Determination of Parallelism and Nonparallelism in

Transcription

1 JOURNAL OF CLINICAL MICROIOLOGY, OCt. 1994, p /94/$4.+ Copyright 1994, Amerian Soiety for Mirobiology Vol. 32, No. 1 Determination of Parallelism and Nonparallelism in ioassay Dilution Curves RIAN D. PLIKAYTIS,l* PATRICIA F. HOLDER,2 LORNA. PATS,2 SUSAN E. MASLANKA,2 LINDA L. GHEESLING,2 AND GEORGE M. CARLONE2 iostatistis and Information Management ranh1 and Childhood and Vaine-Preventable Diseases Immunology Setion,2 Childhood and Respiratory Diseases ranh, Division of aterial and Myoti Diseases, National Center for Infetious Diseases, Centers for Disease Control and Prevention, Atlanta, Georgia 3333 Reeived 15 April 1994/Returned for modifiation 13 June 1994/Aepted 28 June 1994 There is a lak of onsensus among investigators who use a variety of immunoassay tehniques (e.g., enzyme-linked immunosorbent assay [ELISA] and radioimmunoassay) regarding the protools for desribing and forming standard referene or alibration urves and interpolating serum antibody onentrations. This onfounds the issue of deteting the presene or absene of parallelism between standard referene serum and serially diluted serum sample urves. These urves must be parallel to support the assumption that the antibody-binding harateristis are similar enough to allow the determination of antibody levels in the diluted serum sample. There is no universal and widely adopted strategy for assessing parallelism in bioassays, and without an assurane of parallelism, investigators are not able to alulate reliable estimates for antibody onentrations in serum samples. Furthermore, single-point (dilution) serum assays do not provide information related to parallelism and nonparallelism, and this, too, may lead to onsiderable error when alulating antibody onentrations. When assay methodology, tehnique, and preision improve to the extent that standard referene serum and serially diluted serum sample urves are fit with little error, standard analysis of variane tehniques are overly sensitive to negligible departures from parallelism. We present a series of guidelines that ompose a protool for assessing parallelism between bioassay dilution urves that are appliable to data derived from ELISAs. These riteria should be appliable, with minor modifiations, to most immunoassay experimental situations and, most importantly, are not dependent on the mathematial model used to form the standard referene urve. These guidelines have evolved in our laboratories over the past 4 years during the performane of thousands of ELISAs for antibodies to the apsular polysaharides of Neisseria meningitidis groups A and C and Haemophilus influenzae type b. There is a lak of onsensus among investigators who use a variety of immunoassay tehniques (e.g., enzyme-linked immunosorbent assay [ELISA] and radioimmunoassay) regarding the protools for desribing and forming standard referene or alibration urves and interpolating serum antibody onentrations. A wide variety of urve-fitting methods are used to generate standard referene or alibration urves. Rodgers (18) listed many of these methods in a report that inluded an extensive bibliography. The most ommon funtional models for alibration urves range from the simple linear log-log and logit-log transformations to the more omplex four-parameter nonlinear logisti-log models (14). These models also inlude a range of spline funtions (7, 8). The proliferation of existing methods for standard referene urve formation learly indiates that no one desriptive model will work optimally for all experimental situations and assays. Eah model possesses advantages and disadvantages and must be judged on its appliability to the experiment at hand. These onerns onfound the issue of deteting the presene or absene of parallelism between standard referene serum and serially diluted serum sample urves. These urves must be parallel to support the assumption that the antibody-binding harateristis are similar enough to allow the determination of antibody levels in the diluted serum sample. No universal strategy for assessing parallelism in bioassays has been widely * Corresponding author. Mailing address: Division of aterial and Myoti Diseases/NCID, Mailstop C9, Centers for Disease Control and Prevention, Atlanta, GA Phone: (44) FAX: (44) Eletroni mail address: bdpl@iddbdl.em.d.gov. adopted, and without an assurane of parallelism, investigators are not able to alulate reliable estimates for serum antibody onentrations. Furthermore, single-point (dilution) serum assays do not provide information related to parallelism and nonparallelism, and this, too, may lead to onsiderable error when alulating antibody onentrations. The most ommon test for parallelism is an analysis of variane (ANOVA) tehnique applied to sets of straight-line data (2, 5). This tehnique presupposes that the standard referene and serum sample urves are linear and will work with the log-log, logit-log, and any other linearizing transformation. More ompliated ANOVA proedures test for the equivalene of one or more parameters in nonlinear funtions, suh as the four-parameter logisti-log model (15). These tests may be used to examine one parameter whih simply desribes the similarity between slopes of one or more urves or several parameters that, when taken together, desribe the similarity of the overall shape of the nonlinear urves (7, 8, 15). These omputations are omplex, are not readily available in most omputer software pakages dediated to immunoassay data analysis, and are prone to error unless the person performing the alulations knows preisely how to interpret the results. These ANOVA tehniques are also overly sensitive to negligible departures from parallelism when the preision of the given models is high. In an experiment, if there is little differene between the observed outome measure and that predited by the hosen model, the ANOVA tehniques will detet even a slight differene in slopes between the standard referene serum and the serum sample urves as being statistially different. As assay methodology, tehnique, and preision 2441 Downloaded from on Otober 31, 218 by guest

2 2442 PLIKAYTIS ET AL. improve, this will be refleted in a desriptive standard referene urve with muh less variability. In this situation, standard ANOVA tehniques will more frequently detet nonparallelism in near parallel lines (2). These weaknesses in the ANOVA methodology have led many researhers to disard it as a reliable measure for asertaining departures from parallelism. Story et al. (2) adjusted the ANOVA alulations to reflet a more realisti interpretation of parallelism in assays that display high degrees of preision. Other investigators have used alternate nonparametri tehniques, inluding subjetive visual evaluations, for this purpose. These methods may be inadequate if they do not onsider the preision of the standards or serum sample assays. Still others have not addressed parallelism at all when reporting results. In this report we present a series of examples demonstrating the issues involved with the evaluation of parallelism in standard ELISAs. We also present a set of guidelines that omprise a protool for assessing parallelism between bioassay dilution urves that are appliable to ELISAs. These riteria should be appliable, with minor modifiations, to most immunoassay experimental situations and, most importantly, are not dependent on the mathematial model used to form the standard referene urve. These guidelines have evolved in our laboratories over the past 4 years during the performane of thousands of ELISAs for antibodies to the apsular polysaharides of Neisseria meningitidis groups A and C and Haemophilus influenzae type b. MATERIALS AND METHODS ELISA. The ELISA for N. meningitidis group A and C antibody measurement was performed as desribed by Arakere and Frash (1) and as modified by Carlone and olleagues (3, 6). The ELISA for H. influenzae type b antibody measurement was done by the methods of Phipps et al. (11). Statistial methods. referene and serially diluted serum sample urves were desribed by using two of the most frequently used models. (i) Logisti-log model. The four-parameter logisti-log funtion (4, 15-17) desribes, with a high degree of auray, standard referene serum and serum sample urves that display a pronouned sigmoidal shape when plotted on an optial density (OD)-versus-log-dilution sale. The logisti funtion adopts the same general shape and is a reasonable relationship for use in this situation. It is defined as: a -d OD =d + b(1) tdilution\(1 1 + Parameters a and d represent the upper and lower asymptotes, respetively, of the urve and orrespond to the theoretial OD of the assay at infinite and zero onentrations, respetively. is the dilution assoiated with the point of symmetry of the sigmoid and is loated at the midpoint of the assay found at the infletion point of the urve. b is a urvature parameter and is related to the slope of the urve. (ii) Fully speified logit-log model. Rodbard and Hutt (16) and Finney (4) desribed an algebraially equivalent expression for the logisti funtion that effetively linearizes the relationship when plotted on a logit OD-log dilution sale, i.e., logit (OD)fS = a + b log (dilution), where a and b are the y interept and slope, respetively, for the line, and logit (OD)fs is the fully speified (fs) logit of the OD. In this expression, (GO- ODmin Logit (OD)fs = log ODmax-OD (2) ODmin and ODmax are unknown quantities and theoretially orrespond to the lower and upper asymptotes, respetively, of the four-parameter logisti-log model. When the lower and upper asymptotes of the sigmoidal urve are used for these quantities, the slope of the logit-log straight line, b, is equal to the urvature parameter, b, in the four-parameter logisti-log funtion (14). (iii) Partially speified logit-log model. A redued form of the fully speified logit-log model may be derived by assuming that the OD assoiated with a zero onentration should be. This would lead to the ODmi parameter of the fully speified logit transformation being set equal to (14). The resulting partially speified (ps) logit transformation is defined as: GOD Logit (OD)ps = log GODma-OD (3) (iv) Statistial analysis. The data in the present study were displayed on fully or partially speified logit-log sales by using a relative dilution series to failitate visual omparisons between the standard referene serum and the serum sample urves. Frequently, the initial dilution of a serum sample will differ from that of the standard referene serum sample. Dividing eah dilution in a series by the maximum dilution of that series will always plae the standard referene serum sample and the serum sample dilution sequenes on the same relative dilution sale: Relative dilutioni = atual sample dilutioni *1 maximum dilution in (4) series Eah regression line was fit by standard weighted regression tehniques to aount for the heterosedastiity (uneven varianes) in the data by the SAS REG proedure (19). The lower and upper asymptotes of the logisti-log urves desribing these data sets were used for the ODmin and ODma, parameters in the logit alulations and were estimated by using the ELISA software pakage developed at the Centers for Disease Control and Prevention (13). For omparison purposes standard ANOVAs for parallelism between straight-line urves were made by using the SAS GLM proedure (one-way analysis of ovariane) (19). All omputations were made by using a Compaq Deskpro 386s/2 personal omputer equipped with a math oproessor (Compaq Computer Corporation, Houston, Tex.). RESULTS J. CLIN. MICROIOL. Figure 1 displays the results of two typial ELISAs for H. influenzae type b with the standard referene serum sample (FDA 1983) and two serially diluted serum samples. These data are used to illustrate the alulations neessary to interpolate antibody onentrations for a serially diluted serum sample when both the standard referene serum (standard) and the serum sample bioassay urves are parallel (Fig. 1A) and when both urves are learly not parallel (Fig. 1). The datum points assoiated with the three lowest ODs of the serum samples in Fig. 1A had undefined fully speified logits beause their ODs fell below their respetive ODmin parameters (i.e., the lower asymptotes of their logisti-log urves). Although the logit-log transformation was used for this de- Downloaded from on Otober 31, 218 by guest

3 VOL. 32, 1994 PARALLELISM AND NONPARALLELISM IN IOASSAYS 2443 o -1 CO) - -2.) n IL) -3-4 Standar1 4-JQ 3. -i FIG. 1. Antibody onentration alulations by using a standard urve prepared from a serial dilution of H. influenzae type b standard referene serum (FDA 1983) () and two serially diluted serum samples ([) plotted on a fully speified logit-log sale. (A) The parallel ase as determined by ANOVA testing; the slopes for the standard referene and serum sample urves are 1.37 and 1.366, respetively. () The nonparallel ase. The slopes for the standard referene and serum sample urves are and.677, respetively. Serum antibody onentrations for points 1 and 2 are 2. and 9.25,ug/ml, respetively (see text for explanation). sription, this tehnique works for a standard referene urve desribed with any funtional form seleted by the investigator. For the purposes of this example, assume that the initial onentration of the standard referene serum (the alibration fator) is 7 jig/ml and that the standard was diluted 1:1,2 with seven subsequent twofold serial dilutions. The serum sample was initially diluted 1:5 and was also omposed of a series of twofold dilutions. The following formula was used to alulate antibody onentrations: onentration = (interpolated dilution from standard urve atual.dilution sample. alibration fator (5) The alulated antibody onentration for the indiated point from the serum sample series in Fig. 1A would be (.1674/.125) * 7 =.94,ug/ml. Referring to equation - 4,.1674 = (2.1/1) (1/12) and.125 = (6.25/ - 1) (1/5). Repeating these alulations for the remaining serum sample points in Fig. 1A yields a mean antibody onentration of.78,ug/ml, with a standard deviation of.8,ug/ml and a oeffiient of variation (CV; CV = [standard deviation/mean] x 1) of 1.3%. In this ase, with the serum sample urve parallel to the standard referene urve, any point lying lose to the serum sample line yields similar estimates for serum antibody onentrations. The low standard deviation assoiated with the mean as quantified by the small CV undersores this point. The data in Fig. 1 are used to illustrate these priniples when the serum sample urve is learly not parallel to the standard referene urve. Point estimates for two separate observations may be ompared to demonstrate the unertainty of alulating antibody onentrations in this situation. The antibody onentrations assoiated with points 1 and 2 in Fig. 1 were 2. and 9.25,ug/ml, respetively. The antibody onentration for point 2 was more than 4.5 times greater than that for point 1. For these data, the individual antibody onentrations ranged from 1.37 to 25.71,ug/ml; the mean antibody onentration was 6.46 p.g/ml, with a standard deviation of 6.18,ug/ml. The slope of the serum sample urve was less than that of the standard referene urve, and the estimated antibody onentrations steadily inreased from the first point assoiated with the greatest OD to the last point assoiated with the lowest OD. If the slope of the serum sample urve were greater than that of the standard referene urve, the estimated antibody onentrations would steadily derease from the first to the last point. These are positive indiations for the lak of parallelism between standard referene serum and serum sample urves. Additional indiations for nonparallelism are refleted in the large degree of variability about the mean antibody onentration and the inflated CV (95.66%). If one were to eliminate the points assoiated with the two most extreme dilutions beause of their exessive variability, these statistis would still provide strong evidene for nonparallelism; the individual antibody onentrations would 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/ml and a CV of 61.78%. Figure 2 displays the results of a typial ELISA for H. influenzae type b with a ommon standard referene serum sample (FDA 1983) and two serially diluted serum samples and is used to illustrate the diffiulties in applying ANOVA tehniques to sreen for nonparallelism of assay slopes. The datum points assoiated with the three lowest ODs of both serum samples had undefined fully speified logits as their ODs fell below their respetive ODmin parameters (i.e., the lower asymptotes of their logisti-log urves). The slopes for the ommon standard referene and serum sample urves 1 and 2 were 1.37, 1.366, and 1.263, respetively. The differenes in the slopes were.4 in Fig. 2A and.17 in Fig. 2. ANOVAs revealed that the P value assoiated with measuring parallelism or the equivalene of slopes between the two lines in Fig. 2A was less than.75, indiating that the two slopes are not signifiantly different. Figure 2 represents two lines with a slightly larger differene in slopes and yields a P value of <.5 for the test for parallelism. Although the differene in slopes for the lines in Fig. 2 appears negligible, the ANOVA proedures would lead an investigator to onlude that the slopes are signifiantly different. The inongruities with ANOVA testing are also present when it is applied to nonlinear standard referene serum and serum sample urves. These diffiulties are ompounded if the investigator must onsider faets of the standard referene urve other than slope alone. Thus, in a logisti-log model, Downloaded from on Otober 31, 218 by guest

4 2444 PLIKAYTIS ET AL. J. CLIN. MICROIOL (A -2 - A * r -3.-._ 4) - o 4. m. -5-._ o g9 v Serum sample 1-3. a) -4- a~._ev C CE -7 -j -8 O Serum sample FIG. 2. Comparison of a ommon standard urve prepared from a serial dilution of H. influenzae type b standard referene serum (FDA 1983) () and two serially diluted serum samples (O) plotted on a fully speified logit-log sale. The slopes for the ommon standard referene and serum sample urves 1 and 2 are 1.37, 1.366, and 1.263, respetively. (A) The parallel ase as determined by ANOVA testing; () the nonparallel ase. while the slope may be the only parameter of interest in some experimental settings, the equality of one or more of the asymptotes, in addition to the slope, may be important in other situations. In this ase, the investigator must judge whether the "shapes" of the standard referene and serum sample urves are the same. If the ANOVA proves to be too sensitive, visual inspetion may be the only guide avaiable in these situations. Many investigators linearize sigmoid-shaped data to failitate tests for parallelism using the log-log or logit-log transformations. In these ases, the identities of the asymptotes are lost. Figure 3 displays a series of logisti-log urves (Fig. 3A), with their straight-line ounterparts (Fig. 3), omputed by using the fully speified logit-log transformation. In Fig. 3A, urves 1 through 4 were derived with equivalent urvature a, CD A Log(Dilution) parameters (slopes) but different asymptotes; urve 5 was onstruted with a slope less than those of the other four urves. The orresponding lines 1 through 4 in Fig. 3 have idential slopes. The only differenes in slopes of the logit-log lines ourred when the slopes of the respetive logisti-log funtions differed, as shown by urve-line 5. The positions of the asymptotes of the logisti-log urves had no effet on the presene or absene of parallelism between the logit-log lines. The logit-log transformation may be alulated in at least two ways (fully and partially speified) by using a variety of hoies for the ODmin and ODmax These deisions will also affet the nature of parallelism and nonparallelism in the transformed straight lines. Figure 4 displays the results of a typial ELISA for H. influenzae type b with a standard refer- Downloaded from on Otober 31, 218 by guest Log (Dilution) FIG. 3. Comparison of logisti-log urves and their fully speified logit-log transformed ounterparts. (A) Lines 1 and 2, logisti-log urves with idential slopes and asymptotes; lines 3 and 4, logisti-log urves with idential slopes and different asymptotes; line 5, logisti-log urve with different slope and asymptotes. () Corresponding straight lines formed by using the fully speified logit-log transformation.

5 VOL. 32, 1994 PARALLELISM AND NONPARALLELISM IN IOASSAYS A O-.b -2 - t. -3- * -4- <, -5- a-6 -., -7- -J -8 - O, Serum sample -2 C -3. (A -i _, -6 Serum sample -9 - I I l FIG. 4. Comparison of a standard urve prepared from a serial dilution of H. influenzae type b standard referene serum (FDA 1983) () and a serially diluted serum sample ([l). (A) Lines plotted on a fully speified logit-log sale; the slopes for the standard referene and serum sample urves are 1.37 and 1.366, respetively. () Lines plotted on a partially speified logit-log sale; the slopes for the standard referene and serum sample urves are and.96, respetively. ene serum sample (FDA 1983) and a serially diluted serum sample plotted on fully speified (Fig. 4A) and partially speified (Fig. 4) logit-log sales. The datum points assoiated with the three lowest ODs of the serum sample had undefined fully speified logits as their ODs fell below the lower asymptote of the logisti-log urve. These points did, however, have definable partially speified logits. For this reason, these points are absent from Fig. 4A and present in Fig. 4. The degree of parallelism and nonparallelism is strongly influened by the method of linearizing transformation. ANOVA testing revealed that the slopes of the two lines in Fig. 4A were not signifiantly different (P <.75), while the slopes in Fig. 4 were signifiantly different (P <.5). An additional onern that arises when omparing the partially and fully speified logit models is related to the disparity in antibody onentrations derived from eah transformation. The mean antibody onentration for the serum sample in Fig. 4A was.78,ug/ml, with a standard deviation of.8,ug/ml and a CV of 1.3%. The serum sample in Fig. 4 had a mean antibody onentration of 1.72,ugIml, with a standard deviation of 1.32,ug/ml and a CV of 76.89%. DISCUSSION Several biologi fators may aount for nonparallelism between standard referene serum and serially diluted serum sample urves. As an example, the standard referene serum may have an antibody population with binding harateristis different from those of a typial serially diluted serum sample. This ommonly ours when infant serum antibody values are interpolated from standard referene urves onstruted from values for serially diluted adult sera. Investigators need to determine the requirements for judging the parallelism and nonparallelism between bioassay dilution urves as they pertain to their own researh. If the overall shape of the urves is important, inluding the loations of the lower and upper asymptotes, linearization of the data to failitate omparisons between straight lines will be ineffetive and may lead to erroneous onlusions. If the asymptotes are unimportant and the investigator wishes to onentrate on the slopes of the urves alone, linearization of the data may be a viable option. However, the nature of parallelism between straight lines is influened by the methods used to linearize the data, as shown in Fig. 4. The logit-log transformation is one of the most popular of the linearizing transformations, and it may be alulated in several different ways, depending on the hoie of the model (equation 2 or 3) and the seletion of values used for ODmin and ODma,, Generally, the fully speified logit-log transformation will retain the most aurate representation of differenes between slopes when maximum likelihood estimates for d and a from the logisti-log model are used for ODmin and ODmax, respetively, as in equation 2 (14). In atual pratie, these estimates will typially be unknown sine it is unlikely that one would estimate standard referene and serum sample urves by more than one tehnique. This would fore the investigator to substitute less-than-ideal values for these parameters, unduly affeting the slopes of the urves and possibly leading to inorret onlusions regarding parallelism and nonparallelism between the straight lines. Large disrepanies may exist between the mean antibody onentrations derived from the partially and fully speified logit transformations. Jeffoate and Das (9) and Pegg and Miner (1) have shown that differenes in data proessing tehniques aount for a signifiant portion of between-assay variability. Pegg and Miner (1) also demonstrated that different implementations of the same alibration formulas (the logit-log tehnique) gave signifiantly different results. To illustrate this point, the mean antibody onentration for the serum sample in Fig. 4 (partially speified logit-log, 1.72,ug/ml) is more than twie that for the serum sample in Fig. 4A (fully speified logit log,.78,ug/ml). If an investigator used the partially speified logit-log transformation and was unfamiliar with the paralellism and nonparallelism issues disussed here, the reported result ould be in serious error. This arbitrariness of the logit-log transformation argues for keeping the data in its untransformed state when omparing the slopes of bioassay urves and omputing antibody onentrations. It also suggests that greater attention should be paid to the standardization of bioassay data proessing and analysis software (18). This is espeially true for large studies in whih several ollaborating laboratories analyze the same olletion of speimens. Large standard deviations and CVs for mean antibody onentration alulations of serum samples may be attributable to the inadequay of the mathematial model hosen to desribe the dilution urve (14), to exessive variation in ODs related to the serum sample, or to the presene of nonparallelism between the serum sample and the standard referene Downloaded from on Otober 31, 218 by guest

6 2446 PLIKAYTIS ET AL. J. CLIN. MICROIOL. TALE 1. Summary of proedure for evaluating parallelism and nonparallelism between assays and reporting alulated onentrations Within assay Steadily inreasing or dereasing estimates of alulated onn CV (%) No Yes <2 Lines parallel; alulate means, SDs, and CVs Lines may be nonparallel but within aeptable limits; alulate means, SDs, and CVs >2 Unaeptable experimental error; repeat assay; may Lines nonparallel; alulate median values from availalso our if using inadequate mathematial model able onentrations and report probable nonparalto desribe standard urve lel results a CV = - (standard deviation/mean) 1.. urve, as displayed in Fig. 1 and 4. In this regard, one should refrain from blindly aepting the P values generated from various software pakages when testing for parallelism between standard referene serum and serum sample urves. It is ruial that investigators examine the graphi outputs of their dilution urves with the standard referene serum and serum sample urves overlaid onto the datum points omposing the assays. Often, a single outlying observation may adversely affet the slope of a urve, espeially if a linearizing transformation is performed. This is most easily deteted when viewing graphi outputs of the data and the resultant fitted urves. Various robust and weighted fitting tehniques may be used to redue the influene of these outlying observations on the overall fit of the urves to the observed data (12). In view of the many problems assoiated with evaluating bioassay parallelism, we propose a series of guidelines that may be used to assess parallelism between bioassay dilution urves. These guidelines were developed over the past 4 years during the performane of thousands of ELISAs with N. meningitidis groups A and C and H. influenzae type b. Guideline 1. Perform the assay by standardized protools. (i) Monitor for quality ontrol. (ii) Use multipoint assays for standard, quality ontrol, and serum samples. (iii) Use repliate wells or tubes for eah dilution (standard referene urve, quality ontrol, and serum samples). Guideline 2. Construt an assay standard referene urve by using the desired mathematial model. Guideline 3. Chek for the uniformity of the assay standard referene urve with previous standard referene urves. Guideline 4. If a parametri model was seleted for the standard referene urve, hek for the adequay of the fit (ensure that the squared orrelation oeffiient [ri is greater than.95) (12). Guideline 5. Disard serum sample ODs that fall below and above the lower and upper limits, respetively, of detetability for the partiular assay. Guideline 6. Chek that eah 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. Chek 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 pass guideline 6 and are aepted for final analysis. (i) The alulated onentrations for the serum sample do not steadily inrease or derease. If the within-assay CV is <2%, onlude that the lines are parallel; report means, standard deviations, and CVs. If the within-assay CV is >2%, there is an unaeptable experimental error in the assay that may be aused by poor laboratory proedures, exessive variability in OD readings, or an inadequate mathematial model used to form the standard referene urve. Repeat the assay or searh for an alternate mathematial funtion to model the standard urve. (ii) The alulated onentrations for the serum sample do steadily inrease or derease. If the within-assay CV is.2%, the lines may be nonparallel but within aeptable limits; report means, standard deviations, and CVs. If the within-assay CV is >2%, onlude that the lines are nonparallel. Report the median value of the available alulated onentrations and the fat that it is derived from nonparallel assays. Guideline 6 addresses the within-dilution CVs and suggests that 15% be used as an aeptane threshold. In our laboratories, if the within-dilution CV exeeds 15%, the bioassay results are examined to detet any abnormal datum points. One ause for inflated CVs is the presene of outlying observations that may be attributed to data entry or proessing errors and/or exessive variability in the assay. Additionally, if the standard referene serum urve is sigmoid in shape with true lower and upper asymptotes, any serum sample with ODs near the asymptotes will generally result in alulated onentrations with extreme variability, leading to high CVs. These asymptotes orrespond to the limits of detetability for the assay, and eah investigator must examine the points near these limits and deide whether they need to be eliminated or inluded in the overall analysis. We generally do not inlude points near the assay's limits of detetability when the alulated onentrations display exessive variability. Guideline 7 is summarized in Table 1. With minor modifiations, these guidelines should be appliable in most immunoassay experimental situations. In our laboratories, we find the logisti-log model desribes our data with the greatest auray over the widest dilution range (14) when omputing antibody onentrations for serum samples. However, our proposed guidelines are not dependent on the underlying funtional form used to model the standard referene urve, and as suh, they may be applied more generally to a wider body of experimental situations. These guidelines serve as a useful alternative to standard ANOVA tehniques when assessing the parallelism and nonparallelism between bioassay dilution urves. When applied to the data displayed in Fig. 2, the mean alulated H. influenzae antibody onentration for serum sample 1 (Fig. 2A) was.78 p,g/ml, with a standard deviation of.2,ug/ml and a CV of 2.28%; the mean antibody onentration for serum sample 2 (Fig. 2) was.33 plg/ml, with a standard deviation of.5,ug/ml and a CV of 13.72%. While the ANOVA proedures indiated that the slopes for the two lines in Fig. 2 were different with a high degree of statistial signifiane (P <.5), most investigators would view the two lines as being suffiiently parallel to permit valid estimates for the mean antibody onentration. Appliation of the guidelines set forth here results in a mean antibody onentration and assoiated measures of variability that allow a more reasonable assessment of parallelism and nonparallelism between the two lines. The moderate CV of 13.72% would indiate that the slopes of the two lines are lose enough to provide a redible estimate Downloaded from on Otober 31, 218 by guest

7 VOL. 32, 1994 PARALLELISM AND NONPARALLELISM IN IOASSAYS 2447 for the mean antibody onentration, onfirming the visual evaluation of parallelism. These issues undersore the importane of performing multipoint assays for serum samples. Single-point (dilution) serum assays may be inaurate if the assay point does not fall within the most aurate portion of the standard referene urve. If the serum assay point falls outside the range of the standards, the assay must be repeated. If the serum assay point falls near the limits of detetability for the standard referene assay, the alulated onentration for the serum sample will be assoiated with an inflated standard error. This may lead an investigator to question the reliability of the alulated onentration and, possibly, to repeat the assay. Also, single-point assays for serum samples may lead to serious misalulations of onentrations in serum when ertain linearizing transformations (log-log) are used for the standard referene serum urve (14). Finally, single-point assays do not give an investigator the neessary information to determine whether an assay is parallel to the standard referene serum assay, asting doubt on the validity of the alulated onentration in the serum sample. We believe that the issues outlined here will assist investigators in forming an objetive approah for deteting parallelism and nonparallelism between standard referene serum and serum speimen bioassays. The adoption of these guidelines in protools that inlude bioassay analyses an standardize the alulation of serum antibody onentrations, limiting the variability and inreasing the reliability of these measurements. This will failitate the omparison of results within and among laboratories, espeially in large studies in whih several ollaborating laboratories analyze the same olletion of speimens. REFERENCES 1. Arakere, G., and C. E. Frash Speifiity of antibodies to -aetyl-positive and -aetyl-negative group C meningooal polysaharides in sera from vaines and arriers. Infet. Immun. 59: Armitage, P., and G. erry Statistial methods in medial researh, p lakwell Sientifi Publiations, Oxford. 3. Carlone, G. M., C. E. Frash, G. R. Siber, S. Quataert, L. L. Gheesling, S. H. Turner,. D. Plikaytis, L.. Helsel, W. E. DeWitt, W. F. ibb,. Swaminathan, G. Arakere, C. Thompson, D. Phipps, D. Madore, and C. V. roome Multienter omparison of levels of antibody to the Neisseria meningitidis group A apsular polysaharide measured by using an enzymelinked immunosorbent assay. J. Clin. Mirobiol. 3: Finney, D. J Radioligand assay. iometris 32: Finney, D. J Statistial method in biologial assay, p Charles Griffin & Company, London. 6. Gheesling, L. L., G. M. Carlone, L.. Pais, P. F. Holder, S. E. Maslanka,. D. Plikaytis, M. Ahtman, P. Densen, C. E. Frash, H. Kiiyhty, J. P. Mays, L. Nenioni, C. Peeters, D. C. Phipps, J. T. Poolman, E. 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