Biased regression coefficient

A last note refers to the design of the calibration set. Although we do not need to know either the spectra or the concentration of the interferences, we must be sure that the calibration samples contain the analytes and interferences which might contribute to the response of the unknown samples. In this way, the calculated regression coefficients can remove the contribution of the interferences in the predictions. If the instrumental measurement on the unknown sample contains signals from non-modelled interferences, biased predictions are likely to be obtained (as in the CLS model). 

Ridge regression analysis is used when the independent variables are highly interrelated, and stable estimates for the regression coefficients cannot be obtained via ordinary least squares methods (Rozeboom, 1979 Pfaffenberger and Dielman, 1990). It is a biased estimator that gives estimates with small variance, better precision and accuracy. 

We have seen how consideration of theoretical deposition velocities has identified potential biases in economic assessments. An additional consideration is the relative uncertainties in the determination of theoretical vs. experimental deposition velocities. The heat transfer data on which the theoretical deposition velocities are based are generally very precise, within a few percent. In contrast, the damage functions developed by Lipfert et al. (3) for metals from extant corrosion test data are only capable of predicting corrosion losses at a given time and place within a factor of two, although the individual regression coefficients are much better than that. Most of the uncertainty in the experimental approach is felt to be in test site characterization rather than... 

The regression coefEcients are biased by introducing a parameter along the diameter of Z Z [109]. The computation of regression coefficients /3 in Eq. 4.3 is modified by introducing a ridge parameter k ... 

Ridge regression estimators are biased. The trade-off for stabilization and variance reduction in regression coefficient estimators is the bias in the estimators and the increase in the squared error. 

The concept of squared distances has important functional consequences on how the value of the correlation coefficient reacts to various specific arrangements of data. The significance of correlation is based on the assumption that the distribution of the residual values (i.e., the deviations from the regression line) for the dependent variable y follows the normal distribution and that the variability of the residual values is the same for all values of the independent variable. However, Monte Carlo studies have shown that meeting these assumptions closely is not crucial if the sample size is very large. Serious biases are unlikely if the sample size is 50 or more normality can be assumed if the sample size exceeds 100. 

Ridge regression is also used extensively to remedy multicollinearity between the X, predictor variables. It does this by modifying the least-squares method of computing the coefficients with the addition of a biasing component. 

If one accepts the plausible assumption that h[ is negatively correlated with both ji and k, it is clear that when h[ is omitted from the regression, as in equation (A.l), the estimated coefficient on lead pipes will reflect the effects of both h and k, and will be biased upward so that ( ) > ( ). ... 

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