A comparison of five curve-fitting procedures in radioimmunoassay
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1 Ann Clin Biochem 1986; 23: A comparison of five curve-fitting procedures in radioimmunoassay JENNIFER A NISBET, J A OWEN AND GAIL E WARD From the Departmentof Chemical Pathology, St George's Hospital Medical School, London SW170RE, UK SUMMARY. Data obtained from routine analytical radioimmunoassays were processed using five curve-fitting procedures, viz. 'Arnersharn", single binding site, four parameter logistic, a linear logit-log and a polynomial logit-log. The polynomial logit-log procedure gave the best fit, but this was probably due to the inherent flexibility of this curve-fitting process since the analytical precision achieved with it was no better than what was obtained with most of the other procedures. A limited study failed to show that statistical weighting of data before curve fitting had any practical advantage. Much has been written on the subject of fitting curves to radioimmunoassay data,i-ill but often the potential advantages of a particular method have been stated in terms of theory rather than in everyday clinical laboratory practice. To try and shed further light on the matter, we have compared five different curve-fitting procedures applied to a number of sets of raw data obtained in routine radioimmunoassay work. Four of the fitting procedures are in common use, and the fifth is one we have used routinely for many years. Methods ANALYTICAL Plasma cortisol and free tri-iodothyronine (T)) were determined using Amerlex kits (Amersham International, Aylesbury, Bucks, UK) and parathormone (PTH), progesterone, prolactin, testosterone and thyroxine with in-house methods, using mi as radiolabel and polyethylene glycol to accelerate second antibody separation of bound and free fractions. Testosterone was extracted into ether before the radioimmunoassay was carried out. The range of standards used in addition to the zero standard were, respectively, cortisol: nmol/l, free T): pmol/l, PTH: ug/l, progesterone: nmol/l, prolactin: mull, testosterone: nmol/l and thyroxine: lq-400 nmol/l. A multihead gamma counter (LKB Wallac 1260 Multigamma, Croydon, Surrey, UK) was used to measure radioactivity and counting times were chosen as far as possible to give counts exceeding 2500 (except in some instances in measuring non-specific binding). The main data used in this study were obtained from 10 routine analytical batches of each test, run over several weeks. The same standard preparations were used throughout and, in most cases, the same operator performed the tests. Various serum-based commercial preparations were used in precision studies. CURVE-FITTING All calculations were performed on a Rainbow (Digital Equipment Company, Reading, Berks, UK) microprocessor using programs written in interpretative BASIC (listings are available on request). The microprocessor can be connected on-line to the counter but the raw data used in these analyses were entered at the keyboard since they were being examined in retrospect. The models on which the fitting procedures were based were as follows: 'Amersham' model This is the univalent equilibrium model as formulated by Wilkins et al. II Parameters were obtained using the algorithm for use with T] and T 4 radioimmunoassays published by the RadioChemical Centre, Amersham (1976). Single binding site model This is the univalent equilibrium model as formulated by Edwards and Ekins. 12 Parameters 694
2 Five curve-fitting procedures in RIA 695 were determined using an algorithm kindly supplied by Dr P R Edwards. This fits a curve exactly to all the different combinations of four data points, seeking the combination which minimises the sum of the deviations of the curve from other data points. Four parameter logistic model This is the "model described by Youden and Steiner.P Parameters were obtained using an algorithm kindly supplied by Dr P R Edwards, based essentially on the proposals of Zivitz and Hildalgo. 14 Linear logit-log model This is the procedure proposed by Rodbard and Lewald.P Parameters were obtained using a standard linear regression algorithm. Polynomial logit-log model This is an elaboration of the logit-iog procedure in which the log of the dose is expressed as a third order polynomial of the logit. Parameters were obtained using the orthogonal polynomial algorithm described by Forsythe." ASSESSMENT OF CURVE-FITTING In comparing the different curve-fitting procedures, we have used the following statistics: (i) the 'fit' of the curve, expressed as: (a) the mean difference between calculated dose and nominal dose, as a percentage of the nominal value, and (b) the mean difference between the observed binding and the value, calculated from the nominal dose as a percentage of the calculated value. (ii) The precision of the assay determined from 'quality control' samples at two or three different levels expressed as coefficient of variation. (iii) The mean results given by the different procedures. Results A summary of the goodness of fit achieved by the five procedures is given in Table 1. The best fit, that is the lowest mean difference of the calculated dose from the nominal dose, was obtained with the polynomial logit-log fit. This was the case whether fit was assessed in terms of dose or of binding. Calculation of curve parameters took on average less than 2 s processor time with the logit-iog procedure, s with the single-binding site procedure and s with the Amersham and four parameter logistic procedures. In no instance did the re-iterative algorithms fail to converge. The analytical precision obtained with the different curve fitting is presented in Table 2. All five procedures gave essentially the same precision, in spite of the apparently superior curve-fitting given by the polynomial logit-iog procedure. Seeking an explanation, we examined the shapes of some of the polynomial logit-log curves. Plotting the data points [logit (response) v. log (dose)] on graph paper gave what appeared to the eye mostly to be straight lines, but examination of the first and second differentials showed that in practically every instance there was a point of inflexion in the working part of the curve. The position of the point of inflexion varied from run to run even TABLE 1. Fit of curves obtained by different curve-fitting procedures Mean difference-dose (%) Mean difference-binding (%) Test A S F L P A S F L P cortisol Free T PTII Progesterone Prolactin Testosterone Thyroxine H Overall Data were obtained from 10 routine analytical batches. Fit is expressed as: (a) mean difference between calculated and nominal dose, as a percentage and (b) mean difference between observed binding and that calculated from the nominal dose, as a percentage. Fitting procedures: A, Amersham; S, single-binding site; F, four-parameter logistic; L, linear logit-log; P, polynomial logit-iog.
3 696 Nisbet, Owen and Ward TABLE 2. Analytical precision with different curve-fitting procedures Lose dose OC Middle dose OC High dose OC Test A S F L P A S F L P A S F L P Cortisol Free T~ PTII Progesterone Prolactin ) Testosterone Thyroxine Overall Data comprise coefficients of variation for single estimations (n=20). Routinely, assays are performed in duplicate. Fitting procedures: A, Amersham; S, single-binding site, F, four-parameter logistic, L, linear logitlog; P, polynomial logit-log. with the same test and was presumably due to random error in the data rather than to the nature of the relation between logit (response) and log (dose). Results obtained by each procedure for each test at each dose level were compared. Except in the case of cortisol assays, no significant differences were noted. With the Amersham procedure the mean value for cortisol (501 nmol/l) was significantly lower than the grand mean for all procedures (522 nmol/l), but in no other instance did the mean value obtained with a particular procedure differ significantly from the grand mean. We carried out a limited examination of the effect of weighting data on curve fitting. Weighting factors proportional to the reciprocal of the square root of the variance calculated from all the relevant data were determined and applied to observed binding data before applying the algorithm. We chose three tests for which there appeared to be considerable dependence of variance on binding. In comparing the fitting obtained with and without weighting, the deviation of points from the fitted line were not weighted even when the curve had been obtained from weighted data. We found that weighting, as far as we tested it, had negligible effect on the goodness of fit (Table 3) or on the analytical precision or the result obtained. All the results so far presented were obtained by analysing data from what were termed 'good runs', i.e. routine analytical batches in which the fit of the calibration line (obtained by the polynomiallogit-iog procedure) was considered satisfactory. To see how the different curvefitting processes dealt with data from 'bad runs', we selected 13 sets of data (covering all tests) from runs which were deemed unsatisfactory on the grounds that our routine procedure indicated a poor fit of the calibration curve to the raw data. We obtained essentially the same ranking of the curve fitting procedures as was obtained with data from 'good' runs. The use of weighted data did not improve the fits obtained with 'bad' data. TABLE 3. Effect of weighting data on curve fitting Single-binding site Four-parameter logistic Range of % dose % binding % dose % binding weighting Test factors' Wt U W U W U W U Cortisol Free T PTII Assessment of fit as described in Table 1. See text for explanation of weighting procedure. 'Lowest binding received the greatest weight. tw curve fitted to weighted data. U curve fitted to unweighted data.
4 Five curve-fitting procedures in RIA 697 Discussion The polynomial logit-iog procedure gave on average the best fit, i.e. the smallest mean deviation of data points from the calculated curve. It seems likely, however, that the fit obtained with the polynomial logit-iog procedure was due to the process selecting curve parameters which gave a point of inflexion in the working part of the curve so that a line was obtained which was nearest to the experimental points but not expressing optimally the overall relation between the dose and response. In theory, the procedure could produce a curve with a maximum and minimum giving meaningless results and in routine use one would have to guard against this. We have, however, used this procedure for many years in the routine calculation of RIA results on a mini-computer, ensuring no extrapolation of results beyond the range of non-zero standards, and have never been misled by this theoretical possibility. With many curve fitting procedures there is a possibility that extrapolation outside the range of data points can be subject to gross error.. A good example of this was illustrated by Challand, Spencer and Ratcliffe.i who use this possibility as part of their argument in favour of a linear interpolation. Forbidding extrapolation, however, avoids this error. Linear interpolation may be satisfactory with error-free data, but the smoothing effect of taking all data points into account in making a calibration curve is lost. This, we feel, is a major advantage of curve-fitting procedures. In this study the Amersham and linear logit-iog procedures gave the poorest fit. While the legit-log transformation has been widely used, others also have found that it does not I. I' d d 2~.7.11I a ways give a inear stan ar curve.,- In contrast to the variable goodness of fit, precision given by all five procedures at the levels measured was essentially identical. This meant that each curve-fitting procedure worked consistently from batch to batch and that the precision was determined almost entirely by the precision of the chemistry and of the counting. Also in contrast to the goodness of fit was the similarity between results obtained for a particular QC material. This implies that the relatively poor fit with some procedures affected mainly the ends of the calibration curves so that interpolation to obtain results which occurred away from the ends of curve was little affected. In considering a curvilinear relation between logit-response and log of the dose, it might be theoretically sounder to consider the logit of the response as the dependent variable rather than the log of the dose as we have done. In other words, to express the relation as a polynomial in terms of log dose. We chose to treat the log dose as the dependent variable because this was the approach used in creating the curve-fitting programme we have used routinely for a great many years. A practical advantage of either logit-iog procedure is the short processor time required for calculation of the curve parameters evenwith an uncompiled program. The singlebinding site procedure took longer because the algorithm involves systematic examination of every possible combination of four data points. The Amersham and four-parameter logistic procedures took longer still because both involve reiterative convergent algorithms. The time could be reduced by using compiled programs or, possibly, improved algorithms. A theoretical disadvantage of either the linear or the polynomial logit-iog procedure is the fact that the parameters of a particular curve have no readily discernable meaning in terms of chemistry. Those of the other curvefitting procedures can, with greater or lesser ease, be interpreted in terms of chemistry and, accordingly, are of potential value in monitoring analytical work over a period; for example, the parameter 'd' in the single binding site model expresses anti'b0 d y concentration. 12 and this could be used in checking a procedure in routine use. The value of interpreting parameters ofindividual curves in practice, however, remains to be determined. In a limited study, we failed to show that weighting data had an important effect either on the goodness of fit, or on the precision of data measured over a series of batches run over a series of weeks. We would not dispute the logic of statistical weighting but conclude that the practical advantages are small, if any. Arguments for use of weighted data are usually expressed in terms of statistical theory and the procedure is designed to get the best overall fit to the data. The effect is to force the curve nearer points on the dose response curve which have a high weighting and away from points with a low weighting. This could be a disadvantage, however, in the clinical usage of results in circumstances where the maximum accuracy is required at a point on the curve which would not merit maximum weighting on statistical grounds.
5 698 Nisbet, Owen and Ward We conclude that a polynomial logit-log curve-fitting procedure is marginally superior to the other procedures we have tested mainly in terms of the processor time required. In practice, the algorithm should guard against bad data producing maxima and minima on the curve which could result in gross errors. Acknowledgements We are most grateful to Dr P R Edwards, Department of Molecular Endocrinology, Middlesex Hospital Medical School, for supplying us with two of the algorithms used in our calculation and for most helpful discussions. References Hidalgo JU, Maduell CR, Bloch T et al. Precision of radioimmunoassay with emphasis on curve fitting procedures. Sem Nucl Med 1975; 5: Challand GS, Spencer CA, Ratcliffe JG. Observations on the automated calculation of radioimmunoassay results. Ann Clin Biochem 1976; 13: Naus AJ, Kuppens PS, Borst A. Calculations of radioimmunoassay standard curves. Clin Chem 1977; 23: Schoneshofer M. Computer programme for evaluation, physicochemical characterisation and optimization of competitive protein binding assays: comparison of four curve-fitting models in peptide and steroid radioimmunoassays. Clin Chim Acta 1977; 77: Samols E, Barrows GH. Automated data processing and radioassays. Sem Nucl Med 1978; 8: Challand GS (ed). Automated calculation of radioimmunoassay results. Ann Clin Biochem 1978; 15: Vogt D, Sandel P, LangfeIder C, Knedel M. Performance of various mathematical methods for computer-aided processing or radioimmunoassay results. Clin Chim Acta 1978; 87: Keilacker H, Becker G, Ziegler M, Gottschling HD. Radioligand assays-methods and applications IV. Uniform regression of hyperbolic and linear radioimmunoassay curves. J Biochem Biophys Meth 1980; 3: Hawker FJ, Challand GS. Effect of outlying standard points on curve fitting in radioimmunoassay. Clin Chem 1981; 27: Geier T, Rohde W. Comparison of four mathematical models for the calculation of radioimmunossay data of LH, FSH and GH. Endokrinologie 1981; 78: Wilkins TA, Chadney DC, Bryant J et al. Nonlinear least-squares curve fitting of a simple theoretical model to radioimmunoassay doseresponse data using a minicomputer. In Radioimmunoassay and Related Procedures in Medicine Vienna: International Atomic Energy Agency, 1978; Edwards PR, Ekins RP. Mass action model based microprocessor program for RIA data processing. WM Munten, JET Corrie, eds. Immunoassays for Clinical Chemistry. Edinburgh: Churchill Livingstone, 1983; 64(} Youden WJ, Steiner EH. Statistical Manual of the Association of Official Analytical Chemists. Washington, DC: Association of Official Analytical Chemists, 1975; Zivitz M, Hidalgo JU. A linearization of the parameters in the logistic function curve fitting radioimmunoassays. Computer Prog Biomed 1977; 7: Rodbard R, Lewald JE. Computer Analysis of Radioligand Assay and Radioimmunoassay Data. 2nd Karolinska Symposium on Research Methods in Reproductive Endocrinology, March Forsythe GE. Generation and use of orthogonal polynomials for data fitting with a digital computer. J Soc Ind App Math 1957; 5: Accepted for publication 22 July 1986
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