Approximating Exact Probabilities by χ2 and Continuity-Corrected χ2 in 2×2 Tables

Computations have been performed to find an adequate definition of exact two-sided probabilities in 2times2 contingency tables. It turns out, that both uncorrected χ² and Yates' correction for continuity give only unsatisfactory approximations to the exact probabilities of the hypergeometric distribution. The latter are therefore recommended for general use.     

Asymptotic Relative Efficiency of Some Jackknife Estimators of a Common Odds Ratio

In the situation of several 2 × 2 tables the asymptotic relative efficiencies of certain jackknife estimators of a common odds ratio are investigated in the case that the number of tables is fixed while the sample sizes within each table tend to infinity. The estimators show very good results over a wide range of parameters. Some situations in which the estimators have low asymptotic relative efficiency are pointed out:.     

Prediction Models in Configural Frequency Analysis

Configural frequency analysis models are presented for predicting a discrete criterion variable from combining 2 or more discrete predictor variables nonparametrically. Predictions are made via so-called prediction types which allow to predict a criterion class from combined predictor classes. Prediction types are implicitely defined as cells of a 2-dimensional contingeney table with rows as predictor combinations and columns as criterion classes. The CFA is extended for two criteria, and for stratified sampling.     

Analysis of Trends in Proportions with an Ordinally Scaled Determinant

In this paper the analysis of several proportions in a comparative study setting is discussed. The case of ordinal grouping variates is considered. F statistics are formulated to test for trend in the proportions over the scored values of the determinant variate. The null chi-square or F (t square) functions are presented separately for the unstratified and stratified analysis, and in either situation the corresponding functions with angular transformed proportions are also expressed. Generalizations to deal with the parameters in the nonnull range are outlined. Throughout, the intimate relation between the presented statistics and standard methods is pointed out.     

On Conditions for Validity of the Approximations to Fisher's Exact Test

A 2 × 2 table is analyzed from the conditional viewpoint, using Fisher's exact test, for which there is abundant software nowadays (StatXact, SPSS with the module “Exact Test”, …). Nevertheless, because it is well-nigh impossible to work the test out “by hand”, it is customary in many books to analyze the table, in an approximate fashion, by using the classic chi-square test with the appropriate continuity correction, and this is what many researchers do in practice. Unfortunately little research has been carried out on the validity conditions of the test (remember the classic advice that the minimum expected quantity E should be larger than or equal to 5), so that it is applied indiscriminately with the obvious danger of obtaining erroneous results. In this paper the exact validity conditions (which depend not only on E, but also on the real P-value, on the size of the sample, on the number of tails of the test and on the continuity correction used) are determined, showing that the classic condition E > 5 can be either very strict or quite liberal (depending on the circumstances) and it can even be necessary for the test to have E > 20. A similar study is made of the case of using the binomial approximation, while the Poisson approximation is also discussed. The article ends with a list of rules and precautions to be borne in mind by researchers using the approximate methods. The strict rules are complex enough to make the researcher want to use the exact test, but simplified (and conservative) rules are also given for those unable to do so.     

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.     

濒危植物缙云卫矛群落种间联结分析

采用方差比率法（VR）、X^2检验、无中心指数等方法对缙云卫矛（Euonymus chloranthoides）3个群落进行了群落物种间关联性分析。结果表明,3个群落内物种的总体关联均表现为不显著的负关联性;北温泉群落6个主要物种的15个种对中有7个正关联,8个负关联;鸡公山群落8个主要物种的28个种对中有11个正关联,16个负关联,1个相对独立的关系;东温泉群落4个主要物种的6个种对中2个正关联,4个负关联。这反映了整个群落处于不稳定状态,这与生境片断化是密切相关的。种对间的关联性表明缙云卫矛与慈竹间显示出较强的正关联性,这表明缙云卫矛的生长需要慈竹为其提供适宜的生境。     

Analysis of Proportions in One- or Two-Way Tables with Spoiled Data

This paper discusses the analysis of data on proportions in contingency tables. The X² ‘index of dispersion test’ (e.g. Fisher, 1954) is developed in these situations and compared with the use of the ‘logit’ transformation. An example using Osborn's (1979) data is given, illustrating the estimation of one or more missing observations.     

On the Equivalence of the Information-Theoretic Transmission-Measure to the Common χ2 Statistic

The results of a simulation study are presented in which the χ²-and the information theoretic approach to association measurement were compared to one another. It could be shown that neither the maximal values nor the minimal values obtainable for a given sample size coincide in all cases. The underlying association concepts are defined; implications are discussed.     

Eine neue Interpretation des HELLMICHschen Maßes K und seine Anwendungen

From the examination of the properties of symmetry of the correlation functions it appears that the asymmetrical functions contain information capability to be used with advantage for the characterization of causal effects. The paper describes a new interpretation of the K measure of HELLMICH as well as its application in the field of investigation of connections. Computer calculations carried out on plant and human populations justify the use of the introduced measure. Concrete results are presented with respect to two maize diseases “stalk rot” and “ear rot”; further a completely analogous problem to examine the relation between diabetes and high blood pressure is outlined. Finally some remarks are made on further investigation of measuring numbers and application in a broader sense.     

Some New Aspects of 2 × 2 Table Measures and Their Applications

The paper deals with the analysis of 2 × 2 table measures and describes several new possibilities for applying them. An estimate is given for HELLMICH's asymmetric K-measure and a possible extension of the contingency concept in a special 2×2 case is presented. The concept introduced and the recommended computer application of the 2×2 table measures is supported by new results obtained for the relationship between jaw joint pains and noises examined earlier by F. SCHMID and C. ZSCHEGE. With a larger number of observations the opportunity arises for a deeper analysis of this dental problem using path analysis. The plan of evaluation and the conncetion between the variables examined is illustrated by a path-diagram. The association tables determined on the basis of the (C) values introduced during the examination of jaw joint disturbances using association analysis draw attention to the special role of the left side of mastication. The paper raises ideas for further generalisations and indicates where further research is needed.     

A 2-step procedure for comparing binomial populations available when given differences are to find out

A two-step procedure for comparing two binomial probabilities is presented. The method is compared with the exact test of FISHER. In average it needs essentially smaller sample sizes. In situations like drug screening it is desired to minimize the type II error rates for great differences of the binomial probabilities and to maximize it for small differences, simultaneously. In this point the method is superior to the FISHER test. Examples from drug screening demonstrate the advantages of the method.     

太白山独叶草及其伴生种的种间关联研究

基于2×2列联表,通过一系列种间关联测定方法[方差比率(variance ratio)法检验多物种间总体关联性,X2检验各种对之间的关联性,关联指数(Ochiai,Dice,Jaccard)测度各种对之间的关联程度]对太白山独叶草及16种伴生植物,共136个种对的种间关联分别进行检验与测度。结果表明:独叶草及其伴生种的总体关联表现为显著正关联,17个种对呈极显著正关联(p<0.01),6个种对表现为显著正关联(p<0.05),6个种对呈极显著负关联(p<0.01),3个种对表现为显著负关联(p<0.05)。多数种对关联程度较低,仅有7个种对正关联Ochiai指数高于0.5,说明物种各自趋于独立。这种\"松散\"的状况可能与群落的发展阶段和物种的生态学特性有关。     

Jeffreys' Likelihood Ratio for the 2×2-Table — A Monte Carlo Study —

Jeffreys' approach for analyzing a 2×2-table is discussed via a Monte Carlo study. The main findings are reported.     

The donation game with roles played between relatives

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.     

Nonparametric Evaluation of Learning Curves by Hierarchical Contingeney Analysis

Hierarchical contingency analysis (HCA) is derived from the Perli-Hommel-Lehmacher (1986) closed test procedure for nonparametrical evaluation of learning curves of a 2 x 2-factorial experiment. By HCA, univariate main effects are detected without Bonferroni alpha adjustment, as is shown by a numerical example from gold fish shock avoidance conditioning. Alternative approaches to nonparametrical evaluation of MANOVA designs with and without repeated measurements are discussed.