Cross-classification

Table 2. Cross-classification of individuals for true versus observed sero-status (+, —) in terms of sensitivity Se, specificity Sp, and probability p of positive sero-status True...

Each column i, i = 1,2,..., s, in this table represents the values of the i-th characteristic, and y denotes the corresponding p component vector. If now Y = y, ..., y is the characteristics set where y is from then yf is the value of the fc-th set of characteristics with respect to the object xL As indicated, some practical problems require simultaneous classification (or cross-classification) of the objects and their corresponding characteristics. Therefore we need a fuzzy partition of X compatible with a fuzzy partition of the set Y of characteristics. 

We are now able to state the hierarchical cross-classification procedure. The root node of the partition tree corresponds to the pair (X,Y). At the first level a fuzzy partition P = A, A2 of X is computed. Let Y = y, y, ...,y be the characteristics set induced by A and A2- We then have... 

FIGURE 5 The first decomposition level of hierarchical cross-classification. 

The fuzzy hierarchical cross-classification algorithm was used to classify eight mud samples. Each sample was characterized as a vector with 23 components representing chemical analysis. The fuzzy partition tree obtained by using simultaneous classification of muds and their characteristics is shown in Fig. 8. There are six final fuzzy classes in this hierarchy. The classical partition corresponding to the final fuzzy classes of the muds is 111 Krinides Lisbori Ai 2z> Argilla Solare A 2i> Pnkolimni ... 

FIGURE 8 Fuzzy cross-classification hierarchy of therapeutic muds. 

The fuzzy cross-classification algorithm produces both a fuzzy partition and a fuzzy partition of characteristics compatible with the former. The advantages of this algorithm include the ability to observe not only the fuzzy classes obtained and their relationship, but also the characteristics corresponding to each final class of objects. Each object class may be well described using the corresponding characteristics. These are the characteristics that have contributed to the separation of the respective fuzzy class. Fuzzy divisive hierarchical cross-classification of therapeutic muds based on their physicochemical characteristics allowed an objective interpretation of their origin and maturation and helped in their classification. It also permitted quantitative and qualitative identification of the compo-... 

If, in the context of the survival distribution we consider all of the times at which an event occurred and index them as t(l) < t(2) < t(3). . . < t(H), it is possible to create a 2 X 2 classification table for event times t h), where h = 1, 2, 3,. . ., H in which the numbers of individuals with and without the event of interest are displayed for each group. Table 11.10 is a sample cross-classification table for time h. 

Table 11.10 Cross-classification table and event at time h of treatment...

Contrary to what is commonly supposed, it is not necessary for clinical trials to be balanced in order to produce a valid comparison between treatments. Consider Table 3.2, which shows the cross-classification of numbers of patients by sex and treatment for a clinical trial. (Suppose, for argument s sake that the trial is in asthma and that we shall be measuring forced expiratory volume in one second, FEVi.)... 

Table 11.1 Cross-classification of two methods of analysis of 202 contrasts from 8 trials of duloxetine in depression. Based on Mallinckrodt et al. (2004b) and Mallinckrodt et al. (2004c).

For the data in Example 8.5.1 carry out a linear discriminant analysis working with the standardized variables. Hence identify the two variables which are most effective at discriminating between the two groups. Repeat the discriminant analysis with these two variables. Use the cross-classification success rate to compare the performance using two variables with that using all four variables. 

The cross-classification success rate with just these two variables is ... 

The discrimination between varieties is good (87.5% success). Results suggest that P and K are most effective at discriminating between varieties. Using just these two elements, a cross-classification rate of 15/16 is achieved. 

FUZZY HIERARCHICAL CROSS-CLASSIFICATION OF CHEMICAL ELEMENTS BASED ON TEN PHYSICAL PROPERTIES... 

Cross-Classification of Chemical Elements, Based on Their Physical, Chemical and Structural Features. 

Cross-Classification Algorithm. Improvements and Comparative Study. 

Fig. 10 illustrates the results of a multifactor cross-classification analysis from the same study, for white men age 45-64, with five risk factors dichotomized (including plasma glucose 1-hour post-50-gm-oral-load). With exclusion from the analysis of hypertensives on treatment and diabetics on treatment, glucose and rate of major ECG abnormalities were significantly related in two cases (noted by asterisks), but not in two others, and in only one of four comparisons for white women age 45-64 (Fig. 11). Similar inconsistent results were obtained with the more elegant multiple logistic regression technique (Fig. 12) where in only two of four analyses (after exclusion of hypertensives on treatment) were the values greater than 2.00 obtained indicating a significant relationship between post-load plasma glucose and major ECG abnormalities.

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