Abstract: This paper discusses the use of PP-plots for survival distributions where for a pair of survival distributions, one is plotted against the other. This is another way of visualizing the nature of the relationship between the two survival distributions along with typical Kaplan-Meier plots. For three survival distributions, the PPP-plot is introduced where the survival distributions are plotted against each other in three-dimensions. At the population level, measures of divergence between distributions are introduced based on areas and lengths associated with the PP- and PPP- plots. At the sample level, two test statistics are defined, based on these areas and lengths, to test the null hypothesis of equivalent survival curves. A simulation exercise showed that, overall, the new tests are worthy competitors to the log-rank and Wilcoxon tests and also to a Levine-type test and a Kolmogorov-Smirnov type test for the case of crossing survival curves. The paper also shows how the PP-plot can be used to estimate the hazard ratio and to assess the ratio of hazard functions if proportional hazards are not appropriate. Finally, the methods introduced are illustrated on two cancer data sets.

Keywords: Crossing survival curves, Hazard ratio, Kaplan-Meier, Log-rank test, PP-plot, Wilcoxon test.

Abstract: The article focuses on the discussion of basic approaches to hypotheses testing, which are Fisher, Jeffreys, Neyman, Berger approaches and a new one proposed by the author of this paper and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered on the basis of computed examples. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses a posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibilityof making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical decision. Moreover, for any information on the basis of which the decision is made, the set of error probabilities is defined for which the decision with given reliability is possible.

Keywords: Hypotheses testing, -value, likelihood ratio, frequentist approaches, Bayesian approach, constrained Bayesian method, decision regions.

Abstract: The Hardy-Weinberg Principle explains how random mating can produce and maintain a population in equilibrium, that is: with constant genotypic proportions. The Hardy-Weinberg formula is in constant use as a basis for developing population genetics theory. Here we give a complete description of a model which can sustain equilibrium but with a general mating system, thereby giving a much broader basis on which to develop population genetics. It was S. N. Bernstein who first showed how Mendel’s first law could be justified simply on the basis of observations of populations in equilibrium. We show how the model can be applied to exploring the change in incidence of a genetic disorder.

Keywords: Hardy-Weinberg Equilibrium, Non-random Mating, Mendel’s First Law, Population Genetics Theory, Tay-Sachs disease.