Elementary Analysis of Multidimensional Contingency Tables. Application to a Medical Example1/2

By the aid of analysing a medical example a three—step procedure for analysing multi—dimensional contingency tables is introduced. This procedure has some good properties. Step one is due to catch the relationship structure between the variables connected by the contingency table. Hereby only so—called graphical models, a subclass of hierarchical models in regard of the parameters of the log—linear model, are admitted. The models can be generated by combination of hypotheses of pairwise conditional independence. Hereby a so-called Extended Combination Procedure is proposed using the position of the Chain of (hierarchical) Hypotheses. A useful symbolic notation for ‘Dependence Models’ in addition to that in form of ‘Independence Models’ and ‘Minimal Sets’ is proposed. Step two analyses the significant conditional pairs in regard to the question for what attribute level combinations of the condition complexes the relations remain significant. Step three investigates those tables recognised as significant in step two more closely to get ideas about the ‘sources’ of dependencies and possibilities of collapsing parts of the table. The procedure is mostly used in explorative data analysis although the simple steps can be used to test hypotheses, too.     

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.     

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.     

Decompositions for 2-Ratios-Parameter Symmetry Model in Square Contingency Tables with Ordered Categoriesat

For square contingency tables with ordered categories, Agresti (1983) considered the linear diagonals-parameter symmetry model. An extended model including that model is proposed which has only one more parameter than that model. The model also includes the conditional symmetry model considered by McCullagh (1978). Decompositions for the proposed model and Agresti's model are given.     

Testing for conditional multiple marginal independence

Survey respondents are often prompted to pick any number of responses from a set of possible responses. Categorical variables that summarize this kind of data are called pick any/c variables. Counts from surveys that contain a pick any/c variable along with a group variable (r levels) and stratification variable (q levels) can be marginally summarized into an r x c x q contingency table. A question that may naturally arise from this setup is to determine if the group and pick any/c variable are marginally independent given the stratification variable. A test for conditional multiple marginal independence (CMMI) can be used to answer this question. Since subjects may pick any number out of c possible responses, the Cochran (1954, Biometrics 10, 417-451) and Mantel and Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) tests cannot be used directly because they assume that units in the contingency table are independent of each other. Therefore, new testing methods are developed. Cochran's test statistic is extended to r x 2 x q tables, and a modified version of this statistic is proposed to test CMMI. Its sampling distribution can be approximated through bootstrapping. Other CMMI testing methods discussed are bootstrap p-value combination methods and Bonferroni adjustments. Simulation findings suggest that the proposed bootstrap procedures and the Bonferroni adjustments consistently hold the correct size and provide power against various alternatives.     

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.     

Selection of Log-Linear Models with Orthonormal Basis for Contingency Tables

A model selection criterion for log-linear models with orthonormal basis for contingency tables is developed using the Gauss discrepancy between the logarithms of the frequencies. The contribution of each parameter to the criterion may be determined separately. A test for the hypothesis that the use of a certain parameter increases the expected discrepancy is given.     

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.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.     

Exact Calculation of Permutational Distributions for Two Independent Samples

A simple Fortran Subroutine is given for the exact calculation of permutational distributions for two independent samples. Important special cases are the Fisher-Pitman randomization test, the Mann-Whitney U-test, the Mantel-Haenszel-test and the exact test for 2×k contingency tables. The algorithm needs polynomial time and is of use even for personal- or microcomputers.     

Configural Frequency Analysis (CFA) Revisited — a New Look at an Old Approach

Configural frequency analysis (CFA) is a widely used method for the identification of types and syndromes in contingency tables. However, the type model of CFA shows some major deficiencies. In this paper, we propose an alternative modeling of types eliminating the shortcomings of CFA. Basically, a type is modeled as a combination of traits or symptoms that deviates from the pattern of association holding true for the complementary configurations of the contingency table. The new approach is formulated in terms of a log-linear model. It is shown that parameter estimation can be performed with methods known from the analysis of incomplete contingency tables. Test procedures for confirmatory analysis and methods for exploratory search for type configurations are developed. We illustrate the methodology with two practical examples.     

On the use of the classical tests for detecting linkage.

Some drawbacks of the classical Mather's linkage text XL2 are considered, and the simple contingency analysis is suggested as an alternative method. The former test is conditional on Mendelian segregation at both loci, whereas the simple contingency test is not. Furthermore, the contingency test and the test for Mendelian segregation at each locus are orthogonal when performed using the G statistic. Simulation results show that, when the XL2 is used, the actual type I error probability (alpha a) can be dramatically perturbed. As expected, no alpha a perturbation is observed when the G contingency test is used. On the other hand, when segregation is Mendelian at both loci, the power of the XL2 method is larger than that of the contingency G test when sample size is small and strong marginal distortion is observed. Because strong marginal distortion may suggest that segregation may be non-Mendelian, the XL2 is in general discouraged in favor of the simple contingency analysis.     

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.     

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.     

Systematic detection of epistatic interactions based on allele pair frequencies

Epistatic genetic interactions are key for understanding the genetic contribution to complex traits. Epistasis is always defined with respect to some trait such as growth rate or fitness. Whereas most existing epistasis screens explicitly test for a trait, it is also possible to implicitly test for fitness traits by searching for the over- or under-representation of allele pairs in a given population. Such analysis of imbalanced allele pair frequencies of distant loci has not been exploited yet on a genome-wide scale, mostly due to statistical difficulties such as the multiple testing problem. We propose a new approach called Imbalanced Allele Pair frequencies (ImAP) for inferring epistatic interactions that is exclusively based on DNA sequence information. Our approach is based on genome-wide SNP data sampled from a population with known family structure. We make use of genotype information of parent-child trios and inspect 3×3 contingency tables for detecting pairs of alleles from different genomic positions that are over- or under-represented in the population. We also developed a simulation setup which mimics the pedigree structure by simultaneously assuming independence of the markers. When applied to mouse SNP data, our method detected 168 imbalanced allele pairs, which is substantially more than in simulations assuming no interactions. We could validate a significant number of the interactions with external data, and we found that interacting loci are enriched for genes involved in developmental processes.     

Closed Form Estimators of Latent Cell Frequencies

One-stage and two-stage closed form estimators of latent cell frequencies in multidimensional contingency tables are derived from the weighted least squares criterion. The first stage estimator is asymptotically equivalent to the conditional maximum likelihood estimator and does not necessarily have minimum asymptotic variance. The second stage estimator does have minimum asymptotic variance relative to any other existing estimator. The closed form estimators are defined for any number of latent cells in contingency tables of any order under exact general linear constraints on the logarithms of the nonlatent and latent cell frequencies.     

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.     

A score test for the linkage analysis of qualitative and quantitative traits based on identity by descent data from sib-pairs

We propose a general likelihood-based approach to the linkage analysis of qualitative and quantitative traits using identity by descent (IBD) data from sib-pairs. We consider the likelihood of IBD data conditional on phenotypes and test the null hypothesis of no linkage between a marker locus and a gene influencing the trait using a score test in the recombination fraction theta between the two loci. This method unifies the linkage analysis of qualitative and quantitative traits into a single inferential framework, yielding a simple and intuitive test statistic. Conditioning on phenotypes avoids unrealistic random sampling assumptions and allows sib-pairs from differing ascertainment mechanisms to be incorporated into a single likelihood analysis. In particular, it allows the selection of sib-pairs based on their trait values and the analysis of only those pairs having the most informative phenotypes. The score test is based on the full likelihood, i.e. the likelihood based on all phenotype data rather than just differences of sib-pair phenotypes. Considering only phenotype differences, as in Haseman and Elston (1972) and Kruglyak and Lander (1995), may result in important losses in power. The linkage score test is derived under general genetic models for the trait, which may include multiple unlinked genes. Population genetic assumptions, such as random mating or linkage equilibrium at the trait loci, are not required. This score test is thus particularly promising for the analysis of complex human traits. The score statistic readily extends to accommodate incomplete IBD data at the test locus, by using the hidden Markov model implemented in the programs MAPMAKER/SIBS and GENEHUNTER (Kruglyak and Lander, 1995; Kruglyak et al., 1996). Preliminary simulation studies indicate that the linkage score test generally matches or outperforms the Haseman-Elston test, the largest gains in power being for selected samples of sib-pairs with extreme phenotypes.     

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.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.