Statistical analysis of the results

Once the best estimates of the adjustable parameters have been computed, an analysis of the results allows one to evaluate the quality of the correspondence between experimental data and mathematical model and to identify the best model among the available alternatives. This analysis consists of different steps, mainly based on 

the hypothesis that the residuals Tmj represent a sample of the universe of the experimental errors can be tested via different methods in order to single out the presence of systematic errors deriving from the inadequacy of the mathematical model. In particular, it is possible to test whether, for any measured component m, the mean of the residuals Tmj (for j = 1. IVd) is significantly different from zero (which is the expected value) and whether, for any measured component, the corrected residual variance s m is significantly different from the universe variance er2, which can be computed by resorting to repeated measurements, as shown by (3.8). 

Interestingly, any detected nonrandom behavior of the residuals may provide useful suggestions to improve the adopted model. In the case of chemical kinetics, by plotting the residuals against the reaction time, it is possible to single out the need for including a new reaction and/or a new intermediate component in the kinetic scheme. 

Since the best estimate 6 must be considered as a random variable, a third step of this analysis allows one to evaluate the accuracy of the estimate. In fact, when the estimation of the kinetic parameters is characterized by a large variance a, the relevant reaction is not supported by the experimental data, and thus, the experimental campaign must be extended, and/or the model must be changed. 

For example, let us consider the simple scalar linear model 

To determine whether an effect is statistically significant or not, a statistical interpretation method is used. Different possibilities have been described. An overview of them is given below. 

Normal probability plots or half normal probability plots (Bimbaun plots) [24,29] are graphical methods that help to decide which factors are significant. Effects that are normally distributed around zero are effects 

Interpretation methods that use t-tests The test statistic [11,13,24,27,28,39] is  

In this type of application, one often does not try to determine the degrees of liberty correctly. Some authors [11,39] simply use the value 2 for tcritical- The tcritical value (a=0.05) really depends on the number of degrees of freedom but tends to 2, especially when the number of degrees of freedom becomes larger. 

All hypothesis tests can be represented in two ways, i.e. using critical values, as described above, or using confidence intervals. The confidence limits are given by  

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In this experiment students synthesize basic copper(ll) carbonate and determine the %w/w Gu by reducing the copper to Gu. A statistical analysis of the results shows that the synthesis does not produce GUGO3, the compound that many predict to be the product (although it does not exist). Results are shown to be consistent with a hemihydrate of malachite, Gu2(0H)2(G03) I/2H2O, or azurite, GU3(0H)2(G03)2. 

In all antiseptic testing, it is recognized that skin and mucous membranes to which products ate appHed cannot be disinfected or sterilized but it is possible to significantly reduce the population of transient and resident pathogenic bacterial flora. AH in vivo test methods requite a deterrnination of the bacteria on the skin before and after treatment. Because of the normal variation in bacterial population of the skin of different people, a number of people must be tested in order to make a statistical analysis of the results. Different parts of the body are used for different tests. In aH of the tests the details of the protocol ate extremely important and must be strictly adhered to in order to obtain reproducible results. 

Besides, the statistical analysis of the results obtained confirmed that the xylan samples did not present a significant effect on the cell viability and cell proliferation rate when in direct contact with HeLa cells at the concentrations used in this study and compared to the control. 

In any book, there are relevant issues that are not covered. The most obvious in this book is probably a lack of in-depth statistical analysis of the results of model-based and model-free analyses. Data fitting does produce standard deviations for the fitted parameters, but translation into confidence limits is much more difficult for reasonably complex models. Also, the effects of the separation of linear and non-linear parameters are, to our knowledge, not well investigated. Very little is known about errors and confidence limits in the area of model-free analysis. 

As can be seen from the mean absolute deviations from experiment, all levels of theory give good overall performance for bond lengths. The poorest result for most of the theoretical procedures is observed for the O-Q bond length in 0C1, which is overestimated (by 0.023 - 0.061 A) by all the methods listed in Tables 6.2 and 6.3. This appears to be a consequence of basis set deficiencies, with improved geometries being obtained at all levels of theory with larger basis sets [28], The OCl radical has therefore been excluded from the statistical analysis of the results. 

Statistical analysis of the results was performed using applied program Microsoft Excel 98 . 

An inter-laboratory study to determine benzoie acid in orange juices was reported, where a Hamilton PRP-1 column and a mobile phase mixture of 4 + 6 of aeetonitrile / phosphate buffer was used [36]. In this work, the juices were fortified with 0.5-10 ppm of benzoic acid. Statistical analysis of the results showed a relative reproducibility between laboratories ranged in the range of 6.92% to 15.97% for juice fortified with 1-10 ppm and 27.90% for juice fortified with 0.5 ppm. Mean recoveries ranged from 94.5% to 114%. 

Statistical analysis of the results. In this part of the test statistically significant effects are identified ... 

The equations were performed on the data obtained from the 416 FC0J samples previously discussed. Qualitatively the absorption and fluorescence spectra did not indicate the presence of orange pulpwash. Therefore, the juices were examples of Case 3, and orange juice content reported as % TCM (equation 7). Statistical analysis of the results showed a mean of 99.6% TCM and standard deviation of 8.5. 

This comparison is performed on the basis of an optimality criterion, which allows one to adapt the model to the data by changing the values of the adjustable parameters. Thus, the optimality criteria and the objective functions of maximum likelihood and of weighted least squares are derived from the concept of conditioned probability. Then, optimization techniques are discussed in the cases of both linear and nonlinear explicit models and of nonlinear implicit models, which are very often encountered in chemical kinetics. Finally, a short account of the methods of statistical analysis of the results is given. 

Acute effects of 2,3,7,8-TCDD on thyroid function have been also reported in mice. In contrast with observations in rats, in which 2,3,7,8-TCDD appears to have independent effects on T4 and T3 levels, serum T4 and T3 levels were decreased in a dose-dependent fashion in male C57BL/6J mice 8 days after a single gavage dose of 0.03-235 g 2,3,7,8-TCDD/kg (Weber et al. 1995). A similar effect was observed in male DBA/2J mice treated with a single dose of 1-3,295 g/kg. In C57BL/6J mice, maximum depression of thyroid hormones (35% of controls) was achieved with a dose of 133 g/kg. In male DBA/2J mice, maximum reductions in T3 and T4 levels (40 and 20% of controls, respectively) were attained with the highest dose level (Weber et al. 1995). It should be noted that the Weber et al. (1995) study did not include statistical analysis of the results. 

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. 

Salem et al. [48] reported simple and accurate methods for the quantitative determination of flufenamic, mefenamic and tranexamic acids utilizing precipitation reactions with cobalt, cadmium and manganese. The acidic drugs were precipitated from their neutral alcoholic solutions with cobalt sulfate, cadmium nitrate or manganese chloride standard solutions followed by direct determination of the ions in the precipitate or indirect determination of the ions in the filtrate by atomic absorption spectroscopy (AAS). The optimum conditions for precipitation were carefully studied. The molar ratio of the reactants was ascertained. Statistical analysis of the results compared to the results of the official methods revealed equal precision and accuracy. The suggested procedures were applied for determining flufenamic, mefenamic and... 

High and low in this context refer to TDF exposures for which the TPD spectra were dominated by the low (P) and high (a) temperature features, respectively. The uncertainties in the magnitude of the fragment ratios derived from a statistical analysis of the results of many independent measurements does not exceed 7% for all species for example, for Ag at high exposure the area under the mle = 30 peak was 49 3, except for mle = 28, for which the error was much greater (ca. 50%). 

Although considerable progress has been made in the metabolic profile approach, a number of problems remain to be overcome. Many of these centre around the fluctuations in component composition, not from metabolic disorders, but brought about by other influences. These are principally due to diet and the metabolic variations in individuals in relation to activity. Drugs can also affect the excretion levels of compounds, in addition to the production of their own metabolites. These factors all make quantitative data difficult to obtain and evaluate. Careful statistical analysis of the results are necessary and a population of 500 subjects, grouped in age and sex, has been studied with a view to obtaining a suitable data base for urinary organic acids [370]. 

The kg and kp values for the investigated wastewaters have been calculated by statistical analysis of the results. Dependence of these coefficients on the mixing ratio of the oil refinery and municipal wastewater has been illustrated in Fig. 7. 

Within each grid cell we could take three random samples and calculate an overall average for the target area (Figure 2.2) from the analysis results for the 75 samples. This stratified random sampling design allows statistical analysis of the results. We can calculate an upper confidence limit of the overall average concentration and test the null hypothesis that the soil is contaminated above the action threshold of 1%. 

With regard to the statistical analysis of the results, what will be your null hypothesis ... 

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