Simple algorithms for nonlinear calibration by the classical and standard additions methods

Abstract

In univariate calibration by both the classical method and the standard additions method, calibration data are fitted to a response function y=f(x), from which the amount of an unknown x(0) is estimated by solving an equation of form y(0)=f(x(0)). Most such calibrations are limited to linear response functions f, for which the uncertainty in x(0) can be estimated from well-known expressions. The present work describes and illustrates one-step algorithms, in which x(0) is treated as an adjustable parameter in a nonlinear fit of the calibration data, with its standard error thus obtained numerically as a direct outcome of the fit. The computations are easily implemented with a number of data analysis programs and are illustrated here for a representative one, KaleidaGraph. The methods handle heteroscedastic data as easily as homoscedastic and nonlinear functions as easily as linear, permitting the analyst to experiment with different response functions in the quest for an optimum calibration. The estimates of sigma (x0) are obtained from the variance-covariance matrix V for the fit, so for weighted fitting with commercial programs, it is important to know which V--a priori or a posteriori--is being used.

Van't Hoff analysis of K°(T): How good…or bad?

“…An obvious example is Monte Carlo computations on a fit model, where the computationalist sets the data error. However, strong arguments can be made for the use of V prior in many experimental situations where each run involves a small number of data points, but where archival data for the same equipment and procedures arguably provide a better determination of the data error than does the single experiment in question [15]. Nonetheless, the firmly ensconced default in most of physical science is V post , defined as…”

Fitting Straight Lines with Replicated Observations by Linear Regression. III. Weighting Data

“…Although a first-order equation is appropriate for many analysis (e.g., absorption measurements at low absorbances), for many others the calibration graph is non-linear and usually convex (33)(34)(35)(36)(37)(38)(39)(40)(41)(42)250). Many chromatographic detector exhibit a linear response over a limited concentration range, particularly spectroscopic methods of detection and deviation from linearity can be expected (36,37,251) for many applications.…”

“…Although calibration can be a complex procedure involving sophisticated statistical methods, most analytical work still relies on that workhorse of analysis, the straight line univariate classical calibration (38). Classical, i.e., non-weighted linear regression is by far the most widely used regression method (118).…”

“…Tellinghuisen indicates (38) that, in most real problems, the response becomes non-linear when the range of the calibration data becomes sufficiently large. Sometimes, the standard response of the analyst to this situation is to curtail the range in order to employ linear method, being introduced clearly bias in the determination of this way as the selection of the "linear region" is arbitrary.…”

Validação da metodologia da avaliação de incerteza em curvas de calibração melhor ajustadas por polinômios de segundo grau

“…O caso mais simples e mais largamente utilizado é o seguinte modelo linear: (y = b 0 + b 1 x), onde os valores da variável independente x e incerteza dos padrões utilizados na construção da curva de calibração são considerados com incerteza desprezível, além da variável de resposta y assumir ter erros aleatoriamente distribuídos de desvio padrão constante -homocedás-tica. 1 1 ), são determinadas pelo método dos mínimos quadrados; a variância em x, var(x o ), obtida usando a expansão da série de Taylor e desprezando os termos de ordem superior, é dada pela Equação 1:…”