Parametric modal regression with varying precision

In this paper, we propose a simple parametric modal linear regression model where the response variable is gamma distributed using a new parameterization of this distribution that is indexed by mode and precision parameters, that is, in this new regression model, the modal and precision responses are related to a linear predictor through a link function and the linear predictor involves covariates and unknown regression parameters. The main advantage of our new parameterization is the straightforward interpretation of the regression coefficients in terms of the mode of the positive response variable, as is usual in the context of generalized linear models, and direct inference in parametric mode regression based on the likelihood paradigm. Furthermore, we discuss residuals and influence diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the results. Finally, we illustrate the usefulness of the new model by two applications, to biology and demography.     

Comparability of segmented line regression models

Segmented line regression models, which are composed of continuous linear phases, have been applied to describe changes in rate trend patterns. In this article, we propose a procedure to compare two segmented line regression functions, specifically to test (i) whether the two segmented line regression functions are identical or (ii) whether the two mean functions are parallel allowing different intercepts. A general form of the test statistic is described and then the permutation procedure is proposed to estimate the p-value of the test. The permutation test is compared to an approximate F-test in terms of the p-value estimation and the performance of the permutation test is studied via simulations. The tests are applied to compare female lung cancer mortality rates between two registry areas and also to compare female breast cancer mortality rates between two states.     

Bias correction of bounded location error in binary data

Binary regression models for spatial data are commonly used in disciplines such as epidemiology and ecology. Many spatially referenced binary data sets suffer from location error, which occurs when the recorded location of an observation differs from its true location. When location error occurs, values of the covariates associated with the true spatial locations of the observations cannot be obtained. We show how a change of support (COS) can be applied to regression models for binary data to provide coefficient estimates when the true values of the covariates are unavailable, but the unknown location of the observations are contained within nonoverlapping arbitrarily shaped polygons. The COS accommodates spatial and nonspatial covariates and preserves the convenient interpretation of methods such as logistic and probit regression. Using a simulation experiment, we compare binary regression models with a COS to naive approaches that ignore location error. We illustrate the flexibility of the COS by modeling individual-level disease risk in a population using a binary data set where the locations of the observations are unknown but contained within administrative units. Our simulation experiment and data illustration corroborate that conventional regression models for binary data that ignore location error are unreliable, but that the COS can be used to eliminate bias while preserving model choice.     

Hierarchical partitioning as an interpretative tool in multivariate inference

Abstract Much of biogeography, conservation and evolutionary biology, and ecology involves very large spatial and temporal extents. Direct manipulation to test hypotheses is usually almost impossible at appropriate scales so that multivariate modelling and especially regression are used to draw causal inferences about which ‘independent’ variables influence the distribution and abundances of species. Such inferences clearly are crucial for the successful management of biological resources and for conserving threatened species. A succession of regression approaches has arisen, many of which yield inconsistent implications. The main problem has been the quest for one (the ‘best’ or the ‘optimal’) regression model from which the impacts of independent variables are inferred. This note is to draw the attention of ecologists to a relatively recent method, hierarchical partitioning, that does not aim to identify a best regression model as such but rather uses all models in a regression hierarchy to distinguish those variables that have high independent correlations with the dependent variable. Such variables are likely to be most influential in controlling variation in the dependent variable. Hierarchical partitioning is not to be regarded as a substitute for experimental manipulation when that is appropriate, but it is likely to produce better deductions than common regression approaches in the many ecological situations in which manipulation is impossible or of doubtful value.     

Multiple regression for molecular-marker,quantitative trait data from large F2 populations

Molecular marker-quantitative trait associations are important for breeders to recognize and understand to allow application in selection. This work was done to provide simple, intuitive explanations of trait-marker regression for large samples from an F₂ and to examine the properties of the regression estimators. Beginning with a(- 1,0,1) coding of marker classes and expected frequencies in the F₂, expected values, variances, and covariances of marker variables were calculated. Simple linear regression and regression of trait values on two markers were computed. The sum of coefficient estimates for the flanking-marker regression is asymptotically unbiased for an included additive effect with complete interference, and is only slightly biased with no interference and moderately close (15 cM) marker spacing. The variance of the sum of regression coefficients is much more stable for small recombination distances than variances of individual coefficients. Multiple regression of trait variables on coded marker variables can be interpreted as the product of the inverse of the marker correlation matrix R and the vector a of simple linear regression estimators for each marker. For no interference, elements of the correlation matrix R can be written as products of correlations between adjacent markers. The inverse of R is displayed and used to illustrate the solution vector. Only markers immediately flanking trait loci are expected to have non-zero values and, with at least two marker loci between each trait locus, the solution vector is expected to be the sum of solutions for each trait locus. Results of this work should allow breeders to test for intervals in which trait loci are located and to better interpret results of the trait-marker regression.     

基于Median函数的分段回归模型及其在生物学上的应用

在生物学科研工作中,经常会遇到因变量和自变量之间存在着多种不同的趋势,分段回归模型可以很好的拟合变量间这种非线性趋势.本文介绍了基于Median函数的分段回归模型,可以同时对各项回归参数和转折点进行估计,最后,本文结合生物学上的实例运用SAS统计软件进行了分段回归模型的拟合.     

Some Improved Regression Estimators of Heritability

Three new improved regression estimators of heritability viz. modified range restricted estimator, minimum quadratic loss estimator and minimax linear restricted estimator are proposed. In addition, these estimators are illustrated and compared numerically with the existing restricted estimator based on linear stochastic constraint.     

Covariate-adjusted varying coefficient models

Covariate-adjusted regression was recently proposed for situations where both predictors and response in a regression model are not directly observed, but are observed after being contaminated by unknown functions of a common observable covariate. The method has been appealing because of its flexibility in targeting the regression coefficients under different forms of distortion. We extend this methodology proposed for regression into the framework of varying coefficient models, where the goal is to target the covariate-adjusted relationship between longitudinal variables. The proposed method of covariate-adjusted varying coefficient model (CAVCM) is illustrated with an analysis of a longitudinal data set containing calcium absorbtion and intake measurements on 188 subjects. We estimate the age-dependent relationship between these two variables adjusted for the covariate body surface area. Simulation studies demonstrate the flexibility of CAVCM in handling different forms of distortion in the longitudinal setting.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

Mixed Model Discrete Regression

Models and estimention procedures are given for linear regression models in discrete distributions when the regression contains both fixed and random effects. The methods are developed for discrete variables with typically a small number of possible outcomes such as occurs in ordinal regression. The method is applied to a problem arising in the comparison of microbiological test methods.     

Discretizing a Normal Prior for Change Point Estimation in Switching Regressions

Bayes decision procedures are considered for change point estimation in the simple bilinear segmented model. A discretized normal prior density is employed as the prior distribution for the change point index. Posterior probability functions are developed for this index under a vague prior formulation on the regression parameters. The procedure is applied to an example involving mercury toxicity data.     

When is Piece-Wise Regression Really Necessary?

Results are obtained showing that when a response surface can be modelled as a single function, then a single regressor is more efficient and less biased than a segmented regression. However, if the surface is segmented, a segmented regressor is less biased than a single regressor. Areas of application are indicated.     

Interpreting genotype-by-environment interaction using redundancy analysis

Summary Methods for the interpretation of genotype-by-environment interaction in the presense of explicitly measured environmental variables can be divided into two groups. Firstly, methods that extract environmental characterizations from the data itself, which are subsequently related to measured environmental variables, e.g., regression on the mean or singular value decomposition of the matrix of residuals from additivity, followed by correlation, or regression, methods. Secondly, methods that incorporate measured environmental variables directly into the model, e.g., multiple regression of individual genotypical responses on environmental variables, or factorial regression in which a genotype-by-environment matrix is modelled in terms of concomitant variables for the environmental factor. In this paper a redundancy analysis is presented, which can be derived from the singular-value decomposition of the residuals from additivity by imposing the restriction on the environmental scores of having to be linear combinations of environmental variables. At the same time, redundancy analysis is derivable from factorial regression by rotation of the axes in the space spanned by the fitted values of the factorial regression, followed by a reduction of dimensionality through discarding the least explanatory axes. Redundancy analysis is a member of the second group of methods, and can be an important tool in the interpretation of genotype-by-environment interaction, especially with reference to concomitant environmental information. A theoretical treatise of the method is given, followed by a practical example in which the results of the method are compared to the results of the other methods mentioned.     

A Comparison of Regression Models for Small Counts

ABSTRACT Count data with means <2 are often assumed to follow a Poisson distribution. However, in many cases these kinds of data, such as number of young fledged, are more appropriately considered to be multinomial observations due to naturally occurring upper truncation of the distribution. We evaluated the performance of several versions of multinomial regression, plus Poisson and normal regression, for analysis of count data with means <2 through Monte Carlo simulations. Simulated data mimicked observed counts of number of young fledged (0, 1, 2, or 3) by California spotted owls (Strix occidentalis occidentalis). We considered size and power of tests to detect differences among 10 levels of a categorical predictor, as well as tests for trends across 10-year periods. We found regular regression and analysis of variance procedures based on a normal distribution to perform satisfactorily in all cases we considered, whereas failure rate of multinomial procedures was often excessively high, and the Poisson model demonstrated inappropriate test size for data where the variance/mean ratio was <1 or >1.2. Thus, managers can use simple statistical methods with which they are likely already familiar to analyze the kinds of count data we described here.     

Comparison of random-effects meta-analysis models for the relative risk in the case of rare events: A simulation study

Pooling the relative risk (RR) across studies investigating rare events, for example, adverse events, via meta-analytical methods still presents a challenge to researchers. The main reason for this is the high probability of observing no events in treatment or control group or both, resulting in an undefined log RR (the basis of standard meta-analysis). Other technical challenges ensue, for example, the violation of normality assumptions, or bias due to exclusion of studies and application of continuity corrections, leading to poor performance of standard approaches. In the present simulation study, we compared three recently proposed alternative models (random-effects [RE] Poisson regression, RE zero-inflated Poisson [ZIP] regression, binomial regression) to the standard methods in conjunction with different continuity corrections and to different versions of beta-binomial regression. Based on our investigation of the models' performance in 162 different simulation settings informed by meta-analyses from the Cochrane database and distinguished by different underlying true effects, degrees of between-study heterogeneity, numbers of primary studies, group size ratios, and baseline risks, we recommend the use of the RE Poisson regression model. The beta-binomial model recommended by Kuss (2015) also performed well. Decent performance was also exhibited by the ZIP models, but they also had considerable convergence issues. We stress that these recommendations are only valid for meta-analyses with larger numbers of primary studies. All models are applied to data from two Cochrane reviews to illustrate differences between and issues of the models. Limitations as well as practical implications and recommendations are discussed; a flowchart summarizing recommendations is provided.     

On the Estimation of Intra-Sire Regression Heritability Subject to Linear Constraint

The derivation of the restricted intra-sire regression heritability estimator is provided. Procedures for obtaining a stable estimate of residual error variance σ² are outlined. A small illustration based on live data is given.     

Estimation of heritability by offspring-parent regression when observations do not have a common mean

Summary Offspring-parent regression is a simple method for estimating heritability. This method yields unbiased estimates even when parents are selected. The usual model in offspring-parent regression assumes that observations have the same mean. This assumption, however, is not met in many situations. A method for estimating heritability by offspring-parent regression when observations do not have a common mean is presented. The estimator is distributed as a multiple of a t random variable centered at its parametric value and is unbiased even when the parents are selected. When observations have a common mean, the method reduces to the usual regression estimator.     

Boosted regression trees with errors in variables

In this article, we consider nonparametric regression when covariates are measured with error. Estimation is performed using boosted regression trees, with the sum of the trees forming the estimate of the conditional expectation of the response. Both binary and continuous response regression are investigated. An approach to fitting regression trees when covariates are measured with error is described, and the boosting algorithms consist of its repeated application. The main feature of the approach is that it handles situations where multiple covariates are measured with error. Some simulation results are given as well as its application to data from the Framingham Heart Study.     

Offspring-parent regression for a binary trait

Summary Offspring-parent regression is often used to estimate the heritability of a quantitative trait. It is shown that for a purely binary trait, the regression of offspring on one parent is always linear, while that on both parents or mid-parent is generally nonlinear. However, the regressions are linear on a logistic scale.     

An Improved Goodness of Fit Statistic for Probability Prediction Models

We consider the general case of probability prediction models having two or more outcomes and propose an adjusted χ² statistic which can be used to assess the goodness of fit of these models. We present a simulation study to show that our proposed statistic has an approximate χ² distribution under the null hypothesis. Two applications are provided to illustrate the use of the new statistic. The first application examines the fit of a logistic regression model using both the proposed statistic and the popular Hosmer-Lemeshow statistic and we compare and contrast these two methods. The second application evaluates the goodness of fit of a polychotomous regression model.