Fitting Individual Data

As a last comment, caution should be exercised when fitting small sets of data to both structural and residual variance models. It is commonplace in the literature to fit individual data and then apply a residual variance model to the data. Residual variance models based on small samples are not very robust, which can easily be seen if the data are jackknifed or bootstrapped. One way to overcome this is to assume a common residual variance model for all observations, instead of a residual variance model for each subject. This assumption is not such a leap of faith. For GLS, first fit each subject and then pool the residuals. Use the pooled residuals to estimate the residual variance model parameters and then iterate in this manner until convergence. For ELS, things are a bit trickier but are still doable. 

FIGURE 11.14 Data set consisting of a control dose-response curve and curves obtained in the presence of three concentrations of antagonist. Panel a curves fit to individual logistic functions (Equation 11.29) each to its own maximum, K value, and slope. Panel b curves fit to the average maximum of the individual curves (common maximum) and average slope of the curves (common n) with only K fit individually. The F value for the comparison of the two models is 2.4, df = 12,18. This value is not significant at the 95% level. Therefore, there is no statistical support for the hypothesis that the more complex model of individual maxima and slopes is required to fit the data. In this case, a set of curves with common maximum and slope can be used to fit these data. 

Multistranded cables and ribbons were also observed for calcitonin (Bauer et al, 1995). These appeared to contain laterally associated 8-nm-wide fibrils, whereas the diameter of the individual strands within the multistranded cables could not be measured directly from the images. To estimate this diameter, the authors plotted the helical crossover spacing of the cables as a function of their diameter. Using a linear regression fit, the data were extrapolated to zero, yielding a width of 4.1 nm, similar to the width of the single protofibrils (Fig. 2A Bauer et al., 1995). 

The sums of squares of the individual items discussed above divided by its degrees of freedom are termed mean squares. Regardless of the validity of the model, a pure-error mean square is a measure of the experimental error variance. A test of whether a model is grossly adequate, then, can be made by acertaining the ratio of the lack-of-fit mean square to the pure-error mean square if this ratio is very large, it suggests that the model inadequately fits the data. Since an F statistic is defined as the ratio of sum of squares of independent normal deviates, the test of inadequacy can frequently be stated... 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

One way of linearizing the problem is to use the method of least squares in an iterative linear differential correction technique (McCalla, 1967). This approach has been used by Taylor et al. (1980) to solve the problem of modeling two-dimensional electrophoresis gel separations of protein mixtures. One may also treat the components—in the present case spectral lines—one at a time, approximating each by a linear least-squares fit. Once fitted, a component may be subtracted from the data, the next component fitted, and so forth. To refine the overall fit, individual components may be added separately back to the data, refitted, and again removed. This approach is the basis of the CLEAN algorithm that is employed to remove antenna-pattern sidelobes in radio-astronomy imagery (Hogbom, 1974) and is also the basis of a method that may be used to deal with other two-dimensional problems (Lutin et al., 1978 Jansson et al, 1983). 

Finally, the CECDC mechanism with more than one rate-determining step is considered. By a procedure similar to that described in the previous section, but now in terms of the second-order Taylor expansions of the individual rate equations, explicit expressions for the three second-order parameters in eqn. (140) have indeed been derived [7]. Although they are not as surveyable as their first-order counterparts, the expressions compiled in Table 7 are sufficiently simple to be suitable for fitting experimental data. 

Tables I and II provide additional statistical data that can be used to qualify the estimates derived from the fitting process. C is the standard deviation of y, its numerical value is largely determined by the sampling error arising from the selection of test specimens. C is the standard deviation of the S fs, which is a measure of theSinhomogeneity of the lot of SRM material. C is the standard deviation of the residuals from the fit, which is a measure of the extent to which individual data values depart from the model in equation 6. We have chosen not to construct the usual confidence or tolerance intervals because we do not have enough data on the distribution of the S s.

From these constants, vapor pressure can be estimated for any desired temperature, and this is done for 25°C. or the melting point if that is higher. Since individual data points were lost in the curve fitting, some indication of variability is needed this is provided as the standard deviation for the regression and the 95% confidence interval for the calculated value at 25°C, (or the melting point). 

In a situation with many data from each individual drawn in an inter subject balanced manner, a two stage method is very often used each individual is fitted individually without considering the inter individual dependencies. In a second step, the parameters are resumed as population mean and standard deviation, often considered as inter... 

Alternatively, response additivity (RA) for independently acting chemicals as a mathematical null model for testing observed responses (associated with the pharmacological concept of independent joint action) and with an assumed correlation of sensitivities of 0 also often fits the data well. Again, misfits occur (e.g., when the test mixture consists of compounds with the same MOA at concentrations below the individual compound s no-effect concentrations), and when they occur, they are often in the tails of the response curves. 

Fig. 9.17. Upper curves are photoemission data for a) the valence band and (6) the conduction band of thin a-Si3N4 H layers on a-Si H. The dotted curves are fits to the spectra using the measurements of the individual bands shown in the lower curves. The band offsets, A y and used to fit the data, are indicated (Iqbal et at. 1987).

Different values of the constant and exponent r were used for different system geometries and operating conditions. Thus, for the turbine agitators at a Reynolds number less than 7400, was 0.00058 and r was 1.4. For the turbine agitators at a Reynolds number above 7400, was 0.62 and r was 0.62. For the propellers, over a range of Reynolds numbers from 3300 to 330,000, was 0.0043, and r was 1.0. In every case, the Schmidt number exponent s was 0.5. Although these equations fit the data fairly well, there is a scatter of the individual points. Hence, for any particular solid-liquid combination at constant temperature in a... 

This is because although 0 = (10), in general, cr(10) oQ (it will usually be less). In principle, the quantities we have defined, E(t), Dit), Gif), and J(i), provide a complete description of tensile and shear properties in creep and stress relaxation (and equivalent functions can be used to describe dynamic mechanical behavior). Obviously, we could fit individual sets of data to mathematical functions of various types, but what we would really like to do is develop a universal model that not only provides a good description of individual creep, stress relaxation and DMA experiments, but also allows us to relate modulus and compliance functions. It would also be nice to be able formulate this model in terms of parameters that could be related to molecular relaxation processes, to provide a link to molecular theories. 

One method is simple visual inspection of the generated master curve. If the fit of the individual data sets is poor, subsequent extrapolation is open to significant error. Another method is to determine an activation energy plot (of the shift of T with frequency) for the data set and compare temperatures or frequencies calculated from the resulting Arrhenius equation to those read from the nomograph. 

Analysis of the shape of error surfaces. To conclude this section, we consider a more quantitative approach to error estimation. The first step is to estimate the accuracy of the individual data points this can either be done by analysis of the variability of replicate measurements, or from the variation of the fitted result. From that, one can assess the shape of the error surface in the region of the minimum. The procedure is straightforward the square root of the error, defined as the SSD, is taken as a measure of the quality of the fit. A maximum allowed error is defined which depends on the reliability of the individual points, for example, 30% more than with the best fit, if the points are scattered by about 30%. Then each variable (not the SSD as before) is minimised and also maximised. A further condition is imposed that the sum of errors squared (SSD) should not increase by more than the fraction defined above. This method allows good estimates to be made of the different accuracy of the component variables, and also enables accuracy to be estimated reliably even in complex analyses. Finally, it reveals whether parameters are correlated. This is an important matter since it happens often, and in some extreme cases where parameters are tightly correlated it leads to situations where individual constants are effectively not defined at all, merely their products or quotients. Correlations can also occur between global and local parameters. 

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