Alternative Linear Regression Models

If bothy and x are subject to measurable random errors, one can derive parameters starting from the following equation to obtain a/ first  

This is the standard situation for the ordinary least squares estimation (OLS) of ax. 

In this situation the error behavior of the two variables is reflected in the slope. The model is called the reduced major axis (RMA) model 

Here the errors made in measuring y and x are assumed to be equal in magnitude. Then is obtained by solving Eq. 2-52. The procedure is called orthogonal (ORTH) regression). 

Other models and results of simulation studies may be appropriate in method comparisons [HARTMANN et al., 1993], 

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There have also been attempts to describe the temporal aspects of perception from first principles, the model including the effects of adaptation and integration of perceived stimuli. The parameters in the specific analytical model derived were estimated using non-linear regression [14]. Another recent development is to describe each individual TI-curve,/j(r), i = 1, 2,..., n, as derived from a prototype curve, S t). Each individual Tl-curve can be obtained from the prototype curve by shrinking or stretching the (horizontal) time axis and the (vertical) intensity axis, i.e. fff) = a, 5(b, t). The least squares fit is found in an iterative procedure, alternately adapting the parameter sets (a, Zi, for 1=1,2,..., n and the shape of the prototype curve [15],... 

As a possible alternative to in vitro metabolism studies, QSAR and molecular modelling may play an increasing role. Quantitative stracture-pharmacokinetic relationships (QSPR) have been studied for nearly three decades [42,45-52]. These are often based on classical QSAR approaches based on multiple linear regression. In its most simple form, the relationship between PK properties and lipophilicity has been discussed by various workers in the field [36, 49, 50]. 

An alternative approach for improving predicted pA(a values has been suggested by Klicic et al. (2002), who developed functional-group-specific linear regression corrections for pA a values computed from a particular DFT SCRF PB formalism. Correction of the raw computed pA(aS increases the model s accuracy to about 0.5 pA" units for those acidic functional groups well represented in their parameterization set. 

Mathematical optimization always requires a deterministic process model to predict the future behavior of a process. However, as previously mentioned, it is difficult to construct mathematical models that can cover the entire range of fermentation due to the complexity of intracellular metabolic reactions. As an alternative to the deterministic mathematical models, Kishimoto et al. proposed a statistical procedirre that uses linear multiple regression models [7], as shown below, instead of a deterministic mathematical model such as a Monod equation. 

In contrast to the explicit analytical solution of least-squares fit used in linear regression, our present treatment of data analysis relies on an iterative optimization, which is a completely different approach as a result of the operations discussed in the previous section, theoretical data are calculated, dependent on the model and choice of parameters, which can be compared with the experimental results. The deviation between theoretical and experimental data is usually expressed as the sum of the errors squared for all the data points, alternatively called the sum of squared deviations (SSD) ... 

Transformed Variables Sometimes an alternative to a simple linear model is suggested by a theoretical relationship or by examining residuals from a linear regression. In some cases, linear least-squares analysis can be used after the simple transformations shown in Table 8-3. 

These later two models of bioavailability as a continuous variable are linear since they used stepwise multiple linear regression (M LR) as the modeling tool. An obvious alternative, which may offer improved performance, is a nonlinear technique and such a model using an artificial neural network (ANN) was reported by Turner and colleagues [30], This study employed 167 compounds characterized by several descriptor types, ID, 2D, and 3D, and resulted in a 10-term model. Although the predictive performance was judged adequate, it was felt that the model was better able to differentiate qualitatively between poorly and highly bioavailable compounds. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

Quantitative Structure - Activity Relationships (QSARs) are estimation methods developed and used to predict certain effects or properties of chemical substances, which are primarily based on the structure of the chemicals. The development of QSARs often relies on the application of statistical methods such as multiple linear regression (MLR) or partial least squares regression (PLS). However, since toxicity data often include uncertainties and measurements errors, when the aim is to point out the more toxic and thus hazardous chemicals and to set priorities, order models can be used as alternative to statistical methods such as multiple linear regression. 

Alternatively, instead of using the EBE of the parameter of interest as the dependent variable, an estimate of the random effect (t ) can be used as the dependent variable, similar to how partial residuals are used in stepwise linear regression. Early population pharmacokinetic methodology advocated multiple linear regression using either forward, backwards, or stepwise models. A modification of this is to use multiple simple linear models, one for each covariate. For categorical covariates, analysis of variance is used instead. If the p-value for the omnibus F-test or p-value for the T-test is less than some cut-off value, usually 0.05, the covariate is moved forward for further examination. Many reports in the literature use this approach. 

The coefficients in the model equation 3.4 may be estimated as before, as linear combinations or contrasts of the experimental results, taking the columns of the effects matrix as described in section III.A.5 of chapter 2. Alternatively, they may be estimated by multi-linear regression (see chapter 4). The latter method is more usual, but in the case of factorial designs both methods are mathematically equivalent. 

We demonstrate below the alternative method of analysing the data. The first-order, second-order, and reduced cubic models (equations 9.1, 9.2, and 9.4) can be estimated from the 10 data points by multiple linear regression. The statistical significance and the goodness of fit of the models can then be determined. 

A class of algorithms which is specialized for multilinear problems is known as alternating least-squares (ALS). Multilinear models are all conditionally linear in a function of each of the three or so independent variables for example, spectral intensity is linear in concentration if the other variables are fixed. Each step of an ALS algorithm fixes the vectors for all but one independent variable, then applies linear regression to select the vectors for the one variable to minimize the error sum of squares. The algorithm cycles among the sets of parameters to be estimated, updating each in turn. Most applications of multilinear models use ALS code. ... 

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