Chi-square statistics

Each element ,y of a contingency table X can be thought of as a random variate. Under the assumption that all marginal sums are fixed, we can derive the expected values E(x,y) for each of the random variates [1]  

For example, the observed value for Clonazepam in anxiety has been recorded as 0 in Table 32.4. The corresponding expected value in Table 32.5 has been computed from eq. (32.3) as follows  

The generalization of Pearson s chi-square statistic for 2x2 contingency tables, which has been discussed in Section 16.2.3, can be written as  

In the case of the 4x3 contingency Table 32.4 we obtain a chi-square value of 

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The global distance of chi-square 8 of a contingency table X is derived from the chi-square statistic as follows ... 

Principal component analysis can be further extended to study the chi-square statistic, since... 

Xa v) chi-square statistic, critical at probability a, with degree of freedom v... 

To apply White s test, we first obtain the residuals from the regression of Yon a constant, X, and W2. Then, we regress the squares of these residuals on a constant, X, X2, X2, X2, and XxX2. The R1 in this regression is. 78296, so the chi-squared statistic is 50x0.78296 = 39.148. The critical value from the table of chi-squared with 6 degrees of freedom is 12.5916, so we would conclude that there is evidence of heteroscedasticity. 

Referring back to the ordinary least squares regression, we now compute the mean squared residual, 1911.9275/50 = 38.23855. Then, we compute v = (1 /38.23855)t",2 for each observation. In the regression of v on a constant, Xx, and X2. the regression sum of squares is 145.551, so the chi-squared statistic is 145.551/2 = 72.775. We reach the same conclusion as in the previous paragraph. In this case, the degrees of freedom for the test are only two, so the conclusion is somewhat stronger. 

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. 

Ratio of likelihood scores for selected tree and star phylogeny 2, 8, 9 is treated as a chi-square statistic with one degree of freedom. Alternatively, standard normal test of the mean and variance of the difference of their likelihood scores can be used to compare one tree to another... 

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. 

This is especially true because there will be an inherent direct improvement in the fit in any case, on account of additional variables being added in the simulation. It is therefore useful to measure the fit quality using a reduced chi-squared statistic (f ) that weights the quality of the fit relative to the number of degrees of freedom in the data, as outlined in equation (7). ... 

Example 3.5 Evaluation of Chi-Squared Statistics Consider that, for a given measurement, regression of a model to real and imaginan/ parts of impedance data yielded = 130. Measurements were conducted at 70 frequencies. The regressed parameters needed to model the data included the solution resistance and 9 Voigt elements, resulting in use of 19 parameters. Under assumption that the variances used in the evaluation ofxf were obtained independently, evaluate the hypothesis that the x value cannot be reduced by refinement of the model. 

CHI-square statistics —> statistical indices (0 concentration indices)... 

Chi-square statistics. It is the ratio of the square difference between the observed and theoretical expected values over the expected values ... 

ANACONDA then calculates the value of the Pearson s chi-squared statistic and the adjusted Pearson residual values. Pearson s statistic represents a global measure of the difference between observed and expected codon frequencies (20). 

In stable, medically-treated patients with confirmed CAD, Iskandrian et al. (32) found that the extent of perfusion defect conveyed incremental and independent prognostic information above clinical, exercise, and catheterization data combined. Event-free survival over a period of 28 15 months was 95% in those with a defect <15% of the myocardium versus 75% for those with a >15% defect (P <0.001). Interestingly, combining gender, clinical, and SPECT data appeared to provide equivalent predictive power (chi-square statistic) regardless of whether or not catheterization data was added. As expected, an exercise capacity of >6 METs, an ejection fraction of >50%, and a lesser extent of CAD were all also associated with improved survival. 

This is the familiar least-squares form, with the statistical weight of each point being the reciprocal of its variance. The value of F is a measure of the goodness of fit. [For constant ai( F is the chi-square statistic]. 

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