Linear regression applications

Chapter 2 covers simple linear regression applications in detail. 

This experiment describes the application of multiwavelength linear regression to the analysis of two-component mixtures. Directions are given for the analysis of permanganate-dichromate mixtures, Ti(IV)-V(V) mixtures and Cu(II)-Zn(II) mixtures. 

Blanco and co-workers" reported several examples of the application of multiwavelength linear regression analysis for the simultaneous determination of mixtures containing two components with overlapping spectra. For each of the following, determine the molar concentration of each analyte in the mixture. 

If a standard method is available, the performance of a new method can be evaluated by comparing results with those obtained with an approved standard method. The comparison should be done at a minimum of three concentrations to evaluate the applicability of the new method for different amounts of analyte. Alternatively, we can plot the results obtained by the new method against those obtained by the approved standard method. A linear regression analysis should give a slope of 1 and ay-intercept of 0 if the results of the two methods are equivalent. 

Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitaole dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linear-regression techniques such as least-sqiiares methods. However, details concerning the procedures utihzed in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the apphcation. 

Application of IP and NCS in conjunction with specification tolerance limits enables to substantiate acceptance criteria for linear regression metrological characteristics (residual standard deviation, correlation coefficient, y-intercept), accuracy and repeatability. Acceptance criteria for impurity influence (in spectrophotometric assay), solution stability and intermediate precision are substantiated as well. 

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

Light filters for colorimeters, see Filters, optical Limiting cathode potential 509 see also Controlled potential electro-analysis Linear regression 145 Lion intoximeter 747 Liquid amalgams applications of, 412 apparatus for reductions, 413 general discussion, 412 reductions with, (T) 413 zinc amalgam, 413 Liquid ion exchangers structure, 204 uses, 204, 560... 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

Application of equation 10 to the experimental D vs. [HSOIJ] data determined at 25°C and both 1 and 2 M acidity yielded straight line plots with slopes indistinguishable from zero and reproduced the Bi values determined in a non-linear regression fit of the data. This result implies no adsorption of PuSO by the resin and justifies use of the simpler data treatment represented by equation 2. A similar analysis of the Th(IV)-HSOiJ system done by Zielen (9) likewise produced results consistent with no adsorption of ThS0 + by Dowex AG50X12 resin. 

Aqueous solubility is selected to demonstrate the E-state application in QSPR studies. Huuskonen et al. modeled the aqueous solubihty of 734 diverse organic compounds with multiple linear regression (MLR) and artificial neural network (ANN) approaches [27]. The set of structural descriptors comprised 31 E-state atomic indices, and three indicator variables for pyridine, ahphatic hydrocarbons and aromatic hydrocarbons, respectively. The dataset of734 chemicals was divided into a training set ( =675), a vahdation set (n=38) and a test set (n=21). A comparison of the MLR results (training, r =0.94, s=0.58 vahdation r =0.84, s=0.67 test, r =0.80, s=0.87) and the ANN results (training, r =0.96, s=0.51 vahdation r =0.85, s=0.62 tesL r =0.84, s=0.75) indicates a smah improvement for the neural network model with five hidden neurons. These QSPR models may be used for a fast and rehable computahon of the aqueous solubihty for diverse orgarhc compounds. 

The relationship between average ADFR and application rate (AR) was tested using linear regression analysis and the model ADFR = a + b AR. All data concerning high-volume applications (n = 8) and low-volume applications (n = 4) were used, separately or together. The results of these regression analyses are presented in Table 2. 

A possible relationship between DFR and the application rate, as well as the crop volume estimate (CrV), was investigated using a multiple linear regression model (ADFR = a + b AR + c CrV). No significant contribution of crop volume to the variation of ADFR was observed (p = 0.19 and p = 0.87 for high-volume applications and all applications, respectively). 

The scope of this chapter-formatted mini-series is to provide statistical tools for comparing two columns of data, X and Y. With respect to analytical applications such data may be represented for simple linear regression as the concentration of a sample (X) versus an instrument response when measuring the sample (Y). X and Y may also denote a comparison of the reference analytical results (X) versus predicted results (Y) from a calibrated instrument. At other times one may use X and Y to represent the instrument response (X) to a reference value (Y). Whatever data pairs one is comparing as X and Y, there are several statistical tools that are useful to assess the meaning of a change in... 

One must note that probability alone can only detect alikeness in special cases, thus cause-effect cannot be directly determined - only estimated. If linear regression is to be used for comparison of X and Y, one must assess whether the five assumptions for use of regression apply. As a refresher, recall that the assumptions required for the application of linear regression for comparisons of X and Y include the following (1) the errors (variations) are independent of the magnitudes of X or Y, (2) the error distributions for both X and Y are known to be normally distributed (Gaussian), (3) the mean and variance of Y depend solely upon the absolute value of X, (4) the mean of each Y distribution is a straight-line function of X, and (5) the variance of X is zero, while the variance of Y is exactly the same for all values of X. 

Without having conducted a full elasticity analysis across the entire portfolio, the analysis helps to prove market perceptions such as a higher elasticity exist in one market compared to another market or comparing elasticity between products being perceived to have a different elasticity. The statistical quality of the linear regression analysis in selected months is considered as good in terms of the number of customers involved and the R-squared value proving the applicability of the approach. 

Since most quantitative applications are on mixtures of materials, complex mathematical treatments have been developed. The most common programs are Multiple Linear Regression (MLR), Partial Least Squares (PLS), and Principal Component Analyses (PCA). While these are described in detail in another chapter, they will be described briefly here. 

When linear regression does not yield a good correlation, application of a non-linear function may be feasible (see Chapter 10). The parameter estimates for higher-order or polynomial equations may prove to be more difficult to interpret than for a linear relationship. Nevertheless, this approach may be preferable to using lower-order levels of correlation (B or C) for evaluating the relationship between dissolution and absorption data. 

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