Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Understanding Recession and Self-Rated Health with the Partial Proportional Odds Model: An Analysis of 26 Countries

IntroductionSelf-rated health is demonstrated to vary substantially by both personal socio-economic status and national economic conditions. However, studies investigating the combined influence of individual and country level economic indicators across several countries in the context of recent global recession are limited. This paper furthers our knowledge of the effect of recession on health at both the individual and national level.MethodsUsing the Life in Transition II study, which provides data from 19,759 individuals across 26 European nations, we examine the relationship between self-rated health, personal economic experiences, and macro-economic change. Data analyses include, but are not limited to, the partial proportional odds model which permits the effect of predictors to vary across different levels of our dependent variable.ResultsHousehold experiences with recession, especially a loss of staple good consumption, are associated with lower self-rated health. Most individual-level experiences with recession, such as a job loss, have relatively small negative effects on perceived health; the effect of individual or household economic hardship is strongest in high income nations. Our findings also suggest that macroeconomic growth improves self-rated health in low-income nations but has no effect in high-income nations. Individuals with the greatest probability of “good” self-rated health reside in wealthy countries ($23,910 to $50, 870 GNI per capita).ConclusionBoth individual and national economic variables are predictive of self-rated health. Personal and household experiences are most consequential for self-rated health in high income nations, while macroeconomic growth is most consequential in low-income nations.     

Confidence Intervals for the Difference of Median Survival Times Using the Stratified Cox Proportional Hazards Model

The stratified Cox proportional hazards model is introduced to incorporate covariates and involve nonproportional treatment effect of two groups into the analysis and then the confidence interval estimators for the difference in median survival times of two treatments in stratified Cox model are proposed. The one is based on baseline survival functions of two groups, and the other on average survival functions of two groups. I illustrate the proposed methods with an example from a study conducted by the Radiation Therapy Oncology Group in cancer of the mouth and throat. Simulations are carried out to investigate the small‐sample properties of proposed methods in terms of coverage rates.     

Analysis of Dichotomized Factorial Data Using Cox's Proportional Hazards Model and Related Tests

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are derived for any number of prognostic factors, but for the sake of simplicity, the mathematical details are presented for the case of two factors. This method is not based on the life table procedure. Kaplan-Meier estimates are obtained for the survival function of the internal control population, Which are in turn used to determine the expected number of deaths in the study population. The asymptotic (large sample) joint sampling distribution of the relative mortality rates is derived and some relevant simultaneous and conditional statistical tests are discussed. The relative mortality rates of several prognostic factors may be jointly considered as the multivariate extension of the familiar standard mortality ratio (SMR) of epidemiological studies. A numerical example is discussed to illustrate the method.     

Modeling Multiallelic Selection Using a Moran Model

We present a Moran-model approach to modeling general multiallelic selection in a finite population and show how it may be used to develop theoretical models of biological systems of balancing selection such as plant gametophytic self-incompatibility loci. We propose new expressions for the stationary distribution of allele frequencies under selection and use them to show that the continuous-time Markov chain describing allele frequency change with exchangeable selection and Moran-model reproduction is reversible. We then use the reversibility property to derive the expected allele frequency spectrum in a finite population for several general models of multiallelic selection. Using simulations, we show that our approach is valid over a broader range of parameters than previous analyses of balancing selection based on diffusion approximations to the Wright–Fisher model of reproduction. Our results can be applied to any model of multiallelic selection in which fitness is solely a function of allele frequency.NATURAL selection has long been a topic of interest in population genetics, yet the stochastic theory of genes under selection remains underdeveloped compared to the theory of neutral genes. Due to the interplay of stochastic and deterministic forces, models of selection present analytical challenges beyond those of neutral models, although a great deal of progress has been made with models that use diffusion approximations to a Wright–Fisher model of reproduction. Diffusion approximations with selection are, however, sometimes difficult to employ and always require assumptions about population parameters for tractability. These limitations suggest that there may be value in developing new methods of solving the problem of selection in a finite population, and here we do so using a Moran model of reproduction in place of the familiar Wright–Fisher model. Our approach has two major advantages over previous models: general applicability to a wide variety of selection models and accuracy over a broad range of parameter values. In this work, we propose new expressions for the full stationary distributions of allele frequencies under multiallelic selection, as well as expressions for average allele frequency distributions.We restrict our attention to exchangeable models of selection, meaning that relabeling the alleles will not change selective outcomes and thus that selection will be a function of allele frequency rather than allele identity. Many models of selection can be transformed into frequency-dependent forms (Denniston and Crow 1990), and some common models of selection have the desired property of exchangeability. For example, symmetric overdominant selection, in which heterozygotes have a selective advantage over homozygotes but the specific genotype of homozygote or heterozygote has no further selective effect, can be expressed as frequency-dependent selection on individual (exchangeable) alleles, although the direct selection is actually on diploid genotypes. Many other proposed models of multiallelic balancing selection, in which substantial variation is maintained by selection, can be viewed in this way. Such models have been of particular interest because of the potential application to highly multiallelic systems found in nature, such as self-incompatibility (SI) loci in plants and the major histocompatibility complex (MHC) loci in vertebrates, and the desire to analyze these systems is a motivation for the present work. We now review some of the population genetic theory related to these systems.Early in the history of population genetics, Wright (1939) presented a somewhat controversial stochastic model of gametophytic self-incompatibility (GSI) genes, sparking much further theoretical and empirical work. An analytic theory of multiallelic symmetric overdominance was developed along similar lines to this early model (Kimura and Crow 1964; Takahata 1990) and has been used as an approximation to the unknown mode of selection in the MHC (Takahata et al. 1992). Drawing insights from these first two applications, other biological systems where balancing selection was posited, including sex determination in honeybees (Yokoyama and Nei 1979), fungal mating systems (May et al. 1999), and heterokaryon incompatibility in fungi (Muirhead et al. 2002), have also been modeled successfully using closely related approaches. Progress has been made in using these models to address genealogical (Takahata 1990; Vekemans and Slatkin 1994) and demographic (Muirhead 2001) questions, as well as extending the models into more complex modes of selection (Uyenoyama 2003) and reproduction (Vallejo-Marin and Uyenoyama 2008).Models of genetic variation under balancing selection have traditionally been focused on specific systems, such that extensions require entirely new analyses, and have also included a number of simplifying assumptions in the interest of mathematical tractability. For example, the symmetric overdominance model has been strongly criticized as an unrealistic approximation of MHC evolution (Paterson et al. 1998; Hedrick 2002; Penn et al. 2002; Ilmonen et al. 2007; Stoffels and Spencer 2008), and yet it has proved difficult to make finite-population models of any of the more realistic frequency dependence schemes using the same approaches. A constraint on further progress is the fact that the standard model of stochastic population genetics, the Wright–Fisher model, is in fact quite difficult to analyze.The Wright–Fisher model of reproduction employs nonoverlapping generations, so that for a diploid population of size N, all 2N allele copies are chosen simultaneously when forming a new generation of individuals. While it is straightforward to describe this reproduction scheme mathematically as a discrete-time Markov chain, that chain unfortunately appears intractable even in simple cases (Ewens 2004). Traditionally, then, diffusion approximations have been used to obtain quantities of interest, such as the equilibrium expected number of alleles, allele frequency spectra, and fixation probabilities and times. Diffusion approximations are derived in the limit , but are applicable to problems of finite N, provided that the strengths of other forces such as mutation and selection can be assumed to be weak, of O(N⁻¹) (Ewens 2004). Watterson (1977) derived such a diffusion approximation for multiallelic symmetric overdominance using these assumptions. More recently, as interest in population genetics has turned to problems of inference, Grote and Speed (2002) considered sampling probabilities under the diffusion approximation for symmetric overdominance, while Donnelly et al. (2001) and Stephens and Donnelly (2003) proposed computational methods for some asymmetric models.Although strong selection can be modeled using diffusion approximations by making the product of the population size and the selection coefficient (Ns) large, the assumption of weak selection is not in fact appropriate for the canonical biological systems of balancing selection. Specifically, selection coefficients are defined by the differences in fitness (the expected number of offspring) among individuals in the population at a given time. These differences may be large in systems such as GSI, where the fitness of a very common allele may be very small while the fitness of other alleles may be greater than one.In an attempt to deal with the extremely strong selection of gametophytic self-incompatibility, Wright''s (1939) original model focused attention on the dynamics of a single representative allele. He collapsed the influence of all other alleles into a single summary statistic: the homozygosity, F, which is a function of the frequencies of all alleles, and which Wright (1939) assumed to be constant. The analysis is essentially that of a two-allele system, using a one-dimensional diffusion analysis. This approach, while shown by simulation to be very effective in the appropriate parameter range (Ewens and Ewens 1966), received substantial criticism on mathematical grounds (Fisher 1958; Moran 1962; Ewens 1964b). Ewens (1964b), in particular, objected to the use of diffusion theory for GSI, pointing out that strong frequency-dependent selection violates the diffusion requirement that both the mean and the variance of the change in allele frequencies be small and of O(N⁻¹). Ewens (1964a) then applied Wright''s basic one-dimensional diffusion approach to modeling symmetric overdominance, but assumed that selection was weak and of O(N⁻¹) to stay within the strict limits of the diffusion approximation.Kimura and Crow (1964) and Wright (1966), on the other hand, presented alternative one-dimensional diffusion approximations to symmetric overdominance, closer in spirit to Wright''s original model of GSI, that did not make the weak-selection assumption. Watterson (1977) was concerned about both the inconsistencies of the approximations used in these models and the treatment of F as a constant rather than as a random variable dependent upon allele frequencies. Using his own multiallelic diffusion approximation for symmetric overdominance (Watterson 1977), he derived an alternative (small-Ns) approximation to the frequency of a single representative allele. We consider this approximation, as well as the best-known one-dimensional symmetric overdominance diffusion, the strong-selection approximation of Kimura and Crow (1964), in comparison with our alternative approach to deriving allele frequency spectra under general multiallelic selection with exchangeable alleles.To avoid the approximations required to employ Wright–Fisher/diffusion-based methods, we turn to an alternative model of reproduction in a finite population: the overlapping-generations model of Moran (1962). In the Moran model, a single allele copy dies and another reproduces in each time step, rather than all 2N allele copies simultaneously being replaced by offspring each generation. As in the Wright–Fisher model, this reproduction scheme is represented mathematically by a Markov chain. Unlike the Wright–Fisher model, however, the Moran model can sometimes yield tractable, exact solutions to the underlying Markov chain, without the need to resort to diffusion approximations. We exploit this trait to develop a new stochastic theory of multiallelic selection with minimal dependence on assumptions about population parameter values. Our method has the additional benefit of being flexible: it can accommodate any exchangeable model of multiallelic selection and either of two general models of parent-independent mutation, the infinite-alleles and k-allele models of mutation. Our Moran-model predictions agree well with the results of Wright–Fisher simulations.     

Unbiased Estimation of Odds Ratio by Using Negative Binomial Plans

The use of the negative binomial distribution in both the numerator and denominator in prospective studies leads to an unbiased estimate of the odds ratio and an exact expression for its variance. Sample sizes that minimize the variance of odds ratio estimates are specified. The variance of the odds ratio estimate is shown to be close to the Cramér-Rao lower bound.     

Relating Cox's Proportional Hazard Model to the Exponential Model for Survival

An approximate representation is given for the partial likelihood estimate of the regression coefficient in Cox's proportional hazard model which indicates how it measures the association between survival time and covariate. The case of a single covariate is concentrated on. The representation is closely related to the first step of a Newton-Raphson iteration, i.e. to the score test. A similar representation for the Feigl-Zelen exponential model shows that a similar type of association is being measured, if observed lifetimes are interpreted as expected lifetimes of ordered exponentials. Necessary and sufficient conditions for the existence of Cox's estimate in the simple case are also written down.     

Joint Modeling of the Clinical Progression and of the Biomarkers' Dynamics Using a Mechanistic Model

Summary Joint models are used to rigorously explore the relationship between the dynamics of biomarkers and clinical events. In the context of HIV infection, where the multivariate dynamics of HIV‐RNA and CD4 are complex, a mechanistic approach based on a system of nonlinear differential equations naturally takes into account the correlation between the biomarkers. Using data from a randomized clinical trial comparing dual antiretroviral therapy to a single drug regimen, a full maximum likelihood approach is proposed to explore the relationship between the evolution of the biomarkers and the time to a clinical event. The role of each marker as an independent predictor of disease progression is assessed. We show that the joint dynamics of HIV‐RNA and CD4 captures the effect of antiretroviral treatment; the CD4 dynamics alone is found to capture most but not all of the treatment effect.     

L1 Penalized Estimation in the Cox Proportional Hazards Model

This article presents a novel algorithm that efficiently computes L₁ penalized (lasso) estimates of parameters in high‐dimensional models. The lasso has the property that it simultaneously performs variable selection and shrinkage, which makes it very useful for finding interpretable prediction rules in high‐dimensional data. The new algorithm is based on a combination of gradient ascent optimization with the Newton–Raphson algorithm. It is described for a general likelihood function and can be applied in generalized linear models and other models with an L₁ penalty. The algorithm is demonstrated in the Cox proportional hazards model, predicting survival of breast cancer patients using gene expression data, and its performance is compared with competing approaches. An R package, penalized , that implements the method, is available on CRAN.     

基于熵准则的鲁棒的RBF谷胱甘肽发酵建模

在谷胱甘肽的发酵过程建模中, 当试验数据含有噪音时, 往往会导致模型预测精度和泛化能力的下降。针对该问题, 提出了一种新的基于熵准则的RBF神经网络建模方法。与传统的基于MSE准则函数的建模方法相比, 新方法能从训练样本的整体分布结构来进行模型参数学习, 有效地避免了传统的基于MSE准则的RBF网络的过学习和泛化能力差的缺陷。将该模型应用到实际的谷胱甘肽发酵过程建模中, 实验结果表明: 该方法具有较高的预测精度、泛化能力和良好的鲁棒性, 从而对谷胱甘肽的发酵建模有潜在的应用价值。     

基于熵准则的鲁棒的RBF谷胱甘肽发酵建模

在谷胱甘肽的发酵过程建模中, 当试验数据含有噪音时, 往往会导致模型预测精度和泛化能力的下降。针对该问题, 提出了一种新的基于熵准则的RBF神经网络建模方法。与传统的基于MSE准则函数的建模方法相比, 新方法能从训练样本的整体分布结构来进行模型参数学习, 有效地避免了传统的基于MSE准则的RBF网络的过学习和泛化能力差的缺陷。将该模型应用到实际的谷胱甘肽发酵过程建模中, 实验结果表明: 该方法具有较高的预测精度、泛化能力和良好的鲁棒性, 从而对谷胱甘肽的发酵建模有潜在的应用价值。     

A Measure of Dependence for the Stratified Cox Proportional Hazards Regression Model

Kent and O'Quigley (1988) apply the concept of information gain to measure both global and partial dependence between explanatory variables and a censored response within the framework of the proportional hazards regression model of Cox (1972). The definition of this measure is extended to cover also the stratified Cox model.     

Estimating Attributable Life Expectancy Under the Proportional Mean Residual Life Model

In population-based health research, the so-called population attributable fraction is an important quantity that calculates the percentage of excess risk of morbidity and mortality associated with modifiable risk factors for a given population. While the concept of “risk” is usually measured by event probabilities, in practice it may be of a more direct interest to know the excess life expectancy associated with the modifiable risk factors instead, particularly when mortality is of the ultimate concern. In this paper, we thus propose to study a novel quantity, termed “attributable life expectancy,” to measure the population attributable fraction of life expectancy. We further develop a model-based approach for the attributable life expectancy under the Oakes–Dasu proportional mean residual life model, and establish its asymptotic properties for inferences. Numerical studies that include Monte-Carlo simulations and an actual analysis of the mortality associated with smoking cessation in an Asia Cohort Consortium are conducted to evaluate the performance of our proposed method.

     

苹果园光能截获率的数学模型

应用气象学原理推导了不同纬度、不同栽植行向、不同树形的苹果园光能截获率的数学模型,给出在保证树冠基部外围日照时间大于25％总日照时间的条件下,生长季中充分截获光能的最佳树形、行距、冠高、冠径的优化组合方案。实例分析表明,计算结果与专家经验基本一致。本研究为果园合理栽植,充分利用光能,合理整形修剪提供理论依据与参考方案。     

Estimation of the Proportional Mean Residual Life Model with Internal and Longitudinal Covariates

Statistics in Biosciences - The proportional mean residual life model has been discussed by many authors and provides a useful alternative to the commonly used proportional hazards model for...     

Evaluation of Genetic Diversity in Chinese Wild Apple Species Along with Apple Cultivars Using SSR Markers

China, one of the primary centers of genetic diversity for the genus Malus, is very rich in wild apple germplasm. In this study, genetic diversity in 29 Malus accessions, including 12 accessions from 7 Chinese Malus species, 4 Chinese landraces, and 13 introduced apple cultivars, was assessed using a set of 19 single-locus simple sequence repeat (SSR) markers distributed across all 17 linkage groups of the apple genome. The number of alleles detected at each locus ranged from 2 to 11, with an average of 5.3 per SSR marker. In some accessions, 16 unique alleles were identified. Ten out of these 16 unique alleles (62.5%) were detected exclusively in wild species, indicating that these Chinese wild apple species have considerable genetic diversity and can be used in breeding programs to increase the genetic diversity of apple cultivars. Using 19 SSRs, an unweighted pair-group method with arithmetic average cluster analysis was conducted, and the resulting dendrogram revealed that all cultivars, except for E??peMeBckoe, were clustered together in the same group. The Russian cultivar E??peMeBckoe was closely related to the Chinese crabapple Baihaitang (M. prunifolia), with a high similarity coefficient value of 0.94. Of the two M. sieversii accessions used, one accession showed a close relationship to apple cultivars, while the other accession was closely related to wild apple species, suggesting the presence of a wider genetic diversity in Chinese M. sieversii species. The influence of SSR marker selection on genetic diversity analysis in this Malus collection was also discussed.     

基于苹果EST-SSR的梨种质资源遗传多样性分析

利用来自苹果的8对EST-SSR标记对48份梨(Pyrus)种质资源进行遗传多样性研究,以分析其在梨属植物上的通用性.结果表明,8对EST-SSR引物在供试材料上均能扩增出与苹果大小相似的产物,所有引物共检测到140个基因位点,其中多态性位点129个,多态性比例为92.14%,并且可成功区分不同品种.根据EST-SSR标记所揭示的多态性和UP-GMA法聚类分析,48份梨种质资源在相似系数0.62处可分为东方梨和西方梨两个种群,而中国的白梨(Pyrus bretschneideri Rehd.)、砂梨(P.pyrifolia Burm.f.Nakai)和秋子梨(P.ussuriensis Maxim.)相互交错在一起,没有独自成组.可见,苹果的EST-SSR标记在梨上具有高度的可转移性,可应用于梨属植物的资源评价及遗传关系研究.     

Application of the Principal Components Method and the Proportional Hazards Regression Model to Analysis of Survival Data

One of factor analysis techniques, viz. the principal components method, and the proportional hazards regression model (Cox, 1972) are applied in this work to study the significance of various factors characterizing the patient, the disease, and the method of treatment in the survival. The application of these methods to analysis of survival data for cervical cancer patients has shown, in particular, the tumor growth rate to be the crucial factor in distribution of the patients survival time and to be even more important than the therapy characteristics.     

Synthesis of Silver and Gold Nanoparticles Using Cashew Nut Shell Liquid and Its Antibacterial Activity Against Fish Pathogens

This study reveals a green process for the production of multi-morphological silver (Ag NPs) and gold (Au NPs) nanoparticles, synthesized using an agro-industrial residue cashew nut shell liquid. Aqueous solutions of Ag⁺ ions for silver and chloroaurate ions for gold were treated with cashew nut shell extract for the formation of Ag and Au NPs. The nano metallic dispersions were characterized by measuring the surface plasmon absorbance at 440 and 546 nm for Ag and Au NPs. Transmission electron microscopy showed the formation of nanoparticles in the range of 5–20 nm for silver and gold with assorted morphologies such as round, triangular, spherical and irregular. Scanning electron microscopy with energy dispersive spectroscopy and X-ray diffraction analyses of the freeze-dried powder confirmed the formation of metallic Ag and Au NPs in crystalline form. Further analysis by Fourier transform infrared spectroscopy provided evidence for the presence of various biomolecules, which might be responsible for the reduction of silver and gold ions. The obtained Ag and Au NPs had significant antibacterial activity, minimum inhibitory concentration and minimum bactericidal concentration on bacteria associated with fish diseases.