Considering of certain relevant information in the analysis of contingency tables. A three-step procedure

A heuristic three-step procedure for analysing multidimensional contingency tables is given to meet the requirements of a mixed analysis from both hypotheses-ruled and data-ruled type. The first-step provides the structure of relationships among the attributes by fitting an appropriate unsaturated log-linear model to the data of the given contingency table. Restriction to elementary hierarchical models allows to get them by combining pairs of conditional independence. The result of the first step may be regarded as a certain validisation of real model ideas. In the second step the significant pairs of conditional dependence are analysed in regard to the levels of the condition complex. Only such significant pairs are to be considered, in general, where the condition complex does not include the response variable. The third-step may test special subtests in that significant two-dimensional tables found in step two or may extend the general statements by partitioning, the corresponding test statistics in additive components. Application examples demonstrate the general line of action.     

To some reasonable test procedures in multiple contingency tables to investigate certain epidemiological or medicin-sociological relationships

One of the most important tasks of the application of mathematical-statistical methods consists in giving help in the search for possible relationships, and connected with this, the specification of new hypotheses. The progress of both the special diciplines of sciences and mathematical statistics itself leads to the application of more and more complex, that means multivariate, methods. In medical fields, especially in epidemiological and medicin-sociological studies, this fact means the necessity of analysing multidimensional contingency tables. The above formulated problem is equivalent to the problem of fitting an appropriate mathematical model (for contingency tables is this a log-linear model) to the data in a way which makes the structural relationships clear to us. In this paper it is shown that one is able to get to well-interpretable models of independence with relatively simple means. Two stepwise test procedures are described yielding essentially the same results: a so called reduction procedure which is particularly profitable in sparsely occupied tables and a procedure which uses a combination of hypotheses of conditional pairwise independence.     

The Power Function of the Test for ‘No Three Factor Interaction’ in 2×2×2 Contingency Tables

Three approximations to the power function of the chi-square test for the hypotheses of ‘no three factor interaction’ in a 2 × 2 × 2 contingency table are introduced and compared. The first method is based on the sampling distribution of the logarithm of the odds ratio, the second-on the non-central X² distribution and the third—on the conditional distribution of a cell entry. The last method is found to provide the closest approximation under various alternative hypotheses and sample sizes.     

On the Analysis of Special Incomplete Three-dimensional Contingency Tables

Incomplete contingency tables, i.e. tables with structurally caused empty cells, are analysed by means of so-called quasilog-linear models. In general the expected values can be calculated by means of iterative cyclic adaption to corresponding marginals of the empirical contingency tables (in the same way as in complete tables) under different hierarchical hypotheses concerning the parameters of the models. For important cases of 2-dimensional contingency tables it is possible to demonstrate that expected values and test statistics are to find in a closed form. If all 2-dimensional sub or partial tables of a 3-dimensional table can be assigned to such cases then the hypotheses of classes (AB×C) (??), (B×C)/A(??), (A??B)/A(??) etc. are testable in closed form. But the expected values to (A×B×C) (×) have to be calculated iteratively. An example shows that some definite additive decompositions of the test statistic 2 I are no longer valid while some others remain valid in spite of incompleteness of the tables.     

One Parameter Model for Three-Factor Interactions in Contingency Tables

This paper shows that the sum of products models for three and higher order interactions in contingency tables can be reparameterized in the spirit of TUKEY (1949) to yield chi-square tests with one degree of freedom. The merits of this new test over the other known tests for the same hypotheses are discussed.     

Use of log-linear analysis to construct explanatory models for TDBP- and AFB1-induced mutation spectra in lacI transgenic animals

Mutation spectra recovered from lacI transgenic animals exposed in separate experiments to tris-(2,3-dibromopropyl)phosphate (TDBP) or aflatoxin B1 (AFB1) were examined using log-linear analysis. Log-linear analysis is a categorical procedure that analyses contingency table data. Expected contingency table cell counts are estimated by maximum likelihood as effects of main variables and variable interactions. Evaluation of hierarchical models of decreasing complexity indicates when significant explanatory power is lost by the sequential omission of interactions between variables. Use of this technique allows construction of the most parsimonious models to account for mutation spectra obtained in the two experiments. The resulting statistical models are consistent with previous analyses of these data and with biological explanations for causes of the observed spectra.     

A Modified Chi-squared Approach for Fitting WEIBULL Models to Synthetic Life Tables1

WEIBULL models are fitted to synthetic life table data by applying weighted least squares analysis to log log functions which are constructed from appropriate underlying contingency tables. As such, the resulting estimates and test statistics are based on the linearized minimum modified X²₁-criterion and thus have satisfactory properties in moderately large samples. The basic methodology is illustrated in terms of an example which is bivariate in the sense of involving two simultaneous, but non-competing, vital events. For this situation, the estimation of WEIBULL model parameters is described for both marginal as well as certain conditional distributions either individually or jointly.     

Symmetry and Asymmetry Models for Rectangular Tables

The models of complete, quasi and conditional symmetry as well as marginal homogeneity and diagonal asymmetry, applicable to square contingency tables are extended to non-square ones. Several properties of the models are investigated and estimation and test theory is presented. The utility of the new proposed models is discussed and illustrated by reanalyzing classical data sets.     

Two-act transitions,partitioned contingency tables,and the ‘significant cells’ problem

When contingency tables of data on sequences, social relationships, feeding, habitat use, or other behaviour exhibit significant associations between variables, ethologists may analyse the residuals in the table in order to test more precise hypotheses about the associations found. This paper critically evaluates currently used and potentially available statistical methods for performing such tests. Specific examples of use are given and recommendations made.     

A Comparison of Tests for Marginal Homogeneity in Square Contingency Tables

Several asymptotic tests were proposed for testing the null hypothesis of marginal homogeneity in square contingency tables with r categories. A simulation study was performed for comparing the power of four finite conservative conditional test procedures and of two asymptotic tests for twelve different contingency schemes for small sample sizes. While an asymptotic test proposed by STUART (1955) showed a rather satisfactory behaviour for moderate sample sizes, an asymptotic test proposed by BHAPKAR (1966) was quite anticonservative. With no a priori information the performance of (r - 1) simultaneous conditional binomial tests with a Bonferroni adjustment proved to be a quite efficient procedure. With assumptions about where to expect the deviations from the null hypothesis, other procedures favouring the larger or smaller conditional sample sizes, respectively, can have a great efficiency. The procedures are illustrated by means of a numerical example from clinical psychology.     

Interactions in Contingency Table Analysis

Proceeding from Lancaster's definition of interactions between random variables, the authors set up a model for contingency tables of any dimension. Three-dimensional contingency tables are used as an example to discuss first and second order interaction effects, and the conventional independence are expressed by hypotheses concerning interaction effects. The opinions of other authors regarding second order interaction effects are discussed.     

An exact test for all-way interaction in a 2M contingency table: application to interval capture-recapture estimation of population size

An exact conditional test for an M-way log-linear interaction in a fully observed 2M contingency table is formulated. From this is derived a procedure for interval estimation of the total count N in a 2M contingency table, one of whose entries is unobserved. This procedure has an immediate application to interval estimation of the size of a closed population from incomplete, overlapping lists of records, as in capture-recapture analysis of epidemiological data. Data on the prevalence of spina bifida in live births in upstate New York in 1969-1974 illustrate this application.     

Loglinear Symmetry and Quasi-Symmetry Models for the Analysis of Change

Loglinear symmetry and quasi-symmetry models are proposed as tools for investigating various hypotheses about change. First, a survey of model representations is provided, including model specification in terms of hierarchical loglinear models and in design matrix notation. Secondly, the range of symmetry and quasi-symmetry models is extended to the joint analysis of several groups. Parameter constraints are discussed which allow one to test specific hypotheses about group differences in symmetric frequency distributions. Finally, symmetry and quasi-symmetry models are considered for multiway contigency tables. In this context, loglinear total score models are proposed for the analysis of symmetry in several marginal distributions. The proposed models reflect cross-sectional as well as longitudinal facets of development.     

Hierarchical partitioning as an interpretative tool in multivariate inference

Abstract Much of biogeography, conservation and evolutionary biology, and ecology involves very large spatial and temporal extents. Direct manipulation to test hypotheses is usually almost impossible at appropriate scales so that multivariate modelling and especially regression are used to draw causal inferences about which ‘independent’ variables influence the distribution and abundances of species. Such inferences clearly are crucial for the successful management of biological resources and for conserving threatened species. A succession of regression approaches has arisen, many of which yield inconsistent implications. The main problem has been the quest for one (the ‘best’ or the ‘optimal’) regression model from which the impacts of independent variables are inferred. This note is to draw the attention of ecologists to a relatively recent method, hierarchical partitioning, that does not aim to identify a best regression model as such but rather uses all models in a regression hierarchy to distinguish those variables that have high independent correlations with the dependent variable. Such variables are likely to be most influential in controlling variation in the dependent variable. Hierarchical partitioning is not to be regarded as a substitute for experimental manipulation when that is appropriate, but it is likely to produce better deductions than common regression approaches in the many ecological situations in which manipulation is impossible or of doubtful value.     

A Bayesian hierarchical model for categorical data with nonignorable nonresponse

Log-linear models have been shown to be useful for smoothing contingency tables when categorical outcomes are subject to nonignorable nonresponse. A log-linear model can be fit to an augmented data table that includes an indicator variable designating whether subjects are respondents or nonrespondents. Maximum likelihood estimates calculated from the augmented data table are known to suffer from instability due to boundary solutions. Park and Brown (1994, Journal of the American Statistical Association 89, 44-52) and Park (1998, Biometrics 54, 1579-1590) developed empirical Bayes models that tend to smooth estimates away from the boundary. In those approaches, estimates for nonrespondents were calculated using an EM algorithm by maximizing a posterior distribution. As an extension of their earlier work, we develop a Bayesian hierarchical model that incorporates a log-linear model in the prior specification. In addition, due to uncertainty in the variable selection process associated with just one log-linear model, we simultaneously consider a finite number of models using a stochastic search variable selection (SSVS) procedure due to George and McCulloch (1997, Statistica Sinica 7, 339-373). The integration of the SSVS procedure into a Markov chain Monte Carlo (MCMC) sampler is straightforward, and leads to estimates of cell frequencies for the nonrespondents that are averages resulting from several log-linear models. The methods are demonstrated with a data example involving serum creatinine levels of patients who survived renal transplants. A simulation study is conducted to investigate properties of the model.     

Gouldian arguments and the sources of contingency

‘Gouldian arguments’ appeal to the contingency of a scientific domain to establish that domain’s autonomy from some body of theory. For instance, pointing to evolutionary contingency, Stephen Jay Gould suggested that natural selection alone is insufficient to explain life on the macroevolutionary scale. In analysing contingency, philosophers have provided source-independent accounts, understanding how events and processes structure history without attending to the nature of those events and processes. But Gouldian Arguments require source-dependent notions of contingency. An account of contingency is source-dependent when it is indexed to (1) some pattern (i.e., microevolution or macroevolution) and (2) some process (i.e., Natural Selection, species sorting, etc.). Positions like Gould’s do not turn on the mere fact of life’s contingency—that life’s shape could have been different due to its sensitivity to initial conditions, path-dependence or stochasticity. Rather, Gouldian arguments require that the contingency is due to particular kinds of processes: in this case, those which microevolutionary theory cannot account for. This source-dependent perspective clarifies both debates about the nature and importance of contingency, and empirical routes for testing Gould’s thesis.     

A More Powerful Simultaneous Test Procedure in Configural Frequency Analysis

In the Configural Frequency Analysis (CFA) of KRAUTH and LIENERT (1973 a, b), overfrequented (or underfrequented) cells in multivariate contingency tables are identified by simultaneous binomial tests. As an alternative, finite and asymptotic tests are proposed, which are derived from the (exact conditional) generalized hypergeometrical distribution of the cell frequencies. These tests allow for considerably more powerful decisions than do the conservative binomial tests.     

Exploring contingency tables with correspondence analysis

An algorithm for correspondence analysis is described and implementedin SAS/IML (SAS Institute, 1985a). The technique is shown, throughthe analysis of several biological examples, to supplement thelog-linear models approach to the analysis of contingency tables,both in the model identification and model interpretation stagesof analysis. A simple two-way contingency table of tumor datais analyzed using correspondence analysis. This example emphasisesthe relationships between the parameters of the log-linear modelfor the table and the graphical correspondence analysis results.The technqiue is also applied to a three-way table of surveydata concerning ulcer patients to demonstrate applications ofsimple correspondence analysis to higher dimensional tableswith fixed margins. Finally, the diets and foraging behaviorsof birds of the Hubbard Brook Forest are each analyzed and thena simultaneous display of the two separate but related tablesis constructed to highlight relationships between the tables. Received on August 29, 1988; accepted on April 25, 1989     

用列联表研究纬度和海拔高度对红杉分布的影响

 本文目的在于探讨列联表用于植物地理学研究。通过用列联表研究垂直高度与纬度对红杉(Larix potaninii)分布的影响,找出垂直高度、纬度及其交互作用等的效应,并且找出了红杉的分布中心。由于卡方检验,剩余分析,有序表连带测度,Log-线性模型拟合联合运用,Log-线性模型的运用有一些改进。本文初步证明,列联表用于植物地理学研究是可以成功的。     

Continued fraction representation for expected cell counts of a 2 x 2 table: a rapid and exact method for conditional maximum likelihood estimation

The expected cell count for a 2 x 2 contingency table, governed by the noncentral (extended) hypergeometric distribution, is expressed as a terminating continued fraction. The coefficients in the continued fraction are better behaved than the multinomial coefficients required for the usual moment calculation. The expected cell count must be calculated repeatedly in a conditional maximum likelihood analysis of K2 x 2 contingency tables. Since the continued fraction can be easily evaluated, a rapid and numerically stable computational algorithm results. Once this first moment is known, higher moments can be obtained as shown by Harkness (1965, Annals of Mathematical Statistics 36, 938-945). A BASIC program to implement the continued fraction algorithm is given in an appendix.