Testing Quasi-Independence in Stratified Triangular Contingency Tables

A special class of incomplete contingency tables has structural zeroes in one or more cells. Some of these tables have a triangular appearance in the sense that they are square and their cells are known a priori to contain zeroes above (or below) their main diagonals. We propose methods of analyzing log-linear models and testing quasi-independence in these triangular tables. We also offer a method for combining such tables that are stratified by a concomitant categorical variable. This strategy follows the same approach used in a Cochran-Mantel-Haenszel test.     

A Note on Binary Contingency Tables

This paper considers contingency tables in which the marginal frequencies for one variable are all 1. This could occur with two-category binary data or when a continuous variable is treated in categorical fashion. Some results concerning the expectation of goodness-of-fit statistics are reported. In particular it is noted that the expectation of the Pearson statistic is independent of the model being fitted.     

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.     

Effect Sizes for 2×2 Contingency Tables

Sample size calculations are an important part of research to balance the use of resources and to avoid undue harm to participants. Effect sizes are an integral part of these calculations and meaningful values are often unknown to the researcher. General recommendations for effect sizes have been proposed for several commonly used statistical procedures. For the analysis of tables, recommendations have been given for the correlation coefficient for binary data; however, it is well known that suffers from poor statistical properties. The odds ratio is not problematic, although recommendations based on objective reasoning do not exist. This paper proposes odds ratio recommendations that are anchored to for fixed marginal probabilities. It will further be demonstrated that the marginal assumptions can be relaxed resulting in more general results.     

Decomposition and Model Selection for Large Contingency Tables

Large contingency tables summarizing categorical variables arise in many areas. One example is in biology, where large numbers of biomarkers are cross‐tabulated according to their discrete expression level. Interactions of the variables are of great interest and are generally studied with log–linear models. The structure of a log–linear model can be visually represented by a graph from which the conditional independence structure can then be easily read off. However, since the number of parameters in a saturated model grows exponentially in the number of variables, this generally comes with a heavy computational burden. Even if we restrict ourselves to models of lower‐order interactions or other sparse structures, we are faced with the problem of a large number of cells which play the role of sample size. This is in sharp contrast to high‐dimensional regression or classification procedures because, in addition to a high‐dimensional parameter, we also have to deal with the analogue of a huge sample size. Furthermore, high‐dimensional tables naturally feature a large number of sampling zeros which often leads to the nonexistence of the maximum likelihood estimate. We therefore present a decomposition approach, where we first divide the problem into several lower‐dimensional problems and then combine these to form a global solution. Our methodology is computationally feasible for log–linear interaction models with many categorical variables each or some of them having many levels. We demonstrate the proposed method on simulated data and apply it to a bio‐medical problem in cancer research.     

A Note on Contingency Tables with one Structural Zero

Based on a definition of Quasi-Independence, which is equivalent to the usual one but easier to interpret, rules for the calculation of ML-estimates in Contingency Tables with one structural zero are given. These rules are much simpler than the usual ones; even explicit solutions exist up to five dimensions. Possible extensions for other hypotheses and for more than one structural zero are suggested.     

A Modified Exact Test for 2 × 2 Contingency Tables

A modified exact test is proposed for 2×2 contingency tables. This test, which is based on a less conservative definition of the concept of significance (STONE, 1969) is compared with a modified form of Pearson's X² test and with Tocher's randomized exact (UMPU) test. The sizes of the new test lie near the nominal 0.05 levels while those of the X² test usually exceed the nominal level, sometimes by a factor of 2 or more. The power of the modified test is usually close to that of the UMPU test.     

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.     

The Decompositions for Point Symmetry Models in Two-way Contingency Tables

The decomposition for the complete point symmetry model in a rectangular contingency table is shown. Also the respective decompositions for the local point symmetry model and the reverse local point symmetry model in a square contingency table are given. Moreover test procedures for the decomposed models and an example are given.     

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.     

The Linear Discriminant Function: Sampling from the Truncated Normal Distribution

A study is made of the behaviour of the linear discriminant function in the classification of an observation when sampling from a truncated normal distribution. It is shown that the truncation proves ‘beneficial’ in that it reduces the error rates.     

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.     

A Multiplicative Model Imposed More Restrictions on the Quasi-Symmetry Model for the Analysis of Square Contingency Tables

For the analysis of square contingency tables, this note proposes a multiplicative model imposed more restrictions on CAUSSINUS ' (1965) quasi-symmetry (QS) model. The model is not so strict as in the quasi-double symmetry model considered by TOMIZAWA (1985b). The proposed model has the structure of the QS about the main diagonal plus an extended QS about the reverse diagonal of the square contingency table. The model is applied to the data on unaided vision of students in a university in Japan.     

Pre-post Treatment Evaluation by Symmetry Testing in Square Contingency Table

Starting from an r × r contingency table derived from ordinal pre- and post treatment observations, the traditional Bowker symmetry test is suggested to be replaced by one of the following alternatives (1) a globals test of general improvement, (2) a set of simultaneous symmetry tests of differential improvement without assuming general improvement under H₀ and (3) a set of simultaneous symmetry tests beyond assuming general improvement under H₀. The tests are illustrated by a bivariate example of observing neuroticism and introversion binarily (+, -) before and after Rogerian psychotherapy in a sample of clients.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

A Monte Carlo Study to Evaluate the Performance of Selected Tests of Homogeneity in Sparse Contingency Tables

We address the problem of tests of homogeneity in two-way contingency tables in case-control studies when the case category is subdivided into k subcategories. In this situation, we have two cells with large frequencies and 2 X k cells with frequencies that become small as k increases. We propose two ad hoc statistics in which a statistic for the sparse cells is combined with a statistic for the cells with large frequencies. We will study these tests along with the Pearson test (using a chi-square approximation) in a Monte Carlo simulation study. Two sets of null hypothesis models and two sets of alternative hypothesis models are considered. The best test for the models considered is the usual Pearson test (using an approximate chi-square distribution) although the ad hoc models are more powerful under one alternative model considered.