Tests for establishing compatibility of an observed genotype distribution with Hardy-Weinberg equilibrium in the case of a biallelic locus

The classical chi(2)-procedure for the assessment of genetic equilibrium is tailored for establishing lack rather than goodness of fit of an observed genotype distribution to a model satisfying the Hardy-Weinberg law, and the same is true for the exact competitors to the large-sample procedure, which have been proposed in the biostatistical literature since the late 1930s. In this contribution, the methodology of statistical equivalence testing is adopted for the construction of tests for problems in which the assumption of approximate compatibility of the genotype distribution actually sampled with Hardy-Weinberg equilibrium (HWE) plays the role of the alternative hypothesis one aims to establish. The result of such a construction highly depends on the choice of a measure of distance to be used for defining an indifference zone containing those genotype distributions whose degree of disequilibrium shall be considered irrelevant. The first such measure proposed here is the Euclidean distance of the true parameter vector from that of a genotype distribution with identical allele frequencies being in strict HWE. The second measure is based on the (scalar) parameter of the distribution first introduced into the present context by Stevens (1938, Annals of Eugenics 8, 377-383). The first approach leads to a nonconditional test (which nevertheless can be carried out in a numerically exact way), the second to an exact conditional test shown to be uniformly most powerful unbiased (UMPU) for the associated pair of hypotheses. Both tests are compared in terms of the exact power attained against the class of those specific alternatives under which HWE is strictly satisfied.     

A program for maximum likelihood estimation and likelihood ratio tests in one-locus ABO-like systems based upon population phenotype data.

Phenotypes in an ABO-like system of a number of genetically-independent persons from a number of populations are supposed to be observed. The program which is written in FORTRAN calculates maximum likelihood estimates of gene frequencies and their standard errors in each population and in the populations taken together. Furthermore the program calculates expected values and likelihood ratio and goodness of fit chi-square tests of Hardy-Weinberg equilibrium. If several subpopulations are pooled together a likelihood ratio test of homogeneity is performed.     

Goodness of Fit Tests with Misclassified Data Based on φ‐Divergences

The well known χ² goodness of fit test for a multinomial distribution is generally biased when observations are subject to misclassification. In this paper, based on a double sampling scheme, the family of φ‐divergence test statistics is introduced for testing goodness of fit under misclassification of the data. The case of binomial data is discussed and an illustrative example is also given.     

Mu1: a very high affinity subtype of enkephalin binding sites in rat brain

Displacement studies of [3H]-[D-Ala2-MePhe4-Gly-ol5]-enkephalin ([3H]-DAGO) and [3H]-[D-Ala2-D-Leu5]-enkephalin ([3H]-DADL) by the corresponding unlabeled ligands show that there are at least three classes of sites which bind these enkephalin analogs with high affinity. Using computer modeling, the introduction of the third site significantly improved the goodness of fit in ten consecutive experiments. These sites appear to correspond to the mu, delta and mu 1 sites, with mean dissociation constants of 11, 1.3 and 0.9 nM for DADL and 2.5, 300 and 0.3 nM for DAGO, respectively.     

A possibility to predict the severity of individual damage following the exposure to supralethal radiation dosage. Prediction of working capacity of irradiated rats and dogs by response to pre-exposure physical load

The experiments on rats and dogs exposed to radiation doses which cause intestinal and cerebral forms of radiation sickness have shown that, within the first 24 hours post-exposure, efficiency impairment can be predicted by changes in some indices of the cardiovascular and endocrine systems after physical load used as a non-radiation testing agent. The goodness of fit of experimental data to the expected result is up to 50-70%, still rising if a number of indices for different sections of the neuroendocrine system are used at a time.     

S-Sample Smooth Goodness of Fit Testing: Rederivation and Monte Carlo Assessment

S-sample smooth goodness of fit tests may be constructed using components from one sample goodness of fit testing. Each sample could be assessed for consistency with a target distribution using these components, although that is not our objective here. Contrasts in the components may be used to assess consistency of the samples with each other. If all the samples are consistent, we could then conveniently perform a one-sample goodness of fit test for the target distribution. If the samples are not consistent, an LSD-type analysis can be performed on the one-sample components to identify where the differences between occur. This approach gives a detailed and informative scrutiny of the data.     

Goodness of Fit Tests for Proportional Hazards Regression Models

We present a test of goodness of fit for the proportional hazard regression model. The test is based on a score statistic for testing against local mixture alternatives. Contrary to the findings of several other authors, we detect a significant lack of fit in Freireich's leukemia data.     

Testing goodness of fit for stochastic models of carcinogenesis.

This paper considers the utility of statistical goodness of fit testing in the context of mechanistic models of carcinogenesis. Two stochastic models of carcinogenesis were tested with several sets of experimental and epidemiological data using a formal goodness of fit test specially designed to accommodate censored observations: these were the two-stage model allowing for clonal expansion of initiated cells and its simpler version with gamma distributed promotion time. The results of this application, supplemented by visual examination of local likelihood kernel estimates of the hazard function and the corresponding model-based estimates, show that mechanistic models of carcinogenesis provide a good fit to the data in the majority of cases under study.     

Modified GEE and Goodness of the Marginal Fit (GOMF) Test with Correlated Binary Responses for Contingency Tables

This paper addresses testing the goodness of fit of models for marginal probabilities estimated by generalized estimating equations. We develop a modified version of generalized estimating equation and a goodness‐of‐fit test based on the fitted marginal means. The test statistic is easy to compute and has a simple reference distribution. Its performance is evaluated asymptotically and in small samples. It is also compared to the deviance and Pearson X² statistics. Example applications are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

U‐CARE: Utilities for performing goodness of fit tests and manipulating CApture–REcapture data

In any statistical analysis, assessing the goodness of fit of a model to the data is crucial to avoid drawing incorrect conclusions. U‐CARE is a computer application that deals with the mandatory first steps of the analyses of capture–recapture data: the preparation of the data set and the assessment of the fit of a general model (Cormack‐Jolly‐Seber and variants for single‐state data; Jolly‐Move and variants for multi‐state data). U‐CARE implements the current state of the art in goodness‐of‐fit testing by incorporating components aimed at detecting the most likely departures from assumptions ( Pradel et al. 2003, 2005 ). It is a free and stand‐alone application for Windows.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

A Comparison Between Several Goodness of Fit Tests for Cox's Model

A comparison is made between two approaches to testing goodness of fit of Cox's regression model for survival data. The first approach is based on the inclusion of time dependent covariates, whereas the second one is based on the autocovariance of successive contributions to the derivative of the loglikelihood. It appears that the second test is most appropriate for testing in situations where the structure of the departure from proportional hazards is not known a priori. An approximate expression for the relative efficiency of the two test procedures is presented.     

The Performance of Goodness of Fit Tests for Logistic Regression with Discrete Covariates

This paper considers the distribution of previously proposed goodness of fit tests when some or all of the covariates are dichotomous variables. The simulations show that of the statistics suggested for testing fit only one appears suitable for use with discrete covariates. This statistic uses conditional maximum likelihood estimates and groups the estimated probabilities into groups of equal size or into groups based on the patterns of the covariates when these are few in number.     

Assessing the goodness of fit of logistic regression models in large samples: A modification of the Hosmer-Lemeshow test

Evaluating the goodness of fit of logistic regression models is crucial to ensure the accuracy of the estimated probabilities. Unfortunately, such evaluation is problematic in large samples. Because the power of traditional goodness of fit tests increases with the sample size, practically irrelevant discrepancies between estimated and true probabilities are increasingly likely to cause the rejection of the hypothesis of perfect fit in larger and larger samples. This phenomenon has been widely documented for popular goodness of fit tests, such as the Hosmer-Lemeshow test. To address this limitation, we propose a modification of the Hosmer-Lemeshow approach. By standardizing the noncentrality parameter that characterizes the alternative distribution of the Hosmer-Lemeshow statistic, we introduce a parameter that measures the goodness of fit of a model but does not depend on the sample size. We provide the methodology to estimate this parameter and construct confidence intervals for it. Finally, we propose a formal statistical test to rigorously assess whether the fit of a model, albeit not perfect, is acceptable for practical purposes. The proposed method is compared in a simulation study with a competing modification of the Hosmer-Lemeshow test, based on repeated subsampling. We provide a step-by-step illustration of our method using a model for postneonatal mortality developed in a large cohort of more than 300 000 observations.     

种子植物区系属大小与种多度分布格局的相似性

对属大小的分析是植物区系研究的重要组成,属大小的分布格局可以在一定程度上反映植物区系的类群多样性与系统发生多样性。在植物区系研究中,平均属大小指标与植物区系包含的物种数有关,使不同植物区系的研究结果难以进行整合分析。对中国8个地区种子植物区系属大小的分布格局进行研究,对数级数模型取得了良好的拟合结果,拟合优度R²＞0.99,属大小表现出与种—多度相似的分布格局。稀疏分析和相关分析结果表明：抽样大小较小时,平均属大小在稀疏分析前后的相关性不显著;而对数级数模型中α指数与稀疏分析前表现出稳定的相关性。同时,对于相同的抽样大小α指数与平均属大小之间存在显著的相关性。因此,α指数可以作为稳定的指标来描述植物区系的丰富程度。     

THE BETA-BINOMIAL MODEL: ACCOUNTING FOR INTER-TRIAL VARIATION IN REPLICATED DIFFERENCE AND PREFERENCE TESTS

Binomial tests are commonly used in sensory difference and preference testing under the assumptions that choices are independent and choice probabilities do not vary from trial to trial. This paper addresses violations of the latter assumption (often referred to as overdispersion) and accounts for variation in inter-trial choice probabilities following the Beta distribution. Such variation could arise as a result of differences in test substrate from trial to trial, differences in sensory acuity among subjects or the existence of latent preference segments. In fact, it is likely that overdispersion occurs ubiquitously in product testing. Using the Binomial model for data in which there is inter-trial variation may lead to seriously misleading conclusions from a sensory difference or preference test. A simulation study in this paper based on product testing experience showed that when using a Binomial model for overdispersed Binomial data, Type I error may be 0.44 for a Binomial test specification corresponding to a level of 0.05. Underestimation of Type I error using the Binomial model may seriously undermine legal claims of product superiority in situations where overdispersion occurs. The Beta-Binomial (BB) model, an extension of the Binomial distribution, was developed to fit overdispersed Binomial data. Procedures for estimating and testing the parameters as well as testing for goodness of fit are discussed. Procedures for determining sample size and for calculating estimate precision and test power based on the BB model are given. Numerical examples and simulation results are also given in the paper. The BB model should improve the validity of sensory difference and preference testing.     

Testing Differentiation in Diploid Populations

We examine the power of different exact tests of differentiation for diploid populations. Since there is not necessarily random mating within populations, the appropriate hypothesis to construct exact tests is that of independent sampling of genotypes. There are two categories of tests, F(ST)-estimator tests and goodness of fit tests. In this latter category, we distinguish ``allelic statistics', which account for the nature of alleles within genotypes, from ``genotypic statistics' that do not. We show that the power of F(ST)-estimator tests and of allelic goodness of fit tests are similar when sampling is balanced, and higher than the power of genotypic goodness of fit tests. When sampling is unbalanced, the most powerful tests are shown to belong to the allelic goodness of fit group.     

The suitability of using cryopreservation of spermatozoa for the conservation of genetic diversity in African catfish (Clarias gariepinus)

• 1.1. The effect of cryopreservation of semen on expected Hardy-Weinberg proportions in the f₁ progeny of selected breeding pairs was evaluated.
• 2.2. Four different African catfish breeding pairs were selected, each pair displaying different heterozygous alleles at the glucose-6-phosphate isomerase-1 or 2 loci.
• 3.3. Equal volumes of ova from each female were artificially inseminated with cryopreserved or fresh semen obtained from males possessing corresponding genotypes.
• 4.4. A comparison of growth rates between f₁ groups of offspring produced from fresh and cryopreserved semen was made.
• 5.5. The G-test for goodness of fit showed no significant differences from expected Hardy-Weinberg proportions for allele frequencies obtained in the f₁ progeny.
• 6.6. The application of cryopreservation for the conservation of genetic diversity is discussed.

     

A possibility to predict the severity of individual damage following the exposure to supralethal radiation dosage. Prediction of the clinical manifestation of radiation injury in laboratory animals by response to pre-exposure hypobaric hypoxia

The experiments on rats, dogs and monkeys exposed to radiation doses which cause intestinal and cerebral forms of radiation sickness have shown that the severity of a clinical state can be predicted by cardiovascular and endocrine changes when a pre-exposure test with moderate hypobaric hypoxia is performed. The goodness of fit of experimental data to the exposed result is up to 60-80%, still rising if a set of indices of the neuroendocrine status is used.     

Time variations in the risk of cancer following irradiation in childhood

The Japanese atomic bomb survivors and three other cohorts of children exposed to radiation are analyzed, and evidence is found for a reduction in the radiation-induced relative risk of cancers other than leukemia with time following exposure. Multiplicative adjustments to the excess risk either of the form exp[-delta.(time since exposure)] or of the form [time since exposure] gamma give equivalent goodness of fit. Using the former type of adjustment an annual overall reduction of 6.9-8.6% in excess relative risk is indicated (depending on the year after which this reduction might take effect). Using the second type of multiplier an adjustment to the excess relative risk varying between [time after exposure]-2.0 and [time after exposure]-3.2 fits best overall. All these reductions are statistically significant at the 5% level. There is no significant variation by cohort, by sex, by cancer type, or by age at exposure group in the degree of annual reduction in excess relative risk. Although time-adjusted relative and absolute risk models give equivalently good fits within each cohort, there is significant variation between cohorts in the degree of increase of risk with time in the absolute risk formulation, in contrast to the lack of such heterogeneity for the relative risk formulation. It is shown that if the range of observed reductions in relative risk is assumed to operate 40 or more years after exposure in the youngest age groups, the calculated UK population risks would be reduced by 30-45% compared to those based on a constant relative risk model.