Standard error, regression analysis

A linear regression analysis should not be accepted without evaluating the validity of the model on which the calculations were based. Perhaps the simplest way to evaluate a regression analysis is to calculate and plot the residual error for each value of x. The residual error for a single calibration standard, r , is given as... 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

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]. 

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. 

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... 

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. 

As shown in Table VII there appears to be no significant change of k with respect to temperature. These data were plotted using Equation 3 and from linear regression analysis, the heat of solution was tO.IE Kcal/mole. Since Ah should be negative, this low value is obviously caused by experintental error. Furthermore, the Ah calculated from the standard error of the estimate (t1 standard deviation units) of the linear regression line is +0.17 Kcal/mole. Since Ah is zero or is very close to zero. Equation 3 reduces to... 

Rate constants were determined by linear least-squares regression analysis, and error limits are reported as standard deviations (S.D.). 

Data from Dzvinchuk and Lozinskii (88ZOR2167). Determined by H-NMR at 25°C (in CHCI3 at 20°C). The values (Xt)o were directly measured, rather than obtained from regression analysis. The standard errors of the slope are in the range 0.01-0.03. n = 8 (X = H, 3-NO2, 4-NO2, 4-Br, 4-Ph, 4-Me, 4-MeCONH, 4-MeO). Molar ratio. This is the coefficient of the Yukawa-Tsuno equation, not the correlation coefficient. The correlation coefficients lie in the range 0.997-0.999. 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

Regression analysis assumes that all error components are independent, have a mean of zero and have the same variance throughout the range of POM values. Through an examination of residuals, serious violations in these assumptions can usually be detected. The standardized residuals for each of the fitted models were plotted against the sequence of cases in the file and this scatterplot was examined visually for any abnormalities (10, 14). 

Ordinary least squares technique, used for treatment of the calibration data, is correct only when uncertainties in the certified value of the measurement standards or CRMs are negligible. If these uncertainties increase (for example, close to the end of the calibration interval or the shelf-life), they are able to influence significantly the calibration and measurement results. In such cases, regression analysis of the calibration data should take into account that not only the response values are subjects to errors, but also the certified values. 

Calibration curves were constructed with the NIST albumin (5 concentrations in triplicate) and with the FLUKA albumin (5 concentrations in duplicate) in the concentration range of 50 250 mg/1. The measured values of individual concentrations fluctuated around the fitted lines, with a standard error of 0.007 of the measured absorbance. The difference between FLUKA and NIST albumin calibration lines was statistically insignificant, as evaluated by the t-test P=0.14 > 0.05. The calibration lines differed only in the range of a random error. The FLUKA albumin was, thus, equivalent to that of NIST. Statistical evaluation was carried out using the regression analysis module of the statistical package SPSS, version 4.0. 

Figure 2.3. Linear regression analysis with Excel. Simple linear regression analysis is performed with Excel using Tools -> Data Analysis -> Regression. The output is reorganized to show regression statistics, ANOVA residual plot and line fit plot (standard error in coefficients and a listing of the residues are not shown here).

Error in Calculated Rate Constants. The slope of the straight-line section in the reaction rate curves should be equal to the calculated values of k2. To obtain an idea of the probable error in the calculated values for k2 (Table I), they can be compared with the slopes of these lines. The slopes (and standard errors) were determined by least-squares regression analysis. Table II lists the values for k> from Table I, the least-squares slope, and the standard error of the least-squares value. In most cases, k2 agrees with the least-squares slope to within the standard error (4% or less). 

---