Chi-square fitting

For comparison, the corresponding tfb values for the nonchiral selectand DNB-Gly obtained by ITC are included. A relative error, AK /K, estimated on the basis of the chi-square fitting in which an error of 5% of the measured quantity was assigned for each titration point is about 30% (ITC) and 50% (CD, UV). 

If the standard deviations Oy for all elements of the matrix Y are known or can be estimated, it makes sense to use this information in the data analysis. Instead of the sum of squares as defined in Equation 7.6, it is the sum over all appropriately weighted and squared residuals that is minimized. This is known as chi-square fitting [15, 16],... 

An advantage of the cumulants approach is that it is computationally very fast. A chi-squared fitting error parameter serves to test whether the assumed Gaussian shape in diffusivities is reasonable. The calculated values of mean size and polydispersity are reasonable (chi-squared approaching unity) for approximately symmetrical distributions having a coefficient of variation within 25% of mean size. 

A similar chi-square fitting procedure was carried out in ref. [Ra 85] using the RIA model, where the neutron vector density was parametrized as in eq. (5.1) and the neutron scalar density was obtained from eq. (4.35), where p y(0 this equation was replaced by p v"(r). This method of generating the scalar density prevented the neutron-proton density variation from affecting the very sensitive... 

The chi-square distribution can be applied to other types of apph-catlon which are of an entirely different nature. These include apph-cations which are discussed under Goodness-of-Fit Test and Two-Way Test for Independence of Count Data. In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. 

In order to determine the quality (or the validity) of fit of a particular function to the data points given, a comparison of the deviation of the curve from the data to the size of the experimental error can be made. The deviations (i.e., the scatter off the curve) should be of the same order of magnitude as the experimental error, so that the quantity chi-squared is defined as... 

As regards the x-ray data, the conclusion in the text was reached by applying the chi-square goodness-of-fit test. See C. A. Bennett and N. L. Franklin, Statistical Analysis, page 620, for details of the test. 

The goodness-of-fit between the experimental and theoretically calculated CBED rocking curves is described by a merit function, and in the present study we use the chi-square merit function defined as... 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

It is estimated that, for reasonably good fits, the Unear results hold, i.e., c2 is distributed as a chi-squared variable with (m—n) degrees of freedom and... 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

The quality of fits to simulated decays wasjudged by the usual criteria the reduced chi-square, x2 and visual inspection of the residual, r, and weighted residuals, rWj, plots. 56,57lXri is given by... 

Figure 4.11. Reduced chi-square for fitting a single Gaussian distribution function of decays with either a discrete single or double exponential model as a function of dis tribud on width (/f)...

A statistical term referring to a monoparametric distribution used to obtain confidence intervals for the variance of a normally distributed random variable. The so-called chi-square (x ) test is a protocol for comparing the goodness of fit of observed and theoretical frequency distributions. 

Fig. 3 Comparison of the surface tension for nonionic surfactant CnEg as measured at T = 298.15 K, data points [45], with improved models considering orientational states of surfactant molecules at the surface. The data shown are obtained by regression analysis minimizing the revised chi-square The calculation with fi = 0 represents the best fit of the improved Szyszkowski-Langmuir model described by Eqs. 21 and 22. The other calculated curve with =- 3.921 shows the best fit of the improved Frumkin adsorption model described by Eqs. 23 and 24...

The XPS spectra were recorded on a Surface Science Laboratories small spot system using a monochromatized A1K X-ray radiation source. The take-off angle used for these measurements was 35°. Full details of the methods used in interpreting the XPS data have been described elsewhere [14], Data reduction was done using Surface Science Laboratories software version 8.0. This software utilizes a least squares curve fitting approach with only chi square statistics for goodness of the calculated fit to the experimental data. 

The decay parameters [a (X) and rj are recovered from the experimentally measured phase shift and demodulation factor by the method of non-linear least squares (24,25). The goodness-of-fit between the assumed model (c subscript) and the experimentally measured (m subscript) data is determined by the chi-squared (x2) function ... 

Attenuation Coefficient Measured in the Calibration Experiment The Uncertainty is the One-Sigma Fit Error. The Reduced-Chi-Square for Each Fit was of Order 1. 

The primary statistical tests used in the studies described in this text are based on the chi-square tests which are in turn derived from the chi-square distribution which is based on the chi distribution. These tests include the chi-square test for goodness of fit, the chi-square test of independence, and Fisher s Exact Test. There are also corrections to some of the tests that account for small number deviations, Yates Correction for Continuity, and for multiple studies attempting to verify the same procedures or processes, Bonferroni s correction. 

The number of degrees of freedom used with the chi-square distribution associated with the 2-dimensional distribution would be (N- 1)(M- l)-m, where m is the number of independent parameters estimated from the measurements. For the -dimensional case, the degrees of freedom used with the chi-square distribution would be (N1- )(/V, - I)... (Nf - I) - m, where m is the number of independent parameters estimated from the measurements. The steps used in the implementation of the chi-square test of independence are essentially the same as those listed for the chi-square test for goodness of fit. The only difference is that the expected values must be calculated for all NxM cases in the two-dimensional distribution and for all. .. Nk cases in the -dimensional distribution. The expected values for the cells are often arranged in a table that resembles the contingency table or are sometimes included, inside parentheses, within the same cell of the contingency table as the measurement. 

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