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Statistical package for analysis of competition ELISA results
Journal of Immunological Methods, 47 (1981) 375--385 Elsevier/North-Holland Biomedical Press
375
STATISTICAL PACKAGE FOR ANALYSIS OF COMPETITION ELISA RESULTS
PEDRO F. CANELLAS and ALEXANDER E. KARU Department of Biochemistry, University of California, Riverside, CA 92521, U.S.A. (Received 18 May 1981, accepted 17 July 1981)
A computation and analysis program has been assembled to facilitate the use of competition ELISA and similar assays for studies of antigen regulation and turnover. The program fits sigmoidal standard curves using a 4-parameter logistic function, determines amounts of antigen from the equation which defines the standard curve, and calculates specific activities by linear regression of the levels of antigen in varying amounts of total protein. An optional weighting function is provided to adjust for systematic non-uniform variance. Outlying points are identified during the linear regression, and the user may delete them or redefine the acceptable working range of the standard curve. The program provides a complete print-out of the data and optional plots of the fitted standard curve and the regression analysis of the samples, as well as statistics which are useful for quality control. It also provides the option of storing the data points from the standard curve on magnetic diskettes. The package is written in BASIC for a Wang Model 2200 computer equipped with a magnetic diskette drive, line printer, and flat-bed X--Y plotter, but it is readily adaptable to other systems and input/output devices.
INTRODUCTION C o m p e t i t i o n E L I S A m e t h o d s have been gaining increasing a c c e p t a n c e as an alternative t o r a d i o i m m u n e assay ( R I A ) . When p r o p e r l y o p t i m i z e d , b o t h m e t h o d s m a y have essentially identical range, sensitivity, a n d precision. The t w o t e c h n i q u e s used m o s t f r e q u e n t l y f o r c o m p e t i t i o n E L I S A involve: (a) c o m p e t i t i o n o f a small a m o u n t o f a n t i g e n - e n z y m e c o n j u g a t e with an u n k n o w n a m o u n t o f native antigen for b i n d i n g to a limiting n u m b e r o f antib o d y m o l e c u l e s w h i c h are a t t a c h e d t o t h e solid phase; o r (b) c o m p e t i t i o n b e t w e e n a k n o w n small a m o u n t o f a n t i g e n affixed t o t h e solid phase, and an u n k n o w n a m o u n t o f t h e native antigen, f o r b i n d i n g a limiting a m o u n t o f a n t i b o d y . In this m e t h o d t h e a m o u n t o f a n t i b o d y t h a t associates w i t h the i m m o b i l i z e d antigen is q u a n t i t a t e d b y a d d i t i o n o f an e n z y m e - s e c o n d antib o d y ( a n t i - i m m u n o g l o b u l i n ) c o n j u g a t e . Each o f these m e t h o d s has u n i q u e merits a n d d r a w b a c k s , b u t b o t h e x h i b i t t h e sigmoidal c o m p e t i t i o n r e s p o n s e as a f u n c t i o n o f l o g [ a n t i g e n ] w h i c h is characteristic o f t h e R I A and o t h e r i m m u n o l o g i c a l assays ( R o d b a r d and H u t t , 1 9 7 4 ) . R e c e n t l y we a d a p t e d t h e t y p e (b) c o m p e t i t i o n E L I S A described a b o v e 0022-1759/81/0000--0000/$02.75 © 1981 Elsevier/North-Holland Biomedical Press
376 to quantitate two recombination p r o t e i n s - the recA and recBC e n z y m e s in crude extracts of E. coli. To facilitate use of the competition ELISA in studies of induction, repression, and turnover of antigens, we have designed a computerized statistical processing package which is rapid, flexible, and provides antigen specific activity values and estimates of their significance. This paper presents details of the program and guidelines for its application. The most critical element in a routine for processing data from any immunological assay is the m e t h o d used to fit the standard curve. The relevant criteria include directness of computation, sufficient m o n o t o n i c i t y to generate a smooth function through a practical number of standards, flexibility adequate to accommodate changes in sensitivity and range of the response, and relevance of the fitting parameters to the reliability, sensitivity, and precision of the assay. Several algorithms provide approximations of the sigmoidal dose-response (Rodbard et al., 1969; Wellington, 1980). We have compared two of the most frequently used fitting methods -- linear regression using the classical 'logit' transformation, and non-linear least-squares approximation of the sigmoid using a 4-parameter exponential function with optional weighting. Although we found the two methods to be similar in accuracy and working range, fitting of the sigmoid appears preferable for competition ELISA analysis of limited numbers of samples that may differ widely in specific activity of the antigen. MATERIALS AND METHODS Samples for demonstration analysis Homogeneous E. coli recA protein and rabbit antiserum were the generous gift of Dr. Kevin McEntee, University of California, Los Angeles, and strains of E. coli K-12 were provided by Dr. Alvin J. Clark, University of California, Berkeley. Growth of the cells, conditions for induction, and preparation of extracts will be described in detail elsewhere (Karu and Belk, 1981, in preparation). Competition ELISA The composition ELISA assay was performed in polystyrene EIA cuvettes (Gilford Instruments, Palo Alto, CA), essentially as described by Voller et al. (1976). Gilford EIA cuvettes coated for 18 h with 50 ng of homogeneous E. coli recA protein were challenged for 2 h at 22°C with 0.3 ml of PBSTween containing 0.2--100 ng of recA protein (or cell extracts containing 0.1--200 pg of total protein) and 0.05 ml of rabbit antiserum diluted 1 : l 0 s. After an additional 2 h incubation with calf intestine alkaline phosphatase-goat anti-rabbit IgG conjugate (Miles-Yeda, L o t S-751) diluted 1 : 2000, the substrate solution was applied; color development at 22°C was terminated after 60 min by addition of 0.05 ml 3 N NaOH. Absorbances at 405 nm were read on a Gilford Manual EIA reader or a Bausch and Lomb
377 Spectronic 710 spectrophotometer with rapid sampler. The usable range of the assay was generally 0.5--50 ng.
Components o f the statistical package Fitting of the sigmoid competition curve employed the 4-parameter 'logistic function': a--d +d 1 + (X/c) b
R-
where R is defined as the response, i.e., the ratio of observed to m a x i m u m difference in absorbance, a is the absorbance asymptotically approached at limiting low antigen doses, d is the absorbance at the high
1
~ n--li=l
(Ri -- ~2) (Wellington, 1980).
The logit transformation was c o m p u t e d as y = ln[. 1 (B/B0) -- (B/B0)J
where B = O . D . ( s a m p l e ) - O.D.(limiting high dose), and B0 = O.D.(limiting low dose) --O.D.(limiting high dose). The standard curve (Y versus log X) was fitted by linear least-squares calculation. Plotting subroutines were originally formulated by Dr. David W. Craig (1977) and modified to provide the necessary options. RESULTS
Function o f the statistics package The flow chart in Fig. 1 summarizes the steps used to obtain statistically valid estimates of the a m o u n t of antigen per unit of total protein from the competition ELISA data. If the molecular weight of the antigen and number
378
> ~ E n t e r ELISA o.d. of Standards; Estimate of a, b, c, d
Compute Rcalculated Display Rcalculated + Robserved
--No
Compare Rcalculated + Robserved: Good Correlation?
i<
Yes
I
Perform iterative fit of standard data to 4-parameter equation using a, b, c, d.
l
Options: I Store standard data, Plot standard Data and Curve
P r i n t b e s t e s t i m a t e s o f a, b, c , d, X + R for standard data, standard d e v i a t i o n s , and c o o r d i n a t e s o f t h e m i d p o i n t . !
Enter sample data: ELISA o.d. and associated parameter, e.g., total protein or cell number
i Print amounts o f a n t i g e n r e a d from s t a n d a r d c u r v e
l
Options: I Plot X vs. total protein Set upper and lower limits of R
Linear r e g r e s s i o n f i t t i n g of a n t i g e n amount v s . a s s o c i a t e d p a r a m e t e r : P r i n t s regression equation, specific activity ( s l o p e of r e g r e s s i o n l i n e ) , s t a n d a r d e r r o r s , mean s q u a r e d e v i a t i o n , d e v i a t i o n c o r r e l a tion coefficient, F-value.
Additional Options:
<
I
Plot data & regression line Perform T-test on slope Repeat linear regression on the same data, with new limits on R, or on a new set of data using the same standard curve Fig. 1. F l o w c h a r t i n d i c a t i n g m a j o r f u n c t i o n s a n d u s e r o p t i o n s ( e n c l o s e d in r e c t a n g l e s ) for p r o c e s s i n g r e g u l a t i o n d a t a o b t a i n e d b y c o m p e t i t i o n E L I S A . G u i d e l i n e s for d e t e r m i n i n g w h e t h e r d a t a p o i n t s c a n be l e g i t i m a t e l y d e l e t e d are d e s c r i b e d in t h e t e x t .
379 o f cells is k n o w n and a reasonable estimate can be made of recovery of antigen in the extracts, one can calculate the n u m b e r of antigen molecules per cell. Fitting the standard curve T h e program is initiated by entering the am ount s of each standard and the corresponding absorbance values from the assay. Up to 20 different or replicate samples are a c c o m m o d a t e d . The user must also provide estimates o f the 4 constants a, b, c and d; if no estimate is available, a, b and c m ay be set equal to 1, and d equal to 0. However, if values from a previously c o m p u t e d standard curve are used, the n u m b e r of iterations and the time required for convergence of the fitting rout i ne will be greatly reduced. The program displays the actual and calculated response values for each a m o u n t o f antigen, and the user may either accept the initial estimates of a, b, c and d values or change t he m to improve the initial fit. Normally, it is advisable n o t t o change the values of b and c, but the upper and lower a s y m p t o t e s a and d m a y be adjusted t o improve the fit near the highest and lowest standards. The iterative fitting r o u t i n e then operates until the most significant fit is achieved, i.e., t he sum o f the deviations squared is minimized and the correlation coefficient is maximized. The fitted standard curve and the data points m a y be pl ot t ed, the parameters o f the fitted line and the data points are printed, and the data can be stored on the magnetic disk for future use. The coordinates of t he m i dpoi nt of the best-fit curve m a y be used to com pa r e the 'relative p o t e n c y ' of antigen preparations, or for quality control purposes (Burger et al., 1972; Pekary, 1979). The slope parameter provides an index of the assay's sensitivity within the working range. When p er f o r m ed as described above, the c o m p e t i t i o n ELISA demonstrates the systematic n o n - u n i f o r m i t y of variance observed in most ot her c o m p e t i t i o n immunoassays. Consequently the quadratic weighting funct i on for sigmoid curves described by Rodbard and H ut t (1974) was included as an o p t io n in th e r o ut i ne for fitting the standard curve. In the present study, the responses for 6 replicates run at 14 dosages were used to determine the coefficients a0, al, and a2 which are used by the program as constants in the e q uatio n a 2 = a0 + ai R + a2R 2. The variance may differ for di fferent ELISA systems, experimental conditions, and antisera, and we r e c o m m e n d t hat the coefficients be experimentally d e t e r m i n e d by each user. T h e results, summarized in Table 1, indicate t hat bot h m e t h o d s fit the data equally well. In b o t h cases, the ratio of calculated to actual values is nearly the same, and neither ratio differs f r om 1.0 to a confidence level b e t t e r than 0.5. After the standard curve equation is c o m p u t e d , the program requests the absorbance values of the ' u n k n o w n ' samples, and the corresponding a m o u n t o f antigen is calculated f r om the equation t hat describes the standard curve. These values are printed along with their corresponding 'R' value
380 TABLE 1 Comparison of standard curve fitting procedures for competition ELISA results. Six separate competition ELISA assays were performed with known amounts of E. coli recA protein between 0.5 and 50 ng, and the data (number of points denoted by n) were fitted with the functions as defined in Materials and Methods. A linear least-squares regression analysis was performed between the actual values and those calculated using each fitting function. A t-test was used to compare the slopes of the regression lines to the ideal slope of 1.0, with the result noted in the text. Fitting function
4-parameter (n = 36) Logit(n=50)
Regression equation Slope
Intercept
1.104 + 0.04 1.116-+0.04
0.278 + 0.95 --1.22 -+0.95
Correlation coefficient
Standard error of estimate
0.966 0.950
4.729 5.582
( p o s i t i o n on the o r d i n a t e o f the s t a n d a r d curve, w h i c h ranges f r o m 0 to 1) a n d the a m o u n t o f t o t a l p r o t e i n , cell n u m b e r , or o t h e r p a r a m e t e r with w h i c h t h e antigen is to be c o m p a r e d . The user m a y t h e n set ' a c c e p t a b l e ' u p p e r and l o w e r R values, i.e., limits defining t h e reliable region on the s t a n d a r d curve. To assist the user in deciding u p o n R values and r e c o g n i z i n g o u t l y i n g points, the a m o u n t o f antigen is a u t o m a t i c a l l y p l o t t e d as a f u n c t i o n o f the correlate variable, e.g., t o t a l p r o t e i n . The p r o g r a m t h e n p e r f o r m s a linear least-squares regression analysis o n the data and the correlate variable, e x c l u d i n g a n y values w h i c h fall o u t s i d e o f the a c c e p t a b l e limits. The p r o g r a m prints the regression e q u a t i o n , t h e c o r r e l a t i o n coefficients, and t h e s t a n d a r d errors o f its slope a n d i n t e r c e p t . The regression line as well as the individual d a t a p o i n t s m a y be p l o t t e d , and the user m a y elect t o r e p e a t t h e analysis with d i f f e r e n t limits o n the range o f R values. The slope o f the regression line is an estimate o f t h e specific activity o f the antigen, e.g., ng o f recA p r o t e i n per pg o f t o t a l p r o t e i n in t h e e x a m p l e o f Fig. 2. The user can o b t a i n an e s t i m a t e o f t h e significance o f a slope as c o m p a r e d with a n y o t h e r , determ i n e d b y S t u d e n t ' s t-test. Guidelines for data r e d u c t i o n Fig. 2 shows a t y p i c a l p l o t f r o m a c o m p e t i t i o n E L I S A d e t e r m i n a t i o n o f recA p r o t e i n in various a m o u n t s o f a c r u d e E. coli e x t r a c t . The stepwise testing strategy for o b t a i n i n g t h e best regression line is o u t l i n e d in the figure legend. As with a n y b i o c h e m i c a l assay, a m o u n t s o f antigen d e t e r m i n e d b y c o m p e t i t i o n E L I S A are directly p r o p o r t i o n a l to t h e a m o u n t o f sample, and the regression line m u s t t h e o r e t i c a l l y e x t r a p o l a t e t o zero at zero dose. However, in practice t h e l o w e r limit o f p r o p o r t i o n a l i t y is o f t e n i n f l u e n c e d b y c o m p o n e n t s o t h e r t h a n antigen in the sample. Zero antigen s t a n d a r d s are
381 16 14 12 e-v
z
10
LLJ bO EE Q_
8 6
q~ t,, 4
'
0
2
4
6
8
o 0
20
40
60
80
100
TOTAL PROTEIN (/..Lg) REGRESSION
EQUATION
NO. OF POINTS
SLOPE
i0
0.I16
3.54
0.905
120.44
1.822
9
0.210
2.921
0.922
63.78
1.269
8
0.305
2.607
0.896
30.4219
I.i]4
INTERCEPT
CORRELATION COEFFICIENT
HEAN ~QUARE DEVIATION
STANDARD ERROR OP E ST I~DhTE
7
0.642
1.998
0.985
20.835
0.359
6
0.705
1.936
0.952
5.709
0.394
Fig. 2. E x a m p l e o f data r e d u c t i o n f r o m c o m p e t i t i o n E L I S A o f recA p r o t e i n in an E. coli cell e x t r a c t . T e n dilutions o f t h e e x t r a c t c o n t a i n i n g f r o m 0.1 to 100 pg o f t o t a l p r o t e i n were a n a l y z e d , and the a m o u n t o f recA p r o t e i n in each was d e t e r m i n e d f r o m a s t a n d a r d curve f i t t e d b y t h e 4 - p a r a m e t e r f u n c t i o n . Regression lines and statistics were c o m p u t e d for all 10 values and r e - c o m p u t e d w i t h s a m p l e s d e l e t e d in o r d e r o f decreasing total protein. T h e a m o u n t o f recA p r o t e i n in t h e t w o s a m p l e s o f highest p r o t e i n c o n c e n t r a t i o n was calculated f r o m a r e s p o n s e (R) value less t h a n 0.1, and t h e s e s a m p l e s were a u t o m a t i c a l l y d e l e t e d as ' u n r e l i a b l e ' w h e n the w o r k i n g range o n the s t a n d a r d curve was set b e t w e e n R = 0.9 and 19. = 0.1. The criteria d e s c r i b e d u n d e r ' G u i d e l i n e s for data r e d u c t i o n ' were best satisfied w i t h t h e 7 l o w e s t values; t h e t h i r d highest p o i n t was d e l e t e d because its inclusion p r e v e n t s c o n v e r g e n c e o f t h e regression slope and i n t e r c e p t , and significantly increases t h e s t a n d a r d error o f e s t i m a t e . Data are always t e s t e d in this o r d e r ; values at low total p r o t e i n c o n c e n t r a t i o n s are never d e l e t e d unless t h e y e x c e e d t h e u p p e r R value or have an o b v i o u s gross r a n d o m error. The inset s h o w s t h e final regression fit o n an e x p a n d e d scale.
382 c o n s e q u e n t l y misleading, a n d we r a r e l y use t h e m * . Similarly, v e r y c o n c e n t r a t e d s a m p l e s m a y deviate f r o m p r o p o r t i o n a l i t y b e c a u s e t h e y Saturate t h e c o m p e t i t i o n r e a c t i o n , or b e c a u s e o f p r o t e o l y s i s , aggregation e f f e c t s , and n o n - s p e c i f i c i n t e r a c t i o n o f antigen or a n t i b o d y w i t h o t h e r m a t e r i a l . This generally is v e r y o b v i o u s in t h e initial p r i n t - o u t or p l o t o f a m o u n t o f antigen versus t o t a l p r o t e i n or cell n u m b e r , a n d such s a m p l e s m a y l e g i t i m a t e l y b e d e l e t e d f r o m t h e regression analysis. Serious r a n d o m errors w h i c h o c c u r i n d e p e n d e n t l y o f t h e p r e d i c t a b l e d e v i a t i o n s can also b e d e t e c t e d a n d e d i t e d o u t as t h e user m o n i t o r s t h e p r i n t e d or p l o t t e d results. T h e a d v a n t a g e o f o u r d a t a regression r o u t i n e is t h a t it enables t h e user to edit or re-calculate a c c o r d i n g t o 6 d e f i n e d statistical criteria: (a) e l i m i n a t i o n o f s a m p l e s w h i c h are b e y o n d t h e w o r k i n g limits o f t h e s t a n d a r d curve; (b) m a x i m i z a t i o n o f t h e c o r r e l a t i o n c o e f f i c i e n t ; (c) m i n i m i z a t i o n o f t h e m e a n s q u a r e d e v i a t i o n ; {d) m i n i m i z a t i o n o f t h e m e a n e r r o r o f t h e e s t i m a t e ; (e) c o n v e r g e n c e o f t h e regression slope t o w a r d a c o n s t a n t value; a n d (f) conv e r g e n c e o f t h e i n t e r c e p t t o w a r d zero. N o r m a l l y , t h e b e s t analysis is o b t a i n e d w h e n t h e s e criteria are a p p l i e d in t h e o r d e r i n d i c a t e d . T h e correlat i o n c o e f f i c i e n t , t h e m e a n s q u a r e d e v i a t i o n a n d t h e m e a n e r r o r o f t h e estim a t e are n o t i n d e p e n d e n t a n d criteria b, c a n d d will a l w a y s b e fulfilled s i m u l t a n e o u s l y . In p r a c t i c e , o n e or m o r e o f t h e o t h e r criteria o f t e n c a n n o t b e satisfied as t h e s e are m e t ; w h i c h o n e s will d e v i a t e d e p e n d s o n t h e relative a m o u n t s o f r a n d o m a n d c o n s i s t e n t error. DISCUSSION T h e statistical p a c k a g e d e s c r i b e d h e r e was designed f o r s i m p l i c i t y a n d c o n v e n i e n c e o f use, as well as statistical rigor. Unlike m o s t p r o g r a m s p r e s e n t l y available f o r E L I S A , o u r s goes b e y o n d s t a n d a r d c u r v e - f i t t i n g a n d d a t a printo u t , to p r o v i d e a statistical analysis a n d m e a n s f o r editing, graphical pres e n t a t i o n , a n d reliability a s s e s s m e n t o f t h e t e s t s a m p l e s . T h e p l o t s as well as t h e p r i n t e d statistics e n a b l e t h e user to t r u n c a t e t h e w o r k i n g region o f t h e s t a n d a r d curve a n d t o i d e n t i f y o u t l y i n g s a m p l e s in a s y s t e m a t i c , statistically valid w a y . S a m p l e s w h i c h are n o t i m m u n o l o g i c a l l y identical t o t h e s t a n d a r d can q u i c k l y be i d e n t i f i e d , a n d t h e p r e s e n c e o f s u b s t a n c e s w h i c h d e g r a d e t h e antigen or i n t e r f e r e w i t h t h e E L I S A generally b e c o m e s evid e n t as a d i s p r o p o r t i o n a l i t y o f antigen t o t o t a l p r o t e i n . No m a n i p u l a t i o n * Zero standards generally give absorbance values significantly below the value at the lowdose asymptote, which in our assays involves samples containing 0.2--1 pg of total protein. This seems to be due mainly to the antigen, antibody and complex concentrations approaching the dissociation constant at very low doses (Miiller, 1980), but it may also reflect adventitious binding of antibody to material which is in large excess of the antigen in crude extracts. Even 'pure' antigen samples may contain some denatured or weakly immunoreactive material. It is often difficult or unwieldy to attempt to compensate accurately for this by adding inert protein or 'control extracts;' in any case, the zero standard would not necessarily improve the sensitivity or precision of the assay.
383 of the raw data is necessary before it is entered into the program, and the program can be adapted to accept data from any compatible input device. Requests for decisions by the user are virtually self-explanatory, and include safeguards against mistakes in data input, erasure of stored data files, improper plotting commands, etc. Standards, data, and the statistical information useful for quality control can be saved for future comparison, and the plotting routines simplify scale changes and representation of related curves on a single plot. With appropriately designed experiments, the 50% competition values provided by the program could be used for comparisons of detection limits for different antisera, estimates of average antibody affinities (Muller, 1980) or as an index of antigenic cross-reactivity, as well as for quality control. This program package was designed specifically for the problems encountered in detailed studies of regulation and turnover of antigens in crude extracts and biological fluids. In experiments of this type, the a m o u n t of each sample is often very limited, but the number of samples may be large. Pure antigen and antisera may be scarce as reagents for the ELISA, and the experimenter is forced to coat the ELISA wells and perform the competition with amounts of protein far below saturation. This tends to maximize the combined errors of individual samples. Specific activity differences greater than 10-fold are c o m m o n , and inevitably a significant proportion of samples is near the limit of the acceptable range of the standard curve. Furthermore, because the ELISA is run on individual 96-well plates or 50-cuvette Gilford trays and the timing of additions is important, the variance between different plates or trays m a y become significant, particularly if the assays are performed manually. It is generally inconvenient, if n o t impossible, to assay enough replicates of the samples to get adequate data on sample variance, because the additional manipulations and delays would contribute adversely to the variance. Single-point assays at several dilutions are thus preferable, and variance is best taken into account by a pre-determined weighting function. ' Our program does n o t assign confidence limits to the standard curve. This means t h a t sample values read from any portion of the standard curve enter the linear regression analysis as though t h e y were absolute values. Consequently, the linear regression is forced to fit each sample datum point as though it had zero variance. This causes the over-all error estimate to be larger than it would be if the regression fit were allowed to make use of the variance of each sample. Thus, the error estimate for any specific activity calculated by our program will be a m a x i m u m value. The relative advantages of the 4-parameter logistic function for fitting bioassay, radioimmune, and immunoradiometric assay data have been summarized by Rodbard and Hutt {1974) and Grotjan and Steinberg (1977) among others; this function clearly has excellent correspondence with competition ELISA data as well. It was chosen in preference to 4 others which have been widely applied, because it requires the least computation, offers
384 very good accuracy with small numbers of samples, and provides useful diagnostic i n f o r m a t i o n for the user. In our experience, c o m p e t i t i o n ELISA data f r eq u en tly show significant deviations (heteroscedasticity) in logit vs. log[dose] plots over a considerable range of working dosages. Although the 4-parameter function and the logit f unct i on are algebraically equivalent, the logit t r a n s f o r m a t i o n in effect weights all responses by fixing the highest and lowest experimentally det er m i ned values {Burger et al., 1972). Various forms o f rectangular-hyperbola fitting have been described (Bliss, 1974), but these generally provide an optimal fit only for a particular region of the dose-response curve (Rodbard et al., 1969). Weighting functions used to e x t e n d their useful range have n o t y e t been proven to be valid for competition ELISA. Scatchard analysis is m os t precise when used for univalent antigen-antibody interactions ( R odba r d, 1971) and Scatchard plots are more complicated to fit if the data are n o t linear. Fitting with spline functions (Marschner et al., 1974) forces the curve most closely toward the means of the replicate points that have the least scatter, i.e., the values are weighted by their standard deviations. The iterative fitting routine for this procedure would be more c o m p l e x and require longer c o m p u t a t i o n time, and the spline f u n c t i o n parameters are unrelated t o mechanisms of antigen-antibody interaction. Fitting by spline functions might improve mathematical 'smoothness' at the expense of quality control i nf or m a ti on in the working range of the standard curve, and the major s m oot hi ng effect would occur at the limits of the working range, making t h e m harder to define. Upon written request, persons interested in using our statistical package m a y obtain co mp l et e copies of the program and instructions for its use. ACKNOWLEDGEMENTS We wish to t h a n k Dr. Siong Wie for advice on ELISA t echnol ogy, and Dr. Randolph Wedding for valuable discussions and critique of our statistical methods. This w or k was s uppor t ed by Grant PCM79-22987 from the National Science F o u n d a t i o n . REFERENCES Bliss, C.I., 1974, Statistics in Endocrinology, eds. J.W. McArthur and T. Colton (MIT Press, Cambridge, MA) pp. 431--447. Burger, H.G., V.W.K. Lee and G.C. Rennie, 1972, J. Lab. Clin. Med. 80,302. Craig, D.W., 1977, Ph.D. Dissertation, University of California, Riverside, p. 184. Grotjan, H.E. and E. Steinberg, 1977, Comput. Biol. Med. 7,159. Marquardt, D.W., 1963, J. Soc. Ind. Appl. Math. 11,431. Marschner, I., F. Erhardt and P.C. Scriba, 1974, Radioimmunoassa.y and Related Procedures in Biology and Medicine, Vol. I (International Atomic Energy Agency, Vienna) pp. 111--122. M/iller, R., 1980, J. Immunol. Methods 34,345. Pekary, A.E., 1979, Comput. Biol. Med. 9, 355. Rodbard, D., W. Bridson and P.L. Rayford, 1969, J. Lab. Clin. Med. 74,770.
385 Rodbard, D., 1971, Principles of Competitive Protein-Binding Assays, eds. W.D. Odell and W.H. Daughaday (J.B. Lippincott, Philadelphia) pp. 204--259. Rodbard, D. and D.M. Hutt, 1974, Radioimmunoassay and Related Procedures in Biology and Medicine, Vol. I (International Atomic Energy Agency, Vienna) pp. 165--192. Voller, A., D. Bidwell and A. Bartlett, 1976, Manual of Clinical Immunology, eds. N. Rose and H. Friedman (American Society for Microbiology, Washington, D.C.) pp. 506-512. Wellington, D., 1980, Enzyme-Immunoassay, ed. E.T. Maggio (CRC Press, Boca Raton) pp. 249--273.
375
STATISTICAL PACKAGE FOR ANALYSIS OF COMPETITION ELISA RESULTS
PEDRO F. CANELLAS and ALEXANDER E. KARU Department of Biochemistry, University of California, Riverside, CA 92521, U.S.A. (Received 18 May 1981, accepted 17 July 1981)
A computation and analysis program has been assembled to facilitate the use of competition ELISA and similar assays for studies of antigen regulation and turnover. The program fits sigmoidal standard curves using a 4-parameter logistic function, determines amounts of antigen from the equation which defines the standard curve, and calculates specific activities by linear regression of the levels of antigen in varying amounts of total protein. An optional weighting function is provided to adjust for systematic non-uniform variance. Outlying points are identified during the linear regression, and the user may delete them or redefine the acceptable working range of the standard curve. The program provides a complete print-out of the data and optional plots of the fitted standard curve and the regression analysis of the samples, as well as statistics which are useful for quality control. It also provides the option of storing the data points from the standard curve on magnetic diskettes. The package is written in BASIC for a Wang Model 2200 computer equipped with a magnetic diskette drive, line printer, and flat-bed X--Y plotter, but it is readily adaptable to other systems and input/output devices.
INTRODUCTION C o m p e t i t i o n E L I S A m e t h o d s have been gaining increasing a c c e p t a n c e as an alternative t o r a d i o i m m u n e assay ( R I A ) . When p r o p e r l y o p t i m i z e d , b o t h m e t h o d s m a y have essentially identical range, sensitivity, a n d precision. The t w o t e c h n i q u e s used m o s t f r e q u e n t l y f o r c o m p e t i t i o n E L I S A involve: (a) c o m p e t i t i o n o f a small a m o u n t o f a n t i g e n - e n z y m e c o n j u g a t e with an u n k n o w n a m o u n t o f native antigen for b i n d i n g to a limiting n u m b e r o f antib o d y m o l e c u l e s w h i c h are a t t a c h e d t o t h e solid phase; o r (b) c o m p e t i t i o n b e t w e e n a k n o w n small a m o u n t o f a n t i g e n affixed t o t h e solid phase, and an u n k n o w n a m o u n t o f t h e native antigen, f o r b i n d i n g a limiting a m o u n t o f a n t i b o d y . In this m e t h o d t h e a m o u n t o f a n t i b o d y t h a t associates w i t h the i m m o b i l i z e d antigen is q u a n t i t a t e d b y a d d i t i o n o f an e n z y m e - s e c o n d antib o d y ( a n t i - i m m u n o g l o b u l i n ) c o n j u g a t e . Each o f these m e t h o d s has u n i q u e merits a n d d r a w b a c k s , b u t b o t h e x h i b i t t h e sigmoidal c o m p e t i t i o n r e s p o n s e as a f u n c t i o n o f l o g [ a n t i g e n ] w h i c h is characteristic o f t h e R I A and o t h e r i m m u n o l o g i c a l assays ( R o d b a r d and H u t t , 1 9 7 4 ) . R e c e n t l y we a d a p t e d t h e t y p e (b) c o m p e t i t i o n E L I S A described a b o v e 0022-1759/81/0000--0000/$02.75 © 1981 Elsevier/North-Holland Biomedical Press
376 to quantitate two recombination p r o t e i n s - the recA and recBC e n z y m e s in crude extracts of E. coli. To facilitate use of the competition ELISA in studies of induction, repression, and turnover of antigens, we have designed a computerized statistical processing package which is rapid, flexible, and provides antigen specific activity values and estimates of their significance. This paper presents details of the program and guidelines for its application. The most critical element in a routine for processing data from any immunological assay is the m e t h o d used to fit the standard curve. The relevant criteria include directness of computation, sufficient m o n o t o n i c i t y to generate a smooth function through a practical number of standards, flexibility adequate to accommodate changes in sensitivity and range of the response, and relevance of the fitting parameters to the reliability, sensitivity, and precision of the assay. Several algorithms provide approximations of the sigmoidal dose-response (Rodbard et al., 1969; Wellington, 1980). We have compared two of the most frequently used fitting methods -- linear regression using the classical 'logit' transformation, and non-linear least-squares approximation of the sigmoid using a 4-parameter exponential function with optional weighting. Although we found the two methods to be similar in accuracy and working range, fitting of the sigmoid appears preferable for competition ELISA analysis of limited numbers of samples that may differ widely in specific activity of the antigen. MATERIALS AND METHODS Samples for demonstration analysis Homogeneous E. coli recA protein and rabbit antiserum were the generous gift of Dr. Kevin McEntee, University of California, Los Angeles, and strains of E. coli K-12 were provided by Dr. Alvin J. Clark, University of California, Berkeley. Growth of the cells, conditions for induction, and preparation of extracts will be described in detail elsewhere (Karu and Belk, 1981, in preparation). Competition ELISA The composition ELISA assay was performed in polystyrene EIA cuvettes (Gilford Instruments, Palo Alto, CA), essentially as described by Voller et al. (1976). Gilford EIA cuvettes coated for 18 h with 50 ng of homogeneous E. coli recA protein were challenged for 2 h at 22°C with 0.3 ml of PBSTween containing 0.2--100 ng of recA protein (or cell extracts containing 0.1--200 pg of total protein) and 0.05 ml of rabbit antiserum diluted 1 : l 0 s. After an additional 2 h incubation with calf intestine alkaline phosphatase-goat anti-rabbit IgG conjugate (Miles-Yeda, L o t S-751) diluted 1 : 2000, the substrate solution was applied; color development at 22°C was terminated after 60 min by addition of 0.05 ml 3 N NaOH. Absorbances at 405 nm were read on a Gilford Manual EIA reader or a Bausch and Lomb
377 Spectronic 710 spectrophotometer with rapid sampler. The usable range of the assay was generally 0.5--50 ng.
Components o f the statistical package Fitting of the sigmoid competition curve employed the 4-parameter 'logistic function': a--d +d 1 + (X/c) b
R-
where R is defined as the response, i.e., the ratio of observed to m a x i m u m difference in absorbance, a is the absorbance asymptotically approached at limiting low antigen doses, d is the absorbance at the high
1
~ n--li=l
(Ri -- ~2) (Wellington, 1980).
The logit transformation was c o m p u t e d as y = ln[. 1 (B/B0) -- (B/B0)J
where B = O . D . ( s a m p l e ) - O.D.(limiting high dose), and B0 = O.D.(limiting low dose) --O.D.(limiting high dose). The standard curve (Y versus log X) was fitted by linear least-squares calculation. Plotting subroutines were originally formulated by Dr. David W. Craig (1977) and modified to provide the necessary options. RESULTS
Function o f the statistics package The flow chart in Fig. 1 summarizes the steps used to obtain statistically valid estimates of the a m o u n t of antigen per unit of total protein from the competition ELISA data. If the molecular weight of the antigen and number
378
> ~ E n t e r ELISA o.d. of Standards; Estimate of a, b, c, d
Compute Rcalculated Display Rcalculated + Robserved
--No
Compare Rcalculated + Robserved: Good Correlation?
i<
Yes
I
Perform iterative fit of standard data to 4-parameter equation using a, b, c, d.
l
Options: I Store standard data, Plot standard Data and Curve
P r i n t b e s t e s t i m a t e s o f a, b, c , d, X + R for standard data, standard d e v i a t i o n s , and c o o r d i n a t e s o f t h e m i d p o i n t . !
Enter sample data: ELISA o.d. and associated parameter, e.g., total protein or cell number
i Print amounts o f a n t i g e n r e a d from s t a n d a r d c u r v e
l
Options: I Plot X vs. total protein Set upper and lower limits of R
Linear r e g r e s s i o n f i t t i n g of a n t i g e n amount v s . a s s o c i a t e d p a r a m e t e r : P r i n t s regression equation, specific activity ( s l o p e of r e g r e s s i o n l i n e ) , s t a n d a r d e r r o r s , mean s q u a r e d e v i a t i o n , d e v i a t i o n c o r r e l a tion coefficient, F-value.
Additional Options:
<
I
Plot data & regression line Perform T-test on slope Repeat linear regression on the same data, with new limits on R, or on a new set of data using the same standard curve Fig. 1. F l o w c h a r t i n d i c a t i n g m a j o r f u n c t i o n s a n d u s e r o p t i o n s ( e n c l o s e d in r e c t a n g l e s ) for p r o c e s s i n g r e g u l a t i o n d a t a o b t a i n e d b y c o m p e t i t i o n E L I S A . G u i d e l i n e s for d e t e r m i n i n g w h e t h e r d a t a p o i n t s c a n be l e g i t i m a t e l y d e l e t e d are d e s c r i b e d in t h e t e x t .
379 o f cells is k n o w n and a reasonable estimate can be made of recovery of antigen in the extracts, one can calculate the n u m b e r of antigen molecules per cell. Fitting the standard curve T h e program is initiated by entering the am ount s of each standard and the corresponding absorbance values from the assay. Up to 20 different or replicate samples are a c c o m m o d a t e d . The user must also provide estimates o f the 4 constants a, b, c and d; if no estimate is available, a, b and c m ay be set equal to 1, and d equal to 0. However, if values from a previously c o m p u t e d standard curve are used, the n u m b e r of iterations and the time required for convergence of the fitting rout i ne will be greatly reduced. The program displays the actual and calculated response values for each a m o u n t o f antigen, and the user may either accept the initial estimates of a, b, c and d values or change t he m to improve the initial fit. Normally, it is advisable n o t t o change the values of b and c, but the upper and lower a s y m p t o t e s a and d m a y be adjusted t o improve the fit near the highest and lowest standards. The iterative fitting r o u t i n e then operates until the most significant fit is achieved, i.e., t he sum o f the deviations squared is minimized and the correlation coefficient is maximized. The fitted standard curve and the data points m a y be pl ot t ed, the parameters o f the fitted line and the data points are printed, and the data can be stored on the magnetic disk for future use. The coordinates of t he m i dpoi nt of the best-fit curve m a y be used to com pa r e the 'relative p o t e n c y ' of antigen preparations, or for quality control purposes (Burger et al., 1972; Pekary, 1979). The slope parameter provides an index of the assay's sensitivity within the working range. When p er f o r m ed as described above, the c o m p e t i t i o n ELISA demonstrates the systematic n o n - u n i f o r m i t y of variance observed in most ot her c o m p e t i t i o n immunoassays. Consequently the quadratic weighting funct i on for sigmoid curves described by Rodbard and H ut t (1974) was included as an o p t io n in th e r o ut i ne for fitting the standard curve. In the present study, the responses for 6 replicates run at 14 dosages were used to determine the coefficients a0, al, and a2 which are used by the program as constants in the e q uatio n a 2 = a0 + ai R + a2R 2. The variance may differ for di fferent ELISA systems, experimental conditions, and antisera, and we r e c o m m e n d t hat the coefficients be experimentally d e t e r m i n e d by each user. T h e results, summarized in Table 1, indicate t hat bot h m e t h o d s fit the data equally well. In b o t h cases, the ratio of calculated to actual values is nearly the same, and neither ratio differs f r om 1.0 to a confidence level b e t t e r than 0.5. After the standard curve equation is c o m p u t e d , the program requests the absorbance values of the ' u n k n o w n ' samples, and the corresponding a m o u n t o f antigen is calculated f r om the equation t hat describes the standard curve. These values are printed along with their corresponding 'R' value
380 TABLE 1 Comparison of standard curve fitting procedures for competition ELISA results. Six separate competition ELISA assays were performed with known amounts of E. coli recA protein between 0.5 and 50 ng, and the data (number of points denoted by n) were fitted with the functions as defined in Materials and Methods. A linear least-squares regression analysis was performed between the actual values and those calculated using each fitting function. A t-test was used to compare the slopes of the regression lines to the ideal slope of 1.0, with the result noted in the text. Fitting function
4-parameter (n = 36) Logit(n=50)
Regression equation Slope
Intercept
1.104 + 0.04 1.116-+0.04
0.278 + 0.95 --1.22 -+0.95
Correlation coefficient
Standard error of estimate
0.966 0.950
4.729 5.582
( p o s i t i o n on the o r d i n a t e o f the s t a n d a r d curve, w h i c h ranges f r o m 0 to 1) a n d the a m o u n t o f t o t a l p r o t e i n , cell n u m b e r , or o t h e r p a r a m e t e r with w h i c h t h e antigen is to be c o m p a r e d . The user m a y t h e n set ' a c c e p t a b l e ' u p p e r and l o w e r R values, i.e., limits defining t h e reliable region on the s t a n d a r d curve. To assist the user in deciding u p o n R values and r e c o g n i z i n g o u t l y i n g points, the a m o u n t o f antigen is a u t o m a t i c a l l y p l o t t e d as a f u n c t i o n o f the correlate variable, e.g., t o t a l p r o t e i n . The p r o g r a m t h e n p e r f o r m s a linear least-squares regression analysis o n the data and the correlate variable, e x c l u d i n g a n y values w h i c h fall o u t s i d e o f the a c c e p t a b l e limits. The p r o g r a m prints the regression e q u a t i o n , t h e c o r r e l a t i o n coefficients, and t h e s t a n d a r d errors o f its slope a n d i n t e r c e p t . The regression line as well as the individual d a t a p o i n t s m a y be p l o t t e d , and the user m a y elect t o r e p e a t t h e analysis with d i f f e r e n t limits o n the range o f R values. The slope o f the regression line is an estimate o f t h e specific activity o f the antigen, e.g., ng o f recA p r o t e i n per pg o f t o t a l p r o t e i n in t h e e x a m p l e o f Fig. 2. The user can o b t a i n an e s t i m a t e o f t h e significance o f a slope as c o m p a r e d with a n y o t h e r , determ i n e d b y S t u d e n t ' s t-test. Guidelines for data r e d u c t i o n Fig. 2 shows a t y p i c a l p l o t f r o m a c o m p e t i t i o n E L I S A d e t e r m i n a t i o n o f recA p r o t e i n in various a m o u n t s o f a c r u d e E. coli e x t r a c t . The stepwise testing strategy for o b t a i n i n g t h e best regression line is o u t l i n e d in the figure legend. As with a n y b i o c h e m i c a l assay, a m o u n t s o f antigen d e t e r m i n e d b y c o m p e t i t i o n E L I S A are directly p r o p o r t i o n a l to t h e a m o u n t o f sample, and the regression line m u s t t h e o r e t i c a l l y e x t r a p o l a t e t o zero at zero dose. However, in practice t h e l o w e r limit o f p r o p o r t i o n a l i t y is o f t e n i n f l u e n c e d b y c o m p o n e n t s o t h e r t h a n antigen in the sample. Zero antigen s t a n d a r d s are
381 16 14 12 e-v
z
10
LLJ bO EE Q_
8 6
q~ t,, 4
'
0
2
4
6
8
o 0
20
40
60
80
100
TOTAL PROTEIN (/..Lg) REGRESSION
EQUATION
NO. OF POINTS
SLOPE
i0
0.I16
3.54
0.905
120.44
1.822
9
0.210
2.921
0.922
63.78
1.269
8
0.305
2.607
0.896
30.4219
I.i]4
INTERCEPT
CORRELATION COEFFICIENT
HEAN ~QUARE DEVIATION
STANDARD ERROR OP E ST I~DhTE
7
0.642
1.998
0.985
20.835
0.359
6
0.705
1.936
0.952
5.709
0.394
Fig. 2. E x a m p l e o f data r e d u c t i o n f r o m c o m p e t i t i o n E L I S A o f recA p r o t e i n in an E. coli cell e x t r a c t . T e n dilutions o f t h e e x t r a c t c o n t a i n i n g f r o m 0.1 to 100 pg o f t o t a l p r o t e i n were a n a l y z e d , and the a m o u n t o f recA p r o t e i n in each was d e t e r m i n e d f r o m a s t a n d a r d curve f i t t e d b y t h e 4 - p a r a m e t e r f u n c t i o n . Regression lines and statistics were c o m p u t e d for all 10 values and r e - c o m p u t e d w i t h s a m p l e s d e l e t e d in o r d e r o f decreasing total protein. T h e a m o u n t o f recA p r o t e i n in t h e t w o s a m p l e s o f highest p r o t e i n c o n c e n t r a t i o n was calculated f r o m a r e s p o n s e (R) value less t h a n 0.1, and t h e s e s a m p l e s were a u t o m a t i c a l l y d e l e t e d as ' u n r e l i a b l e ' w h e n the w o r k i n g range o n the s t a n d a r d curve was set b e t w e e n R = 0.9 and 19. = 0.1. The criteria d e s c r i b e d u n d e r ' G u i d e l i n e s for data r e d u c t i o n ' were best satisfied w i t h t h e 7 l o w e s t values; t h e t h i r d highest p o i n t was d e l e t e d because its inclusion p r e v e n t s c o n v e r g e n c e o f t h e regression slope and i n t e r c e p t , and significantly increases t h e s t a n d a r d error o f e s t i m a t e . Data are always t e s t e d in this o r d e r ; values at low total p r o t e i n c o n c e n t r a t i o n s are never d e l e t e d unless t h e y e x c e e d t h e u p p e r R value or have an o b v i o u s gross r a n d o m error. The inset s h o w s t h e final regression fit o n an e x p a n d e d scale.
382 c o n s e q u e n t l y misleading, a n d we r a r e l y use t h e m * . Similarly, v e r y c o n c e n t r a t e d s a m p l e s m a y deviate f r o m p r o p o r t i o n a l i t y b e c a u s e t h e y Saturate t h e c o m p e t i t i o n r e a c t i o n , or b e c a u s e o f p r o t e o l y s i s , aggregation e f f e c t s , and n o n - s p e c i f i c i n t e r a c t i o n o f antigen or a n t i b o d y w i t h o t h e r m a t e r i a l . This generally is v e r y o b v i o u s in t h e initial p r i n t - o u t or p l o t o f a m o u n t o f antigen versus t o t a l p r o t e i n or cell n u m b e r , a n d such s a m p l e s m a y l e g i t i m a t e l y b e d e l e t e d f r o m t h e regression analysis. Serious r a n d o m errors w h i c h o c c u r i n d e p e n d e n t l y o f t h e p r e d i c t a b l e d e v i a t i o n s can also b e d e t e c t e d a n d e d i t e d o u t as t h e user m o n i t o r s t h e p r i n t e d or p l o t t e d results. T h e a d v a n t a g e o f o u r d a t a regression r o u t i n e is t h a t it enables t h e user to edit or re-calculate a c c o r d i n g t o 6 d e f i n e d statistical criteria: (a) e l i m i n a t i o n o f s a m p l e s w h i c h are b e y o n d t h e w o r k i n g limits o f t h e s t a n d a r d curve; (b) m a x i m i z a t i o n o f t h e c o r r e l a t i o n c o e f f i c i e n t ; (c) m i n i m i z a t i o n o f t h e m e a n s q u a r e d e v i a t i o n ; {d) m i n i m i z a t i o n o f t h e m e a n e r r o r o f t h e e s t i m a t e ; (e) c o n v e r g e n c e o f t h e regression slope t o w a r d a c o n s t a n t value; a n d (f) conv e r g e n c e o f t h e i n t e r c e p t t o w a r d zero. N o r m a l l y , t h e b e s t analysis is o b t a i n e d w h e n t h e s e criteria are a p p l i e d in t h e o r d e r i n d i c a t e d . T h e correlat i o n c o e f f i c i e n t , t h e m e a n s q u a r e d e v i a t i o n a n d t h e m e a n e r r o r o f t h e estim a t e are n o t i n d e p e n d e n t a n d criteria b, c a n d d will a l w a y s b e fulfilled s i m u l t a n e o u s l y . In p r a c t i c e , o n e or m o r e o f t h e o t h e r criteria o f t e n c a n n o t b e satisfied as t h e s e are m e t ; w h i c h o n e s will d e v i a t e d e p e n d s o n t h e relative a m o u n t s o f r a n d o m a n d c o n s i s t e n t error. DISCUSSION T h e statistical p a c k a g e d e s c r i b e d h e r e was designed f o r s i m p l i c i t y a n d c o n v e n i e n c e o f use, as well as statistical rigor. Unlike m o s t p r o g r a m s p r e s e n t l y available f o r E L I S A , o u r s goes b e y o n d s t a n d a r d c u r v e - f i t t i n g a n d d a t a printo u t , to p r o v i d e a statistical analysis a n d m e a n s f o r editing, graphical pres e n t a t i o n , a n d reliability a s s e s s m e n t o f t h e t e s t s a m p l e s . T h e p l o t s as well as t h e p r i n t e d statistics e n a b l e t h e user to t r u n c a t e t h e w o r k i n g region o f t h e s t a n d a r d curve a n d t o i d e n t i f y o u t l y i n g s a m p l e s in a s y s t e m a t i c , statistically valid w a y . S a m p l e s w h i c h are n o t i m m u n o l o g i c a l l y identical t o t h e s t a n d a r d can q u i c k l y be i d e n t i f i e d , a n d t h e p r e s e n c e o f s u b s t a n c e s w h i c h d e g r a d e t h e antigen or i n t e r f e r e w i t h t h e E L I S A generally b e c o m e s evid e n t as a d i s p r o p o r t i o n a l i t y o f antigen t o t o t a l p r o t e i n . No m a n i p u l a t i o n * Zero standards generally give absorbance values significantly below the value at the lowdose asymptote, which in our assays involves samples containing 0.2--1 pg of total protein. This seems to be due mainly to the antigen, antibody and complex concentrations approaching the dissociation constant at very low doses (Miiller, 1980), but it may also reflect adventitious binding of antibody to material which is in large excess of the antigen in crude extracts. Even 'pure' antigen samples may contain some denatured or weakly immunoreactive material. It is often difficult or unwieldy to attempt to compensate accurately for this by adding inert protein or 'control extracts;' in any case, the zero standard would not necessarily improve the sensitivity or precision of the assay.
383 of the raw data is necessary before it is entered into the program, and the program can be adapted to accept data from any compatible input device. Requests for decisions by the user are virtually self-explanatory, and include safeguards against mistakes in data input, erasure of stored data files, improper plotting commands, etc. Standards, data, and the statistical information useful for quality control can be saved for future comparison, and the plotting routines simplify scale changes and representation of related curves on a single plot. With appropriately designed experiments, the 50% competition values provided by the program could be used for comparisons of detection limits for different antisera, estimates of average antibody affinities (Muller, 1980) or as an index of antigenic cross-reactivity, as well as for quality control. This program package was designed specifically for the problems encountered in detailed studies of regulation and turnover of antigens in crude extracts and biological fluids. In experiments of this type, the a m o u n t of each sample is often very limited, but the number of samples may be large. Pure antigen and antisera may be scarce as reagents for the ELISA, and the experimenter is forced to coat the ELISA wells and perform the competition with amounts of protein far below saturation. This tends to maximize the combined errors of individual samples. Specific activity differences greater than 10-fold are c o m m o n , and inevitably a significant proportion of samples is near the limit of the acceptable range of the standard curve. Furthermore, because the ELISA is run on individual 96-well plates or 50-cuvette Gilford trays and the timing of additions is important, the variance between different plates or trays m a y become significant, particularly if the assays are performed manually. It is generally inconvenient, if n o t impossible, to assay enough replicates of the samples to get adequate data on sample variance, because the additional manipulations and delays would contribute adversely to the variance. Single-point assays at several dilutions are thus preferable, and variance is best taken into account by a pre-determined weighting function. ' Our program does n o t assign confidence limits to the standard curve. This means t h a t sample values read from any portion of the standard curve enter the linear regression analysis as though t h e y were absolute values. Consequently, the linear regression is forced to fit each sample datum point as though it had zero variance. This causes the over-all error estimate to be larger than it would be if the regression fit were allowed to make use of the variance of each sample. Thus, the error estimate for any specific activity calculated by our program will be a m a x i m u m value. The relative advantages of the 4-parameter logistic function for fitting bioassay, radioimmune, and immunoradiometric assay data have been summarized by Rodbard and Hutt {1974) and Grotjan and Steinberg (1977) among others; this function clearly has excellent correspondence with competition ELISA data as well. It was chosen in preference to 4 others which have been widely applied, because it requires the least computation, offers
384 very good accuracy with small numbers of samples, and provides useful diagnostic i n f o r m a t i o n for the user. In our experience, c o m p e t i t i o n ELISA data f r eq u en tly show significant deviations (heteroscedasticity) in logit vs. log[dose] plots over a considerable range of working dosages. Although the 4-parameter function and the logit f unct i on are algebraically equivalent, the logit t r a n s f o r m a t i o n in effect weights all responses by fixing the highest and lowest experimentally det er m i ned values {Burger et al., 1972). Various forms o f rectangular-hyperbola fitting have been described (Bliss, 1974), but these generally provide an optimal fit only for a particular region of the dose-response curve (Rodbard et al., 1969). Weighting functions used to e x t e n d their useful range have n o t y e t been proven to be valid for competition ELISA. Scatchard analysis is m os t precise when used for univalent antigen-antibody interactions ( R odba r d, 1971) and Scatchard plots are more complicated to fit if the data are n o t linear. Fitting with spline functions (Marschner et al., 1974) forces the curve most closely toward the means of the replicate points that have the least scatter, i.e., the values are weighted by their standard deviations. The iterative fitting routine for this procedure would be more c o m p l e x and require longer c o m p u t a t i o n time, and the spline f u n c t i o n parameters are unrelated t o mechanisms of antigen-antibody interaction. Fitting by spline functions might improve mathematical 'smoothness' at the expense of quality control i nf or m a ti on in the working range of the standard curve, and the major s m oot hi ng effect would occur at the limits of the working range, making t h e m harder to define. Upon written request, persons interested in using our statistical package m a y obtain co mp l et e copies of the program and instructions for its use. ACKNOWLEDGEMENTS We wish to t h a n k Dr. Siong Wie for advice on ELISA t echnol ogy, and Dr. Randolph Wedding for valuable discussions and critique of our statistical methods. This w or k was s uppor t ed by Grant PCM79-22987 from the National Science F o u n d a t i o n . REFERENCES Bliss, C.I., 1974, Statistics in Endocrinology, eds. J.W. McArthur and T. Colton (MIT Press, Cambridge, MA) pp. 431--447. Burger, H.G., V.W.K. Lee and G.C. Rennie, 1972, J. Lab. Clin. Med. 80,302. Craig, D.W., 1977, Ph.D. Dissertation, University of California, Riverside, p. 184. Grotjan, H.E. and E. Steinberg, 1977, Comput. Biol. Med. 7,159. Marquardt, D.W., 1963, J. Soc. Ind. Appl. Math. 11,431. Marschner, I., F. Erhardt and P.C. Scriba, 1974, Radioimmunoassa.y and Related Procedures in Biology and Medicine, Vol. I (International Atomic Energy Agency, Vienna) pp. 111--122. M/iller, R., 1980, J. Immunol. Methods 34,345. Pekary, A.E., 1979, Comput. Biol. Med. 9, 355. Rodbard, D., W. Bridson and P.L. Rayford, 1969, J. Lab. Clin. Med. 74,770.
385 Rodbard, D., 1971, Principles of Competitive Protein-Binding Assays, eds. W.D. Odell and W.H. Daughaday (J.B. Lippincott, Philadelphia) pp. 204--259. Rodbard, D. and D.M. Hutt, 1974, Radioimmunoassay and Related Procedures in Biology and Medicine, Vol. I (International Atomic Energy Agency, Vienna) pp. 165--192. Voller, A., D. Bidwell and A. Bartlett, 1976, Manual of Clinical Immunology, eds. N. Rose and H. Friedman (American Society for Microbiology, Washington, D.C.) pp. 506-512. Wellington, D., 1980, Enzyme-Immunoassay, ed. E.T. Maggio (CRC Press, Boca Raton) pp. 249--273.