Regression measurements, error

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Weighted Regression) requires the user to dehne a signal-dependent model of the measurement error, e.g., sy = a + b x, which is then used to calculate the weighting factors 1/Vy at every abscissa x,-. For an example on how to enter the model, see Algebraic Function, ... 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

Faber K, Kowalski BR (1997b) Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares. J Chemom 11 181... 

In the error-in-variable method (EVM), measurement errors in all variables are treated in the parameter estimation problem. EVM provides both parameter estimates and reconciled data estimates that are consistent with respect to the model. The regression models are often implicit and undetermined (Tjoa and Biegler, 1992), that is,... 

In the calibration problem two related quantities, X and Y, are investigated where Y, the response variable, is relatively easy to measure while X, the amount or concentration variable, is relatively difficult to measure in terms of cost or effort Furthermore, the measurement error for X is small compared with that of Y The experimenter observes a calibration set of N pairs of values (x, y ), i l,...,N, of the quantities X and Y, x being the known standard amount or concentration values and y the chromatographic response from the known standard The calibration graph is determined from this set of calibration samples using regression techniques Additional values of the dependent variable Y, say y., j l,, M, where M is arbitrary, are also observed whose corresponding X values, x. are the unknown quantities of interest The statistical literature on the calibration problem considers the estimation of these unknown values, x, from the observed and the... 

The second possibility is called cross-validation. The test samples are measured by a reference method. Howevep the reference method cannot provide true values because measurement error occurs here as well. Nevertheless, well-characterized methods can provide generally accepted values, which are then compared to the ones obtained using the test calibration. Note that this comparison must be done using particularly suited regression methods, because the error for both methods will be in the same order of magnitude. Especially, cross-validation of CE and HPLC has been frequently reported. 

Designed Experiments Produce More Precise Models. In the context of linear regression, this is demonstrated by examining the statistical uncertainties of the regression coefficients. Equation 2.1 is the regression model where the response for the th sample (r ) of an instrument is shown as a linear function of the sample concentration (c.) with measurement error... 

Since tar concentrations are also corrupted by measurement errors, and since we do not know which variable is more reliable, it is equally meaningful to fit the inverse model x = Ay + to the data, thereby regarding the nicotine concentration as independent variable. Show that the two regression... 

J. A. Fernandez Pierna, L. Jin, F. Wahl, N. M. Faber and D. L. Massart, Estimation of partial least squares regression prediction uncertainty when the reference values carry a sizeable measurement error, Chemom. Intell. Lab. Syst., 65, 2003, 281-291. 

One assumption of regression analysis is the increased precision of measuring or fixing a factor. When measuring or fixing a factor, such conditions are recommended where a factor measurement error is incomparably smaller when compared to an error in determining a response. 

The regression coefficients obtained for potassium and sodium have to be proved for significance. Firstly the average measurement error sy is calculated according to ... 

ICE was developed for estimating acute toxicity of chemicals to species where data are lacking. Interspecies correlations were created for 95 aquatic and terrestrial organisms using least squares regression where both variables are random (i.e., both variables are independent and subject to measurement error Asfaw et al. 2004). The correlation coefficient (r) is used to describe the linear association amongst the... 

Least squares is used to determine the model parameters for concentration prediction of unknown samples. This is achieved by minimizing the usual sum of the squared errors, (y-y)T(y-y). As stated before, the errors in y are assumed to be much larger than the errors in X for these models. Because the regression parameters are determined from measured data, measurement errors propagate into the estimated coefficients of the regression vector b and the estimated values in y. In fact, we can only estimate the residuals, e, in the y measurements, as shown in Equation 5.12 through Equation 5.14. Summarizing previous discussions and equations, the model is defined in Equation 5.11 as... 

Nonlinear mixed-effects modeling methods as applied to pharmacokinetic-dynamic data are operational tools able to perform population analyses [461]. In the basic formulation of the model, it is recognized that the overall variability in the measured response in a sample of individuals, which cannot be explained by the pharmacokinetic-dynamic model, reflects both interindividual dispersion in kinetics and residual variation, the latter including intraindividual variability and measurement error. The observed response of an individual within the framework of a population nonlinear mixed-effects regression model can be described as... 

Hence, a plot of ln(C) versus t will give a straight line with a slope of — k and an intercept of ln(Co). Because each number will have some measurement error, you will need to use statistical regression techniques to get the best values of Co and k. The technique is simple Just... 

Figure 23.1 Schematic flowchart showing the relationship between impedance measurements, error analysis, supporting observations, model development, and weighted regression analysis. (Taken from Orazem and Tribollet and reproduced with permission of Elsevier, Inc.)...

Unlike the individual model discussed above, a more elaborate statistical model is required to deal with sparse PK data. In formulating the model, it is recognized that overall variability in the measured (response) data in a sample of individuals reflects both measurement error and intersubject variability. The observed response (e.g., concentration) in an individual within the framework of population (regression) nonlinear random mixed effects models can be described as... 

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