Empirical regression as a conditional expected value of a special distribution-mixture for a modelfree quantitative recording of stochastical relations

An “empirical” distribution function F?(x, y) is estimated from measured points (xi, yi), i =1(1)n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y). The density function F?(x, y) of F?(x, y) is a mixture of n two-dimensional normal densities. The first order moments of F?(x, y) are the sample means x and y, whilst the second order moments are only proportional to the sample variances and the sample covariance. This “empirical” distribution F?(x, y) is used for evaluation of an empirical regression curve where a free parameter has to be fixed by an optimality criterion. The procedure is demonstrated by an example from morphometrical research.     

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.     

Equivalence of regression curves sharing common parameters

In clinical trials, the comparison of two different populations is a common problem. Nonlinear (parametric) regression models are commonly used to describe the relationship between covariates, such as concentration or dose, and a response variable in the two groups. In some situations, it is reasonable to assume some model parameters to be the same, for instance, the placebo effect or the maximum treatment effect. In this paper, we develop a (parametric) bootstrap test to establish the similarity of two regression curves sharing some common parameters. We show by theoretical arguments and by means of a simulation study that the new test controls its significance level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters, a considerably more powerful test can be constructed compared with the test that does not use this assumption. Finally, we illustrate the potential applications of the new methodology by a clinical trial example.     

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.     

半散放条件下赤麂卧息地特征分析

2006年4～8月,在海南省大田国家级自然保护区对位于E108°47′05·8″～108°47′12·0″、N19°05′55·7″～19°05′58·1″,面积为10400m2的围栏内6只半散放条件下赤麂(Muntiacus muntjak)的卧息生境进行了研究。将围栏内的面积划为104个10m×10m的样方,作为收集数据的基本单元。将有赤麂卧息的样方定义为卧息单元,反之为非卧息单元。赤麂选择在14个样方卧息,卧息地平均长(1·6643±0·6197)m、宽(1·1679±0·3123)m、高(0·8143±0·1445)m。采用逻辑斯蒂回归模型作为赤麂卧息地特征分析的数学模型,分析了在14个卧息单元中收集的11类生态因子。结果表明,赤麂卧息地的选择与草本高度、隐蔽度、灌木高度及灌木密度正相关。赤麂卧息地选择的预测模型可用以下数学表达式表示:ln[P/(1-P)]=-26·888 0·089×灌木密度 0·253×草本高度 0·114×隐蔽度 0·079×灌木高度。     

Application of Random Coefficient Regression Model to Myopia Data: A Case Study

A one-year birth cohort from Northern Finland has been followed up since 1966. As a part of this study, we are in this paper concerned with analysing the progression of myopia (nearsightness) up to the age of 20 years. The random coefficient regression model was chosen for the analysis because of the large individual variation in the development of myopia. Maximum likelihood estimates for the parameters in the model were obtained via the expectation maximization (EM) algorithm. It is shown how the estimated model can be used to predict future observations for an individual using the previously recorded refractive error measurements as well as other relevant data on the patient in question.