Log-linear models for correlated marginal totals of a contingency table

The marginal totals of a contingency table can be rearranged to form a new table. If at least twoof these totals include the same cell of the original table, the new table cannot be treated as anordinary contingency table. An iterative method is proposed to calculate maximum likelihood estimators for the expected cell frequencies of the original table under the assumption that some marginal totals (or more generally, some linear functions) of these expected frequencies satisfy a log-linear model.In some cases, a table of correlated marginal totals is treated as if it was an ordinary contingency table. The effects of ignoring the special structure of the marginal table on thedistributionof the goodness-of-fit test statistics are discussed and illustrated, with special reference to stationary Markov chains.     

The Standard Error of the Mean and the Difference between Means for Finite Populations

This article discusses a representation of Pearson's chi-square for independence in two-way contingency tables in terms of conditional probabilities of two categorical random variables and proposes a functional interpretation of Pearson's chi-square. This representation is suggested for use in the teaching of statistical independence between categorical variables.     

On the Computation of Non-Central Chi-Square Distribution Function

The cumulative distribution function of the non-central chi-square is very important in calculating the power function of some statistical tests. On the other hand it involves an integral which is difficult to obtain. In literature some workers discussed the evaluation and the approximation of the c.d.f. of the non-central chi-square [see references (2)]. In the present work two computational formulae for computing the cumulative distribution function of the non-central chi-square distribution are given, the first one deals with the case of any degrees of freedom (odd and even), and the second deals with the case of odd degrees of freedom. Numerical illustrations are discussed.     

New representations of the noncentral chi-square density and cumulative

Representations of noncentral chi-square cumulative distribution function and probability density function are reviewed and new repre¬sentations are given. One representation of the cdf in terms of an integral is easily computed on any machine which has an accurate algorithm for computing the normal cdf.     

A nonparametric procedure for testing partially ranked data

In consumer preference studies, it is common to seek a complete ranking of a variety of, say N, alternatives or treatments. Unfortunately, as N increases, it becomes progressively more confusing and undesirable for respondents to rank all N alternatives simultaneously. Moreover, the investigators may only be interested in consumers’ top few choices. Therefore, it is desirable to accommodate the setting where each survey respondent ranks only her/his most preferred k (k?N) alternatives. In this paper, we propose a simple procedure to test the independence of N alternatives and the top-k ranks, such that the value of k can be predetermined before securing a set of partially ranked data or be at the discretion of the investigator in the presence of complete ranking data. The asymptotic distribution of the proposed test under root-n local alternatives is established. We demonstrate our procedure with two real data sets.     

Resolving the conflict over fisher's exact test

Fisher's exact test for two-by-two contingency tables has repeatedly been criticized as being too conservative. These criticisms arise most frequently in the context of a planned experiment for which the numbers of successes in each of two experimental groups are assumed to be binomially distributed. It is argued here that the binomial model is often unrealistic, and that the departures from the binomial assumptions reduce the conservatism in Fisher's exact test. Further discussion supports a recent claim of Barnard (1989) that the residual conservatism is attributable, not to any additional information used by the competing method, but to the discrete nature of the test, and can be drastically reduced through the use of Lancaster's mid-p-value. The binomial model is not recommended in that it depends on extra, questionable assumptions.     

A New Look at an Old Test

Fisher's exact test for 2 × 2 frequency tables is presented in a new format which, we believe, is easier to use and which permits a substantial saving in tabulation-space of critical values. Tables are provided here for total frequency N ≤ 25. Tables for N ≤ 100 are obtainable on request to the author     

Asymptotic properties of the score test for varying dispersion in exponential family nonlinear models

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].     

A generalization of the log-series distribution

A new generalized logarithmic series distribution (GLSD) with two parameters is proposed.The proposed model is flexible enough to describe short-tailed as well as long-tailed data.Some recurence relations for its probabilities and the factorial moments are presente.These recurrence relations are utilized to obtain the minimum chi-square estimators for the parmaters.Maximum likelihood estimators and some other estimators based on first few moments and probabilities are also suggested.Asymptotic relative efficiency of some of these estimators is also obtained and compared.Two test statistics based on the minimum chi-square estimators fo testing some hypotheses regarding the GLSD are proposed.The fit of the model and the application of the test statistics are exemplified by some data sets.Finally, a graphical method is suggested for differentiating between the ordinary logarithmic series distribution and the GLSD.     

Minimum chi-square estimator and test for incomplete multidimensional contingency tables

A modified transformed chi-square statistic is defined for testing hypotheses of quasi-independence in the incomplete multi-dimensional contingency table and a simple method for determining degrees of freedom is given. A modified transformed chi-squareestimator of the expected cell frequencies is given in closed form for a general class of exact linear constraints. The co-variance matrix of estimated cell frequencies is derived under the assumption of a conditional Poisson distribution.