Experimental error estimating

Statistical analysis of the results was performed using the software Statistica 5.5 (Stat Soft). Maximum lipase activities and time to reach the maximum were calculated through fitting of kinetic curves. The maximum was estimated by derivation of the fits. Empirical models were built to fit maximum lipase activity in the function of incubation temperature (T), moisture of the cake (%M), and supplementation (%00). The experimental error estimated from the duplicates was considered in the parameter estimation. The choice of the best model to describe the influence of the variables on lipase activity was based on the correlation coefficient (r2) and on the x2 test. The model that best fits the experimental data is presented in Table 2. 

For the application of this test, the experimental error must be known. One way of determining the experimental error is to carry out a set of tests, conducted in identical conditions. The relationship between the error from model fitting and the experimental error estimate is given by Equation 90 ... 

Another solution, by far the more general one, is to carry out the experiments in a random order. If there are the same number of experiments in the design as there are parameters in the postulated model, the effect of time is to perturb the different estimations in a random fashion. If the number of experiments exceeds the number of parameters to be estimated then the experimental error estimated by multi-linear regression (see chapter 4) includes the effect of time and is therefore overestimated with respect to its true value. 

Exercise 6.9. Use the data in Table 6.10 to calculate an experimental error estimate with more than 79 degrees of freedom. 

The output from the modeling tools is often not used to the extent possible. Often, the FITEQL results used are the numerical values of the optimized parameters and the overall goodness of fit. Sometimes, also the standard deviations are considered. The numerical value of the goodness-of-fit parameter and the standard deviations are dependent on the defined experimental error estimates. The values for these, which are most frequently used, are the standard values, which may not at all be reasonable for the actual equilibrium problem treated [36]. 

The consistency between experimental error estimates used in inverse modeling and actual experimental errors... 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

An analytical method for the prediction of compressed liquid densities was proposed by Thomson et al. " The method requires the saturated liquid density at the temperature of interest, the critical temperature, the critical pressure, an acentric factor (preferably the one optimized for vapor pressure data), and the vapor pressure at the temperature of interest. All properties not known experimentally maybe estimated. Errors range from about 1 percent for hydrocarbons to 2 percent for nonhydrocarbons. 

The goal of any statistical analysis is inference concerning whether on the basis of available data, some hypothesis about the natural world is true. The hypothesis may consist of the value of some parameter or parameters, such as a physical constant or the exact proportion of an allelic variant in a human population, or the hypothesis may be a qualitative statement, such as This protein adopts an a/p barrel fold or I am currently in Philadelphia. The parameters or hypothesis can be unobservable or as yet unobserved. How the data arise from the parameters is called the model for the system under study and may include estimates of experimental error as well as our best understanding of the physical process of the system. 

Similarly, the network predicted data must be unsealed for error estimation with the experimental output data. The unsealing was performed using a simple linear transformation to each data point. 

The numbers in parentheses correspond to 1 estimated standard deviation (esd), except those for the Cpf Cn ratio, which correspond to 3 esd. The values of n and m shown are the smallest integers for which the ratio (n + m)/n lies within the experimental error limits of the measured axial ratio < Fe-<=R — t + e. The c parameters in the bottom line refer to the supercells corresponding to the underlined (n + ro)/n values ... 

The experimental error was estimated as 1-4% in k (208). In Figure 11, the more skeptical value of 5% is shown in Figure 12, the corresponding errors in E and log A are pictured at the point 29. In the coordinates logkj versus log ki, the correlation coefficient can be computed as r =. 9919 or. 9991, with or without the point 4, respectively. The corresponding standard deviations from the regression lines are. 068 and. 022 log unit, respectively. Their difference justifies the exclusion of point 4 the latter value compares favorably with the estimated error of 5% in k =. 021 log unit. 

In order to formulate the statistical problem generally, let us return to the Arrhenius graph (Figure 5) and ask the question of how to estimate the position of the common point of intersection, if it exists (162). That is, in the coordinates x = T and y = log k, a family of 1 straight lines is given with the slopes bj (i = 1,2,..3) and with a common point of intersection (xq, yo). The ith line is determined by mj points (m > 2) with coordinates (xy, yjj) where j = 1,2,..., mj. Instead of the true coordinates yy, only the values yy = yy + ey are available, ey being random variables with a zero average value and a constant variance,. If the hypothesis of a common point of intersection is accepted, ey may be identified with the experimental error. 

Of course, Sqo Sq if the difference is significant, the hypothesis of a common point of intersection is to be rejected. Quite rigourously, the F test must not be used to judge this significance, but a semiquantitative comparison may be sufficient when the estimated experimental error 6 is taken into consideration. We can then decide whether the Arrhenius law holds within experimental error by comparing Soo/(mi-21) with 6 and whether the isokinetic relationship holds by comparing So/(ml — i— 2) with 5. ... 

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. 

The derivation outlined may thus serve to explain such scattered graphs however, no possibility is offered of estimating pi and 2- The situation is complicated by the known fact that the plot of AH versus AS is statistically erroneous. The same objections apply to Leffler s special case (153) when experimental error is formally treated as an additional interaction mechanism with P2 - Texp Even in this case, no possibility is given of estimating the real Pi. 

However, because of the high price, MALDI-TOF mass spectrometers have not come into wide use. Vapor pressure osmometry (VPO), an old and traditional method for estimating molecular weight, is useful in the field of CPO chemistry. The experimental error of this measurement is approximately 10% however, the obtained data are sufficiently useful to estimate the number of porphyrins in a molecule. 

The experimental errors on the %DE measurements are estimated to be between 1 and 2 %, taking into account a relative long time span and the involvement of different lab-workers. As indicated by Table 2 the best models converge to an RMSEP of 1.5 % to refine the models further the experimental chemical errors have to be thoroughly investigated. 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

The object of these comments is, first of all, to draw attention to the fact that experimental limits of inflammability present a big experimental error and have to be handled with caution. The methods enable an estimation of the relevance of these experimental values to be made, and when these are not sufficiently reliable or are unknown - an estimation based on calculation models is made. 

The study is based on four iinear hydrocarbons (in Ci, Ce to Ca) and the model uses Antoine and Clapeyron s equations. The flashpoints used by the author do not take into account all experimental values that are currently available the correlation coefficients obtained during multiple linear regression adjustments between experimental and estimated values are very bad (0.90 to 0.98 see the huge errors obtained from a correlation study concerning flashpoints for which the present writer still has a coefficient of 0.9966). The modei can be used if differences between pure cmpounds are still low regarding boiling and flashpoints. 

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