe n d e n t a n d d e p e n d e n t va r ia b le s a n d th e m o re d is to r -

t ion

is

in troduced in to th e s lo p e e s tim a te . W h e n th e

ana ly t i -

ca l variat ion is a cons tant, th e D e min g a n d w eig h ted

D e min g

e s tim a te s a re n o t

c o rre la te d to th e tru e s lo p e a n d

ar e accura te th rou gh a w ide ra n g e (0 .5 0 7

±

0 .1 1 5 fo r 0 .5

ac tua l ,

1 .5 0 0

±

0 .1 8 1 fo r 1 .5 actual b y D em in g re g re s s io n).

W ith in -m e th o d ca rryo v e r in tro du c e s c o rre la tio n b e tw e e n

th e in d e p e n d e n t a n d d e p e n d e n t v a r ia b le s in th is a n a ly s is

a n d c o u ld th e re fo re c a u s e

prob lems

in th e a c c u ra c y an d

prec ision

o f th e e s tim atio n

process .

W h e n th e

s imu la ted

with in -method carryover w as varied o v e r a range o f 0 .1 to

1 0 .0 % , th e re w a s

ve ry litt le e ffec t

o n th e a c c u ra c y o r

precision o f th e s lo p e

es t imates

b y e ith e r m e th o d a n d

th e re fo re

th is

sh o u ld n o t p re se n t a p ro b le m .

Increas ing th e n u m b er o f d a ta p o in ts p e r ru n is n o t

expec ted to in flu e n c e th e s lo p e e s tim a te s . H o w ev e r , th e

leas t - squares an d weighted le a s t-sq ua res m eth od s g a v e

m o r e b iased lo w re s u lts a s th e n u m b e r of s a mp le s p e r ru n

inc reased

f rom 1 2 to 6 2

(F ig u re 8 ).

Increasing th e n u m b e r o f

p o in ts d id n o t in f luence th e a c c u ra c y o f th e D e m i n g an d

w e ig h te d D e m in g e s tim a te s a n d s ig n ifica n tly im p ro v e d th e

p re c is io n fo r a ll fo u r m eth o d s o f e s tim atio n .

D is c u s s io n

O ne o f th e su rp r ises in th is s tu d y w as th e a lm os t cons tan t

b ia s o f th e tra d itio n a l m e th o d o f w ith in -m eth o d ca rryo v e r

e s t i m a t i on .

T h is b ia s w a s

re d u c e d b y in c re a s in g th e length

o f a c a rry o v er e s tim atio n ru n a n d th e u s e o f d u p lic a te a s s a y s

in e a c h group , b u t it w as n o t e lim in a te d u n d e r a n y c o n d i-

t io n s . In c o n tras t, th e M C S S m e th o d o f carryover e s t ima t io n

is v e ry a c c u ra te u n d e r m o s t circumstances e x c e p t w he n

ana ly t ica l p re c is io n is v e ry p o o r.

T h e p re c is io n o f th e M C S S m e th o d w a s

b e tte r un d er

a lm o s t a ll c o n d itio n s a n d , in a n in h e re n tly im p re c is e e s ti- i

m a tio n p ro c e s s , th is is a

qu i te

d e s i ra b le attr ibu te . T h e

p re c is io n o f carryover e s tim a tio n is n e v e r v e ry g o o d , h o w e v -

e r, a n d s e v e ra l a n a ly tic a l ru n s o f 4 0 d a ta p o in ts o r b e tte r a re

p ro b a b ly necessa ry to ob tain a n e s tim ate o f c a rry o ve r tha t

c o u ld b e c o n s id e re d typ ic a l

o f a la rg e p o p u la tio n o f d a ta .

W h e n a n aly tic a l p re c is io n is p o o r

in co m p a ris o n to

c a r -

ry o v e r (i.e ., th e ra tio o f c o e ff ic ie n t o f v a ria tio n to percen t

c a rryo v er e xc ee d s

1 0 ), it is p ro b a b ly

im p rac tic a l to g e t a n

accura te

an d

pre c is e es tim ate o f

ca r ryove r . F or perspec t ive ,

s e ve ra l ex a m p le s o f

p u b l is h e d

ra t io s o f c o e ffic ie n t o f varia-

tio n to p e rc en t

c a rryo ve r a re fro m 0 .5 to 1 .0 (3 ), 0 .6 to 3 .7 (4 ),

an d 1 .4 to 90.0 (12).

Th e

leas t -squa res

a n d w eig hte d le a s t-s q ua re s me th o d s o f

estimating s lo p e a n d

in te rce p t a re

n o t

accep tab le

fo r th e

de te rmina t ion

of constant a nd p ro p or ti on a l e rr or in tr od uc ed

by b e tw e e n -m e th o d ca rry o ve r b e c a u se o f th e p re s e n c e o f

var ia t ion in both th e in de pe nd en t an d d ep en de nt v aria ble s.

Th is re s u lts in th e underestimation o f s lo pe a nd o ve re stim a-

tion o f i nt er ce p t. T h e D e m in g a n d weighted Deming r egres-

sion m e t h o d s w e re fo u n d to correc t fo r this p ro b le m a n d to

b e q u ite a c c e p ta b le u n d e r th e s e co n d it io n s , w ith th e w e ig h t-

e d D em in g p ro c e d u re g iv in g s lig h tly m ore p re c is e e s tim ate s

fo r s lo p e a n d in te rce p t w h e n th e v a ria tio n in th e d e p e n d e n t

Fig . 8 . E ffe c t o f th e

n u m be r o f d a ta

po in ts p er ru n on est imat ing

p ro po rt io na l e rr or

caused

by b etw ee n-m e th od c arr yo ve r

M e a n

(A )

an d s ta nd ard de via tio n (

for

th e s lo pe . S ym bo ls as

in

F ig .

6


7/7

en h an ce th e a b ility o f a m e th o d to d iscr iminate b e t w e e n

C L IN IC A L C H E M IS T R Y , V o l. 3 4 , N o . 9 ,

1988 1 8 1 1

( or i nd ep e nd e nt

var iable ,

in th is c as e ) w as p ro p o r tio n a l to

the m ea n . S in ce th e a s s u mp t io n c a n b e m ad e tha t th e ra tio

o f va ria n ce s is eq u a l to o n e , th e D e m in g re g re ss io n m e th o d

c an b e ap p lied w ith o u t an y a d d itio n a l re q u irem e n t fo r th e

d e term in a tio n o f va r ia n ce s .

Th is

m ay le ad to o n ly s lig h t

inaccu racy a t v e ry la rg e prop o rtion a l e rro rs w h e n th e

u nw e igh ted p ro ce du re is a pp lie d an d

th e v ar ia t io n in th e

d e p e n d e n t (o r in d e p e n d e n t v a ria b le ) is p ro p o rtio n a l to th e

m e a n .

Th e a p p lica tio n o f w e ig h te d re g re s s io n re qu ire s es tim at-

in g e ith e r a var iance a t e a c h

c o n ce n tra tio n o f

ana ly te o r th e

determinat ion o f a n e q u a tio n

th a t fits

th e re la tio n s h ip o f

va ria n ce to

th e m e a n . T h e va ria n ce w a s

k n o w n

in

th is

s tudy

so th at w e ig h te d reg re ss io n c o u ld b e a p p lied w ith o u t th e

independen t determination of a variance-vs-mean re la tion-

sh ip . Th e d e te rm in a tio n o f va ria n ce a t e ac h v a lu e

co u ld b e a

ted io u s ta sk ; h o w e v e r , th is

is u s u a lly re q u ired

in th e e va lu a-

t io n o f a n ana ly t ica l m eth o d . T h e a d d it io n a l re q u i re me n t of

determin ing ho w th e va r ia tio n va rie s w ith th e m e a n w o u ld

a ls o be u se fu l in th e e v a lu a tio n o f a m e th o d , b ec au se th is

in fo rmat ion is actual ly necessa ry fo r a n y s ta t is t ica l c o m p a r -

ison

of

m e an s , If

th e var iance is n o t v e ry d e p e n d e n t o n th e

m e a n ,

th e n w eig h te d re g re ss io n is u n n ec es s ary

a n d p ro ba-

b ly unde s i r a b l e .

T h e c o m b in a tio n o f w ith in - a n d b e tw e e n -m e th o d c a r-

ry o v e r e stim a tio n sh o u ld m a ke it p o ss ib le to p re d ic t th e

degree of th is typ e o f error in

m o st a na ly tica l s itu a tion s . O n -

line c a rry ov e r c o rrec tion

in a n a u to ma te d ana lyzer c ou ld b e

u s e d

to imp ro v e th e a iu ra c y of a resu l t o r to fla g a result fo r

w h ic h accuracy is q u e s tio n a b le . S u c h co rre c tio n s h o u ld , in

add i t ion , im pro v e th e prec is ion o f a n ana ly t ica l re s u lt a n d

c o n ce n tra tio n s o f ana ly t ica l s a mp le s .

R e fe re n c e s

1. Brough ton P M G , B utto iph M A , G ow enlock A H , N eill D W ,

Skente lbery R G . A gu ide to a u to ma t io n in c li ni ca l c he m is tr y. J Cl in

P a th o l 1 9 69 ; 22 :2 7 8- 8 5.

2. D ix on K . A

theo re t ica l

study of carryover in d is cre te an d

con t inuous-f low s ys te m s. A n n C lin B io ch e m 1982 ;19 :224-6 .

3.

M o s e s

G C, L igh tle G O ,

T u ck e rm a n JF ,

H en d e r so n A R .

T he

E P O S A utomated S ele ctive C hem istry Ana lyze r e va lu ate d . C lin

C h e m 1986 ;32 :165-9 .

4,

Nolan JP , D iB enede t to B ,

Ta r s a

N J. Continuous-flow e n z y me

immunoassay

fo r

thy rox in

in seru m . C lin C h e m 1981 ;27 :738-41 .

5. B ro u g h to n P M G . C a rry -o v er in a u to m atic a n a ly z e rs . J A u t o m a t -

ed C hem 1984;6 :94-5 .

6.

R o u t h

M W , S w a rtz P A , D en t o n

M B .

Perfo rmance of th e super

mod i f ied

s imp lex .

A n a l

C h em 1 97 7; 49 :1 42 2- 8.

7 . S ca le s , L E . In t roduc t ion to n o n -lin ea r o p tim iza t io n . N ew Y o rk :

S p r in g er -Ver la g , 1 9 8 5 ; ch a p ter 2 .

8 . A d b y PR , D em ps te r M AIl. In tro d u c tio n to o p tim iza tio n m eth -

o d s . N ew Y o rk : J o h n W iley S o n s, 1 9 7 4 ; ch a p ter 2 .

9 . F letch er R .

Prac t ica l

m eth o d s o f

op t im iza t ion .

N ew Y o rk :

John

W iley S o n s, 1 9 8 0 ; sect io n 2 .6 .

10 . D r a p e r N R , S m ith H . A p p lied reg ress ion

ana lys is .

2n d ed . N ew

Y o rk : J o h n W iley S o n s, 1 9 8 1 :7 0 -1 4 0 .

11 .

C ornb lee t

P J , G o ch m a n N . In co rrec t

leas t-sq uares reg ression

co e ffic ien ts in m eth o d co m p a r iso n a n a ly s is . C lin C h e m

1 9 7 9 ;2 5 :4 3 2 -8 .

12 .

K n e de l M , H a ec ke l

R , S e id e l D , e ta . A naly tica l p erfo rm ance

of

th e random

access Hitach i

7 3 7 : a

mult icen t re

ev a lu a tio n . J C lin

C h e m

Clin

B io ch e m 1 98 6; 24 :4 09 -3 2.

Documents

Stephens Performance of Two New Algorithms