Contingency tables

Each column of a measurement table can be expanded into an indicator table. The rows of an indicator table refer to the same objects and in the same order as in the measurement table. The columns of the indicator table represent non-over-lapping categories of the selected measurement. Table 32.1 has been expanded into the indicator Table 32.2 for compounds and into the indicator Table 32.3 for disorders. In the indicator table for compounds, a value of one in a particular row is recorded if a person has used the corresponding compound. In the indicator table for disorders, one in a particular row indicates that the person has been treated for the corresponding disorder. All other elements of the row are set to zero. Note that the order of the columns in the indicator tables is not relevant. 

A contingency table X can be constructed by means of the matrix product of two indicator tables  

Each element in an nxp contingency table X represents the number of objects that can be associated simultaneously with category i of the first measurement and with category j of the second measurement. The element Xy can be interpreted as 

Measurement table describing the use of four compounds (C, D, L, T) for the treatment of three disorders (A, E, S) by 30 persons. For explanation of symbols, see text. 

Contingency table X derived from the indicator Tables 32.2 and 32.3 

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A 2 X 2 contingency table was used to evaluate the frequency of anomalies and resorptions within the fetal population and between litters. Body weight and body measurements were statistically analyzed by an Analysis of Variance and Tukey s test (13). In all cases, the level of significance was P < 0.05. 

Significantly different from control by an Analysis of Variance and Tukey s test (measurements) or the 2X2 contingency table (resorptions), P <0.05. 

Incidence among litters % (number of affected litters/number examined). Significantly different from control by 2 X 2 contingency table, P <0.05. 

A special type of table is the contingency table, which has been introduced in Chapter 16 of Part A. In Part B the 2x2 contingency table is extended to the general case (Chapter 32) which can be analyzed in a multivariate way. The above... 

One of the air of multivariate analysis is to reveal patterns in the data, whether they are in the form of a measurement table or in that of a contingency table. In this chapter we will refer to both of them by the more algebraic term matrix . In what follows we describe the basic properties of matrices and of operations that can be applied to them. In many cases we will not provide proofs of the theorems that underlie these properties, as these proofs can be found in textbooks on matrix algebra (e.g. Gantmacher [2]). The algebraic part of this section is also treated more extensively in textbooks on multivariate analysis (e.g. Dillon and Goldstein [1], Giri [3], Cliff [4], Harris [5], Chatfield and Collins [6], Srivastana and Carter [7], Anderson [8]). 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

A measurement table is different from a contingency table. The latter results from counting the number of objects that belong simultaneously to various categories of two measurements (e.g. molar refractivity and partition coefficient of chemical compounds). It is also called a two-way table or cross-tabulation, as the total number of objects is split up in two ways according to the two measurements that are crossed with one another. The analysis of contingency tables is dealt with specifically in Chapter 32. 

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) ... 

L. A. Goodman, Some useful extensions of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables. Int. Statistical Rev., 54 (1986) 243-309. 

Summation row-wise (horizontally) of the elements of a contingency table produces the vector of row-sums with elements Summation column-wise (vertically) yields the vector of column-sums with elements x j. The global sum is denoted by. These marginal sums are defined as follows ... 

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] ... 

Table 32.5 presents the expected values of the elements in the contingency Table 32.4. Note that the marginal sums in the two tables are the same. There are, however, large discrepancies between the observed and the expected values. Small discrepancies between the tabulated values of our illustrations and their exact values may arise due to rounding of intermediate results.

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... 

In the literature we encounter three common transformations of the contingency table. These can be classified according to the type of closure that is involved. By closure we mean the operation of dividing each element in a row or column of a table by its corresponding marginal sum. We reserve the word closure for the specific operation where the elements in a row or column of the table are reduced to unit sum. This way, we distinguish between closure and normalization, as the latter implies an operation which reduces the elements of a table to unit sums of squares. In a strict sense, closure applies only to tables with non-negative elements. 

Comparison between rows of a contingency table X is made easier after dividing each element of the table by its corresponding row-sum. This operation is called row-closure as it forces all rows of the table to possess the same unit sum. After closure, the rows of the table are called row-profiles. These can be represented in the form of stacked histograms such as shown in Fig. 32.1. 

Double-closure is the joint operation of dividing each element of the contingency table X by the product of its corresponding row- and column-sums. The result is multiplied by the grand sum in order to obtain a dimensionless quantity. In this context the term dimensionless indicates a certain synunetry in the notation. If x were to have a physical dimension, then the expressions involving x would appear as dimensionless. In our case, x represents counts and, strictly speaking, is dimensionless itself. Subsequently, the result is transformed into a matrix Z of deviations of double-closed data from their expected values ... 

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. 

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... 

We assume that Z is a transformed nxp contingency table (e.g. by means of row-, column- or double-closure) with associated metrics defined by W and W. Generalized SVD of Z is defined by means of ... 

In CFA we can derive biplots for each of the three types of transformed contingency tables which we have discussed in Section 32.3 (i.e., by means of row-, column- and double-closure). These three transformations produce, respectively, the deviations (from expected values) of the row-closed profiles F, of the column-closed profiles G and of the double-closed data Z. It should be reminded that each of these transformations is associated with a different metric as defined by W and W. Because of this, the generalized singular vectors A and B will be different also. The usual latent vectors U, V and the matrix of singular values A, however, are identical in all three cases, as will be shown below. Note that the usual singular vectors U and V are extracted from the matrix. ... 

The two plots can be superimposed into a biplot as shown in Fig. 32.7. Such a biplot reveals the correspondences between the rows and columns of the contingency table. The compound Triazolam is specific for the treatment of sleep disturbances. Anxiety is treated preferentially by both Lorazepam and Diazepam. The latter is also used for treating epilepsy. Clonazepam is specifically used with epilepsy. Note that distances between compounds and disorders are not to be considered. This would be a serious error of interpretation. A positive correspondence between a compound and a disorder is evidenced by relatively large distances from the origin and a common orientation (e.g. sleep disturbance and Triazolam). A negative correspondence is manifest in the case of relatively large distances from the origin and opposite orientations (e.g. sleep disturbance and Diazepam). 

The reconstruction Z of the transformed contingency table Z in a reduced space of latent vectors follows from ... 

CFA can also be defined as an expansion of a contingency table X using the generalized latent vectors in A, B and the singular values in A ... 

In the case when one of the two measurements of the contingency table is divided in ordered categories, one can construct a so-called thermometer plot. On this plot we represent the ordered measurement along the horizontal axis and the scores of the dominant latent vectors along the vertical axis. The solid line in Fig. 32.9 displays the prominent features of the first latent vector which, in the context of our illustration, is called the women/men factor. It clearly indicates a sustained progress of the share of women doctorates from 1966 onwards. The dashed line corresponds with the second latent vector which can be labelled as the chemistry/ other fields factor. This line shows initially a decline of the share of chemistry and a slow but steady recovery from 1973 onwards. The successive decline and rise are responsible for the horseshoe-like appearance of the pattern of points representing... 

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