Underlying Regression Model Research Articles

Kernel continuum regression

The continuum regression technique provides an appealing regression framework connecting ordinary least squares, partial least squares and principal component regression in one family. It offers some insight on the underlying regression model for a given application. Moreover, it helps to provide deep understanding of various regression techniques. Despite the useful framework, however, the current development on continuum regression is only for linear regression. In many applications, nonlinear regression is necessary. The extension of continuum regression from linear models to nonlinear models using kernel learning is considered. The proposed kernel continuum regression technique is quite general and can handle very flexible regression model estimation. An efficient algorithm is developed for fast implementation. Numerical examples have demonstrated the usefulness of the proposed technique.

The admixture maximum likelihood test to test for association between rare variants and disease phenotypes

BackgroundThe development of genotyping arrays containing hundreds of thousands of rare variants across the genome and advances in high-throughput sequencing technologies have made feasible empirical genetic association studies to search for rare disease susceptibility alleles. As single variant testing is underpowered to detect associations, the development of statistical methods to combine analysis across variants – so-called “burden tests” - is an area of active research interest. We previously developed a method, the admixture maximum likelihood test, to test multiple, common variants for association with a trait of interest. We have extended this method, called the rare admixture maximum likelihood test (RAML), for the analysis of rare variants. In this paper we compare the performance of RAML with six other burden tests designed to test for association of rare variants.ResultsWe used simulation testing over a range of scenarios to test the power of RAML compared to the other rare variant association testing methods. These scenarios modelled differences in effect variability, the average direction of effect and the proportion of associated variants. We evaluated the power for all the different scenarios. RAML tended to have the greatest power for most scenarios where the proportion of associated variants was small, whereas SKAT-O performed a little better for the scenarios with a higher proportion of associated variants.ConclusionsThe RAML method makes no assumptions about the proportion of variants that are associated with the phenotype of interest or the magnitude and direction of their effect. The method is flexible and can be applied to both dichotomous and quantitative traits and allows for the inclusion of covariates in the underlying regression model. The RAML method performed well compared to the other methods over a wide range of scenarios. Generally power was moderate in most of the scenarios, underlying the need for large sample sizes in any form of association testing.

No effect tests in regression on functional variable and some applications to spectrometric studies

Recent advances in structural tests for regression on functional variable are used to construct test of no effect. Various bootstrap procedures are considered and compared in a simulation study. These tests are finally applied on real world datasets dealing with spectrometric studies using the information collected during this simulation study. The results obtained for the Tecator dataset are relevant and corroborated by former studies. The study of a smaller dataset concerning corn samples shows the efficiency of our method on small size samples. Getting information on which derivatives (or which parts) of the spectrometric curves have a significant effect allows to get a better understanding of the way spectrometric curves influence the quantity to predict. In addition, a better knowledge of the structure of the underlying regression model may be useful to construct a relevant predictor.

Model based period analysis of absolute and relative survival with R: Data preparation, model fitting and derivation of survival estimates

Period analysis is increasingly employed in analyses of long-term survival of patients with chronic diseases such as cancer, as it derives more up-to-date survival estimates than traditional cohort based approaches. It has recently been extended with regression modelling using generalized linear models, which increases the precision of the survival estimates and enables to assess and account for effects of additional covariates.This paper provides a detailed presentation how model based period analysis may be used to derive population-based absolute and relative survival estimates using the freely available R language and statistical environment and already available R programs for period analysis.After an introduction of the underlying regression model and a description of the software tools we provide a step-by-step implementation of two regression models in R and illustrate how estimates and a test for trend over time in relative survival may be derived using data from a population based cancer registry.

Comparing ROC curves derived from regression models

In constructing predictive models, investigators frequently assess the incremental value of a predictive marker by comparing the ROC curve generated from the predictive model including the new marker with the ROC curve from the model excluding the new marker. Many commentators have noticed empirically that a test of the two ROC areas often produces a non-significant result when a corresponding Wald test from the underlying regression model is significant. A recent article showed using simulations that the widely used ROC area test produces exceptionally conservative test size and extremely low power. In this article, we demonstrate that both the test statistic and its estimated variance are seriously biased when predictions from nested regression models are used as data inputs for the test, and we examine in detail the reasons for these problems. Although it is possible to create a test reference distribution by resampling that removes these biases, Wald or likelihood ratio tests remain the preferred approach for testing the incremental contribution of a new marker.

An empirical examination of distributional assumptions underlying the relationship between personality disorder symptoms and personality traits.

This article examines the relationship between personality disorder (PD) symptoms and personality traits using a variety of distributional assumptions. Prior work in this area relies almost exclusively on linear models that treat PD symptoms as normally distributed and continuous. However, these assumptions rarely hold, and thus the results of prior studies are potentially biased. Here we explore the effect of varying the distributions underlying regression models relating PD symptomatology to personality traits using the initial wave of the Longitudinal Study of Personality Disorders (N=250; Lenzenweger, 1999), a university-based sample selected to include PD rates resembling epidemiological samples. PD symptoms were regressed on personality traits. The distributions underlying the dependent variable (i.e., PD symptoms) were variously modeled as normally distributed, as counts (Poisson, Negative-Binomial), and with two-part mixture distributions (zero-inflated, hurdle). We found that treating symptoms as normally distributed resulted in violations of model assumptions, that the negative-binomial and hurdle models were empirically equivalent, but that the coefficients achieving significance often differ depending on which part of the mixture distributions are being predicted (i.e., presence vs. severity of PD). Results have implications for how the relationship between normal and abnormal personality is understood.

Adjusting confounders in ranking biomarkers: a model-based ROC approach

High-throughput studies have been extensively conducted in the research of complex human diseases. As a representative example, consider gene-expression studies where thousands of genes are profiled at the same time. An important objective of such studies is to rank the diagnostic accuracy of biomarkers (e.g. gene expressions) for predicting outcome variables while properly adjusting for confounding effects from low-dimensional clinical risk factors and environmental exposures. Existing approaches are often fully based on parametric or semi-parametric models and target evaluating estimation significance as opposed to diagnostic accuracy. Receiver operating characteristic (ROC) approaches can be employed to tackle this problem. However, existing ROC ranking methods focus on biomarkers only and ignore effects of confounders. In this article, we propose a model-based approach which ranks the diagnostic accuracy of biomarkers using ROC measures with a proper adjustment of confounding effects. To this end, three different methods for constructing the underlying regression models are investigated. Simulation study shows that the proposed methods can accurately identify biomarkers with additional diagnostic power beyond confounders. Analysis of two cancer gene-expression studies demonstrates that adjusting for confounders can lead to substantially different rankings of genes.

Mapping carbon sequestration in forests at the regional scale - a climate biomonitoring approach by example of Germany

BackgroundThe United Nations Framework Convention on Climate Change recognizes carbon (C) fixation in forests as an important contribution for the reduction of atmospheric pollution in terms of greenhouse gases. Spatial differentiation of C sequestration in forests either at the national or at the regional scale is therefore needed for forest planning purposes. Hence, within the framework of the Forest Focus regulation, the aim of this investigation was to statistically analyse factors influencing the C fixation and to use the corresponding associations in terms of a predictive mapping approach at the regional scale by example of the German federal state North Rhine-Westphalia. The results of the methodical scheme outlined in this article should be compared with an already-published approach applied to the same data which were used in the investigation at hand.MethodsSite-specific data on C sequestration in humus, forest trees/dead wood and soil from two forest monitoring networks were intersected with available surface information on topography, soil, climate and forestal growing areas and districts. Next, the association between the C sequestration and the influence factors were examined and modelled by linear regression analyses. The resulting regression equations were applied on the surface data to predicatively map the C sequestration for the entire study area.ResultsThe computations yielded an estimation of 146.7 mio t C sequestered in the forests of North Rhine-Westphalia corresponding to 168.6 t/ha. The calculated values correspond well to according specifications given by the literature. Furthermore, the results are almost identical to those of another pilot study where a different statistical methodology was applied on the same database. Nevertheless, the underlying regression models contribute only a low degree of explanation to the overall variance of the C fixation. This might mainly be due to data quality aspects and missing influence factors in the analyses.DiscussionIn another study, an alternative approach was introduced to map the spatial differentiation of C sequestration in North Rhine-Westphalia based on the combination of geostatistics, decision tree analyses and GIS techniques. As a result, the overall mean of C sequestration amounted for 177 t C/ha which is 8.4 t C/ha higher than what was calculated in the study at hand and 14 t C/ha below the roughly guessed German-wide mean of 191 t C/ha.ConclusionsThe surface estimations of C pools in living forest trees/dead wood, the humus layer and the mineral soil enable to map the fixation of the greenhouse gas CO2 in forests at the regional scale. The estimations that were derived in this study are in good accordance with estimations based on techniques which, in contrast, did neither allow for spatial differentiation nor for mapping. The presented approach should be validated by application of other statistical techniques and by use of German wide inventory data. Furthermore, C sequestration should be modelled according to different climate change scenarios by combining statistical methods and dynamic modelling.

SAS macros for estimation of direct adjusted cumulative incidence curves under proportional subdistribution hazards models

The cumulative incidence function is commonly reported in studies with competing risks. The aim of this paper is to compute the treatment-specific cumulative incidence functions, adjusting for potentially imbalanced prognostic factors among treatment groups. The underlying regression model considered in this study is the proportional hazards model for a subdistribution function [1]. We propose estimating the direct adjusted cumulative incidences for each treatment using the pooled samples as the reference population. We develop two SAS macros for estimating the direct adjusted cumulative incidence function for each treatment based on two regression models. One model assumes the constant subdistribution hazard ratios between the treatments and the alternative model allows each treatment to have its own baseline subdistribution hazard function. The macros compute the standard errors for the direct adjusted cumulative incidence estimates, as well as the standard errors for the differences of adjusted cumulative incidence functions between any two treatments. Based on the macros’ output, one can assess treatment effects at predetermined time points. A real bone marrow transplant data example illustrates the practical utility of the SAS macros.

Partial orders with respect to continuous covariates and tests for the proportional hazards model

Several omnibus tests of the proportional hazards assumption have been proposed in the literature. In the two-sample case, tests have also been developed against ordered alternatives like monotone hazard ratio and monotone ratio of cumulative hazards. Here we propose a natural extension of these partial orders to the case of continuous and potentially time varying covariates, and develop tests for the proportional hazards assumption against such ordered alternatives. The work is motivated by applications in biomedicine and economics where covariate effects often decay over lifetime. The proposed tests do not make restrictive assumptions on the underlying regression model, and are applicable in the presence of time varying covariates, multiple covariates and frailty. Small sample performance and an application to real data highlight the use of the framework and methodology to identify and model the nature of departures from proportionality.

Current strategies for monitoring men with localised prostate cancer lack a strong evidence base: observational longitudinal study

Background:The UK National Institute for Health and Clinical Excellence (NICE) guidance recommends conservative management of men with ‘low-risk’ localised prostate cancer, monitoring the disease using prostate-specific antigen (PSA) kinetics and re-biopsy. However, there is little evidence of the changes in PSA level that should alert to the need for clinical re-assessment.Methods:This study compares the alerts resulting from PSA kinetics and a novel longitudinal reference range approach, which incorporates age-related changes, during the monitoring of 408 men with localised prostate cancer. Men were monitored by regular PSA tests over a mean of 2.9 years, recording when a man's PSA doubling time fell below 2 years, PSA velocity exceeded 2 ng ml–1 per year, or when his upper 10% reference range was exceeded.Results:Prostate-specific antigen doubling time and PSA velocity alerted a high proportion of men initially but became unresponsive to changes with successive tests. Calculating doubling time using recent PSA measurements reduced the decline in response. The reference range method maintained responsiveness to changes in PSA level throughout the monitoring.Conclusion:The increasing unresponsiveness of PSA kinetics is a consequence of the underlying regression model. Novel methods are needed for evaluation in cohorts currently being managed by monitoring. Meanwhile, the NICE guidance should be cautious.

Exact Analysis of Squared Cross-Validity Coefficient in Predictive Regression Models

In regression analysis, the notion of population validity is of theoretical interest for describing the usefulness of the underlying regression model, whereas the presumably more important concept of population cross-validity represents the predictive effectiveness for the regression equation in future research. It appears that the inference procedures of the squared multiple correlation coefficient have been extensively developed. In contrast, a full range of statistical methods for the analysis of the squared cross-validity coefficient is considerably far from complete. This article considers a distinct expression for the definition of the squared cross-validity coefficient as the direct connection and monotone transformation to the squared multiple correlation coefficient. Therefore, all the currently available exact methods for interval estimation, power calculation, and sample size determination of the squared multiple correlation coefficient are naturally modified and extended to the analysis of the squared cross-validity coefficient. The adequacies of the existing approximate procedures and the suggested exact method are evaluated through a Monte Carlo study. Furthermore, practical applications in areas of psychology and management are presented to illustrate the essential features of the proposed methodologies. The first empirical example uses 6 control variables related to driver characteristics and traffic congestion and their relation to stress in bus drivers, and the second example relates skills, cognitive performance, and personality to team performance measures. The results in this article can facilitate the recommended practice of cross-validation in psychological and other areas of social science research.

Sequential Design and Estimation in Heteroscedastic Nonparametric Regression

The paper considers, for the first time in the literature, sharp minimax design of predictors and sharp minimax sequential estimation of regression functions in a classical heteroscedastic nonparametric regression. The suggested methodology of a sharp minimax design of predictors in controlled regression experiments with fixed-size samples is based on minimization of the coefficient of difficulty of an underlying regression model, which is defined as a factor in changing the sample size that makes the estimation problem comparable with estimation in a homoscedastic regression with unit-variance errors and uniform design. It is established that an optimal design density is proportional to an underlying scale function. This makes a sequential design of predictors, based on a sequential choice of design densities, a particularly attractive statistical strategy that allows the statistician to minimize the coefficient of difficulty during a controlled experiment. It is shown that a developed sequential design of predictors asymptotically matches the mean integrated squared error (MISE) of an oracle that knows the optimal design (the scale function). Another considered problem is a sequential estimation of an underlying regression function with a preassigned MISE and sharp minimax mean stopping time in a heteroscedastic regression setting where predictors are generated according to some (in general unknown) underlying design density. For this setting the theory of a sharp minimax sequential estimation of a regression function is developed and a sharp minimax sequential estimator is suggested. Finally, a problem of sequential estimation of a regression function based on a sequential design is considered. A procedure of sequential analysis of heteroscedastic regression is suggested that matches the performance of an oracle that knows the optimal design density and optimal stopping time, which implies a preassigned MISE of estimation of an underlying regression function. A discussion of possible extensions and further developments is also presented.

Inference for covariate adjusted regression via varying coefficient models

We consider covariate adjusted regression (CAR), a regression method for situations where predictors and response are observed after being distorted by a multiplicative factor. The distorting factors are unknown functions of an observable covariate, where one specific distorting function is associated with each predictor or response. The dependence of both response and predictors on the same confounding covariate may alter the underlying regression relation between undistorted but unobserved predictors and response. We consider a class of highly flexible adjustment methods for parameter estimation in the underlying regression model, which is the model of interest. Asymptotic normality of the estimates is obtained by establishing a connection to varying coefficient models. These distribution results combined with proposed consistent estimates of the asymptotic variance are used for the construction of asymptotic confidence intervals for the regression coefficients. The proposed approach is illustrated with data on serum creatinine, and finite sample properties of the proposed procedures are investigated through a simulation study.

Using Wherry's Adjusted R 2 and Mallow's Cp for Model Selection From All Possible Regressions

Selecting a subset of predictors from a pool of potential predictors continues to be a common problem encountered by applied researchers in education. Because of several limitations associated with stepwise variable selection procedures, the examination of all possible regression solutions has been recommended. The authors evaluated the use of Mallow's Cp and Wherry's adjusted R 2 statistics to select a final model from a pool of model solutions. Neither the Cp nor the adjusted R 2 statistic correctly identified the underlying regression model any better and was generally worse than the stepwise selection method, which itself was poor. Using any of the model selection procedures studied here resulted in biased estimates of the authentic regression coefficients and underestimation of their standard errors. The use of theory and professional judgment is recommended for the selection of variables in a prediction equation.

Fitting a regression model for genotype-by-environment data on heading dates in grasses by methods for nonlinear mixed models.

The analysis of agricultural crop variety trials is usually complicated by the presence of genotype-by-environment interaction. A number of methods and models have been proposed to tackle this problem. One of the most common methods is the regression approach due to Yates and Cochran (1938, Journal of Agricultural Science 28, 556-580), in which performances of genotypes in the environments are regressed onto environmental means. The underlying regression model contains a multiplicative term with two unknown parameters (one for genotypes and one for environments). In the present paper, the model is modified by exchanging the role of genotypes and environments. Various diagnostic plots show that this modified model is adequate for a data set on heading dates in the grass species Dactylis glomerata. If environments are considered as a random factor while genotypes are taken as fixed, the model falls into the class of nonlinear mixed models. Recently, a number of procedures have been suggested for this class of models, which are based on first-order Taylor series expansion. Alternatively, the model can be estimated by maximum likelihood. This paper discusses the application of these methods for estimating parameters of the model.

A mixed approach and a distribution-free multiple imputation technique for the estimation of a multivariate probit model with missing values.

In the present paper a mixed generalized estimating/pseudo-score equations (GEPSE) approach together with a distribution-free multiple imputation technique is proposed for the estimation of regression and correlation structure parameters of multivariate probit models with missing values for an ordered categorical time-invariant variable. Furthermore, a generalization of the squared trace correlation (RT2) for multivariate probit models, denoted by pseudo-RT2, is proposed. A simulation study was conducted, simulating a probit model with an equicorrelation structure in the errors of an underlying regression model and using two different missing mechanisms. For a low 'true' correlation the difference between the GEPSE, a generalized estimating equations (GEE) and a maximum likelihood (ML) estimator were negligible. For a high 'true' correlation the GEPSE estimator turned out to be more efficient than the GEE and very efficient relative to the ML estimator. Furthermore, the pseudo-RT2 was close to RT2 of the underlying linear model. The mixed approach is illustrated using a psychiatric data set of depressive in-patients. The results of this analysis suggest that the depression score at discharge from a psychiatric hospital and the occurrence of stressful life events seem to increase the probability of having an episode of major depression within a one-year interval after discharge. Furthermore, the correlation structure points to short-time effects on having or not having a depressive episode, not accounted for in the systematic part of the regression model.

Proxy Data and Income Estimates: Reply to Pammer

Michael Pammer lodges three main criticisms of the proxy approach I used to estimate income levels in pre-World War I central and eastern Europe. I am grateful for two of the criticisms because accounting for them improves the underlying regression model and leads to revised estimates of income levels and growth rates in the region that are more reliable and more plausible than the original estimates. The new estimates do not, however, fundamentally change my earlier conclusions. The third criticism is more problematic. Pammer argues that the proxy approach does not offer a viable alternative to the more standard national-income approach for estimating incomes. His argument makes sense in an ideal world, but it completely ignores the realities of doing economic history in this highly complicated region of Europe.'

Misspecification Testing: A Comprehensive Approach

AbstractMisspecification tests of individual assumptions underlying regression models often lead to erroneous conclusions regarding source of misspecification. Monte Carlo experiments demonstrate that a comprehensive set of individual and joint tests reduces the likelihood of such conclusions. A practical testing strategy is proposed and suggestions made regarding its implementation.

Large Sample Theory of Estimation in Biased Sampling Regression Models. I

Biased sampling regression models were introduced by Jewell, generalizing the truncated regression model studied by Bhattacharya, Chernoff and Yang. If the independent variable takes on only a finite number of values (as does the stratum variable), we show: 1. That if the slope of the underlying regression model is assumed known, then the nonparametric maximum likelihood estimates of the distribution of the independent and dependent variables (a) can be calculated from ordinary $M$ estimates; (b) are asymptotically efficient. 2. How to construct $M$ estimates of the slope which are always $\sqrt n$ consistent, asymptotically Gaussian and are efficient locally, for example, if the error distribution is Gaussian. We support our asymptotics with a small simulation.