Use of nonlinear regression to analyze enzyme kinetic data: application to situations of substrate contamination and background subtraction

In a recent publication, A. Lundin, P. Arner, and J. Hellmér [Anal. Biochem. 177, 125-131 (1989)] describe a method whereby kinetic substrate assays can be performed when the assay mixture includes a significant contaminating levels of substrate. Their method requires various rearrangements of the data, and involves three separate linear regression calculations. We show how the same data may be analyzed directly, and far more simply, by nonlinear regression. Unlike the linear regression method, nonlinear regression allows direct calculation of the actual values for Km, Vmax, and the concentration of contaminating substrate (as well as estimates of their standard errors); the former method gives only apparent values. The nonlinear regression technique is also statistically a more valid means of analysis, as the rearrangements required to give linearized equations will considerably distort the error distribution and render simple unweighted linear regression inappropriate. The ease of incorporating extra parameters into standard equations when nonlinear regression is used is further illustrated by fitting enzyme reaction data which describe a first-order process when a significant nonspecific background is present. For this equation no simple rearranged linear plot is possible, but nonlinear regression is easily applied to determine the kinetic parameters.     

A Technique for the Analysis of Unbalanced Data

The regression methods with dummy variables have been shown to be effective in preventing confusion in the analysis of linear models. In particular, this model simplifies interpretation of parameters and clarifies hypothesis statements. All existing methods have been shown as special cases of the general linear hypothesis in regression setting. Three regression on dummy variables methods are examined critically to bring out the salient features of each method. The choice of a method should be based on the way definitions of the parameters are desired. The linear models are considered in a regression model setting. This has been done by defining appropriate dummy variables in a regression model which often is desirable, if not mandatory, when dealing with unbalanced data involving two or more factors.     

A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

选择回归方程自变量的条件数法及其在RK手术中的应用

选择合适的自变量是确定线性回归模型的首要问题,本文以消除自变量之间的复共线性为目标,介绍了一种选择回归方程自变量的条件数法,并在ＲＫ手术的结果预测问题中采用了这一方法。     

俄罗斯大果沙棘人工林生物量遥感评估与分析

以2011年的Landsat TM为主要遥感数据,借助于RS和GIS技术完成对俄罗斯大果沙棘人工林生物量进行估侧。结果表明：植被指数和生物量的一元线性回归分析模型中,比值植被指数（RVI）和归一化植被指数（NDVI）与俄罗斯大果沙棘具有较高的相关性,相关系数（R²）分别为0.908 6和0.868 5;基于植被指数和生物量的多元线性回归分析模型中,相关系数（R²）为0.909,经过模型检验,多元回归遥感植被指数模型的精度要高于一元遥感植被指数的精度,但是基于遥感指数模型预测生物量值比理论生物量值偏高。     

Bent Line Quantile Regression with Application to an Allometric Study of Land Mammals' Speed and Mass

Summary Quantile regression, which models the conditional quantiles of the response variable given covariates, usually assumes a linear model. However, this kind of linearity is often unrealistic in real life. One situation where linear quantile regression is not appropriate is when the response variable is piecewise linear but still continuous in covariates. To analyze such data, we propose a bent line quantile regression model. We derive its parameter estimates, prove that they are asymptotically valid given the existence of a change‐point, and discuss several methods for testing the existence of a change‐point in bent line quantile regression together with a power comparison by simulation. An example of land mammal maximal running speeds is given to illustrate an application of bent line quantile regression in which this model is theoretically justified and its parameters are of direct biological interests.     

Regression analysis when covariates are regression parameters of a random effects model for observed longitudinal measurements

We consider regression analysis when covariate variables are the underlying regression coefficients of another linear mixed model. A naive approach is to use each subject's repeated measurements, which are assumed to follow a linear mixed model, and obtain subject-specific estimated coefficients to replace the covariate variables. However, directly replacing the unobserved covariates in the primary regression by these estimated coefficients may result in a significantly biased estimator. The aforementioned problem can be evaluated as a generalization of the classical additive error model where repeated measures are considered as replicates. To correct for these biases, we investigate a pseudo-expected estimating equation (EEE) estimator, a regression calibration (RC) estimator, and a refined version of the RC estimator. For linear regression, the first two estimators are identical under certain conditions. However, when the primary regression model is a nonlinear model, the RC estimator is usually biased. We thus consider a refined regression calibration estimator whose performance is close to that of the pseudo-EEE estimator but does not require numerical integration. The RC estimator is also extended to the proportional hazards regression model. In addition to the distribution theory, we evaluate the methods through simulation studies. The methods are applied to analyze a real dataset from a child growth study.     

On the effect of misclassification on bias of perfectly measured covariates in regression

This note clarifies under what conditions a naive analysis using a misclassified predictor will induce bias for the regression coefficients of other perfectly measured predictors in the model. An apparent discrepancy between some previous results and a result for measurement error of a continuous variable in linear regression is resolved. We show that similar to the linear setting, misclassification (even when not related to the other predictors) induces bias in the coefficients of the perfectly measured predictors, unless the misclassified variable and the perfectly measured predictors are independent. Conditional and asymptotic biases are discussed in the case of linear regression, and explored numerically for an example relating birth weight to the weight and smoking status of the mother.     

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.     

A mathematical framework for examining whether a minimum number of chiasmata is required per metacentric chromosome or chromosome arm in human

We introduce a piecewise linear regression called "hockey stick regression" to model the relationship between genetic and physical lengths of chromosomes in a genome. This piecewise linear regression is an extension of the two-parameter linear regression we proposed earlier [W. Li and J. Freudenberg, Two-parameter characterization of chromosome-scale recombination rate, Genome Res., 19 (2009) 2300-2307]. We use this, as well as the one-piece regression with a fixed y-intercept, to compare the two competing hypotheses concerning the minimum number of required chiasmata for meiosis: minimum one chiasma per chromosome (PC) and per chromosome arm (PA). Using statistical model selection and testing, we show that for human genome data, one-piece PC (PC1) is often in a statistical tie with two-piece PA model (PA2). If an upper bound for the segmentation point in two-piece regression is imposed, PC is usually the preferred model. This indicates that a presence of more than one chiasmata is rather caused by the relationship between chromosome size and chiasma formation than by cytogenetic constraints.     

Triaxial accelerometry for assessment of physical activity in young children

Objective: The purpose of the present study was to derive linear and non‐linear regression equations that estimate energy expenditure (EE) from triaxial accelerometer counts that can be used to quantitate activity in young children. We are unaware of any data regarding the validity of triaxial accelerometry for assessment of physical activity intensity in this age group. Research Methods and Procedures: EE for 27 girls and boys (6.0 ± 0.3 years) was assessed for nine activities (lying down, watching a video while sitting and standing, line drawing for coloring‐in, playing blocks, walking, stair climbing, ball toss, and running) using indirect calorimetry and was then estimated using a triaxial accelerometer (ActivTracer, GMS). Results: Significant correlations were observed between synthetic (synthesized tri‐axes as the vector), vertical, and horizontal accelerometer counts and EE for all activities (0.878 to 0.932 for EE). However, linear and non‐linear regression equations underestimated EE by >30% for stair climbing (up and down) and performing a ball toss. Therefore, linear and non‐linear regression equations were calculated for all activities except these two activities, and then evaluated for all activities. Linear and non‐linear regression equations using combined vertical and horizontal acceleration counts, synthetic counts, and horizontal counts demonstrated a better relationship between accelerometer counts and EE than did regression equations using vertical acceleration counts. Adjustment of the predicted value by the regression equations using the vertical/horizontal counts ratio improved the overestimation of EE for performing a ball toss. Discussion: The results suggest that triaxial accelerometry is a good tool for assessing daily EE in young children.     

Prediction of body volume by a stepwise linear regression technique

Body volume and 35 anthropometric measurements were obtained from 88 active soldiers using standard techniques. These anthropometric measurements were examined for their possible relationships to body volume using stepwise linear regression analysis. Four measurements (Body weight, anterior thigh skinfold thickness, subscapular skinfold thickness and suprailiac skinfold thickness) accounted for 99.7% of the variation in body volume and the introduction of each of these measurements in the equation was significant. The regression equation for predicting body volume from these 4 anthropometric measurements had a multiple correlation coefficient of 0.9987 (P less than 0.001). Body weight alone was correlated with body volume to the extent of 0.9966. An attempt has therefore been made to develop a multiple linear regression equation without incorporation of body weight in the regression analysis. Nine measurements were selected by stepwise linear regression analysis for predicting body volume. These nine measurements accounted for 97.1% of the variation in body volume. These equations have been validated on another small sample of 22 soldiers. The analysis has also revealed that a direct regression of body density from the anthropometric variables gives more accurate results than when estimated body volumes are utilized for calculating body density.     

Statistical determination of steady state condition in bioremediation tests

Linear regression and two newly developed statistical techniques were used to determine steady states in the dependent y-variables (effluent concentration or removal rate of pollutants) using correlation coefficients (r) for the relationship between the independent x-variables (reactor operating or treatment time) and the dependent y-variables. The statistical technique applied to chlorophenol bioremediation using a varying number of data points for linear regression analysis was more useful in determining a steady state for six general data patterns from bioremediation tests than the statistical technique using a fixed number of data points for linear regression analysis.     

Improved Predictions in Linear Regression Models with Stochastic Linear Constraints

In this article, we have considered two families of predictors for the simultaneous prediction of actual and average values of study variable in a linear regression model when a set of stochastic linear constraints binding the regression coefficients is available. These families arise from the method of mixed regression estimation. Performance properties of these families are analyzed when the objective is to predict values outside the sample and within the sample.     

降水量和温度对植被覆盖指数影响的空间非平稳性特征——以新疆伊犁河谷地区为例

多变量空间相关分析多基于时间序列数据,对数据时长与统计要求严格,空间非平稳性特征分析可以利用单期数据分析多变量之间的相关性。通过空间变系数回归模型分析了2006年和2011年的新疆伊犁地区降水量和温度对植被覆盖度指数影响的空间变化特征,利用局部线性地理加权回归(GWR)方法估计得到了回归系数曲面,揭示出变量间相互影响的空间异质性,同时利用线性回归最小二乘估计进行了对比。结果表明:(1)空间变系数回归模型可以用于变量间的空间相关分析;(2)局部线性GWR估计方法明显优于线性回归最小二乘估计;(3)拟合结果表明,伊犁地区降水量和温度对植被覆盖指数的影响具有显著的空间非平稳性特征;(4)模型估计误差是降水、气温之外的地形、地貌及人类活动等多种因素造成的,需进一步研究。方法可为具有空间非平稳性特征变量间空间相关性分析以及植被覆盖指数的空间模拟分布提供思路和方法。     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

New QSAR combined strategy for the design of A1 adenosine receptor agonists

Combined discriminant and regression analysis was carried out on a series of 167 A1 adenosine receptor agonists to identify the best linear and nonlinear models for the design of new compounds with a better biological profile. On the basis of the best linear discriminant analysis and both linear and nonlinear Multi Layer Perceptron neural networks regression, we have designed and synthesized 14 carbonucleoside analogues of adenosine. Their biological activities were predicted and experimentally measured to demonstrate the capability of our model to avoid the prediction of false positives. A good agreement was found between the calculated and observed biological activity.     

Methodology for estimating the abundance of rare animals: seabird nesting on north east Herald Cay

We discuss the problem of estimating the number of nests of different species of seabirds on North East Herald Cay based on the data from a 1996 survey of quadrats along transects and data from similar past surveys. We consider three approaches based on different plausible models, namely a conditional negative binomial model that allows for additional zeroes in the data, a weighting approach (based on a heteroscedastic regression model), and a transform-both-sides regression approach. We find that the conditional negative binomial approach and a linear regression approach work well but that the transform-both-sides approach should not be used. We apply the conditional negative binomial and linear regression approaches with poststratification based on data quality and availability to estimate the number of frigatebird nests on North East Herald Cay.     

The mutational rate of Drosophila viability decline: tinkering with old data

In the first 25 generations of his classical mutation accumulation experiment, T. Mukai estimated a large rate of early linear decay for the relative viability of Drosophila melanogaster chromosome II (delta MII = 0.004). Mukai forced through zero the regression of viability decline on generation number, but it has recently been shown (Fry, 2001) that a similar decline (delta MII = 0.006) is obtained from unforced regression even if generation 32 instead of generation 25 (whose validity has been questioned) is included. We show that, from the perspective of the whole long-term experiment. it is hard to decide up to which generation viability can be considered to decline linearly. Depending on this decision, and on whether or not the regression is forced through the origin, very different estimates are obtained. Furthermore, the particular behaviour of the lines used as control suggests that they could have been different from the remaining lines at the beginning of the experiment, and casts doubts on the adequacy of a forced regression. Estimates from the linear unforced regression (delta MII = 0.011) or from the linear term in a quadratic unforced regression (delta MII = 0.001) are very different. The data fit both models very well, and the choice between them should be based on biological grounds.