Regression standard error

Sr Standard deviation about regression = standard error of estimate - - standard error in y (LINEST calculated this for a calibration curve in Figure 3.10)... 

Policies on the l pe of Injury Multinomial Logit Regression (standard error in parentheses)... 

E descriptor related to interaction with the stationary phase through polarizable bonds 5 descriptor related to dipole- or induced dipole-type interactions A descriptor related to the solute acidity B descriptor measuring the solute basicity V descriptor hnked to the size of the solute (McGovan volume) r regression coefficient SE regression standard error n number of experiments in the regression or number of data points. 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

Kittrell et al. (1965a) also performed two types of estimation. First the data at each isotherm were used separately and subsequently all data were regressed simultaneously. The regression of the isothermal data was also done with linear least squares by linearizing the model equation. In Tables 16.7 and 16.8 the reported parameter estimates are given together with the reported standard error. Ayen and Peters (1962) have also reported values for the unknown parameters and they are given here in Table 16.9. 

They include simple statistics (e.g., sums, means, standard deviations, coefficient of variation), error analysis terms (e.g., average error, relative error, standard error of estimate), linear regression analysis, and correlation coefficients. 

The standard error of performance, also termed the standard error of prediction (SEP), which represents an estimate of the prediction error (1 sigma) for a regression line is given as ... 

A complication arises. We learn from considerations of multiple regression analysis that when two (or more) variables are correlated, the standard error of both variables is increased over what would be obtained if equivalent but uncorrelated variables are used. This is discussed by Daniel and Wood (see p. 55 in [9]), who show that the variance of the estimates of coefficients (their standard errors) is increased by a factor of... 

A variety of statistical parameters have been reported in the QSAR literature to reflect the quality of the model. These measures give indications about how well the model fits existing data, i.e., they measure the explained variance of the target parameter y in the biological data. Some of the most common measures of regression are root mean squares error (rmse), standard error of estimates (s), and coefficient of determination (R2). 

Standard error of prediction Standard deviation of the predicted value obtained from linear regression. 

We have already given the equations for the computation of the standard errors in the parameters optimised by linear regression, equation (4.32). The equations are very similar for parameters that are passed through the Newton-Gauss algorithm. In fact, at the end of the iterative fitting, the relevant information has already been calculated. 

FIGURE 4.4 Determination of optimum complexity of regression models (schematically). Measure for prediction errors for instance RMSECv in arbitrary linear units. Left, global and local minimum of a measure for prediction performance. Right, one standard error mle. 

Supernatant BOD5 and TOA concentrations are expressed as g/1. The mean variance of the offensiveness from the regression line and the standard error of the supernatant BOD5 relationship are 0.325 and 0.050 and for the TOA relationship are 0.399 and 0.119 respectively. 

Sivakesava et al. also used Raman (as well as FT-IR and NIR) to perform a simultaneous on-line determination of biomass, glucose, and lactic acid in lactic acid fermentation by Lactobacillus casei.2 Partial least squares (PLS) and principal components regression (PCR) equations were generated after suitable wavelength regions were determined. The best standard errors were found to be glucose, 2.5 g/1 lactic acid, 0.7 g/1 and optical cell density, 0.23. Best numbers were found for FT-IR with NIR and Raman being somewhat less accurate (in this experiment). 

In a paper that addresses both these topics, Gordon et al.11 explain how they followed a com mixture fermented by Fusarium moniliforme spores. They followed the concentrations of starch, lipids, and protein throughout the reaction. The amounts of Fusarium and even com were also measured. A multiple linear regression (MLR) method was satisfactory, with standard errors of prediction (SEP) for the constituents being 0.37% for starch, 4.57% for lipid, 4.62% for protein, 2.38% for Fusarium, and 0.16% for com. It may be inferred from the data that PLS or PCA (principal components analysis) may have given more accurate results. 

Table VIII. Regression Coefficients and Standard Errors for...

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

Estimated standard error of slope of regression line 0.118... 

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