Univariate least-squares regression

Dichromate-permanganate determination is an artificial problem because the matrix of coefficients can be obtained as the slopes of A vs. x from four univariate least squares regression treatments, one on solutions containing only at... 

Because there is only one independent variable, the subscript has been omitted. We now note that Zx/n = x and Zy/n = y, so we find Eqs. (2-75) as the normal equations for unweighted univariate least-squares regression. 

Carrying through the treatment as before yields Eqs. (2-78) as the normal equations for weighted linear univariate least-squares regression. 

More on Simple linear least squares regression (SLLSR), also known as Simple least squares regression (SLSR) or univariate least squares regression... 

Traditionally, the determination of a difference in costs between groups has been made using the Student s r-test or analysis of variance (ANOVA) (univariate analysis) and ordinary least-squares regression (multivariable analysis). The recent proposal of the generalized linear model promises to improve the predictive power of multivariable analyses. 

Calibration is the process by which a mathematical model relating the response of the analytical instrument (a spectrophotometer in this case) to specific quantities of the samples is constructed. This can be done by using algorithms (usually based on least squares regression) capable of establishing an appropriate mathematical relation such as single absorbance vs. concentration (univariate calibration) or spectra vs. concentration (multivariate calibration). 

Univariate and multivariate spectroscopy was applied to the analysis of spironolactone in presence of chlorthalidone [17]. Satisfactory results were obtained by partial least squares regression, with the calibration curve being linear over the range 2.92 -14.6 pg/mL. A kinetic-spectrophotometric method was described for the determination of spironolactone and canrenone in urine that also used a partial least-squares regression method [18]. After the compounds were extracted from urine, the spectra were recorded at 400 - 520 nm for 10 minutes at 30 second intervals. The relative error was less than 5%. 

In order to illustrate how the partial least squares regression components are constructed, regression with a univariate y (/ x 1) is considered. Suppose X (/ x J) and y (/ x 1) are available and centered columnwise. The first partial least squares regression component ti which is calculated to predict y from X solves... 

Note that the algorithms in Equations (3.45) and (3.46) are noniterative, because the y is univariate. For multivariate Y, the PLS algorithms become iterative. There are alternative algorithms for partial least squares regression, depending on the size of X [De Jong Ter Braak 1994, Lindgren et al. 1993], Moreover, it is a matter of choice whether to deflate X and/or y [Burnham et al. 1996],... 

Least squares regression is known as a univariate method. Univariate methods assume that there is a direct relationship between two sets of values that can be described by a single equation. In univariate methods, there is generally one independent variable (in this case, spectral response) and one dependent variable (concentration). By knowing either one of these values, a solution can be calculated for the other. 

Appendix A3 presents a traditional, nonmatrix, approach to performing univariate (one variable), linear, least-squares regression. 

MSC corrects spectra according to a simple linear univariate fit to a standard spectrum a and are estimated by least squares regression using the standard spectrum. As the standard spectrum, a spectrum of a particular sample or an average spectrum is used. 

In the preceding section we showed that linear least-squares regression is limited to single components or simple mixtures where the bands are totally isolated from one another and there is no overlap in absorbance between the components. Under these conditions a separate linear least-squares regression can be completed for each of the components. This is the definition of a univariate system. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

The total residual sum of squares, taken over all elements of E, achieves its minimum when each column Cj separately has minimum sum of squares. The latter occurs if each (univariate) column of Y is fitted by X in the least-squares way. Consequently, the least-squares minimization of E is obtained if each separate dependent variable is fitted by multiple regression on X. In other words the multivariate regression analysis is essentially identical to a set of univariate regressions. Thus, from a methodological point of view nothing new is added and we may refer to Chapter 10 for a more thorough discussion of theory and application of multiple regression. 

Examination of the univariate distribution of 5-FU clearance revealed it to be skewed and not normally distributed suggesting that any regression analysis based on least squares will be plagued by non-normally distributed residuals. Hence, Ln-transformed 5-FU clearance was used as the dependent variable in the analyses. Prior to analysis, age was standardized to 60 years old, BSA was standardized to 1.83 m2, and dose was standardized to 1000 mg. A p-value less than 0.05 was considered to be statistically significant. The results from the simple linear regressions of the data (Table 2.4) revealed that sex, 5-FU dose, and presence or absence of MTX were statistically significant. 

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