Standard error of estimate

The standard deviation gives the accuracy of prediction. If Y is related to one or more predictor variables, the error of prediction is reduced to the standard error of estimate S, (the standard deviation of the errors), where... 

The major disadvantage of the integral method is the difficulty in computing an estimate of the standard error in the estimation of the specific rates. Obviously, all linear least squares estimation routines provide automatically the standard error of estimate and other statistical information. However, the computed statistics are based on the assumption that there is no error present in the independent variable. 

At this point is worthwhile commenting on the computer standard estimation errors of the parameters also shown in Table 16.24. As seen in the last four estimation runs we are at the minimum of the LS objective function. The parameter estimates in the run where we optimized four only parameters (K2, kt, K k3) have the smallest standard error of estimate. This is due to the fact that in the computation of the standard errors, it is assumed that all other parameters are known precisely. In all subsequent runs by introducing additional parameters the overall uncertainty increases and as a result the standard error of all the parameters increases too. 

Once the standard error of estimate of the mean forecasted response has been estimated, i.e., the uncertainty in the total production rate, one can compute the probability level, a, for which the minimum total production rate is below some pre-determined value based on a previously conducted economic analysis. Such calculations can be performed as part of the post-processing calculations. 

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. 

Now we come to the Standard Error of Estimate and the PRESS statistic, which show interesting behavior indeed. Compare the values of these statistics in Tables 25-IB and 25-1C. Note that the value in Table 25-1C is lower than the value in Table 25-1B. Thus, using either of these as a guide, an analyst would prefer the model of Table 25-1C to that of Table 25-1B. But we know a priori that the model in Table 25-1C is the wrong model. Therefore we come to the inescapable conclusion that in the presence of error in the X variable, the use of SEE, or even cross-validation as an indicator, is worse than useless, since it is actively misleading us as to the correct model to use to describe the data. 

Table 33-1 Summary of results obtained from synthetic linearity data using one PCA or PLS factor. We present only those performance results listed by the data analyst as Correlation Coefficient and Standard Error of Estimate...

A graphical comparison of the correlation coefficient (r) and the standard error of estimate (SEE) for a calibration model. 

You may be surprised that for our example data from Miller and Miller ([2], p. 106), the correlation coefficient calculated using any of these methods of computation for the r-value is 0.99887956534852. When we evaluate the correlation computation we see that given a relatively equivalent prediction error represented as (X - X), J2 (X - X), or SEP, the standard deviation of the data set (X) determines the magnitude of the correlation coefficient. This is illustrated using Graphics 59-la and 59-lb. These graphics allow the correlation coefficient to be displayed for any specified Standard error of prediction, also occasionally denoted as the standard error of estimate (SEE). It should be obvious that for any statistical study one must compare the actual computational recipes used to make a calculation, rather than to rely on the more or less non-standard terminology and assume that the computations are what one expected. 

A graphical comparison of the correlation coefficient (r) versus the standard error of estimate (SEE) is shown in Graphic 59-4. This graphic clearly shows that when the Sr is held constant (Sr = 4) the correlation decreases as the SEE increases. 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

Standard Error of Estimate Syx = 0.4328 Slope Confidence Limits Method 1... 

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). 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

Equation 8 has a standard error of estimate of 0.07 fraction gypsum saturation over the saturation range of 0.5-2.0. The equation is useful for monitoring the actual gypsum saturation... 

Equation 10 explains 95 percent of the variation in the data for SO2 removal with a standard error of estimate of 3.2 percent SO2 removal. Values of SO2 removal predicted by Equation 10 are plotted against the corresponding measured values in Figure 7. 

Flexible optimal descriptors have been defined as specific modifications of adjacency matrix, by means of utilization of nonzero diagonal elements (Randic and Basak, 1999, 2001 Randic and Pompe, 2001a, b). These nonzero values of matrix elements change vertex degrees and consequently the values of molecular descriptors. As a rule, these modifications are aimed to change topological indices. The values of these diagonal elements must provide minimum standard error of estimation for predictive model (that is based on the flexible descriptor) of property/activity of interest. 

Measurement of precision. Measurement of data quality is valuable for both the analyst and the data user. Least-squares curve-of-best-fit statistical programs usually provide some information on precision (correlation coefficient, standard error of estimate). However, these are not sufficiently quantitative and often overstate the quality parameters of the data. 

Calibration curve quality. Calibration curve quality is usually evaluated by statistical parameters, such as the correlation coefficient and standard error of estimate, and by empirical indexes, such as the length of the linear range. Using confidence band statistics, curve quality can be better described in terms of confidence band widths at several key concentrations. Other semi-quantitative indexes become redundant. Alternatively, the effects of curve quality can be incorporated into statements of sample analysis data quality. 

Light measurement offers the combined capability of rapidly predicting by nondestructive means dust and trash content in cotton and airborne dust level. Of course, the standard error of estimate is not a practical statistic based on only six cottons and is not reported in this feasibility paper. 

Predictions of log P with regression. As would be expected, the largest values of the explained variation (r squared) and the smallest standard error of estimates found with the regression models were those that Included all 90 variables. These models... 

Studentized concentration residuals are concentration residuals that have been divided by the concentration standard error of estimate and VI— leverage. Sample leverage is a measure of the influence a sample measurement vector has on the model. 

Stat. - Statistical Theor. - Theoretical (S.E.) - Standard error of estimate... 

Multiple regression programs also calculate auxiliary statistics, designed to help decide how well the calibration fits the data, and how well it can be expected to predict future samples. For example, two of these statistics are the standard error of calibration (SEC) and the multiple correlation coefficient (R). The SEC (also called standard error of estimate, or residual standard deviation) and the multiple correlation coefficient indicate how well the calibration equation fits the data. Their formulas are given in Table 3. 

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