An accurate test for homogeneity of odds ratios based on Cochran's Q-statistic

Background: A frequently used statistic for testing homogeneity in a meta-analysis of K independent studies is Cochran's Q. For a standard test of homogeneity the Q statistic is referred to a chi-square distribution with K - 1 degrees of freedom. For the situation in which the effects of the studies are logarithms of odds ratios, the chi-square distribution is much too conservative for moderate size studies, although it may be asymptotically correct as the individual studies become large. Methods: Using a mixture of theoretical results and simulations, we provide formulas to estimate the shape and scale parameters of a gamma distribution to t the distribution of Q. Results: Simulation studies show that the gamma distribution is a good approximation to the distribution for Q. Conclusions: : Use of the gamma distribution instead of the chi-square distribution for Q should eliminate inaccurate inferences in assessing homogeneity in a meta-analysis. (A computer program for implementing this test is provided.) This hypothesis test is competitive with the Breslow-Day test both in accuracy of level and in power

Approximations of the Generalized Log-Logistic Distribution to the Chi-Square Distribution

The main purpose of this article is to do approximations graphically and mathematically the four-parameter generalized log-logistic distribution, denoted by G4LL(α,β,m_1,m_2), to the one-parameter Chi-square distribution with υ degrees of freedom. In order to achieve this purpose, this article creates graphically the probability density functions of both distribution and derives mathematically the MGF of the both distributions. To prove the MGF of Chi-square as a special case of the MGF of G4LL distribution, we utilized an expansion of the MacLaurin series. The results show that graphically, the Chi-square distribution can be approximated by the generalized log-logistic distribution. Moreover, by letting α=1,β=-ln⁡(2m_2 ),m_1=v/2 and m_2→∞, the MGF of the G4LL distribution can be written in the form of the MGF of the Chi-square distribution. Thus, the Chi-square distribution is a limiting or special case distribution of the generalized log-logistic distribution.The main purpose of this article is to do approximations graphically and mathematically the four-parameter generalized log-logistic distribution, denoted by G4LL(α,β,m_1,m_2), to the one-parameter Chi-square distribution with υ degrees of freedom. In order to achieve this purpose, this article creates graphically the probability density functions of both distribution and derives mathematically the MGF of the both distributions. To prove the MGF of Chi-square as a special case of the MGF of G4LL distribution, we utilized an expansion of the MacLaurin series. The results show that graphically, the Chi-square distribution can be approximated by the generalized log-logistic distribution. Moreover, by letting α=1,β=-ln⁡(2m_2 ),m_1=v/2 and m_2→∞, the MGF of the G4LL distribution can be written in the form of the MGF of the Chi-square distribution. Thus, the Chi-square distribution is a limiting or special case distribution of the generalized log-logistic distribution

A plan classifier based on Chi-square distribution tests

To make good decisions in a social context, humans often need to recognize the plan underlying the behavior of others, and make predictions based on this recognition. This process, when carried out by software agents or robots, is known as plan recognition, or agent modeling. Most existing techniques for plan recognition assume the availability of carefully hand-crafted plan libraries, which encode the a-priori known behavioral repertoire of the observed agents; during run-time, plan recognition algorithms match the observed behavior of the agents against the plan-libraries, and matches are reported as hypotheses. Unfortunately, techniques for automatically acquiring plan-libraries from observations, e.g., by learning or data-mining, are only beginning to emerge. We present an approach for automatically creating the model of an agent behavior based on the observation and analysis of its atomic behaviors. In this approach, observations of an agent behavior are transformed into a sequence of atomic behaviors (events). This stream is analyzed in order to get the corresponding behavior model, represented by a distribution of relevant events. Once the model has been created, the proposed approach presents a method using a statistical test for classifying an observed behavior. Therefore, in this research, the problem of behavior classification is examined as a problem of learning to characterize the behavior of an agent in terms of sequences of atomic behaviors. The experiment results of this paper show that a system based on our approach can efficiently recognize different behaviors in different domains, in particular UNIX command-line data, and RoboCup soccer simulationThis work has been partially supported by the Spanish Government under project TRA2007-67374-C02-0

On the computation of moments of the partial non-central chi-squared distribution function

Properties satisfied by the moments of the partial non-central chi-square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.Comment: 6 page

Maximally selected chi-square statistics and umbrella orderings

Binary outcomes that depend on an ordinal predictor in a non-monotonic way are common in medical data analysis. Such patterns can be addressed in terms of cutpoints: for example, one looks for two cutpoints that define an interval in the range of the ordinal predictor for which the probability of a positive outcome is particularly high (or low). A chi-square test may then be performed to compare the proportions of positive outcomes in and outside this interval. However, if the two cutpoints are chosen to maximize the chi-square statistic, referring the obtained chi-square statistic to the standard chi-square distribution is an inappropriate approach. It is then necessary to correct the p-value for multiple comparisons by considering the distribution of the maximally selected chi-square statistic instead of the nominal chi-square distribution. Here, we derive the exact distribution of the chi-square statistic obtained by the optimal two cutpoints. We suggest a combinatorial computation method and illustrate our approach by a simulation study and an application to varicella data