Goodness-of-fit in generalized nonlinear mixed-effects models

In recent years, generalized linear and nonlinear mixed-effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data. To date, much of the work has focused on various methods for estimating and comparing the parameters of mixed-effects models. Very little work has been done in the area of model selection and goodness-of-fit, particularly with respect to the assumed variance-covariance structure. In this paper, we present a goodness-of-fit statistic which can be used in a manner similar to the R2 criterion in linear regression for assessing the adequacy of an assumed mean and variance-covariance structure. In addition, we introduce an approximate pseudo-likelihood ratio test for testing the adequacy of the hypothesized convariance structure. These methods are illustrated and compared to the usual normal theory likelihood methods (Akaike's information criterion and the likelihood ratio test) using three examples. Simulation results indicate the pseudo-likelihood ratio test compares favorably with the standard normal theory likelihood ratio test, but both procedures are sensitive to departures from normality.     

Effects of nonideal input functions on PET measurements of pulmonary blood flow.

Regional pulmonary blood flow (rPBF) can be measured with an intravenous infusion of 15O-labeled water and positron emission tomography (PET). The current method depends on two assumptions related to the input of activity to the lung during the scan: 1) the pulmonary arterial tracer input is constant (i.e., a "step function" in shape), and 2) the scan begins at the instant of arrival of the step function. To determine the effect that departures from these assumptions might have on the measurement of rPBF, we performed a series of mathematical simulations for three different input functions: 1) a step function that arrived either 1 or 2 s before or after scan start; 2) a dispersed input function, with activity rising during the scan period; and 3) a combination of these two errors. Calculated values, based on the standard assumptions, were compared against the "known" values used in generating the simulated data. The results show that timing errors associated with starting the scan late cause an overestimation of rPBF, whereas timing errors due to low regional flow or departures from the assumed step input function both cause an underestimation of true rPBF. Thus, in actual practice, the combined errors probably partially offset one another. Except for states of truly high rPBF and low lung density, the errors remain less than 15% of the true value. We conclude that PET measurements of rPBF are not highly sensitive to these presumably common departures from the assumed pulmonary arterial input function to lung regions of interest.     

Exact tests of Hardy-Weinberg equilibrium and homogeneity of disequilibrium across strata

Detecting departures from Hardy-Weinberg equilibrium (HWE) of marker-genotype frequencies is a crucial first step in almost all human genetic analyses. When a sample is stratified by multiple ethnic groups, it is important to allow the marker-allele frequencies to differ over the strata. In this situation, it is common to test for HWE by using an exact test within each stratum and then using the minimum P value as a global test. This approach does not account for multiple testing, and, because it does not combine information over strata, it does not have optimal power. Several approximate methods to combine information over strata have been proposed, but most of them sum over strata a measure of departure from HWE; if the departures are in different directions, then summing can diminish the overall evidence of departure from HWE. An exact stratified test is more appealing because it uses the probability of genotype configurations across the strata as evidence for global departures from HWE. We developed an exact stratified test for HWE for diallelic markers, such as single-nucleotide polymorphisms (SNPs), and an exact test for homogeneity of Hardy-Weinberg disequilibrium. By applying our methods to data from Perlegen and HapMap--a combined total of more than five million SNP genotypes, with three to four strata and strata sizes ranging from 23 to 60 subjects--we illustrate that the exact stratified test provides more-robust and more-powerful results than those obtained by either the minimum of exact test P values over strata or approximate stratified tests that sum measures of departure from HWE. Hence, our new methods should be useful for samples composed of multiple ethnic groups.     

农作物种子检验中发芽试验的容许误差的统计模型与计算

当代计算手段远远优于计算种子检验规程发芽试验现行标准中容许误差数值的20 世纪60年代,鉴于此,本文在提出发芽试验统计检验模型的基础上,利用MATLAB编程重新计算了容许误差值．过去计算的三种容许误差有较大的正偏差,另一种有很小的负偏差．建议对现行的种子检验规程有关计数型的容许误差数值进行修订．     

A disequilibrium coefficient approach to Hardy-Weinberg testing

A comparison was made of various tests for Hardy-Weinberg equilibrium, with emphasis on methods for multiple alleles. For an overall test of deviations from equilibrium, the classical chi-square goodness-of-fit test generally performs well, with continuity corrections needed for extreme gene frequencies or extreme departures from equilibrium. For small samples, probability tests are preferable and for multiple alleles these probability tests may be performed on a sample of all possible sets of genotypic frequencies having a fixed set of sample gene frequencies. Numerical work showed that the continuity-corrected chi-square was the most conservative test procedure, and the uncorrected chi-square the least conservative. With multiple alleles, a better appreciation of the nature of departures from equilibrium is given by the use of disequilibrium coefficients, defined for each heterozygote as the difference between observed and expected frequencies. Likelihood-ratio tests can be used to test each of these coefficients individually but a satisfactory procedure is to divide the squared estimate of each coefficient by its estimated variance and regard the ratio as a single-degree-of-freedom chi-square. Numerical studies confirmed the validity of this approach, which has the great advantage of not requiring solutions of nonlinear equations.     

Multiple-Locus Departures from Panmictic Equilibrium within and between Village Gene Pools of Amerindian Tribes at Different Stages of Agglomeration

A comparative analysis of departures from multiple-locus Hardy-Weinberg equilibrium is presented for a set of four tribal Indian groups (the Yanomama, Makiritare, Wapishana and Ticuna) from the lowlands of South America. These tribes span a range of agglomeration and acculturation from the most traditional, swidden horticulturalists to frontier townspeople. The small-group social organization typical of traditional horticulturalists leads to substantial departures from tribal panmixia, as manifested by the distribution of multiple-locus genotypes both within and between villages. Within villages, the departures from single-locus Hardy-Weinberg equilibrium are small and nonsignificant, but the departures from gametic equilibrium (independence of loci) are substantial, even for the unlinked loci we have used to characterize these populations. The departures from single-locus homogeneity across villages are also substantial. One of the normal concomitants of increasing acculturation in this setting is an increase in agglomeration. As agglomeration increases, the departures from multiple-locus panmixia decrease, a process that can be very rapid. We discuss both the shifting balance theory of evolution and punctuated evolutionary rates in light of the small group social organization that must have obtained throughout most of human evolution.     

On the Asymptotic Relative Efficiency of the Mantel-Haenszel Test

The Mantel-Haenszel test is optimal when the odds ratio is constant. This paper investigates the effects of departures from the assumption of a constant odds ratio on the behavior of the Mantel-Haenzel test. A simple approximation is proposed for the non-null distribution of the test statistic. Based on this approximation, the asymptotic relative efficiency of the Mantel-Haenszel test, compared to the overall χ² test for no partial association, is calculated. For the case of 2 strata, it is shown that the Mantel-Haenszel test is efficient as long as the logarithms of the odds ratios are of the same sign and their absolute values exceed 1.     

hw‐quickcheck: an easy‐to‐use computer program for checking genotypes for agreement with Hardy–Weinberg expectations

Deviations from Hardy–Weinberg expectations are frequently a sign of genotyping error. hw‐quickcheck is an easy‐to‐use computer program for detecting departures from Hardy–Weinberg equilibrium. hw‐quickcheck uses exact tests for all of its calculations. These tests include a global test for heterozygote excess/deficiency and genotype‐specific tests.     

Threshold and other departures from linear-quadratic curvature in the non-cancer mortality dose-response curve in the Japanese atomic bomb survivors

Recently released data on non-cancer mortality in Japanese atomic bomb survivors are analysed using a variety of generalised relative risk models that take account of errors in estimates of dose to assess the dose-response at low doses. If linear-threshold, quadratic-threshold or linear-quadratic-threshold relative risk models (the dose-response is assumed to be linear, quadratic or linear-quadratic above the threshold, respectively) are fitted to the non-cancer data there are no statistically significant (p>0.10) indications of threshold departures from linearity, quadratic curvature or linear-quadratic curvature. These findings are true irrespective of the assumed magnitude of dosimetric error, between 25%–45% geometric standard deviations. In general, increasing the assumed magnitude of dosimetric error had little effect on the central estimates of the threshold, but somewhat widened the associated confidence intervals. If a power of dose model is fitted, there is little evidence (p>0.10) that the power of dose in the dose-response is statistically significantly different from 1, again irrespective of the assumed magnitude of dosimetric errors in the range 25%–45%. Again, increasing the size of the errors resulted in wider confidence intervals on the power of dose, without marked effect on the central estimates. In general these findings remain true for various non-cancer disease subtypes.     

Robustness of path analysis of family resemblance against deviations from multivariate normality

Path analysis is one of several methods available for quantitative genetic analysis, providing for both tests of hypotheses and estimates of relevant parameters. Central to the theory is the assumption that the observations follow a multivariate normal distribution within families. The purpose of the present investigation is to assess the effects of a certain type of departures from multivariate normality using quantitative family data on lipid and lipoprotein levels. The results show that even large departures produce reasonably unbiased parameter estimates. Whereas moderate departures lead to few inferential errors in hypothesis testing, gross departures from multivariate normality may have considerable effects on likelihood ratio tests.     

Testing separability in spatial-temporal marked point processes

Nonparametric tests for investigating the separability of a spatial-temporal marked point process are described and compared. It is shown that a Cramer-von Mises-type test is very powerful at detecting gradual departures from separability, and that a residual test based on randomly rescaling the process is powerful at detecting nonseparable clustering or inhibition of the marks. An application to Los Angeles County wildfire data is given, in which it is shown that the separability hypothesis is invalidated largely due to clustering of fires of similar sizes within periods of up to 3.9 years.     

The problem of a covariate-time qualitative interaction in a survival study

We introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards. Although this kind of alternative is somewhat rare, designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures, upon which we can call in those few cases where such departures are suspected. Furthermore, the proposed test and an approximate version of the test are seen to suffer only moderate losses in power, when compared with their optimal counterparts, should the alternative be one of proportional hazards. Our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail. The model we use in this work is a special case of the one introduced by Anderson and Senthilselvan (1982. Applied Statistics 31, 44-51). We propose overcoming an inferential problem stemming from their model by using the methods of Davies (1977, Biometrika 64, 247-254; 1987, Biometrika 74, 33-43) backed up by resampling techniques. We also look at an approach relying directly on resampling techniques. The distributional aspects of this approach under the null hypothesis are interesting but, practically, its behaviour is such that its use cannot be generally recommended. Outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory.     

A Simple Geometrical Pattern for the Branching Distribution of the Bronchial Tree, Useful to Estimate Optimality Departures

The design of the bronchial tree has largely been proposed as a model of optimal design from a physical-functional perspective. However, the distributive function of the airway may be more related to a geometrical than a physical problem. The bronchial tree must distribute a three dimensional volume of inspired air on a two dimensional alveolar surface, included in a limited volume. It is thus valid to ask whether an optimal bronchial tree from a physical perspective is also optimum from a geometrical point of view. In this paper we generate a simple geometric model for the branching pattern of the bronchial tree, deducing relationships that permit estimation of the departures from the geometrical optimum of each bifurcation. We also, for comparative purposes, estimate the departures from the physical optimum. From the geometrical assumptions: i) a symmetrical dichotomic fractal design, ii) with minimum volume and iii) maximum dispersion of the terminal points; and several simulations we suggest that the optimality is characterized by a bifurcation angle theta approximately 60 degrees and a length reduction scale gamma = (1/2)(1/3) = 0.7937. We propose distances from the physical and geometrical optimality defined as Euclidean distances from the expected optima. We show how the advanced relationships and the distances can be used to estimate departures from the optimality in bronchographs of four species. We found lower physical and geometrical departures in the distal zone than those of the proximal zones, as well as lower physical than geometrical departures from optimality.     

Measures of Imbalance for Unbalanced Models

In this paper we present a procedure to measure the degree of imbalance of an unbalanced data set. The procedure is based on choosing an appropriate loglinear model for the subclass frequencies of the data. A measure of imbalance is then introduced as some function of the chi-squared statistic used in the goodness-of-fit test for the loglinear model. The proposed procedure can also be used to measure departures from certain types of balance, such as proportionality of subclass frequencies, partial balance, and last-stage uniformity.     

Ancestral inference and the study of codon bias evolution: implications for molecular evolutionary analyses of the Drosophila melanogaster subgroup

Reliable inference of ancestral sequences can be critical to identifying both patterns and causes of molecular evolution. Robustness of ancestral inference is often assumed among closely related species, but tests of this assumption have been limited. Here, we examine the performance of inference methods for data simulated under scenarios of codon bias evolution within the Drosophila melanogaster subgroup. Genome sequence data for multiple, closely related species within this subgroup make it an important system for studying molecular evolutionary genetics. The effects of asymmetric and lineage-specific substitution rates (i.e., varying levels of codon usage bias and departures from equilibrium) on the reliability of ancestral codon usage was investigated. Maximum parsimony inference, which has been widely employed in analyses of Drosophila codon bias evolution, was compared to an approach that attempts to account for uncertainty in ancestral inference by weighting ancestral reconstructions by their posterior probabilities. The latter approach employs maximum likelihood estimation of rate and base composition parameters. For equilibrium and most non-equilibrium scenarios that were investigated, the probabilistic method appears to generate reliable ancestral codon bias inferences for molecular evolutionary studies within the D. melanogaster subgroup. These reconstructions are more reliable than parsimony inference, especially when codon usage is strongly skewed. However, inference biases are considerable for both methods under particular departures from stationarity (i.e., when adaptive evolution is prevalent). Reliability of inference can be sensitive to branch lengths, asymmetry in substitution rates, and the locations and nature of lineage-specific processes within a gene tree. Inference reliability, even among closely related species, can be strongly affected by (potentially unknown) patterns of molecular evolution in lineages ancestral to those of interest.     

A P-Value for an Exact Two-Sided Test of Hardy-Weinberg Equilibrium

A two-tailed P-value is proposed for testing two-sided departures from Hardy-Weinberg equilibrium at a diallelic locus. The calculation of P uses the exact conditional distribution of the test statistic P, the observed number of heterozygotes in the sample. The proposed P-value is always two-tailed, unlike other P-values proposed in the literature.     

An extension of the proportional-hazards model for grouped data

A family of transformations for probabilities is considered for the analysis of grouped survival data. Additive and multiplicative models for the hazard function are compared. A method is suggested for estimating the scale on which an additive representation of the hazard, in terms of explanatory variables, is appropriate. Computations are performed using GLIM. A test for departures from the grouped proportional-hazards model is introduced. An example is analyzed by the methods put forward.     

Deleterious mutations at the mitochondrial ND3 gene in South American marsh rats (Holochilus).

Statistical analyses of DNA sequences have revealed patterns of nonneutral evolution in mitochondrial DNA of mice, humans, and Drosophila. Here we report patterns of mitochondrial sequence evolution in South American marsh rats (genus Holochilus). We sequenced the complete mitochondrial ND3 gene in 82 Holochilus brasiliensis and 21 H. vulpinus to test the neutral prediction that the ratio of nonsynonymous to synonymous nucleotide changes is the same within and between species. Within H. brasiliensis we observed a greater number of amino acid polymorphisms than expected based on interspecific comparisons. This contingency table analysis suggests that many amino acid polymorphisms are mildly deleterious. Several tests of the frequency distribution also revealed departures from a neutral, equilibrium model, and these departures were observed for both nonsynonymous and synonymous sites. In general, an excess of rare sites was observed, consistent with either a recent selective sweep or with populations not at mutation-drift equilibrium.     

Diffractometric measurements of the tonicity volume relations of human red cells in hypotonic systems

Red cell volumes in media sufficiently hypotonic to produce partial hemolysis have been measured with a high speed hematocrit and by an improved diffraction method. A comparison of the results shows that the upward departures from the linear relation based on the van't Hoff-Mariotte law are not observed when the volumes are measured diffractometrically. Downward departures are observed at low tonicities by both methods. These results provide strong evidence that the upward departures are due to the inclusion of semirigid ghosts in the column of packed red cells which is measured when the volumes are found with the hematocrit.