The Chi-Squared Function

It has been shown, in Sect. 4.2.1, how to estimate from a sample mean the limits within which the true mean of a population will fall with a desired level of confidence C. It may be desirable to estimate, now, the limits within which the population standard deviation will fall with a desired level of confidence C from the sample standard deviation s. The target is always the same to estimate true values when the entire population of data is not available. To this purpose, the method of the so called (Chi-squared) function may be used. Again, as in aU preceding sections, it must be assumed that the distribution is normal. The relation between 5 and A is given by 

S 0 o oolo ooor 0-H -.rna Omooolo oa a 0 t ppppppt piOi0 ppfNfNCNfNppppppp t ioio rnrnrnrnrnrnrnrnrnrnrnrnrnrnrnrnrnriririri 

Values of yj corresponding to a given probability a associated to (4.31) are shown in the diagram of Fig. 4.11. The confidence interval is that between a and 1 — a. 

---

The Chi-Squared function was used with the 70% confidence interval and 40(2 X (19 + 1)) degrees of freedom. The +1 is added to account for the possibility that another failure may occur in the next moment of operation. 

Note Impedance spectra were modeled using complex nonlinear least-squares (CNLS) fitting program EQUIVCRT developed by B. A. Boukamp. The Chi-square function gives a good indication of the quality of the fit a value of... 

Limits to the mean and standard deviation have been discussed in the previous sections based upon Student s t function and the Chi-squared function. While the theory of confidence intervals for these two quantities is well developed, such is not the case for the general nonlinear fitting of parameters to a data set. This will be discussed in the next chapter on general parameter estimation. For such cases, about the only approach to confidence intervals is through the Monte Carlo simulation of a number of test data sets. This approach is also applicable to limits on the mean and standard deviation and will be discussed here partly as background for the next chapter and as another approach to obtaining confidence intervals on statistical quantities. 

It may be decided that the gamma prior cannot be greater than a certain value xf. This has the effect of true Ling the normalizing denominator in equation 2.6-10," and leads to equation 2.6-17, where P(x v) is the cumulative integral from 0 to over the chi-squared density function with V degrees of freedom, a is the prescribed confidence fraction, and = 2 A" (t+Tr). Thus, the effect of the truncated gamma prior is to modify the confidence interval to become an effective confidence interval of a ... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

CHIDIST is the Excel function for the one-tailed probability of the chi-squared distribution. 

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... 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. The Microsoft Excel function FDIST(X, dfh df2) gives the upper percent points of Table 3-8, where X is the tabular value. The function FINV(Percent, dfh df2) gives the table value. 

There are several mathematically different ways to condnct the minimization of S [see Refs. 70-75]. Many programs yield errors of internal consistency (i.e., the standard deviations in the calculated parameters are due to the deviations of the measured points from the calculated function), and do not consider external errors (i.e., the uncertainty of the measured points). The latter can be accommodated by weighting the points by this uncertainty. The overall rehabU-ity of the operation can be checked by the (chi square) test [71], i.e., S (L + N - ) should be in the range 0.5-1.5 for a reasonable consistency between the measured points and the calculated parameters. 

The last term of Equation a-7 is the chi-square (x ) probability function and... 

Moreover, because the Mahalanobis distance is a chi-square function, as is the SIMCA distance used to define the class space in the SIMCA method (Sect. 4.3), it is possible to use Coomans diagrams (Sect. 4.3) both to visualize the results of modelling and classification (distance from two category centroids) and to compare two different methods (Mahalanobis distance from the centroids versus SIMCA distance). 

Suppose a polydisperse system is investigated experimentally by measuring the number of particles in a set of different classes of diameter or molecular weight. Suppose further that these data are believed to follow a normal distribution function. To test this hypothesis rigorously, the chi-squared test from statistics should be applied. A simple graphical examination of the hypothesis can be conducted by plotting the cumulative distribution data on probability paper as a rapid, preliminary way to evaluate whether the data conform to the requirements of the normal distribution. 

The log-likelihood function at the maximum likelihood estimates is -28.993171. For the model with only a constant term, the value is -31.19884. The t statistic for testing the hypothesis that (3 equals zero is 5.16577/2.51307 = 2.056. This is a bit larger than the critical value of 1.96, though our use of the asymptotic distribution for a sample of 10 observations might be a bit optimistic. The chi squared value for the likelihood ratio test is 4.411, which is larger than the 95% critical value of 3.84, so the hypothesis that 3 equals zero is rejected on the basis of these two tests. 

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 ... 

The chi-square distribution was discussed briefly in the earlier section on probability distributions. Suppose we have (k+1) independent standard normal variables. We then define % as the sum of the squares of these (k+1) variables. It can be shown that the probability density function of % is ... 

There are two restrictions to the chi-squared test. One is that the test should not be used when expected frequencies are less than five. The other restriction is that when only one degree of freedom is involved, the difference (/ - / ) is reduced by 0.5 before squaring. This is necessary to correct for a bias in the numerical answer when the chi-squared distribution, which is a continuous mathematical function, is applied to discrete numbers. Adjustment is not necessary when the degrees of freedom exceed one. 

All the data analysis methods shown in Fig. 3 involve linear or nonlinear regression of ACF data, (representing data point j of Gi2 g(2K or j 1 ). to fit a proposed model, yjnixlel. The model parameters or amplitudes of a proposed distribution are adjusted until a characteristic function is minimized or maximized. The characteristic function is often the chi-square [Pg.218]

Alternative characteristic functions have also been used. For example, the sum of the absolute values of the residuals (53] has been evaluated for fitting C(11 data and produced results similar to those based on the chi-square statistic 49. When one sets w, = 1, the characteristic function in Eq. (52) is known as the L2 norm. 

By means of eq.(3), we have simulated the observed fluorescence decay curves of CNA in the presence of CHD. In the course of the simulation, the time walk between the response function of the system and the fluorescence decay curve was decided to minimize the chi-square parameter. A small change of the time walk leads to a large difference in the fitting parameter b at the low concentration of CHD. R and D values were obtained from the linear relation of a and b vs. [CHD] with the aid of eqs.(4) and (5). 

The chi-square distribution is used to perform statistical tests on the sample variance. It is highly asymmetric for small values of n, but becomes more symmetric and similar to a normal distribution as n becomes large, such as 20 or 30. The cumulative distribution function of the chi-square distribution is listed in Table 3.4 as a function of v and a, where v = - 1 is the number of degrees of freedom and a is the percentage of the distribution above the particular Microsoft Excel has built-in functions, CHIDIST and CHIINV, that compute a chi-square distribution [5, 6]. 

Figure 14 The chi-square distribution for one and two degrees of freedom. The lower the number, v of degrees of freedom, the higher the fraction of low-intensity transitions. At a higher number of degrees of freedom the function becomes localized in a Gaussian-like fashion.

We also can calculate the parameter confidence intervals. We merely compute the size of the ellipse containing a given probability of the multivariate normal. That can be shown to be the chi-square probability function [5], Given the number of estimated parameters, rip, and the confidence level, then... 

By the two approximations, the probability density function of S)c can be well approximated by the product of the Chi-square distributions ... 

Look up the chi-squared value (x ) from distribution tables, or in spreadsheet applications, use the CHIINV function (if available) ... 

---