Regression, parameter estimation dependent variable

One assumption until now has been that the dependent and independent variables are measured without error. The impact of measurement error on the regression parameter estimates depends on whether the error affects the dependent or independent variable. When Y has measurement error, the effect on the regression model is not problematic if the measurement errors are uncorrelated and unbiased. In this case the linear model becomes... 

Within the procedure, the major part of the code is virtually identical to the code that would be used to obtain the usual least-squares regression parameters for y = mx + b, namely, obtaining the sum of x values, the sum of squares of x values, etc. The difference is that pairs of values are used the mean value of each pair of x values or of y values is used as the independent variable or dependent variable, respectively, and the estimate of the standard errors for the two sets of data is obtained from the differences between pairs. 

By calculating relative partial regression coefficients, the role of solvent acidity and basicity in determining the thermodynamic quantity can be clearly seen [50]. In order to do this, one must estimate the variance for the independent and dependent variable involved in the multiparameter analysis. For the parameter Q, the variance is defined as... 

The same caveats that apply to linear models when the predictor variables are measured with error apply to nonlinear models. When the predictor variables are measured with error, the parameter estimates may become biased, depending on the nonlinear model. Simulation may be used as a quick test to examine the dependency of parameter estimates within a particular model on measurement error (Fig. 3.14). The SIMEX algorithm, as introduced in the chapter on Linear Models and Regression, can easily be extended to nonlinear models, although the computation time will increase by orders of magnitude. 

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 Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

The models in chemical kinetics usually contain a number of unknown parameters, whose values should be determined from experimental data. Regression analysis is a powerful and objective tool in the estimation of parameter values. The task in regression analysis can be stated as follows the value of the dependent variable (y) is predicted by the model a function (/), contains independent variables (x) and parameters (/ ). The independent variable is measured experimentally, at different conditions, i.e. at different values of the independent variables (x). The goal is to find such numerical values of the parameters (/ ) that the model gives the best possible agreement with the experimental data. Typical independent variables are reaction times, concentrations, pressures and temperatures, while molar amounts, concentrations, molar flows... 

In the case of multivariate modeling, several independent as well as several dependent variables may operate. Out of the many regression methods, we will learn about the conventional method of ordinary least squares (OLS) as well as methods that are based on biased parameter estimations reducing simultaneously the dimensionality of the regression problem, that is, principal component regression (PCR) and the partial least squares (PLS) method. 

This latter technique of Himmelblau, Jones, and Bischoff (H-J-B) has proved to be efficient in various practical situations with few, scattered, data available for complex reaction kinetic schemes (see Ex. 1.6.2-1). Recent extensions of the basic ideas are given by Eakman, Tang, and Gay [48,49, 50]. It should be pointed out, however, that the problem has been cast into one of linear regression at the expense of statistical rigor. The independent variables , X jp, do not fulfill one of the basic requirements of linear regression that the Xi p have to be free of experimental error. In fact, the X p are functions of the dependent variables C/tf) and this may lead to estimates for the parameters that are erroneous. This problem will be discussed further in Chapter 2, when the estimation of parameters in rate equations for catalytic reactions will be treated. Finally, all of the methods have been phrased in terms of batch reactor data, but it should be recognized that the same formulas apply to plug flow and constant volume systems, as will be shown later in this book. 

A regression model is a. fitting relationship that allows the estimation of a dependent variable or experimental response for given settings of a specified group of independent variables or factors. The parameters of the model are known as regression coefficients. Typical tests include the following ... 

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