Nonconstant variance regression techniques for calibration-curve-based analysis

A statistical overview on univariate calibration, inverse regression, and detection limits: Application to gas chromatography/mass spectrometry technique

“…For the calculation of a reliable calibration model some recommendations are often reported in the literature: (i) the number of concentration levels must range between seven and ten (Garden, Mitchell, & Mills, 1980); (ii) the replicates must be at least eight to ten to verify the normality of the data by the Shapiro-Wilk test (Shapiro & Wilk, 1965), for instance, and to ascertain their scedasticity; (iii) the calibration design must tune the problem in hand: the estimation of detection limits requires calibration points near the hypothesized values of the limits, whereas for accurate quantitative analysis the standard solutions must bracket the unknown one; (iv) the calibration measurements are to be run in blocks, each block containing one replicate of each standard, randomly chosen to avoid the effect of any systematic error, and blanks to avoid carryover effects; (v) finally, the blank solution response must be inserted in the regression procedure when the detection limits are determined (Mocak et al, 1997;Vial & Jardy, 1999) to decrease the difference between the experimentally measured blank and the intercept of the regression line.…”

“…The analytical application of the calibration curve is the inverse regression, called also discrimination (Miller, 1966;Garden, Mitchell, & Mills, 1980), that is, the obtainment of x from an instrumental response y with the confidence interval for the true value of x (Brownlee, 1960). This interval depends on two factors: the uncertainty of b 0 and b 1 and the uncertainty of the experimental response reading, which can be a single or the mean of m replicate measurements.…”

Harmonization of the quantitative determination of volatile fatty acids profile in aqueous matrix samples by direct injection using gas chromatography and high-performance liquid chromatography techniques: Multi-laboratory validation study

“…With respect to the fitting technique, ordinary least-squares (OLS) regression is often selected for a mathematical fit of the relation between concentration and instrumental response. Linear regression by OLS assumes that each data point in the calibration curve has a constant variance (homoscedasticity).However, many analytical methods produce data with increasing variance as a function of concentration (heteroscedasticity) [24].It is important to note that a systematic error occurs if heterocedastic calibration data are evaluated by using OLS [25]. This error should not cause problems if the concentration range is small, but if the calibration is carried out over a few orders of magnitude, this systematic error should be the cause of biased regression values (slope and intercept), then, sensitive to extreme data points.…”

Multiple Linear Regression: An Overview with Analytical and Physico-Chemical Applications

“…Expressed in matrix form [22,30,33,[40][41][42] GZ   (20) where G is the vector of observations (n x 1), Z is the matrix of independent variables Z 1 and Z 2 of known form (n x p),  is the vector of parameters to be estimated (p x 1) and  is a vector of errors (n…”