JOURNAL OF CLINICAL MICROBIOLOGY, OCt. 1994, p. 2441-24470095-1137/94/$04.00+0Copyright © 1994, American Society for Microbiology

Determination of Parallelism and Nonparallelism inBioassay Dilution Curves

BRIAN D. PLIKAYTIS,l* PATRICIA F. HOLDER,2 LORNA B. PATS,2 SUSAN E. MASLANKA,2LINDA L. GHEESLING,2 AND GEORGE M. CARLONE2

Biostatistics and Information Management Branch1 and Childhood and Vaccine-Preventable Diseases ImmunologySection,2 Childhood and Respiratory Diseases Branch, Division of Bacterial and Mycotic Diseases, National

Center for Infectious Diseases, Centers for Disease Control and Prevention, Atlanta, Georgia 30333

Received 15 April 1994/Returned for modification 13 June 1994/Accepted 28 June 1994

There is a lack of consensus among investigators who use a variety of immunoassay techniques (e.g.,enzyme-linked immunosorbent assay [ELISA] and radioimmunoassay) regarding the protocols for describingand forming standard reference or calibration curves and interpolating serum antibody concentrations. Thisconfounds the issue of detecting the presence or absence of parallelism between standard reference serum andserially diluted serum sample curves. These curves must be parallel to support the assumption that theantibody-binding characteristics are similar enough to allow the determination of antibody levels in the dilutedserum sample. There is no universal and widely adopted strategy for assessing parallelism in bioassays, andwithout an assurance of parallelism, investigators are not able to calculate reliable estimates for antibodyconcentrations in serum samples. Furthermore, single-point (dilution) serum assays do not provide informa-tion related to parallelism and nonparallelism, and this, too, may lead to considerable error when calculatingantibody concentrations. When assay methodology, technique, and precision improve to the extent thatstandard reference serum and serially diluted serum sample curves are fit with little error, standard analysisof variance techniques are overly sensitive to negligible departures from parallelism. We present a series ofguidelines that compose a protocol for assessing parallelism between bioassay dilution curves that are

applicable to data derived from ELISAs. These criteria should be applicable, with minor modifications, to mostimmunoassay experimental situations and, most importantly, are not dependent on the mathematical modelused to form the standard reference curve. These guidelines have evolved in our laboratories over the past 4years during the performance of thousands of ELISAs for antibodies to the capsular polysaccharides ofNeisseria meningitidis groups A and C and Haemophilus influenzae type b.

There is a lack of consensus among investigators who use a

variety of immunoassay techniques (e.g., enzyme-linked immu-nosorbent assay [ELISA] and radioimmunoassay) regardingthe protocols for describing and forming standard reference or

calibration curves and interpolating serum antibody concentra-tions. A wide variety of curve-fitting methods are used togenerate standard reference or calibration curves. Rodgers(18) listed many of these methods in a report that included an

extensive bibliography. The most common functional modelsfor calibration curves range from the simple linear log-log andlogit-log transformations to the more complex four-parameternonlinear logistic-log models (14). These models also include a

range of spline functions (7, 8). The proliferation of existingmethods for standard reference curve formation clearly indi-cates that no one descriptive model will work optimally for allexperimental situations and assays. Each model possessesadvantages and disadvantages and must be judged on itsapplicability to the experiment at hand.These concerns confound the issue of detecting the presence

or absence of parallelism between standard reference serum

and serially diluted serum sample curves. These curves must beparallel to support the assumption that the antibody-bindingcharacteristics are similar enough to allow the determinationof antibody levels in the diluted serum sample. No universalstrategy for assessing parallelism in bioassays has been widely

* Corresponding author. Mailing address: Division of Bacterial and

Mycotic Diseases/NCID, Mailstop C09, Centers for Disease Control

and Prevention, Atlanta, GA 30333. Phone: (404) 639-1364. FAX:

(404) 639-3970. Electronic mail address: bdpl@ciddbdl.em.cdc.gov.

adopted, and without an assurance of parallelism, investigatorsare not able to calculate reliable estimates for serum antibodyconcentrations. Furthermore, single-point (dilution) serum

assays do not provide information related to parallelism andnonparallelism, and this, too, may lead to considerable error

when calculating antibody concentrations.The most common test for parallelism is an analysis of

variance (ANOVA) technique applied to sets of straight-linedata (2, 5). This technique presupposes that the standardreference and serum sample curves are linear and will workwith the log-log, logit-log, and any other linearizing transfor-mation. More complicated ANOVA procedures test for theequivalence of one or more parameters in nonlinear functions,such as the four-parameter logistic-log model (15). These testsmay be used to examine one parameter which simply describesthe similarity between slopes of one or more curves or severalparameters that, when taken together, describe the similarityof the overall shape of the nonlinear curves (7, 8, 15). Thesecomputations are complex, are not readily available in mostcomputer software packages dedicated to immunoassay dataanalysis, and are prone to error unless the person performingthe calculations knows precisely how to interpret the results.These ANOVA techniques are also overly sensitive to negligi-ble departures from parallelism when the precision of the givenmodels is high. In an experiment, if there is little differencebetween the observed outcome measure and that predicted bythe chosen model, the ANOVA techniques will detect even a

slight difference in slopes between the standard reference

serum and the serum sample curves as being statisticallydifferent. As assay methodology, technique, and precision

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2442 PLIKAYTIS ET AL.

improve, this will be reflected in a descriptive standard refer-ence curve with much less variability. In this situation, standardANOVA techniques will more frequently detect nonparallel-ism in near parallel lines (20).These weaknesses in the ANOVA methodology have led

many researchers to discard it as a reliable measure forascertaining departures from parallelism. Story et al. (20)adjusted the ANOVA calculations to reflect a more realisticinterpretation of parallelism in assays that display high degreesof precision. Other investigators have used alternate nonpara-metric techniques, including subjective visual evaluations, forthis purpose. These methods may be inadequate if they do notconsider the precision of the standards or serum sample assays.Still others have not addressed parallelism at all when report-ing results.

In this report we present a series of examples demonstratingthe issues involved with the evaluation of parallelism instandard ELISAs. We also present a set of guidelines thatcomprise a protocol for assessing parallelism between bioassaydilution curves that are applicable to ELISAs. These criteriashould be applicable, with minor modifications, to most immu-noassay experimental situations and, most importantly, are notdependent on the mathematical model used to form thestandard reference curve. These guidelines have evolved in ourlaboratories over the past 4 years during the performance ofthousands of ELISAs for antibodies to the capsular polysac-charides of Neisseria meningitidis groups A and C and Hae-mophilus influenzae type b.

MATERIALS AND METHODS

ELISA. The ELISA for N. meningitidis group A and Cantibody measurement was performed as described by Arakereand Frasch (1) and as modified by Carlone and colleagues (3,6). The ELISA for H. influenzae type b antibody measurementwas done by the methods of Phipps et al. (11).

Statistical methods. Standard reference and serially dilutedserum sample curves were described by using two of the mostfrequently used models.

(i) Logistic-log model. The four-parameter logistic-log func-tion (4, 15-17) describes, with a high degree of accuracy,standard reference serum and serum sample curves that dis-play a pronounced sigmoidal shape when plotted on an opticaldensity (OD)-versus-log-dilution scale. The logistic functionadopts the same general shape and is a reasonable relationshipfor use in this situation. It is defined as:

a -dOD =d + b(1)

tdilution\(11 +

c

Parameters a and d represent the upper and lower asymptotes,respectively, of the curve and correspond to the theoretical ODof the assay at infinite and zero concentrations, respectively. cis the dilution associated with the point of symmetry of thesigmoid and is located at the midpoint of the assay found at theinflection point of the curve. b is a curvature parameter and isrelated to the slope of the curve.

(ii) Fully specified logit-log model. Rodbard and Hutt (16)and Finney (4) described an algebraically equivalent expres-sion for the logistic function that effectively linearizes therelationship when plotted on a logit OD-log dilution scale, i.e.,logit (OD)fS = a + b log (dilution), where a and b are the yintercept and slope, respectively, for the line, and logit (OD)fsis the fully specified (fs) logit of the OD.

In this expression,

(GO-ODminLogit (OD)fs = log ODmax-OD (2)

ODmin and ODmax are unknown quantities and theoreticallycorrespond to the lower and upper asymptotes, respectively, ofthe four-parameter logistic-log model. When the lower andupper asymptotes of the sigmoidal curve are used for thesequantities, the slope of the logit-log straight line, b, is equal tothe curvature parameter, b, in the four-parameter logistic-logfunction (14).

(iii) Partially specified logit-log model. A reduced form ofthe fully specified logit-log model may be derived by assumingthat the OD associated with a zero concentration should be 0.This would lead to the ODmi parameter of the fully specifiedlogit transformation being set equal to 0 (14). The resultingpartially specified (ps) logit transformation is defined as:

GODLogit (OD)ps = log GODma-OD (3)

(iv) Statistical analysis. The data in the present study weredisplayed on fully or partially specified logit-log scales by usinga relative dilution series to facilitate visual comparisons be-tween the standard reference serum and the serum samplecurves. Frequently, the initial dilution of a serum sample willdiffer from that of the standard reference serum sample.Dividing each dilution in a series by the maximum dilution ofthat series will always place the standard reference serumsample and the serum sample dilution sequences on the samerelative dilution scale:

Relative dilutioni =actual sample dilutioni

*100maximum dilution in series

(4)

Each regression line was fit by standard weighted regressiontechniques to account for the heteroscedasticity (uneven vari-ances) in the data by the SAS REG procedure (19). The lowerand upper asymptotes of the logistic-log curves describingthese data sets were used for the ODmin and ODma, parame-ters in the logit calculations and were estimated by using theELISA software package developed at the Centers for DiseaseControl and Prevention (13). For comparison purposes stan-dard ANOVAs for parallelism between straight-line curveswere made by using the SAS GLM procedure (one-wayanalysis of covariance) (19). All computations were made byusing a Compaq Deskpro 386s/20 personal computer equippedwith a math coprocessor (Compaq Computer Corporation,Houston, Tex.).

RESULTS

Figure 1 displays the results of two typical ELISAs for H.influenzae type b with the standard reference serum sample(FDA 1983) and two serially diluted serum samples. Thesedata are used to illustrate the calculations necessary to inter-polate antibody concentrations for a serially diluted serumsample when both the standard reference serum (standard)and the serum sample bioassay curves are parallel (Fig. 1A)and when both curves are clearly not parallel (Fig. 1B). Thedatum points associated with the three lowest ODs of theserum samples in Fig. 1A had undefined fully specified logitsbecause their ODs fell below their respective ODmin parame-ters (i.e., the lower asymptotes of their logistic-log curves).Although the logit-log transformation was used for this de-

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PARALLELISM AND NONPARALLELISM IN BIOASSAYS 2443

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Standard

1.56 6.25 25 100

Log(Relative dilution)

FIG. 1. Antibody concentration calculations by using a standard curve prepared from a serial dilution ofH. influenzae type b standard reference

serum (FDA 1983) (0) and two serially diluted serum samples ([) plotted on a fully specified logit-log scale. (A) The parallel case as determinedby ANOVA testing; the slopes for the standard reference and serum sample curves are 1.370 and 1.366, respectively. (B) The nonparallel case.

The slopes for the standard reference and serum sample curves are 1.361 and 0.677, respectively. Serum antibody concentrations for points 1 and

2 are 2.00 and 9.25 ,ug/ml, respectively (see text for explanation).

scription, this technique works for a standard reference curve

described with any functional form selected by the investigator.For the purposes of this example, assume that the initialconcentration of the standard reference serum (the calibrationfactor) is 70 jig/ml and that the standard was diluted 1:1,200with seven subsequent twofold serial dilutions. The serum

sample was initially diluted 1:50 and was also composed of a

series of twofold dilutions. The following formula was used tocalculate antibody concentrations:

concentration =

(interpolated dilution from standard curve

actual sample .dilution . calibration factor (5)actual sample dilution

The calculated antibody concentration for the indicatedpoint from the serum sample series in Fig. 1A would be(0.00001674/0.00125) * 70 = 0.94 ,ug/ml. Referring to equation4, 0.00001674 = (2.01/100) - (1/1200) and 0.00125 = (6.25/100) - (1/50). Repeating these calculations for the remainingserum sample points in Fig. 1A yields a mean antibodyconcentration of 0.78 ,ug/ml, with a standard deviation of 0.08,ug/ml and a coefficient of variation (CV; CV = [standarddeviation/mean] x 100) of 10.30%. In this case, with the serumsample curve parallel to the standard reference curve, anypoint lying close to the serum sample line yields similarestimates for serum antibody concentrations. The low standarddeviation associated with the mean as quantified by the smallCV underscores this point.The data in Fig. 1B are used to illustrate these principles

when the serum sample curve is clearly not parallel to thestandard reference curve. Point estimates for two separateobservations may be compared to demonstrate the uncertaintyof calculating antibody concentrations in this situation. Theantibody concentrations associated with points 1 and 2 in Fig.1B were 2.00 and 9.25 ,ug/ml, respectively. The antibodyconcentration for point 2 was more than 4.5 times greater thanthat for point 1. For these data, the individual antibodyconcentrations ranged from 1.37 to 25.71 ,ug/ml; the mean

antibody concentration was 6.46 p.g/ml, with a standard devi-ation of 6.18 ,ug/ml. The slope of the serum sample curve wasless than that of the standard reference curve, and the esti-mated antibody concentrations steadily increased from the first

point associated with the greatest OD to the last pointassociated with the lowest OD. If the slope of the serum

sample curve were greater than that of the standard referencecurve, the estimated antibody concentrations would steadilydecrease from the first to the last point. These are positiveindications for the lack of parallelism between standard refer-ence serum and serum sample curves. Additional indicationsfor nonparallelism are reflected in the large degree of variabil-ity about the mean antibody concentration and the inflated CV(95.66%). If one were to eliminate the points associated withthe two most extreme dilutions because of their excessivevariability, these statistics would still provide strong evidencefor nonparallelism; the individual antibody concentrationswould then range from 1.37 to 9.25 ,ug/ml. The revised mean

would be 3.96 ,ug/ml, with a standard deviation of 2.45 ,ug/mland a CV of 61.78%.

Figure 2 displays the results of a typical ELISA for H.influenzae type b with a common standard reference serum

sample (FDA 1983) and two serially diluted serum samplesand is used to illustrate the difficulties in applying ANOVAtechniques to screen for nonparallelism of assay slopes. Thedatum points associated with the three lowest ODs of bothserum samples had undefined fully specified logits as their ODsfell below their respective ODmin parameters (i.e., the lowerasymptotes of their logistic-log curves). The slopes for thecommon standard reference and serum sample curves 1 and 2were 1.370, 1.366, and 1.263, respectively. The differences inthe slopes were 0.004 in Fig. 2A and 0.107 in Fig. 2B. StandardANOVAs revealed that the P value associated with measuringparallelism or the equivalence of slopes between the two linesin Fig. 2A was less than 0.75, indicating that the two slopes are

not significantly different. Figure 2B represents two lines witha slightly larger difference in slopes and yields a P value of<0.0005 for the test for parallelism. Although the difference inslopes for the lines in Fig. 2B appears negligible, the ANOVAprocedures would lead an investigator to conclude that the

slopes are significantly different.The incongruities with ANOVA testing are also present

when it is applied to nonlinear standard reference serum and

serum sample curves. These difficulties are compounded if the

investigator must consider facets of the standard reference

curve other than slope alone. Thus, in a logistic-log model,

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2444 PLIKAYTIS ET AL.

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Serum sample 1

B

Standard

OSerum sample 2

1.56 6.25 25

Log(Relative dilution)

100 1.56 6.25 25 100

Log(Relative dilution)

FIG. 2. Comparison of a common standard curve prepared from a serial dilution of H. influenzae type b standard reference serum (FDA 1983)(0) and two serially diluted serum samples (O) plotted on a fully specified logit-log scale. The slopes for the common standard reference and serumsample curves 1 and 2 are 1.370, 1.366, and 1.263, respectively. (A) The parallel case as determined by ANOVA testing; (B) the nonparallel case.

while the slope may be the only parameter of interest in someexperimental settings, the equality of one or more of theasymptotes, in addition to the slope, may be important in othersituations. In this case, the investigator must judge whether the"shapes" of the standard reference and serum sample curvesare the same. If the ANOVA proves to be too sensitive, visualinspection may be the only guide avaiable in these situations.Many investigators linearize sigmoid-shaped data to facili-

tate tests for parallelism using the log-log or logit-log transfor-mations. In these cases, the identities of the asymptotes arelost. Figure 3 displays a series of logistic-log curves (Fig. 3A),with their straight-line counterparts (Fig. 3B), computed byusing the fully specified logit-log transformation. In Fig. 3A,curves 1 through 4 were derived with equivalent curvature

A

0

CDa,

parameters (slopes) but different asymptotes; curve 5 wasconstructed with a slope less than those of the other fourcurves. The corresponding lines 1 through 4 in Fig. 3B haveidentical slopes. The only differences in slopes of the logit-loglines occurred when the slopes of the respective logistic-logfunctions differed, as shown by curve-line 5. The positions ofthe asymptotes of the logistic-log curves had no effect on thepresence or absence of parallelism between the logit-log lines.The logit-log transformation may be calculated in at least

two ways (fully and partially specified) by using a variety ofchoices for the ODmin and ODmax These decisions will alsoaffect the nature of parallelism and nonparallelism in thetransformed straight lines. Figure 4 displays the results of atypical ELISA for H. influenzae type b with a standard refer-

Log(Dilution)

B

0

0

Log (Dilution)FIG. 3. Comparison of logistic-log curves and their fully specified logit-log transformed counterparts. (A) Lines 1 and 2, logistic-log curves with

identical slopes and asymptotes; lines 3 and 4, logistic-log curves with identical slopes and different asymptotes; line 5, logistic-log curve withdifferent slope and asymptotes. (B) Corresponding straight lines formed by using the fully specified logit-log transformation.

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Log(Relative dilution)

FIG. 4. Comparison of a standard curve prepared from a serial dilution of H. influenzae type b standard reference serum (FDA 1983) (0) anda serially diluted serum sample ([l). (A) Lines plotted on a fully specified logit-log scale; the slopes for the standard reference and serum samplecurves are 1.370 and 1.366, respectively. (B) Lines plotted on a partially specified logit-log scale; the slopes for the standard reference and serum

sample curves are 1.366 and 0.906, respectively.

ence serum sample (FDA 1983) and a serially diluted serum

sample plotted on fully specified (Fig. 4A) and partiallyspecified (Fig. 4B) logit-log scales. The datum points associ-ated with the three lowest ODs of the serum sample hadundefined fully specified logits as their ODs fell below thelower asymptote of the logistic-log curve. These points did,however, have definable partially specified logits. For thisreason, these points are absent from Fig. 4A and present inFig. 4B. The degree of parallelism and nonparallelism isstrongly influenced by the method of linearizing transforma-tion. Standard ANOVA testing revealed that the slopes of thetwo lines in Fig. 4A were not significantly different (P < 0.75),while the slopes in Fig. 4B were significantly different (P <0.0005). An additional concern that arises when comparing thepartially and fully specified logit models is related to thedisparity in antibody concentrations derived from each trans-formation. The mean antibody concentration for the serumsample in Fig. 4A was 0.78 ,ug/ml, with a standard deviation of0.08 ,ug/ml and a CV of 10.30%. The serum sample in Fig. 4Bhad a mean antibody concentration of 1.72 ,ugIml, with astandard deviation of 1.32 ,ug/ml and a CV of 76.89%.

DISCUSSIONSeveral biologic factors may account for nonparallelism

between standard reference serum and serially diluted serumsample curves. As an example, the standard reference serummay have an antibody population with binding characteristicsdifferent from those of a typical serially diluted serum sample.This commonly occurs when infant serum antibody values areinterpolated from standard reference curves constructed fromvalues for serially diluted adult sera.

Investigators need to determine the requirements for judg-ing the parallelism and nonparallelism between bioassay dilu-tion curves as they pertain to their own research. If the overallshape of the curves is important, including the locations of thelower and upper asymptotes, linearization of the data tofacilitate comparisons between straight lines will be ineffectiveand may lead to erroneous conclusions. If the asymptotes are

unimportant and the investigator wishes to concentrate on theslopes of the curves alone, linearization of the data may be a

viable option. However, the nature of parallelism betweenstraight lines is influenced by the methods used to linearize thedata, as shown in Fig. 4. The logit-log transformation is one of

the most popular of the linearizing transformations, and it maybe calculated in several different ways, depending on thechoice of the model (equation 2 or 3) and the selection ofvalues used for ODmin and ODma,, Generally, the fully speci-fied logit-log transformation will retain the most accuraterepresentation of differences between slopes when maximumlikelihood estimates for d and a from the logistic-log model areused for ODmin and ODmax, respectively, as in equation 2 (14).In actual practice, these estimates will typically be unknownsince it is unlikely that one would estimate standard referenceand serum sample curves by more than one technique. Thiswould force the investigator to substitute less-than-ideal valuesfor these parameters, unduly affecting the slopes of the curvesand possibly leading to incorrect conclusions regarding paral-lelism and nonparallelism between the straight lines.

Large discrepancies may exist between the mean antibodyconcentrations derived from the partially and fully specifiedlogit transformations. Jeffcoate and Das (9) and Pegg andMiner (10) have shown that differences in data processingtechniques account for a significant portion of between-assayvariability. Pegg and Miner (10) also demonstrated that differ-ent implementations of the same calibration formulas (thelogit-log technique) gave significantly different results. Toillustrate this point, the mean antibody concentration for theserum sample in Fig. 4B (partially specified logit-log, 1.72,ug/ml) is more than twice that for the serum sample in Fig. 4A(fully specified logit log, 0.78 ,ug/ml). If an investigator used thepartially specified logit-log transformation and was unfamiliarwith the paralellism and nonparallelism issues discussed here,the reported result could be in serious error. This arbitrarinessof the logit-log transformation argues for keeping the data inits untransformed state when comparing the slopes of bioassaycurves and computing antibody concentrations. It also suggeststhat greater attention should be paid to the standardization ofbioassay data processing and analysis software (18). This isespecially true for large studies in which several collaboratinglaboratories analyze the same collection of specimens.

Large standard deviations and CVs for mean antibodyconcentration calculations of serum samples may be attribut-able to the inadequacy of the mathematical model chosen todescribe the dilution curve (14), to excessive variation in ODsrelated to the serum sample, or to the presence of nonparal-lelism between the serum sample and the standard reference

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TABLE 1. Summary of procedure for evaluating parallelism and nonparallelism between assays and reporting calculated concentrations

Within assay Steadily increasing or decreasing estimates of calculated concnCV (%) No Yes

<20 Lines parallel; calculate means, SDs, and CVs Lines may be nonparallel but within acceptable lim-its; calculate means, SDs, and CVs

>20 Unacceptable experimental error; repeat assay; may Lines nonparallel; calculate median values from avail-also occur if using inadequate mathematical model able concentrations and report probable nonparal-to describe standard curve lel results

a CV = (standard deviation/mean) - 100.0.

curve, as displayed in Fig. 1B and 4B. In this regard, one shouldrefrain from blindly accepting the P values generated from

various software packages when testing for parallelism be-tween standard reference serum and serum sample curves. It is

crucial that investigators examine the graphic outputs of theirdilution curves with the standard reference serum and serum

sample curves overlaid onto the datum points composing theassays. Often, a single outlying observation may adverselyaffect the slope of a curve, especially if a linearizing transfor-mation is performed. This is most easily detected when viewinggraphic outputs of the data and the resultant fitted curves.

Various robust and weighted fitting techniques may be used to

reduce the influence of these outlying observations on theoverall fit of the curves to the observed data (12).

In view of the many problems associated with evaluatingbioassay parallelism, we propose a series of guidelines that maybe used to assess parallelism between bioassay dilution curves.

These guidelines were developed over the past 4 years duringthe performance of thousands of ELISAs with N. meningitidisgroups A and C and H. influenzae type b.

Guideline 1. Perform the assay by standardized protocols. (i)Monitor for quality control. (ii) Use multipoint assays forstandard, quality control, and serum samples. (iii) Use repli-cate wells or tubes for each dilution (standard reference curve,

quality control, and serum samples).Guideline 2. Construct an assay standard reference curve by

using the desired mathematical model.Guideline 3. Check for the uniformity of the assay standard

reference curve with previous standard reference curves.

Guideline 4. If a parametric model was selected for thestandard reference curve, check for the adequacy of the fit(ensure that the squared correlation coefficient [ri is greaterthan 0.95) (12).

Guideline 5. Discard serum sample ODs that fall below and

above the lower and upper limits, respectively, of detectabilityfor the particular assay.

Guideline 6. Check that each within-dilution CV for the

serially diluted serum sample is .15%. If all or the majority of

the within-dilution CVs are >15%, reassay the serum sample.Guideline 7. Check the within-assay CV for the serially

diluted serum sample. Four situations are possible when three

or more of the dilution results for a serum sample passguideline 6 and are accepted for final analysis. (i) The calcu-

lated concentrations for the serum sample do not steadilyincrease or decrease. If the within-assay CV is <20%, conclude

that the lines are parallel; report means, standard deviations,and CVs. If the within-assay CV is >20%, there is an unac-

ceptable experimental error in the assay that may be caused bypoor laboratory procedures, excessive variability in OD read-

ings, or an inadequate mathematical model used to form the

standard reference curve. Repeat the assay or search for an

alternate mathematical function to model the standard curve.

(ii) The calculated concentrations for the serum sample do

steadily increase or decrease. If the within-assay CV is .20%,the lines may be nonparallel but within acceptable limits;report means, standard deviations, and CVs. If the within-assayCV is >20%, conclude that the lines are nonparallel. Reportthe median value of the available calculated concentrationsand the fact that it is derived from nonparallel assays.

Guideline 6 addresses the within-dilution CVs and suggeststhat 15% be used as an acceptance threshold. In our labora-tories, if the within-dilution CV exceeds 15%, the bioassayresults are examined to detect any abnormal datum points.One cause for inflated CVs is the presence of outlying obser-vations that may be attributed to data entry or processingerrors and/or excessive variability in the assay. Additionally, ifthe standard reference serum curve is sigmoid in shape withtrue lower and upper asymptotes, any serum sample with ODsnear the asymptotes will generally result in calculated concen-trations with extreme variability, leading to high CVs. Theseasymptotes correspond to the limits of detectability for theassay, and each investigator must examine the points nearthese limits and decide whether they need to be eliminated orincluded in the overall analysis. We generally do not includepoints near the assay's limits of detectability when the calcu-lated concentrations display excessive variability.

Guideline 7 is summarized in Table 1.With minor modifications, these guidelines should be appli-

cable in most immunoassay experimental situations. In ourlaboratories, we find the logistic-log model describes our datawith the greatest accuracy over the widest dilution range (14)when computing antibody concentrations for serum samples.However, our proposed guidelines are not dependent on theunderlying functional form used to model the standard refer-ence curve, and as such, they may be applied more generally toa wider body of experimental situations.These guidelines serve as a useful alternative to standard

ANOVA techniques when assessing the parallelism and non-

parallelism between bioassay dilution curves. When applied to

the data displayed in Fig. 2, the mean calculated H. influenzaeantibody concentration for serum sample 1 (Fig. 2A) was 0.78p,g/ml, with a standard deviation of 0.02 ,ug/ml and a CV of2.28%; the mean antibody concentration for serum sample 2(Fig. 2B) was 0.33 pLg/ml, with a standard deviation of 0.05,ug/ml and a CV of 13.72%. While the ANOVA proceduresindicated that the slopes for the two lines in Fig. 2B were

different with a high degree of statistical significance (P <

0.0005), most investigators would view the two lines as beingsufficiently parallel to permit valid estimates for the mean

antibody concentration. Application of the guidelines set forthhere results in a mean antibody concentration and associatedmeasures of variability that allow a more reasonable assess-

ment of parallelism and nonparallelism between the two lines.The moderate CV of 13.72% would indicate that the slopes ofthe two lines are close enough to provide a credible estimate

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PARALLELISM AND NONPARALLELISM IN BIOASSAYS 2447

for the mean antibody concentration, confirming the visualevaluation of parallelism.These issues underscore the importance of performing

multipoint assays for serum samples. Single-point (dilution)serum assays may be inaccurate if the assay point does not fallwithin the most accurate portion of the standard referencecurve. If the serum assay point falls outside the range of thestandards, the assay must be repeated. If the serum assay pointfalls near the limits of detectability for the standard referenceassay, the calculated concentration for the serum sample willbe associated with an inflated standard error. This may lead aninvestigator to question the reliability of the calculated con-centration and, possibly, to repeat the assay. Also, single-pointassays for serum samples may lead to serious miscalculations ofconcentrations in serum when certain linearizing transforma-tions (log-log) are used for the standard reference serum curve(14). Finally, single-point assays do not give an investigator thenecessary information to determine whether an assay is paral-lel to the standard reference serum assay, casting doubt on thevalidity of the calculated concentration in the serum sample.We believe that the issues outlined here will assist investi-

gators in forming an objective approach for detecting parallel-ism and nonparallelism between standard reference serum andserum specimen bioassays. The adoption of these guidelines inprotocols that include bioassay analyses can standardize thecalculation of serum antibody concentrations, limiting thevariability and increasing the reliability of these measurements.This will facilitate the comparison of results within and amonglaboratories, especially in large studies in which several collab-orating laboratories analyze the same collection of specimens.

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