Distance of chi-square

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

There are three ways by which the global distance of chi-square can be meaningfully rewritten as a weighted sum. These correspond with the three different ways of closing the data in the original contingency table X, such as has been described above in Section 32.3. The metric matrices W and have to be defined differently for each of the three cases. 

In the case of the contingency Table 32.4 we obtained a chi-square of 15.3. Taking into account that the global sum equals 30, this produces a global interaction of 15.3/30 = 0.510. The square root of this value is the global distance of chi-square which is equal to 0.714. 

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In this respect, the weight coefficients are proportional to the column-sums. Distances of Chi-square form the basis of correspondence factor analysis (CFA) which is discussed in Chapter 32. 

The terms in the expression of the global distance of chi-square 8 can be rearranged into ... 

Weighted sums of squares c of the row-profiles in F around the origin of can be expressed as distances of chi-square ... 

The distance 5 - between two row-profiles / and i can also be expressed formally as a distance of chi-square ... 

Using the deviations Z of the double-closed data from their expected values and the column-weights in Table 32.6, we compute the distances of chi-square from... 

Similarly, we compute the distance of chi-square from the origin for epilepsy, using the same deviations Z and the row-weights w ... 

In the following section on the analysis of contingency tables we will relate the distances of chi-square in terms of contrasts. In the present context we use the word contrast in the sense of difference (see also Section 31.2.4). For example, we will show that the distance of chi-square from the origin 5, can be related to the amount of contrast contained in row i of the data tables, with respect to what can be expected. Similarly, the distance 5 can be associated to the amount of contrast in column j, relative to what can be expected. In a geometrical sense, one will find rows and columns with large contrasts at a relatively large distance from the origin of and S", respectively. The distance of chi-square 5- then represents the amount of contrast between rows i and i with respect to the difference between their expected values. Similarly, the distance of chi-square indicates the amount... 

The squared Chi-square distance is appropriate for the analysis contingency tables (when the data represent counts) and for cross-tabulations (when the data represent parts of a whole) ... 

If the data majority is multivariate normally distributed, the squared score distances can be approximated by a chi-square distribution, y2, with a degrees of freedom. 

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). 

Objects do not fall exactly into the inner model space, and a residual error on each variable can be computed. These residuals are uncorrelated variables, because each significant correlation is retained in the linear model. So, the variance of residuals is a chi-square variable, the SIMCA distance, and, multiplied by a suitable coefficient obtained from the F distribution, it fixes the boundary of the class space around the model, called the SIMCA box, that corresponds to the confidents hyperellipsoid of the bayesian method. Objects, both those used and those not used to obtain the... 

Finally, the warm happy feeling benefits strongly from indications of internal consistency and plausibility of the structural results themselves rational values for ordinary distances and angles, benzene rings that are regular and planar, etc. It does no harm to have more than one molecule per asymmetric unit and to show (by chi square testing or half-normal probability plots where appropriate) that the structural results are mutally consistent. 

The resulting value for the distance is 2.1 1 (P-, which is smaller than the chi-squared value for probability 0.01 and 1 degree of freedom. 

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