Estimation of error

Statistics teaches that the deviation of data based on less than 30 measurements is not a normal distribution but Student s t-distribution. So it is suitable to express the binding constant K with 95% confidence interval calculated by applying by Student s t-distribution. Student s t-distribution includes the normal distribution. When the number of measurements is more than 30, Student s t-distribution and the normal distribution are practically the same. The actual function of Student s t-distribution is very complicated so that it is rarely used directly. A conventional way to apply Student s t-distribution is to pick up data from the critical value table of Student s t-distribution under consideration of degree of freedom , level of significance and measured data. It is troublesome to repeat this conventional way many times. Most spreadsheet software even for personal computers has the function of Student s t-distribution. Without any tedious work, namely, picking up data from the table, statistical treatment can be applied to experimental results based on Student s t-distribution with the aid of a computer. In Fig. 2.12, an example is shown. When the measurement data are input into the gray cells, answers can be obtained in the cell D18 and D19 instantaneously. 

When the confidence interval obtained after statistical treatment is very wide, there is high probability that a precise experiment has not been carried out. The experimental conditions and also each procedure should be checked then. 

The method described here includes no approximation at the data treatment step, so it can be used generally. In addition, the required level of mathematical knowledge is not high, only a formula for polynomials of degree 2, therefore the logical basis can be easily understood. Moreover, statistical treatment of the obtained data is understandable with primary statistics. These are the merits to use this method at first in order to understand the way of determination of binding constants. 

When stoichiometry of the complex is not 1 to 1 or when other premises are not satisfied, the way of data treatment should be changed or modified. Nonlinear least 

Practical Course of Action for NMR Spectroscopic Binding Constant Determination 

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One asterisk indicates significance at 95%, two asterisks at 99% level. NS, not significant at 95% level. Calculated by dividing mean square of line by mean square for error in this case deviations from double regression are used as an estimate of error. Significance determined from tables cf., e.g., G. W. Snedecor, Statistical Methods, 4th Edn. Iowa State College Press, Ames, 1946. 

The Hamming method [12] applies a predictor y , then a modifier y which provides a correction for the estimate of error in the predictor and corrector, and then iterates the corrector y" as desired. The procedure is... 

Some important conditions concerning the estimation of error should be pointed out. First, modulus measurements of rectangular bars are made in torsion and the calculations contain assumptions that may depend on geometry. How this influences error, particularly at low torque levels is not known. Second, the strains were kept constant at 0.1% other strains might not yield the same results. On the other hand one would expect an inverse proportionality to exist between the magnitudes of error and strain. Thirdly, these errors were estimated for a frequency of 1Hz. 

Wood, R.H., Estimation of errors in free energy calculations due to the lag between the Hamiltonian and the system configuration, J. Phys. Chem. 1991, 95, 4838 1842... 

Sinusoidal fitting is more flexible than Fourier methods, as it does not require evenly spaced phase steps. There is no special convenience associated with sampling of an angle of 2n and estimation of errors in parameters is somewhat more straightforward. 

It is not the purpose of this paper at this moment to investigate further for more detailed reasons for discrepancies in confidence bands or estimated amount intervals. That will be investigated fully at a later time. I do wish to point out that the assumptions one makes about the information he has and the statistical approaches he makes profoundly affect the resultant error calculations. Far from being a staid and dormant subject matter, statistical estimations of error are currently very actively being studied in order for scientific workers and citizens alike to be informed about the error in their work. 

In references 71 and 72, SST limits are defined based on experience, and the examined responses should fall within these limits. The two papers do not provide much information concerning the robustness test performed. Therefore, it is not evident to comment on the analysis applied, or to suggest alternatives. In reference 73, a graphical analysis of the estimated effects by means of bar plots was performed. In reference 74, a statistical analysis was made in which an estimation of error based on negligible two-factor interaction effects was used to obtain the critical effects between levels [—1,0] and [0,4-1]. 

Derivative of intensity against structure parameters and thickness can be obtained using the first order perturbation method [31]. The finite difference method can also be used to evaluate the derivatives. Estimates of errors in refined parameters can also be obtained by repeating the measurement. In case of CBED, this can also be done by using different... 

Bischoff and Levenspiel (B14) considered this problem, and have presented design charts which allow estimation of errors in the calculated dispersion coefficients for various conditions. It was found that when the ratio of injection to tube diameter is less than 20%, or e < 0.2, then the assumption of a point source, or e-> 0, was good to within 5%. Thus for many cases, the neglect of the finite size of injection tube is justified. 

Error propagation analysis is the estimation of error accumulation in a final result as a consequence of error in the individual components used to obtain the result. Given an equation explicitly expressing a result, the error propagation equation can be used to estimate the error in the result as a function of error in the other variables. 

Estimate of errors is fundamental to all branches of natural sciences that deal with experiments. Very frequently, the final result of an experiment cannot be measured directly. Rather, the value of the final result (u) will be calculated from several measured quantities (x, y, Z-., each of which has a mean value and an error) ... 

For this arrangement, higher-order interactions are assumed to be negligible and their sums of squares are pooled to give an estimate of error. From the ANOVA table it appears that in the sub-plot analysis there... 

A commonly used invalid estimate is called the re-substitution estimate. You use all the samples to develop a model. Then you predict the class of each sample using that model. The predicted class labels are compared to the true class labels and the errors are totaled. It is well known that the re-substitution estimate of error is highly biased for small data sets and the simulation of Simon et al. (14) confirmed that, with a 98.2% of the simulated data sets resulting in zero misclassifications even when no true underlying difference existed between the two groups. 

Figure 12, the correlation between liquor yield and R , shows that the liquor product decreases as the average vitrinite R increases. However, the correlation between R and liquor is poor (r = —0.72 ). The degree of scatter (Sy = 1.26, where Sy = standard estimate of error for t/) is probably a function of moisture differences in the subject coals high percentages of moisture increase liquor yields (17). 

The exchange broadening Awe may be either measured directly from the spectrum or calculated from equation (177) upon substituting for K the value obtained from the iterative analysis. The procedure, after due modifications, can be employed for the estimation of errors for more complicated exchanging systems. 

We have previously introduced the sum of squares due to error as MSE=SSE/(n-2) and said that it is the unbiased estimate of error variance a2 because E(MSe)=o2 no matter whether the null hypothesis H0 Pi=0 is correct or not. It is easy to prove that the expected value of the regression mean square, MSR=SSR/1, is the biased estimate of variance o2 if not Pi=0. This can be written in the form ... 

Here Df is the final, composite particle diameter, Df is the initial (seed) diameter, and Dp is the projected diameter. Df is obtained by using the previously reported (20) interative procedure. The method of calculating Dp can be found in the Appendix of that previous report. The independent estimate of error on y, determined from the present and related studies is. 16 (22 degrees of freedom). Because of random error, y > 1.00 should often be observed when complete association occurs. 

A quite similar consideration was published by INGAMELLS and SWITZER [1973] and by INGAMELLS [1974b 1976]. A sampling constant Cs is proposed which enables estimation of error of subsampling, i.e. the withdrawal of a small portion from a well mixed material. The relationship to GY s equation (Eq. 4-7) is given by ... 

Most recent data sets are accompanied by either estimates of errors of the measurements or observed sample-to-sample fluctuations which were entered into the library, as modern CMB programs (e.g., 13) use uncertainties of the source compositions and in ambient samples in weighted least-squares fitting procedures. Many older data sets do not include estimates of analytical errors or fluctuations of particle composition, so we attached estimates based on the... 

Quantitative estimates of errors in the determined numerical values of substance concentrations at points xeq and xext as a function of the structure and dimension of x and y are obtained with great difficulty. It is only clear that when we are interested in the detailed composition of products, it is desirable to increase this dimension with thorough choice of the set of components x, and y, based on the whole preliminary knowledge about specific features of the studied process. Such an increase will be limited by the possibility to analyze numerous results. However, despite the great sophistication of the problem of specifying a list of substances, it is solved much easier than the problem of specifying a process mechanism. Both the list of elementary reactions (that can include many hundreds and even thousands of elements) and the constants of their rates are hard by far to determine than the list and thermophysical properties of reactants of the studied system. 

Temp., This Literature Estimate of Error 95% Confidence... 

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