Analysis of 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. 

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. 

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... 

B.S. Everitt, The Analysis of Contingency Tables. Chapman and Hall, London, 1977. 

The analysis of contingency tables involves looking for evidence of association between the variables involved, in the example above whether the use of a stylet influenced correct placement. For this we use the x2 (pronounced kh/ -squared ) test. To test the null hypothesis, that correct placement was not associated with use of a stylet, we compare the number of times placement was correct with the number we would expect if there was no association. If there was no association we would expect correct placement in (25/65)x30=11.5 cases when the stylet was used, and (25/65)x35=13.5 cases when no stylet was used. In a similar... 

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. 

Kostov, B., Becue-Bertaut, M. and Husson, E. (2014) An original methodology for the analysis and interpretation of word-count based methods Multiple factor analysis for contingency tables complemented by consensual words, Food Qual Prefer, 32A, 35-40. DOI 10.1016/j.foodqual.2013.06.009. 

In this paper we have considered the analysis on contingency tables for numbers of failures in a reliability context. There will typically be some failure types with low probability and as such small numbers of realised failures in some of the cells in the contingency table. This means that the traditional Chi-Squared Test will be unsuitable and the most common alternative, Fisher s Test, is conservative meaning that statistically significant differences will be missed. 

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. 

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 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) ... 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

The LLM approach described above has been applied to the 35x4 contingency Table 32.10 after some modification. In this case, we have replaced the pooled data in the first five rows by the corresponding average annual values. Further, columns 1 and 2 have been combined with columns 3 and 4 in order to produce the categories of women in all fields and of men in all fields. The reason for the change is our objective to produce an analysis of the ratios between chemistry and all fields rather than between chemistry and the other fields. 

Correspondence factor analysis (CFA) is most appropriate when the data represent counts of contingencies, or when there are numerous true zeroes in the table (i.e. when zero means complete absence of a contingency, rather than a small quantity which has been rounded to zero [47]). A detailed description of the method is found in Section 32.3.6. 

The analysis is started by setting up a 2x2 contingency table to summarize the numbers of positive and negative responses as well as the totals of these, as follows ... 

Though Fisher s Exact Test is preferable for analysis of most 2x2 contingency tables in toxicology, the chi square test is still widely used and is preferable in a few unusual situations (particularly if cell sizes are large yet only limited computational support is available). 

Exploration of the scope of NPS in electrochemical science and engineering has so far been rather limited. The estimation of confidence intervals of population mean and median, permutation-based approaches and elementary explorations of trends and association involving metal deposition, corrosion inhibition, transition time in electrolytic metal deposition processes, current efficiency, etc.[8] provides a general framework for basic applications. Two-by-two contingency tables [9], and the analysis of variance via the NPS approach [10] illustrate two specific areas of potential interest to electrochemical process analysts. 

Kay R (2006) Letter to the Editor on Phase specific analysis of herpes zoster associated pain data a new statistical approach Statistics in Medicine, 25, 359-360 Landis RJ, Heyman ER and Koch GG (1978) Average partial association in three-way contingency tables A review and discussion of alternative tests International Statistical Review, 46, 237-254... 

For the most part, the meta-analyses used in this text are based on fourfold contingency tables, which include the number of responders or nonresponders to a given treatment. An advantage of dichotomous data is that information from each individual subject can be abstracted (i.e., the results come from actual patients). In one sense, this is not strictly a meta-analysis, because calculations are not done on summary statistical parameters but on observations of individual subjects. Such an approach has the advantage of directness, however, because the percentage of patients who respond or do not respond to a new treatment, standard treatment, or placebo is intuitively meaningful to clinicians, whereas a change of 0.8 SD units may not be. 

The use of nuclear family data in association studies was initially developed to avoid possible ethnic mismatching between patients and randomly ascertained controls (76). The parental marker alleles not transmitted to an affected child, or never transmitted to an affected sib pair, form the so-called AFBAC population (19,20) (Fig. 1). In a random mating population, when there is a marker association with disease, the AFBAC population provides an unbiased estimate of the overall population (control) marker alleles when the recombination fraction between the marker and disease genes is sufficiently small that it can be taken as zero (0 = 0), and differences between patient and AFBAC frequencies can be tested for example by a contingency table analysis for heterogeneity. 

Table 16.2 Generic output from a contingency chi-square analysis of rates of expulsion of two designs of IUD...

To investigate relationships between crustacean grazing rates on Phaeocystis and experimental conditions, a multiple correspondence analysis (MCA) followed by a hierarchical cluster analysis (HCA) was performed using SPAD 3.5 software (Lebart et al. 1988). The combination of MCA and cluster analysis is a common way to explore relationships among a large number of variables and to facilitate interpretation of the correspondence analysis results (Lebart et al. 2000). MCA uses a contingency table as data, which provides a simultaneous representation of the observations (rows) and variables (column) in a factorial space. This form of multivariate analysis describes the total inertia (or variability) of a multidimensional... 

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