guidelines immunoassay

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C LI N. C HE M. 3 1/ 8,1 26 4- 12 71 1 98 5)

1264

CLINICALCHEMISTRY,Vol.

31 ,

No.8 ,

1985

G uid elin es for Im m un oa ssa y D ata P ro ce ssing 1

R. A .

Dud ley ,2 P .

Edwards,3

R . P . E kln s,3 D . J .

Flnney,4 I. G . M . M cKenzie,4 G . M . Raab,4 D .

Rodbard,5 and

R . P . C . Rodgers6

T he se gu ide lin es o utline th e m inim um re qu ire men ts fo r a

da ta -pro ce ssing p acka ge to b e u sed in th e im m un oa ssa y

la bo ra to ry . T he y in clu de re co mm e nd atio ns o n h ard wa re ,

s oftw a re , a nd

program

d esig n. W e o utlin e th e sta tistic al

an aly se s th at sh ou ld b e

performed

to obta in the analy te

c on ce ntra tio ns o f u nk no wn s pe cim e ns a nd to e ns ure a de -

q ua te m on ito ring o f w ith in - a nd be tw een -a ssa y erro rs o f

measurement.

AddI t Ional Keyphrasee:

statistics .

quality contro l com puter

program s data

processing

The authors of th is paper were convened as a group by the

International A tom ic Energy Agency to m ake recom menda-

tions for assayists and laboratory m anagers on good practice

in data processing for radioimmunoassay and related tech-

niques. T hey w ere requested to identify those com putational

procedures that are appropriate, especially in a hospita l

laboratory providing a routine service , and to establish

priorities as to their im portance.

It is timely to exam ine this topic. In recent years the

dramatic increase in power and decrease in price of comput-

ing hardware have brought the capability of machine com -

putation w ithin the reach of all laboratories, either as an

integral part of a sample counter or as an independent

device. M any program s have been designed, thus testing a

diversity of approaches but subjecting the user to a bew il-

dering choice among possib ilities. In the opinion of th is

group, most com mercially available program s for analyzing

immunoassays lack essential features. Indeed, several pro-

grains that have been developed for programmable calcula-

tors (1,2) show more sophistication than those supplied as

“black box” system s by many manufacturers of beta- and

gamma-counters.

The group agreed on several general principles. F irst, all

aasayists can benefit from the computational and statistical

1 This article should be regarded as the com posite view of a

comm ittee of experts, convened by D r. B . A . Dudley under the

auspices of the International A tom ic Energy Agency, V ienna,

A ustria. It is n eith er an “official” po sition o f t he IA EA no r a fo rm al

p oli cy o f t h e AACC. H owe ve r, it s ho u ld s erve to s timul at e t hi nk in g

by all in the RIA and immunoassay fields. W e hope it will

contribute to an im provem ent in the overall quality of

software

systems fo r th es e k in ds o f a na ly se s.

2 International A tom ic E n er gy A g en cy , W a gr am e rs tr as se 5 , P .O .

B ox 1 00 , A -1 40 0 Vienna, Austria.

3 De pa rtm en t o f M ole cu la r E nd oc rin olo gy , M id dle se x H osp ita l,

L on do n, U .K .

4

o f E d in bu rg h, E din burgh , U .K .

Institute of C hild H ealth and H um an Development,

Na ti on al I ns titu te s o f H e alt h, B u il din g 10 , Room 8C413, Bethesda,

MD 2 020 5.

6 Un iv er sity o f C alifo rn ia, S an Francisco, CA .

Rece ived Feb rua ry 5 , 1 98 5; a cc ep te d Ma y 3 1, 19 85.

procedures that a good program can offer.ndeed, the less

statistically experienced the user, the more he stands to

gain by such assistance. Second, while one goal of com puta-

tion obviously is to derive the concentration of analyte in the

samples measured, the m ain advantages of m achine com pu -

tation include automation, speed, improved accuracy

(through avoidance of gross errors), and detailed statistical

analysis and accounting of sources and magnitude of errors.

Third, machine computation should never be thought to

relieve the analyst of responsibility for the reliability of his

measurements; all it can do is provide results that are

com putationally sound, that are relevant to the assessm ent

of reliability , and that are displayed in the m ost com prehen-

sib le manner. Fourth, the creation of suitable program s is a

major task that can be accomplished successfully only

through the com bined efforts of professional analysts, sta tis-

ticians, and programmers. F inally, a lthough the user him-

self need seldom know the details of the mathematical

analysis, it is improper that proprietary secrecy should

conceal the basic strategy, algorithm s, equations, and as-

sumptions.

Two restrictions adopted by the group as to the scope of

th is paper should be stressed. F irst, it is not a general

critique of quality control. A lthough data processing is one

key element in quality control, the la tter field embraces

much more than data processing. Second, being directed at

the practic ing assayust and laboratory manager, the paper

does not offer detailed algorithm s such as would be required

by a programmer. Instead, we seek to explain the type of

analysis that is desirable in program s, w ith some approxi-

mate mathematical amplification in appendices to sharpen

th e c on ce pt s.

H a rdwar e and

Software

The choice of a computer system for a particular labora-

tory will depend on several factors. The most important

factor is that the system be able to run a data-analysis

package that at least m eets the minimum requirements set

out here . Secondly , in term s of capacity and speed it must be

able to handle the volume of work done in the laboratory .

In all cases the system should be tailored to fit the needs

of the laboratory and be regarded as a piece of equipment

that is as essentia l as a gamma counter or a centrifuge. Our

minimum recommendation would be a computer w ith 48K

of memory and one or more disk drives. For two reasons we

do not consider programmable calculators in any detail

here. F irst, the best of the existing program s are near the

lim it of their ca pacity (1,2) and, second, the calculators can

now be replaced by very-low -cost m icrocom puters.

The development of adequate softw are is a difficult and

time-consuming task and should not be undertaken lightly .

If possib le, good existing softw are should be adopted. U nless



CLINICAL CHEMISTRY, Vol.31,No. 8,1985

1265

the user is w illing to accept a program that w ill be usable

only during the lifetime of a particular machine, the follow-

ing criteria should be met:

Programming

language: The program should be written

in a standard version of a well-established programming

language (e.g., FORTRAN, BASIC , or PASCAL) and in general

should avoid m achine-specific “enhancem ents.” H ow ever,

some machine-specific features w ill often be necessary and

som e “enhancem ents” m ay be desirable.

Modularity: The program should consist of well-defined

modules, each perform ing a few specific tasks. This allow s

existing modules to be replaced by new modules if better

algorithm s become available, and modules containing new

features can easily be added. A ll m achine-specific features

and “enhancements” should be placed in well-defined and

documented modules so that they can be readily adapted to

new hardware or new operating system s.

O perating system s: The problem of moving from one

machine to another may be m inim ized by a good choice of

operating system . Some operating system s can be used on

machines of different power, and a judicious choice allows a

laboratory to upgrade its computer requirements w ith m ini-

mum dislocation. Examples of such system s are CP /M , MS/

DOS, the ucun p-system , unix , and P ICK .

Documentat ion:

Program documentation should always

be clear and detailed. In ternal documentation should be

adequate to facilita te program m odification. A ll algorithm s

should be public ly available so that they may be exam ined

an d, w here necessary, criticized. A m anufacturer unw illing

to do this should provide a full description of the methods

used, including appropriate references to the literature and

specimen analyses of several real data sets that illustrate all

the features of the program . Good documentation on the use

of the program s and on the interpreta tion of the output is

essential.

Graphics: The use of high-resolution graphics is perhaps

the only place where the use of m achin e-sp ecific “enh ance-

ments” to standard programm ing languages is justifiable,

because good graphical output is frequently clearer than

any other representation of information. However, such

output should always be included in separate modules, and

alternative replacement modules, w ith low-resolution

graphics produced by a standard printer, should be avail-

able. In the absence of an adequate graphical output, the

alternative of alphanumeric output should be made avail-

able.

In pu t a nd O u tp ut

In pu t o f R es po ns es

The word “response” is used in this report for the quanti-

tative measurement obtained for each sample of standard or

test preparation . F or techniques that involve radionuclides,

the response is “counts”; for other techniques, it m ight be

the reading from a spectrophotometer or some other instru-

ment. Input of responses by direct link from the counter or

other device or by machine-readable media, such as paper

tape or data-logger cassette, should be the norm for routine

assays. Such an approach elim inates operator error during

data entry . Even so, some errors can occur, and the program

should check that the data have the expected format and

m agnitude. O ccasionally, m anual data entry is unavoidable;

a thorough check that the entry is correct is then essential.

If the computer is connected directly to the measuring

device, the responses should be accumulated in a ifie and

analyzed as a batch once the assay is complete. This makes

data correction easier and allow s more satisfactory error

a na ly sis a nd q ua li ty -c on tr ol procedures. Mor eo ve r, c om pu t-

ing resources w ill usually be used more effic iently if data

collection can proceed while the computer is being used for

other purposes. A printed copy of the stored data ifies must

alw ays be available for inspection.

In put o f A ss ay C on fig ura tio n a nd In struc tio ns

For the responses to be analyzed, the program w ill need

information on the concentration of the calibration stan-

dards, d ilu tions of the unknowns, and the order in which the

responses are being entered (assay configuration). In addi-

tion, instructions are required on the type of analysis

required . The program should allow some flexibility in the

assay configuration, which includes the identification of

quality-control pool samples (see the section on “between-

b atc h q ua lity c on tr ol” . T he p ro vision of stand ard i.zed c onfig -

urations for certain types of assay is useful, so that only the

number of test samples need be entered (3).

Output

The details of output from specific parts of the program

are dealt w ith in the appropriate section below . C are should

be taken to avoid unnecessary output, which reduces the

impact of important information. Numbers should be w rit-

ten in a readable form w ithout the use of exponential

format, and the use of numeric codes or cryptic mnemonics

should be avoided. Instead, clear, concise messages in ordi-

nary language should be used. If the result obtained from a

statistical test on the data from an assay is satisfactory, a

simple message may be all that is required. However, more

information should be available if a test fails or if the

o perator req uests the in form atio n. It may be useful to store

such detailed information in a disk file for later inspection if

necessary.

A na ly sis o f R es po ns es fro m a S In gle A ssa y

Batch

Genera l

Computer program s for the analysis of assay responses

fall into two classes. Program s of the first type take a

manual (usually graphical) analysis as their starting point.

Th e routines are designed to mim ic the procedure that a

technician m ight use with a ruler or a flexicurve. Examples

are linear interpolation , smoothing techniques, and some

uses of spline functions. P rogram s of the second type base

the analysis on a statistical model of the assay, and thus

lead to an assessment of errors of measurement. W e are

unanimous in recommending a program that uses a statisti-

c al m od el.

The basic statistical model of an assay assumes that the

response from a particular tube has two components:

{149}he expected response for the tube, which depends only

on the amount of analyte in the tube. This is the average

response that would be obtained if a very large number of

measurements were made for a given dose of the analyte.

The relation between expected response and dose is called

the

dose-response

curve or

c alib ra tio n c ur ve .

{149}random com ponent due to variability in experimental

procedures and measurements, which will have an average

value of zero across a large set of measurements. It w ill not

be satisfactory to assume that this component has a con-

stant standard deviation at all levels of the response. In

most cases, the size of th is random component w ill increase

w ith the level of response. A formulation of how the stan-

dard deviation (or some other measure of variability) de-

ponds on the mean response is known as the “response-error



COUNTS

BOUND

B

B o 2.0 00

10,000

e000

6,000

4.000

N 2.0 00

0

COUNTS

FR (

IO000

2,000

4.000

16,000

18.000

20,000

1,

1000 22 ,000.1

1266

CLINICAL

C HE MI ST RY , V ol .3 1,

No.8,

1985

relation,”

or RER .7 Details of how the RER affects immune-

assay results have been published (4 , 5, 8).

This is an idealized model of an assay, in which the

response in one assay tube is unaffected by the responses in

neighboring tubes; there are no mishaps that have produced

com pletely erroneous results for certa in tubes; no system at-

ic effects are present, such as drift due to tim e or carrier

position; and the ex pected respo nse is determ ined only by

the concentration of the analyte being measured. However,

in practice, satisfactory assays can come close to this ideal.

Any serious departure from this idealized model of the

assay system may lead to results that are incorrect and

subject to much greater uncertain ty than the calculations

imply . Thus an essential part

of the

analysis

of an

assay

batch is to check that the responses are consistent w ith

these assum ptions about th e assay. The extent t o w h ic h this

will be possible w ill

depend on the

assay

design , wh ich is

beyond the scope of this paper bu t has been discussed

elsewhere (6 , 7 . T he step s described in th e section on “steps

in th e a na ly sis ...“ below suggest how the analysis of an

assay batch can proceed, incorporating checks on the as-

sumptions. W e have

presented

them in one possib le se-

q uen ce, b ut

alternative

orderings are

possible.

R ec om m e nd ed M o de ls

D ose-respo nse curve. N um erous m odels

have

been

sug-

gested. A s ta nd ar d program sho uld em bod y on e acceptable

general-purpose m od el; a m ore so ph is tic ate d p ro gr am might

include altern atives oc casio nally need ed fo r s pe cia l c ir cum -

stances.

Important features of a model suited to w ide use in

routine assays are:

A bbreviations used in t hi s p ap er :

B0, c ou nts o bs er ve d f or z er o dose;

B/B0,

n orm aliz ed re sp on se

variable, ranging from 0 to 1, representing counts bound

above

non spe c if ic , r el at iv e t o c o un ts bound above nonspec if ic f or z er o dose

of analyte . Also com m on ly exp ressed as a p ercen tag e, B/B0,on a

scale from 0 to 100 .

B/F, b oun d-to -free ratio for lab eled lig and .

BIT,

bo und -to-total ratio fo r lab eled lig an d.

CV, coefficien t of variation , as a percentage of the m ean.

a,

b, c, d: parameters

of the four-param eter logistic m odel, w ith

a = expected response at zero dose o f a naly te ;

b

= slope factor or exponent, w ith absolu te magnitude equal to

th e lo git-lo g s lo pe ;

c

= EC or IC , i.e ., concentration of analy te w ith an expected

response exactly halfw ay betw een a an d d;

d

=

expected response for infinite analy te co ncentratio n (o ften ,

though not alw ays, synonym ous w ith nonspecific counts bound).

nASA, en zyme- li nk ed immunoso rb ent assay.

EMrr’, e nz ym e-m ultip lied imm un olo gica l te ch niq ue .

IRMA. immunoradiometric assay (generic name fo r assays involv-

i ng l ab e le d -a nt ib o dy r ea gen ts ).

J, ex pon en t u ti li ze d in p ow er fu nctio n m od el for resp on se-erro r

relationship.

NSB , n onsp ec if ic b in d in g .

r , n um b er o f r ep li ca te s.

RER, response-error r el at io n sh ip , c ommon ly expressed as van-

ance o f t he response as a function of expected l ev el o f response, e.g.

=

ao y’.

RMS , root m ea n sq ua re error. F or u nw eig hted reg re ssion , th e

standard deviation of a point around th e fitted c ur ve . F or w e ig hte d

regressio n, th e ratio o f observed error t o p re di ct ed error, based on

th e particular we ig hti ng m o de l u til iz ed .

sd ,s ta nd ard d ev ia ti on .

se ,

standard

error.

y ’, s lo p e.

r es id u al me an s qu ar e b etw e en doses.

5, mean s qu are w ith in doses.

w , w eight fo r o bservatio n i.

z , e sti ma te o f lo g( do se ).

1/ A fam ily of curves in which individual members are

defined

b y th e

numerical values of very

few (p refera bly o nly

four) parameters.

2/ F lexibility of shape, slope, and position to suit th e

requirements of standard assay techniques.

3/ M onotonic form (i.e., no reversals of slope), and re-

straint from m aking detours fo r outliers.

The “four-param eter logistic” curve appears to be the

most generally useful an d versatile m od el that will satisfy

th e ab ov e r eq u ir ement s, although nothing said h er e imp lie s

that it is “right” and all others “wrong.” It s parameters

characterize:

(a ) expected count at zero dose, not necessarily identical

with any one

observed count (F igure

1) ;

(b ) “slope” factor, related to rate of change of cou nt w ith

increasing dose;

c th e dose expected to give a coun t halfw ay b etw een a

an d d, i.e., th e EC or IC;

d expected

cou nt at

“infinite”

dose

(high-dose plateau or

asymptote) , not necessarily identical with a coun t for non-

specific binding.

The general form of equation then is:

Expected

response at dose

= d + +(dose/c)’

This curve (F igure 1) satisfies

all the desiderata and is

adequate for the

vast majority

o f e xis tin g

immunoassay

system s. It is continuous and smooth, and the slope factor is

very stable in repeated assay batches. I t c an a pp ro xim a te

closely th e simple m as s-a ctio n e qu atio n (8). On e a dd iti on al

param eter for “asymm etry” is easily incorporated

to give

even greater versatility (9, 10).

R esponse curves are further discussed in Appendix 1, an d

greater detail can be found elsewhere (1 1, 1 2) .

R es po ns e-e rr or r ela tio n (RER):

Detailed modeling of

ex -

perim ental errors and their sources is not requ ired . A

program should include a simp le formu lation of how the

variance o r th e standard d eviatio n o f th e ran dom co mp on en t

of th e response varies w ith the m ean response level. This

will usually require no more than two parameters (e.g., a

power function, a linear or quadratic relationship for the

standard d eviation or th e variance as a function of expected

response).

W e recom mend that th e RER be expressed as a product of

two term s. The first term w ill be a constant-i.e., indepen-

dent of the level of response-for a particular assay batch (o r

for a

subset

of the batch if

standards and unknowns are

to be

treated separate ly). The second term will describe ho w th e

standard deviation of the responses varies with the level of

LOG,0

-I 0

2 3

8/T B /U .

I { 1 76 }

I

4 8

-‘-.,.1eN..

‘ {176}“0

-I

6

6 PEDjo

2 4 le -

1 .2

: -19-

0 0 . ‘ {176} ‘ “‘“ d.O.(flO.Oec,GccJ -

1

. iei

I

0IO I 10 00

D OSE S (1 04 scsI.)

F ig . 1 . Schematic d ra w in g o f a d os e-r es po ns e ( ca lib ra tio n) curve

N o te s m o ot h, s y m m et ri ca l s ig m o id a l s h ap e , c h ar a ct er iz e d b y

four

p a ra m e te r s ( a .

b

C, c . C o nf id en ce limitstaper in a s m o ot h, c o ns is te n t m a n ne r . R e p ro d uc e d,

w i th p e rm i ss io n , f ro m ref.

29



C LI NI CA L C HE MI ST RY , V ol .3 1, N o . 8 , 1985 1267

th e

response, an d this relationship w ill remain fairly con-

stant across a series of b atch es (assays). T his im plies that

th e scale of the random errors in the responses may differ

between batches, but that

th e

shape

of th eir relation sh ip to

th e mean response wou ld be similar in each batch (4, 5 .

An appropriate shape for the dependence of the standard

deviation on the mean response should be determ ined from

th e data for a series of a ss ay s. T his in fo rm atio n is fed back

into th e p ro gra m either:

{149}y the

user entering a

f ew n um b er s ( pr ef er ab ly

just

one)

calculated

from a

series of assays, or

{149}referably, by the com puter perform ing this autom atical-

ly by utilizing stored inform ation from previous batches.

P ois so n e rr or s due to coun ting may be e stim ate d d ire ctly

from the number of counts and th e RER can, if desired, be

specified without this component. The counting error can

then be added to the

estimated RER

to

obtain

th e

to ta l e rro r

1 3 . Details of how to characterize the RER ar e given in

Appendix 2.

S teps in the A nalysis of a S ing le B atch

A djustm ent and verification of responses. C ertain proce-

dures m ay yield responses that require adjustm ent before

c alc ula tio n s ta rts . F or e xa mp le , v ar ia ble c ou nti ng e ff ic ie nc y

as a result

of

variab le quenching in liquid-scintillation

counting gives a response

that must be corrected before

other ca lcu lation s. O ther ad justm ents, however, such as

v ariable reco very

in

preliminary separation procedures,

must be made after th e co ncen tra tion s h ave b e en e st im a te d.

T he co mpu ter

program

should be designed to handle any

s uc h p ro ce du re th at th e lab oratory m etho ds dem an d.

It m ay be usefu l to display any apparen t anom alies in the

data-such as serious d iscrepan cies am ong replicates from

th e sa me sam ple or incons is ten t s ta nd ar d r es po ns es -b ef or e

the main analysis is undertaken. Error correction can be

done at this stage, but a record of any changes shou ld be

sent to the quality-control file (see below) and must be

recorded on the output.

Screening of re plic ate s ets. When some or all of the

specimens

have

b ee n m ea su re d

in replicate , th e

RER

may

be estim ated from

th e

scatter

of th e sets of rep licates, w hich

may thus enab le ind ividual “ou tliers” to be detected. The

term “outlier” is used with two different meanings in

im munoassays. F irstly , the term “o utlier” h as been used to

denote a member of a se t o f rep li cate s that is dramatically

further from the set mean than would be exp ected from th e

estimated HER, and presumably indicates a b lunder. W e

us e it in this sen se here. Secondly, the term is used for the

case

when all the

responses

for a

concentration

o f s tandard

deviate from their expected value. T his case is dealt w ith in

th e section on “tes t ing goodness-o f- fi t.”

The first step is to use the data from the rep licate sets to

estimate the HER. W here the shape of the HER can be

assumed to be constant over a series of assays, only th e

multiply ing constant needs

to

be found. This

is ea sily

estim ated as a weighted combination of the individual

variances.

Practical d ifficu lties can arise if any gross outli-

er s

are presen t

in the data. Robust

techniques,

such as

takin g m ed ians

within

sub-groups

5 or a m od ification of

the m eth od proposed by Healy and Kimber (14), should

prevent these extrem e sets f rom i nf lu e nc in g th e estimates.

After th e RER i s e st ima t ed from all the data, the scatter

of individual sets of replicates about their mean can be

compared with what would be ex pec ted from th e RER. A ll

sets where the ratio of observed to expected variance is

greater than some value that wou ld be very unlikely to

occur by chance (say, p <0.005) can be classified as outliers.

Such sets would ordinarily be discarded. A permanent

record must be kep t of all re je cte d s pe cim en s, an d th eir

occurrence m ust be printed on the ou tpu t. V arious op tion s

f or a ut oma ti c and manual rejection are possib le, and th e

program

m ay allow each laboratory to

specify its ow n rules.

A fter these “outliers” are excluded, it w ill u sually be

desirable to calculate and d isp lay the RER va lu e s s ep a ra te ly

fo r

standards

an d

unknowns

and to

test

whether they are

con sistent. In exceptional circu mstances th e unknowns an d

standards may be subject to different sources of error, fo r

example, when the unknowns undergo

some

preliminary

processing. A sophisticated program w ould handle the two

estim ates of the HER and use wh ichever is appropriate at

each stage of the calculations.

F ittin g th e d os e- re sp on se

curve. F ittin g o f the c alibratio n

curve ought to be perform ed in terms of th e actual quanti -

t ie s i nd epend en t ly measured a nd o bs er ve d, nam ely , doses

an d responses. Outpu t m ay be expressed in term s of”B/T,”

“B /B0 ,” “B /F ,” o r o th er f am ilia r d er iv ed q ua ntitie s.

The response curve should be fitted to the raw counts

obtained from

th e

calibration stan dards

by the

method

of

iter ative ly rew eig hted n on lin ear le ast squares. Th e w e ig ht s

should be obtained from th e H ER . A n a pp ro pria te algorithm

would be the Gauss-New ton method

or the

Marquardt-

Levenberg m od ification of th is techn iqu e (1 5, 16 ). Robust

regression

m ethods are also acceptable.

In itial estim ates of param eters are needed to start th e

calculations;

they are easily built into the program , either

by reference to values in past runs of similar assays or by

rough initial com putations (e.g .,

based

on

th e

logit-log

method or even simp ler approximations, u sing the dose

having a response

close to 50% B/B0

as

th e in itia l e stim ate

of c) .

Te st in g g oodn es s- o f- fi t.

The program must perform an

analysis of variance on the responses for the stan dard s,

weighted in inverse proportion to the variance given by th e

HER . I t m ust conform to standard sta tistical m ethods fo r

w eighted least squares, producing w eigh ted m ean

squares

and degrees of freedom for:

11 th e scatter o f m ean responses ( at i nd iv idu a l standard

doses) abou t the fitted calibration c ur ve , a ve ra ge d over all

standards

(i.e.,

“residual mean square between doses”, sf).

2/

th e

scatter

of responses on replicate

standard tubes

ab ou t th eir respective means, averaged over all standards

(i.e., “m ean squa re w ith in d os es” , 4).

The variance ratio 4/4 exam ines whether the model

cho sen fo r th e d ose-respo nse cu rv e gives an accep tab le fit to

th e

resp onses from

th e stan dard s. T he resid uals for

means

of

individual standard doses help to p inpoin t where the lack of

fit occu rs. T he program should calculate the “S tudentized”

residual 1 7 fo r th e mean count at each dose of the

standard; a residual

>3.0

warns

of

poor

fit at that dose. A

plot of the stand ard resp on ses and the fitted curve resem -

bling F igure 2 will be a useful d iagnostic tool for the

assay ist. It will help to reveal w hether there is a system atic

lack of fit-perhaps caused by an

unsuitable choice of m odel

fo r th e

d os e-r es po ns e c urv e- or

whether responses at one or

tw o dose levels are grossly ou t of lin e. The latter situation

corresponds

to the

second usage

of the term “outlier,”

meaning inconsistent results from

on e

standard dose

rela-

tive to other doses, rather than a

response

in co ns is te nt w it h

other replicates at a single dose.

Au tom at ic r eje ct io n of a ll th e resp onse s fro m o ne s ta nd a rd

dose is n ot recommended, there seld om b ein g en ou gh doses

to make such a procedure reliab le. A m in im um of eigh t dose

levels is recommended. M anual in terven tion shou ld be

allowed to reject an occasional standard dose (but never

more than one) on the basis of a m ixture of statistical

grounds an d lab oratory experien ce. E xcessive m an ual rejec-

tion , h ow ever, w ill lead to biased , over-op tim istic estim ates



-1

Logl(X)

0.1 1 10 100

1268

C LI NI CA L C HE MI ST RY , V ol .3 1, N o . 8 ,

1985

Fig .2 .

Computer

generated

plot ofthestandardcurvedata,thefitted

logisticurv e,andthe 95 confidence lim its fo r a s in gle observation,

excludingncertainty in t he pos it ion ofthe curve)

F ro m I BM - PC R IA program

o fM . L J af fe

o f p r ec is io n .

Finally,

th e

program

should

produce a table

sh ow in g th e

apparent concentration of analyte in the standards when

these are treated as if they were unknow n specim ens.

Comparison of th ese re su lts with th e a c tu a l c on c en tr at io n s

of the

standards

will

suggest

how much b ias a

misfitting

dose-response curve m ight introduce i nt o the estimates fo r

th e u nkn ow ns. A statistically sig nificant lack o f f it , e sp e ci al -

ly in assays of h igh precision , m ay be s uf fic ie ntly sma ll that

th e results w ill rem ain useful fo r t he ir in te nd ed purpose.

Presentation of p re ci sio n p ro fil e.

The

above estim ation of

the HER provides the

program w ith an estimate

of the

random error at any level of response. The combination of

this estim ate with the slope of the f itt ed d os e- re sp on se

c ur ve p ermit s one to ob tain a n estim ate o f t he random error

in th e estimated co ncen tra tion o f a n u nk no wn Appendix 3) .

A rep resen t at ion of this derived variab ility in the estimated

c on ce nt ra tio n o f

unknow ns against analyte

co ncen tra tion is

known as th e precision p rofile. (T he term “im precision

p ro ff le ” is also used , to em phasize that a larger percentage

coefficient of variation indicates poorer precision.) This

s ho uld b e p re se nte d either as a g ra ph ( Fig ur e 3) or as a tab le

of expected precision (e .g ., the percentage coefficient of

variation or standard error for a test preparation) vs concen-

tration.

hC G

ngimi

Fig.

3.

“Precis ion

profi le”:

%CVforan

unknown specimenmeasured

in

dup lic ateordin ate )lot teds serum ana lyt econ cen tra tio nlogsca le

on th e abscissa (3 (

T h e e ff ec ts o f u n ce r ta in t y in the pos i tionof thes tandardcurveare not includedn

this e x am p le . A l so s h ow n : e m p ir ic a l w i th in -b a tc h %CV( {149})ndbetween-batch

% CV o )or three quali ty-controlpools,nalyzedn triplicaten e ac h o f t he p as t

20

batchesor assays

P recision profile s c an be calcu lated

for a sin gle

measure-

ment for the

unknown,

or for the

mean

of duplicate or

triplicate measurements. The error in estimating the re-

s po ns e c ur ve may or m ay not be included in the errors used

to calculate the precision profile: th is option should be

specified. The program should generate a tab le or graph of

the precision proffle for the num ber of rep licate measure-

ments and the sample volume ordinarily used for the

unknowns. T he program shou ld also provide an estimate of

th e lowest le ve l o f r elia b le assay m easurem ent. A statistical

estimate of the m in im al d ete ctab le concentration 1 8 is

recom men ded . A lterna tiv es recen tly describ ed by Oppenh e i-

m er et al. (19) m ay be usefu l.

Estimation of concentration fo r test sam ples un kn ow ns .

The

program m ust p ro vid e a n estim ate o f th e co ncen tra tion

3

o f e ac h

unknow n sam ple

and a

measure

o f its p recisio n. T he

precision may be expressed as an estim ated percentage

error in the result ( CV) from t he p re ci sio n profile, or as

9 5 con fiden ce lim its. A w arning when the estim ated error

e xceed s a certa in th re sh old (e .g ., 10% ) is u seful, an d may be

al l that is requ ired for certain applications.

When a sample is analyzed in rep licate at a single

dilu tion, the concentration corresponding to the mean re-

spon se is read from th e calibration curve an d corrected fo r

sample dilution. The estim ated precision at this level of

response is used to assig n co nfid ence lim its or a % CV to the

result .

l ithe sam ple has been in cluded at two or m ore d ilution s, a

combined

estimate

of con cen tration sh ou ld b e

ob tained from

all th e responses by using a weighted average, which gives

greater influence to the doses that lie within the region of

th e

curve

where

estimation

is m ore precise. Again , esti-

mates of precision shou ld be ob tained for this combined

estim ate. When two or more d ilution s are included , the

program can test w hether th e con cen tration s obtained at

d if fe re nt p ar ts

o f th e

response curve are consistent. This is a

generalized test of “parallelism .” A n o utlin e o f th es e calcula-

tion s is given in

Appendix

4.

For each test sample, th e minimum output from th e

program should be an estim ated concentration, along with

warning messages about outliers, lack of parallelism , or

p oo r p re ci sio n. For ou tliers or lack of parallelism, th e

e stim ates from individu al resp on ses

or in div id ua l d oses

will

help the assayist to in terpret the results.

Evaluating assay drift

or in st abi li ty .

Appreciable drift in

responses from the sam e sam ple placed in d if fe re n t p o si ti on s

in th e assay batch m ay seriously invalidate th e estimates of

precision described above. Inform ation w ill b e available

from r ep li ca te c ou n ts f rom s ta nd ar ds or unknowns placed in

different parts of the sequence

of test

samples.

Tests of

po ss ib le d is to rt io n of results caused by system atic drift can

be based upon these rep licate responses (or upon the corre-

s po nd in g e stim ate s of con cen trations). T he particu lar test

adopted will depend on the assay design, b ut w ill usu ally

consist of a regression analysis

or

a n a na ly sis

of

variance

with

suitable

weigh ting. A com bined test should

make use

of all the available data. Any apparent drift should be

evaluated in term s of its e ff ec t on the estimated concentra-

tion s, so that it s importance m ay be assessed .

1000 Be tween -Ba tch Qua lit y Con tro l

One important goal of data analysis is to reveal w hether

performance is consisten t in a series of assay b atch es.

Automated data analysis is essential if a sufficiently broad

se t of indicators is to be followed withou t excessive labor.

A n archive should be m aintained for the m ost important

indices o f c on sis te nc y, in clu din g the results from quality-

c on tr ol p oo ls , th e parameters o f t he do se-respo nse cu rv e, th e




1985

1269

RER, th e w eighted root mean sq ua re error, and a summary

o f reje cted d ata.

A lthough subsequen t analysis of resu lts fr om q ua lity -

c on tr ol p oo ls may be carried ou t by a s ep ara te p ro gr am , this

shou ld not be op tional. Recom mendations on the p lacing of

the quality-con trol pools in the assay batch an d their

d isp osition at d ifferen t analyte concentrations have been

m ad e elsew here (20), although gu idelines are som ew hat

arbitrary . Use of at least three quality-con trol pools, w ith

different analyte c on ce nt ra tio ns , is r ec om m en de d.

Th e r es ult s from th ese p ools sh ou ld b e analyzed for tren ds

or sudden

shifts between batches. The analysis should

in clude the use of con trol chart methods, m ean

squared

suc c es si ve d if fe re n ce s,

o r o th er

appropriate

t es ts fo r r an do m -

ness. Details of these methods are d iscu ssed in

standard

texts 2 1, 22 ) an d their app lication to immunoassays has

b ee n d is cu ss ed

(23-25).

S im ilar tests can also be ap plied to

other features of the assay such as the parameters of the

stan dard curve, but these w ill be of s ec on da ry im p or ta nc e.

The program should provide tests that combine informa-

tion from a ll t he q u ali ty -c on tr ol pools. T hese test w heth er

al l

th e p ools

are changing in a sim ilar manner. In

their

sim plest form they can be perform ed by applying the tests

ab ove to th e m ean of all the q uality-con trol p ools; w eightin g

according to the estimated precision of each pool is desir-

able.

A ssessm en t of be tw ee n-ba tch precision. The quality-con-

tro l specimens are the only source of information on be-

tween-batch precision , wh ich is the essen tia l m easure that a

clin icia n n eed s fo r co mp arin g re su lts o bta in ed on different

days. The between-batch precision includes com ponen ts

from w ithin-batch errors and b e twe en -ba tch errors.

T he program should compute the estim ated betw een-

batch precision for each quality-control pool. Two compo-

nents of between-batch imprecision can be predicted from

th e responses from a single b atch . T he first is th e within-

batch variab ility for each specim en. A second is due to the

uncertainty in the position of the fitted dose-response curve

as a resu lt of the variab ility in responses fo r th e standards.

Both of these components can be estim ated from the data

from a

single batch, although the

second

c om p on en t, w h ic h

is relatively sm all, is part of the betw een -batch variation as

assessed from th e q ua lity -c on tr ol p oo ls . T he resu lts sh ou ld

b e p re se nte d

in a

manner

that

compares with in- and be-

tw ee n-b atch p re cisio n, estim ate d from th e quality-control

pools, with the precision proffle of the curren t batch or w ith

a pooled estim ate from several recent batches (Figure 3) .

None of these e ff or ts e xc us es th e an alyst fro m p articipa-

tio n in a w ell-d esig ne d in ter-lab oratory q ua lity -co ntro l pro-

gram.

W e thank th e I nte rn at io na l A tom ic Energy A u th or it y f or s po n-

s or in g t he

meeting

that

led to th is paper, and

t he D ep ar tm en t o f

Mo le cu la r E n do cr in olo gy , M id dl es ex H o sp ita l, fo r a ctin g a s h os ts .

W e also thank M . L .

Jaffe

fo r s up ply in g F ig ur e 2 .

AppendIxes

1. R espo nse C urves

The logistic model is app licab le to immunoassays in

w hich bound or free coun ts, or both , are m easured.

(BIT

or

B/B0

may also be used as response variable, although

“counts” are preferab le.) In ad dition , th is m eth od is ap plica-

b le to assays that involve enzym e-labeled an tigens (e.g .,

EMIT) or an t ibod ie s (EusAs); to labe l ed -an tibody assay s (tw o-

site iiu& s, or “sand wich ”-typ e assays); to assays in volvin g

flu orescent, chem ilum inisce nt, elec tron spin resonance, or

bacteriophage lab els; an d to rad io im mun odiffu sion m etho ds.

It is

also

applicab le to m any receptor assay system s and to

several in vivo and in vitro bioassays. Though som e assay

users may be accustomed to-and thus p refer-oth er typ es

of curves an d methods of fitting, they are urged to consider

seriously th e p ro ce du re s d es cr ib ed here as the basis of a

g en er al a pp ro ac h.

The lo gi stic r es po ns e curve is not app licab le when the

curve consists of a sum mation of d iscrete sigm oidal compo-

n en ts, or to n on -m on oton ic

cu rv es (w ith re versals

o f s lo p e) ,

or in cases o f s ev er e asymmetry (when p lo tted as re sp on se

against log dose). Should such

response curves arise,

other

methods, either derived from the mass-act ion law or based

on em pirical m odels (but, unfortunately, w ith more param e-

ters) w ill be necessary. No one method of curve fitting is

likely to be op timal in a ll c ir cum st an ce s, nor is there

sufficient experience w ith som e of th ese a ss ay te ch niq ue s fo r

general proposals to be made. For unusual curves, th e

assayist would need to seek special assistance from col-

leagues experienced in m athem atical m odelling and statis-

tical curve fitting. In add ition , he shou ld

examine

th e

s y st em expe rim en ta ll y, to evaluate w heth er th is an om alou s

behavior could be rem oved w ithou t damage to assay per-

formance.

A program shou ld provide the op tion of reducing the

number of fitted param eters by regarding th e expected

response fo r “infinite” concentration as constant, perhaps

constrained to be equal to the exp erim entally determ ined

mean response fo r nonspecific b ind ing (NSB). In some

assays,

h ow ev er, th e

e xp ec te d r es po ns e

fo r

“infinite”

concen-

tration may differ markedly from th e o bs er ve d r es po ns e fo r

NSB. This indicates that either the model is unsatisfactory

or the method for measurement of NSB does not give an

appropriate estim ate of the corresponding “plateau” or as-

ym ptote. In

b oth cases

the N SB responses should be exclud -

ed . S im ilar prob lem s are less likely to occur for very low

concentrations (approaching zero). However, the assump-

tion that th e infinite concentration param eter is known

exactly-or that the NSB counts can be ignored-should be

made

cau tiously and never m erely as a conven ience. In all

circumstances

th e

program

sh ould preven t an y estim ation

fo r samples beyond the h ighest standard concentration,

other than an explicitly approximate one only for the

purpose of in dicatin g th e need for an ap prop riate d ilution of

sample before re-assay.

A d di tio na l p ar am e te rs may be incorporated in to th e lo gis-

ti c ftmction to allow for marked asymmetry (6 , 9 , 10 .

E stim ation of th e asym metry can be difficult, and it may be

advisab le to fix the value of the asymmetry param eter, on

the basis of data

from several consecutive batches.

The ab ility to d etec t la ck o f sa ti sfactory “goodness-of-fit”

for a sim ple cu rve (e.g ., a fou r-p aram eter logistic) im proves

as the precision of the assay im proves. Thus, failure of th e

four-param eter logistic to fit m ay be encountered in excep-

tio nally p recise assay s.

Likewise,

increasing

the number of

replicates

or increasing the number of dose levels leads

to

significant im provem ent in one’s ability to detect lack of fit,

and hence may introduce th e need to utilize m o re c om p le x

models. Conversely, when experim ental errors ar e la rge,

replicates are few , and the dose levels are few an d fa r

between, sim ple m odels are likely to be “adequate”; i.e., th e

error

of th e estim ate

introduced

by an

in corre ct m od el

fo r

th e

standard

curve is sm all relative to the

uncertainty

in th e

response

of th e

unknow n sam ple itself.

W hen both the low-dose response and h igh -dose p lateaus

ar e fixed at arb itrary values (e.g., m ean values of B0 an d

NSB), then the logistic m ethod becomes identical w ith th e

“ lo g it -l og ” me thod , an d th e magnitude of the slope factor (b )



y

=

Iog,(dose)

X

dose

Fig.4. Illustration of t he p ri nc ip le s i nv o lv ed in the estimation ofthe

standard error

o r % C V

fo r a n u nk no w n: a s a n a pp ro xim a tio n,

CV =

se(Iog6 (i)) = se/Iyi = (sd /V t’)/Iy ’I, w here y’ = slopel =

A1o90(4 ; a lt e rna ti ve l y, s e (s ) = se /Islope(, w he re se =

s4 /VF

an d

slope= dydx

XL X Xu

an d the combined estimate of log dose by

-

,WjZj

-

w1z1 + w2z2 + w3z3 +

-

- w1 - w1 + w + W 3 +

with estimated standard error

1270


1985

is

identical w ith the slope of the logit-log plot.

The “logistic” m ethod is a generalization of , and an

im provem ent on, the “logit-log” m ethod. It w ill be satisfac-

to ry for m an y a ssa ys in w hich th e lo git-lo g is u nsatisfactory .

T hose w ho currently use th e log it-log m eth od sh ou ld co nsid -

er changing to the use of th is more general and flexib le

method.

2 . D eta ils o f R es po ns e-E rro r R ela tio ns hip

Various form s of the response-error relationship ar e

p ossib le, b ut an

exponential

m odel is recom mended that has

th e for m:

variance

a t r es po ns e

( ex pe cte d r es po ns e) ”.

When J

=

0, the va riance is co nstant; when J

=

1, the

variance

is proportional to the response; when J = 2, the

% CV of the response is constant.

Th is m odel is preferred to others because the value of J is

usually quite stable from assay toy . For mos t imm un e-

assays J is very nearly 1. The computational routines

requ ire on ly a knowledge of J and w ill estim ate the propor-

tionality constan t for each assay. One of the published

methods f or e st ima t in g J from a series of assays (4 , 5, 26)

should be available. This method should be implemented in

a r ob us t manner, no t e as ily p er tu rb ed b y o ut li er s.

W hen one is using th is form of response-error relation-

sh ip , the background coun ts or NSB responses must not be

subtracted from the observed counts before processing.

Alternative

models (1 , 4 , 5 , 26)

w ill also be acceptable ,

w ith sim ilar m ethod of calcu lation .

3 . C alc ula tio n o f U nkn ow n C on cen tra tio ns a nd T he ir

C o nfid en ce L im i ts

The ou tline of the calcu lation s given here ignores the

contribution to the error (componen t of variance) in the

estimate of the unknown concentration from the curve-

f itti ng p ro ce du re , an d in volves app roxim ations esp ecially

near the asym ptotes. T hese details are in ten ded to ex plain

the basis of the method . An ideal program shou ld use

m ethods that do not involve these approxim ations (6, 8) .

1. A single dose fo r the

unknown sample:

Calculate the

m ea n r es po ns e for th e (r) replicates o f t he u nk no wn . In te rp o-

late th e correspond ing log d ose

from

th e

fitte d c ur ve

a t th is

point and

add to

this

th e

logarithm of the fraction by which

it has been dilu ted to get the log of the estim ated concentra-

tion (z). Calculate the slope (y’) of the response plotted

against log dose at th is point. The estim ated standard

deviation (ad) of a single response at this point on the curve

is obtained from the predicted RER.

The

standard error of

the estim ated log dose at this poin t is then

sd I\

This standard error can be used to c alcu late a co nfid ence

interval for z and the c or re sp on din g ( ar it hm e ti c) dose, and a

% CV for th e d ose estim ate.

2.

More than one dose

of an

unknown: Proceed as

for a

single dose to g et v alu es z j, ( sd )1 , (y’)1, r1 for each dose. T he

w eight for each

dose

is then given by

- r yl 2

W 1 - (sd),2

= W 2 ± W 3 +

An approximate test of whether the estim ates at the

different d ose levels are com patib le (generalized parallel-

ism ) is obtained by calculating the quantity

=

(z 1

-

z)2 (zj

-

z) 2 + z)2 + (z 3

-

As a first approxim ation , th is quan tity shou ld follow a ch i-

square distribution w ith degrees of freedom

=

n

-

1 , w h er e

n = num ber of doses; a ny larg e departure from it s expected

value of n - 1 wou ld be a

basis

fo r

suspicion.

4 . C alc ula tio n o f P re cis io n P ro file s

For any dose (x), w e can calculate th e e xp ec te d r es po ns e

an d h en ce, from th e response-error relationship , the esti-

m ated standard deviation of a single

response

at th is point

on th e curve. The next step is to divide by

vi

t o o bt ain th e

standard error

of the mean

response

for r rep licates. The

com ponent of error for uncertainty

in the

f itte d c urv e

may

be

added in here (in term s of

variances).This

i s d e si rab le ,

especially

when the curve is

based

on only a few dose levels.

U su ally, this com ponen t of variance should be qu ite sm all.

This standard error is then divided by the slope of the

response curve, and the estim ated coefficient of variation

( C V) calculated (F igure 4)

(23,24,27-30).

it is convenient

to use the approximation that the CV of dose x, expressed as

a decimal fraction, is equal to the standard error of loge (x).

References

1.

D udle y R A. R ad io im mun oassay (R IA ) data p ro cessin g

on

pro-

g ramrnabl e ca lcu la to rs: A n IAEA project. R adioim munoassay and

Related Procedures

in

Med ic in e 1 982 , IAEA , Vienna, 1982 , pp 411-

421.

2. D av is S E, Ja ff e ML , M u ns on PJ, R o db ar d D . Radioimmunoassay

d at a p ro c es si ng w ith a sm all

p ro gr ammabl e c ompu te r. J

Immuno-

assay 1, 15-25 (1980).

3. M cK enzie 1G M, Thom pson RC H.

D es ig n a nd

im plem en ta tio n o f

a software package for an alysis of immu no as sa y d ata . In

Immuno-

assays

fo r C lin ic al

Chemistry, W M H unter, JE T Com e, Eds.,

Churchil l L iv in gs to n, E di nb ur gh , 1983, pp 608-613.

4. F in ne y D ,J . R ad iolig an d assay s. Biometrics 32 , 72 1-7 40 (1 97 6).

5. Rodbard D, Lennox RH, W ray IlL , Ramseth D . S ta ti st ic al

ch aracterization of th e ran dom errors in t he r ad io im m un oa ssa y



CLINICAL CHEMISTRY,

Vol.

31,No. 8,1985 1271

dose-response variable.

Clin

Chem 22 , 350-358 (1976).

6. Raab GM . V alid ity te sts in th e statistical analysis of immunoas-

s ay d at a. Op.

cit.

(ref 3) , pp 614-623.

7.

Walk er WHC. A n a pp ro ac h to immunoassay .

Clin

Chem 23,384-

40 2 (1977).

8. Finney DJ. R es po ns e cu rv es fo r radioimmunoassay.

Clin

Chem

29 , 1 762 -6 (1 98 3).

9. Rodbard D, M unson P, De Lean A. Improved curve fitting,

parallelism

testing,

ch ara cten satio n o f

s en siti vi ty a nd

specificity,

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