The Application of Linear Piecewise Regression to Basal Body Temperature Data

In many situations one wishes to fit a piecewige regression which enables one to obtain estimates of the join points as well as the slopes and intercepts of the fitted submodels. This study developes a technique for fitting piecewise models to data which contain measurement error in an independent variable. The technique developed here combines the HUDSON (1966) procedure for estimating parameters in piecewise regression and the WALD (1940) Grouping Technique which obviates the problem of measurement error. If one assumes some knowledge of the position of the join point in relation to the data, methodology has been developed to estimate the parameters and study the asymptotic properties of the means and variances of the parameter estimates. However, in the more realistic case, when additional knowledge is limited, it is only possible to obtain the parameter estimates using an iterative technique (TEETER, 1982). The general technique for obtaining the join point estimate in the presence of measurement error is presented here and an example is given using data on women's basal body temperature during menstrual cycles.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Determinants of Serum PCBs in Adolescents and Adults: Regression Tree Analysis and Linear Regression Analysis

Regression tree analysis, a non-parametric method, was undertaken to identify predictors of the serum concentration of polychlorinated biphenyls (sum of marker PCB ¹ ABBREVIATIONS: BMI: body-mass index, CV: cross validation, ln: natural logarithm, ns: not significant, PCAHs: polychlorinated aromatic hydrocarbons, PCBs: polychlorinated biphenyls, R² _a: adjusted coefficient of determination, VIF: variance inflation factor. View all notes 138, 153, and 180) in humans. This method was applied on biomonitoring data of the Flemish Environment and Health study (2002–2006) and included 1679 adolescents and 1583 adults. Potential predictor variables were collected via a self-administered questionnaire, assessing information on lifestyle, food intake, use of tobacco and alcohol, residence history, health, education, hobbies, and occupation. Relevant predictors of human PCB exposure were identified with regression tree analysis using ln-transformed sum of PCBs, separately in adolescents and adults. The obtained results were compared with those from a standard linear regression approach. The results of the non-parametric analysis confirm the selection of the covariates in the multiple regression models. In both analyses, blood fat, gender, age, body-mass index (BMI) or change in bodyweight, former breast-feeding, and a number of nutritional factors were identified as statistically significant predictors in the serum PCB concentration, either in adolescents, in adults or in both. Regression trees can be used as an explorative analysis in combination with multiple linear regression models, where relationships between the determinants and the biomarkers can be quantified.     

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.     

首发缺血性脑卒中患者血清Hcy和EPO水平的变化及临床意义

目的:探究首发缺血性脑卒中患者血清同型半胱氨酸(Hcy)和红细胞生成素(EPO)水平的变化和意义。方法:于2013年10月-2015年4月我科收治的首发缺血性脑卒中患者中随机选取98例作为观察组,另选取同期健康体检者98例作为对照组,检测患者的血小板、血浆纤维蛋白原(Fib)以及血白细胞水平,比较两组血清Hcy、EPO、血小板、Fib及血白细胞水平,使用Logistic回归分析法评价缺血性脑卒中病发的危险因素,采用Spearman法对血清Hcy与EPO间相关性进行分析。结果:观察组的Hcy(23.52±12.15)m IU/L与EPO(34.61±11.25)m IU/L水平显著高于对照组的(10.57±2.18)m IU/L、(17.54±5.83)m IU/L;观察组血小板、血浆纤维蛋白原(fibrinogen,Fib)及血白细胞水平均高于对照组;差异均有统计学意义(均P0.05)。经Logistic回归分析法分析可知,Hcy为缺血性脑卒中病发的独立因素,经Spearman相关性分析显示,首发缺血性脑卒中患者EPO水平与Hcy呈正相关。结论:缺血性脑卒中病发与血清Hcy和EPO水平升高密切相关,且Hcy是导致缺血性脑卒中病发的高危因素。     

Remarks upon Empirical Regression Belt

In the presented paper the method of the empirical regression belt is demonstrated. An empirical regression curve r(x), which is determined by the realizations (measured points) (x₁, y₁), i = 1,…., n of a continuous two-dimensional random variable (X, Y), is enclosed by a belt, the local width of which varies dependent on local frequency and variance of the measured points. This empirical regression belt yields certain information for evaluating the empirical regression curve, providing a useful basis for the biomathematical forming of a model. By giving three examples derived from morphometrics the authors discuss important qualities of the empirical regression belt.     

Testing Hypotheses about Regression Parameters,When the Error Term Is Heteroscedastic

The paper considers methods for testing H₀: β₁ = … = β_p = 0, where β₁, … ,β_p are the slope parameters in a linear regression model with an emphasis on p = 2. It is known that even when the usual error term is normal, but heteroscedastic, control over the probability of a type I error can be poor when using the conventional F test in conjunction with the least squares estimator. When the error term is nonnormal, the situation gets worse. Another practical problem is that power can be poor under even slight departures from normality. Liu and Singh (1997) describe a general bootstrap method for making inferences about parameters in a multivariate setting that is based on the general notion of depth. This paper studies the small-sample properties of their method when applied to the problem at hand. It is found that there is a practical advantage to using Tukey's depth versus the Mahalanobis depth when using a particular robust estimator. When using the ordinary least squares estimator, the method improves upon the conventional F test, but practical problems remain when the sample size is less than 60. In simulations, using Tukey's depth with the robust estimator gave the best results, in terms of type I errors, among the five methods studied.     

A Primer on Interpreting Regression Models

Abstract: We perceive a need for more complete interpretation of regression models published in the wildlife literature to minimize the appearance of poor models and to maximize the extraction of information from good models. Accordingly, we offer this primer on interpretation of parameters in single- and multi-variable regression models. Using examples from the wildlife literature, we illustrate how to interpret linear zero-intercept, simple linear, semi-log, log-log, and polynomial models based on intercepts, coefficients, and shapes of relationships. We show how intercepts and coefficients have biological and management interpretations. We examine multiple linear regression models and show how to use the signs (+, -) of coefficients to assess the merit and meaning of a derived model. We discuss 3 methods of viewing the output of 3-dimensional models (y, x₁, x₂) in 2-dimensional space (sheet of paper) and illustrate graphical model interpretation with a 4-dimensional logistic regression model. Statistical significance or Akaike best-ness does not prevent the appearance of implausible regression models. We recommend that members of the peer review process be sensitive to full interpretation of regression models to forestall bad models and maximize information retrieval from good models     

Estimating Litter Decomposition Rate in Single-Pool Models Using Nonlinear Beta Regression

Litter decomposition rate (k) is typically estimated from proportional litter mass loss data using models that assume constant, normally distributed errors. However, such data often show non-normal errors with reduced variance near bounds (0 or 1), potentially leading to biased k estimates. We compared the performance of nonlinear regression using the beta distribution, which is well-suited to bounded data and this type of heteroscedasticity, to standard nonlinear regression (normal errors) on simulated and real litter decomposition data. Although the beta model often provided better fits to the simulated data (based on the corrected Akaike Information Criterion, AIC_c), standard nonlinear regression was robust to violation of homoscedasticity and gave equally or more accurate k estimates as nonlinear beta regression. Our simulation results also suggest that k estimates will be most accurate when study length captures mid to late stage decomposition (50–80% mass loss) and the number of measurements through time is ≥5. Regression method and data transformation choices had the smallest impact on k estimates during mid and late stage decomposition. Estimates of k were more variable among methods and generally less accurate during early and end stage decomposition. With real data, neither model was predominately best; in most cases the models were indistinguishable based on AIC_c, and gave similar k estimates. However, when decomposition rates were high, normal and beta model k estimates often diverged substantially. Therefore, we recommend a pragmatic approach where both models are compared and the best is selected for a given data set. Alternatively, both models may be used via model averaging to develop weighted parameter estimates. We provide code to perform nonlinear beta regression with freely available software.     

Alterations in mRNA expression of ribosomal protein S9 in hydrogen peroxide-treated neurotumor cells and in rat hippocampus after transient ischemia

This study was designed to isolate new genes related to apoptosis in rat pheochromocytoma (PC12) cells treated with hydrogen peroxide (H₂O₂), and to characterize the roles of the genes using both in vitro and in vivo models of oxidative injury. cDNA libraries were prepared from H₂O₂-treated and -untreated PC12 cells, and a ribosomal protein S9 (RPS9) clone was isolated by a differential screening method. Increase of RPS9 expression in both H₂O₂-treated PC12 and neuroblastoma (Neuro-2A) cells was shown by Northern blot analysis. Viability of the antisense-transfected Neuro-2A (RPS9-AS) cells following H₂O₂ treatment was significantly reduced in a dose-dependent manner. In an in vivo model of transient forebrain ischemia, an increase in RPS9 expression was prominent by 1 day postischemia in the granule cell layer neurons of the dentate gyrus. Both activation of caspase-3 and significant recovery of viability following pretreatment with cycloheximide were shown in RPS9-AS cells treated with H₂O₂. These data suggest that RPS9 plays a protective role in oxidative injury of neuronal cells.     

The Impact of Outliers on Net-Benefit Regression Model in Cost-Effectiveness Analysis

Ordinary least square (OLS) in regression has been widely used to analyze patient-level data in cost-effectiveness analysis (CEA). However, the estimates, inference and decision making in the economic evaluation based on OLS estimation may be biased by the presence of outliers. Instead, robust estimation can remain unaffected and provide result which is resistant to outliers. The objective of this study is to explore the impact of outliers on net-benefit regression (NBR) in CEA using OLS and to propose a potential solution by using robust estimations, i.e. Huber M-estimation, Hampel M-estimation, Tukey''s bisquare M-estimation, MM-estimation and least trimming square estimation. Simulations under different outlier-generating scenarios and an empirical example were used to obtain the regression estimates of NBR by OLS and five robust estimations. Empirical size and empirical power of both OLS and robust estimations were then compared in the context of hypothesis testing.Simulations showed that the five robust approaches compared with OLS estimation led to lower empirical sizes and achieved higher empirical powers in testing cost-effectiveness. Using real example of antiplatelet therapy, the estimated incremental net-benefit by OLS estimation was lower than those by robust approaches because of outliers in cost data. Robust estimations demonstrated higher probability of cost-effectiveness compared to OLS estimation. The presence of outliers can bias the results of NBR and its interpretations. It is recommended that the use of robust estimation in NBR can be an appropriate method to avoid such biased decision making.     

Purification of low-abundance Arabidopsis plasma-membrane protein complexes and identification of candidate components

Purification of low-abundance plasma-membrane (PM) protein complexes is a challenging task. We devised a tandem affinity purification tag termed the HPB tag, which contains the biotin carboxyl carrier protein domain (BCCD) of Arabidopsis 3-methylcrotonal CoA carboxylase. The BCCD is biotinylated in vivo , and the tagged protein can be captured by streptavidin beads. All five C-terminally tagged Arabidopsis proteins tested, including four PM proteins, were functional and biotinylated with high efficiency in Arabidopsis. Transgenic Arabidopsis plants expressing an HPB-tagged protein, RPS2::HPB, were used to develop a method to purify protein complexes containing the HPB-tagged protein. RPS2 is a membrane-associated disease resistance protein of low abundance. The purification method involves microsomal fractionation, chemical cross-linking, solubilization, and one-step affinity purification using magnetic streptavidin beads, followed by protein identification using LC-MS/MS. We identified RIN4, a known RPS2 interactor, as well as other potential components of the RPS2 complex(es). Thus, the HPB tag method is suitable for the purification of low-abundance PM protein complexes.     

Regression Analysis for Constraining Free Parameters in Electrophysiological Models of Cardiac Cells

A major challenge in computational biology is constraining free parameters in mathematical models. Adjusting a parameter to make a given model output more realistic sometimes has unexpected and undesirable effects on other model behaviors. Here, we extend a regression-based method for parameter sensitivity analysis and show that a straightforward procedure can uniquely define most ionic conductances in a well-known model of the human ventricular myocyte. The model''s parameter sensitivity was analyzed by randomizing ionic conductances, running repeated simulations to measure physiological outputs, then collecting the randomized parameters and simulation results as “input” and “output” matrices, respectively. Multivariable regression derived a matrix whose elements indicate how changes in conductances influence model outputs. We show here that if the number of linearly-independent outputs equals the number of inputs, the regression matrix can be inverted. This is significant, because it implies that the inverted matrix can specify the ionic conductances that are required to generate a particular combination of model outputs. Applying this idea to the myocyte model tested, we found that most ionic conductances could be specified with precision (R² > 0.77 for 12 out of 16 parameters). We also applied this method to a test case of changes in electrophysiology caused by heart failure and found that changes in most parameters could be well predicted. We complemented our findings using a Bayesian approach to demonstrate that model parameters cannot be specified using limited outputs, but they can be successfully constrained if multiple outputs are considered. Our results place on a solid mathematical footing the intuition-based procedure simultaneously matching a model''s output to several data sets. More generally, this method shows promise as a tool to define model parameters, in electrophysiology and in other biological fields.     

Incorporating Predictor Network in Penalized Regression with Application to Microarray Data

Summary We consider penalized linear regression, especially for “large p, small n” problems, for which the relationships among predictors are described a priori by a network. A class of motivating examples includes modeling a phenotype through gene expression profiles while accounting for coordinated functioning of genes in the form of biological pathways or networks. To incorporate the prior knowledge of the similar effect sizes of neighboring predictors in a network, we propose a grouped penalty based on the L_γ ‐norm that smoothes the regression coefficients of the predictors over the network. The main feature of the proposed method is its ability to automatically realize grouped variable selection and exploit grouping effects. We also discuss effects of the choices of the γ and some weights inside the L_γ ‐norm. Simulation studies demonstrate the superior finite‐sample performance of the proposed method as compared to Lasso, elastic net, and a recently proposed network‐based method. The new method performs best in variable selection across all simulation set‐ups considered. For illustration, the method is applied to a microarray dataset to predict survival times for some glioblastoma patients using a gene expression dataset and a gene network compiled from some Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways.     

Fast and Accurate Registration of the Proximal Femurs in Bilateral Hip Joint Images by Using the Random Sub-Sample Points

Background and ObjectiveAs is known, point clouds representing the objects are frequently used in object registration. Although the objects can be registered by using all the points in the corresponding point clouds of the objects, the registration process can also be achieved with a smaller number of the landmark points selected from the entire point clouds of the objects. This paper introduces a research study focusing on the fast and accurate rigid registration of the bilateral proximal femurs in bilateral hip joint images by using the random sub-sample points. For this purpose, Random Point Sub-sampling (RPS) was analyzed and the reduced point sets were used for an accurate registration of the bilateral proximal femurs in coronal hip joint magnetic resonance imaging (MRI) slices.MethodsIn registration, bilateral proximal femurs in MRI slices were registered rigidly by performing a process consisting of three main phases named as MR image preprocessing, proximal femur registration over the random sub-sample points and MR image postprocessing. In the stage of the MR image preprocessing, segmentation maps of the bilateral proximal femurs are obtained as region of interest (RoI) images from the entire MRI slices and then, the edge maps of the segmented proximal femurs are extracted. In the registration phase, the edge maps describing the proximal femur surfaces are represented as point clouds initially. Thereafter, the RPS is performed on the proximal femur point clouds and the number of points representing the proximal femurs is reduced at different ratios. For the registration of the point clouds, the Iterative Closest Point (ICP) algorithm is performed on the reduced sets of points. Finally, the registration procedures are completed by performing MR image postprocessing on the registered proximal femur images.ResultsIn performance evaluation tests performed on healthy and pathological proximal femurs in 13 bilateral coronal hip joint MRI slices of 13 Legg-Calve-Perthes disease (LCPD) patients, bilateral proximal femurs were successfully registered with very small error rates by using the reduced set of points obtained via the RPS and promising results were achieved. The minimum error rate was observed at RPS rate of 30% as the value of 0.41 (±0.31)% on all over the bilateral proximal femurs evaluated. When the range of RPS rate of 20-30% is considered as the reference, the elapsed time in registration can be reduced by almost 30-40% compared to the case where all the proximal femur points were included in registration. Additionally, it was observed that the RPS rate should be selected as at least 25% to achieve a successful registration with an error rate below 1%.ConclusionIt was concluded from the observed results that a more successful and faster registration can be accomplished by selecting fewer points randomly from the point sets of proximal femurs instead of using all the points describing the proximal femurs. Not only an accurate registration with low error rates was performed, but also a faster registration process was performed by means of the limited number of points that are sub-sampled randomly from the whole point sets.     

Analysis of hypermethylation in the RPS element suggests a signal function for short inverted repeats in de novo methylation

A repetitive DNA sequence (RPS) from Petunia hybrida had previously been shown to enhance expression variegation in petunia and tobacco and to carry a hot spot for de novo DNA methylation. Here we show that a strong de novo hypermethylation site is located within a palindromic segment of the RPS and present indirect evidence, based on sequence homologies to other repeat units within the RPS, for the formation of secondary structures at the methylation site in vivo. We demonstrate that the palindromic RPS element, which is moderately to highly repetitive in petunia, does not predominantly localise to constitutive heterochromatin. To test whether the RPS is subject to de novo methylation due to its repetitive nature or to intrinsic signals within the RPS, we integrated the RPS into the genome of Arabidopsis thaliana, a plant lacking homology to the RPS. Our data indicate that the palindromic element also acts as a de novo hypermethylation site in the non-repetitive genomic background of Arabidopsis, strongly suggesting a signal function of the palindromic RPS element.     

Inference for the Break Point in Segmented Regression with Application to Longitudinal Data

This paper considers inference for the break point in the segmented regression or piece‐wise regression model. Standard likelihood theory does not apply because the break point is absent under the null hypothesis. We use results by Davies for this type of non‐standard set‐up [Biometrika 64 (1977), 247–254 and 74 (1987), 33–43] to obtain a test for the null hypothesis of no break point. A confidence interval can be constructed provided replicate data are available. The methods are exemplified using two longitudinal datasets, the one from ecology, the other from pharmacology.     

Exact Confidence Bounds for Comparing Two Regression Lines with a Control Regression Line on a Fixed Interval

The problem of finding exact simultaneous confidence bounds for comparing simple linear regression lines for two treatments with a simple linear regression line for the control over a fixed interval is considered. The assumption that errors are iid normal random is considered. It is assumed that the design matrices for the two treatments are equal and the design matrix for the control has the same number of copies of each distinct row of the design matrix for the treatments. The method is based on a pivotal quantity that can be expressed as a function of four t variables. The probability point depends on the size of an angle associated with the interval. We present probability points for various sample sizes and angles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)