Use of empirical likelihood to calibrate auxiliary information in partly linear monotone regression models

In statistical analysis, a regression model is needed if one is interested in finding the relationship between a response variable and covariates. When the response depends on the covariate, then it may also depend on the function of this covariate. If one has no knowledge of this functional form but expect for monotonic increasing or decreasing, then the isotonic regression model is preferable. Estimation of parameters for isotonic regression models is based on the pool‐adjacent‐violators algorithm (PAVA), where the monotonicity constraints are built in. With missing data, people often employ the augmented estimating method to improve estimation efficiency by incorporating auxiliary information through a working regression model. However, under the framework of the isotonic regression model, the PAVA does not work as the monotonicity constraints are violated. In this paper, we develop an empirical likelihood‐based method for isotonic regression model to incorporate the auxiliary information. Because the monotonicity constraints still hold, the PAVA can be used for parameter estimation. Simulation studies demonstrate that the proposed method can yield more efficient estimates, and in some situations, the efficiency improvement is substantial. We apply this method to a dementia study. Copyright © 2013 John Wiley & Sons, Ltd.     

A comparison of several methods to test for the existence of a monotonic dose-response relationship in clinical and epidemiological studies.

The Mantel-extension chi-square test for overall trend and an asymptotically equivalent test based on logistic regression are commonly used to test for a monotonic dose-response relationship between exposure and disease in epidemiological and clinical studies. However, these tests present two important disadvantages, as they (i) make the restrictive assumption of a parametric model of linear form on the logit scale and (ii) impose the a priori choice of scores to code for the exposure categories. Indeed, the linear assumption, if made incorrectly, can lead to an invalid conclusion, and the choice of scores lends arbitrariness to the test results. Some alternative tests have been proposed in the literature. We have considered several of these tests, namely one based on isotonic regression, the T-test based on contrasts and a recently published test based on adjacent contrasts (Dosemeci-Benichou test). The aim of our study was to compare the statistical properties (type I error and power) of these tests and of the commonly used Mantel-extension test for overall trend. We generated cohort and case-control data and considered one- and two-sided versions of the tests. Moreover, we studied the tests under the null hypothesis of no relationship between exposure and disease and under various alternative patterns of monotonic or non-monotonic dose-response relationships. This study confirms that the commonly used trend tests can lead to erroneous conclusion of a monotonic dose-response relationship. The test based on isotonic regression does not represent a favourable alternative, as it tends to be too powerful in case of non-monotonic dose-response relationship patterns. The tests based on contrasts seem to possess more favourable properties by combining close to nominal type I error, high power for monotonic alternatives and low power for non-monotonic alternatives.     

Selecting the best dose when a monotonic dose-response relation exists

We propose a method for selecting the best treatment when a monotonic dose—response relationship exists. Because of side effects associated with higher doses, the highest dose may not be the optimum, particularly when a lower dose gives a similar response. Rather than assume a particular functional relationship of dose to response, we use isotonic regression techniques. We consider the case of three treatment levels, which is applicable to many clinical trials. The lowest treatment level may represent a placebo or no treatment control. While we focus primarily on Bernoulli response variables, we also discuss a model for normally distributed data. We suggest a two-stage procedure that we have investigated via simulation.     

Estimation of the general threshold limit values for dust

In 1997 the German MAK Commission set new general threshold limit values for dust. The procedure has recently been assessed (McLaughlin et al. 2001). One of the points raised was the use of inappropriate statistical methods. We want to address this point to a greater extent by discussing several alternatives implied by the already established threshold models, and we present results from a new approach that has been refined in the meantime: the use of the additive isotonic model. The underlined assumption of monotonicity regarding the dose-response relationship has been extensively investigated. One very flexible model, based on smoothing splines, shows in some of the samples a decline in the risk over a certain range of the exposure compared to the risk at baseline. Another approach with fractional polynomials and segmented regression lines shows that this result can be explained by chance. These methods show an increasing risk with increasing exposure. Additionally, permutation tests are used to prove the trend within the isotonic framework. The results from the additive isotonic model confirm previous assessments of the general threshold limit value.     

Non-parametric estimators of a monotonic dose-response curve and bootstrap confidence intervals

In this paper we consider study designs which include a placebo and an active control group as well as several dose groups of a new drug. A monotonically increasing dose-response function is assumed, and the objective is to estimate a dose with equivalent response to the active control group, including a confidence interval for this dose. We present different non-parametric methods to estimate the monotonic dose-response curve. These are derived from the isotonic regression estimator, a non-negative least squares estimator, and a bias adjusted non-negative least squares estimator using linear interpolation. The different confidence intervals are based upon an approach described by Korn, and upon two different bootstrap approaches. One of these bootstrap approaches is standard, and the second ensures that resampling is done from empiric distributions which comply with the order restrictions imposed. In our simulations we did not find any differences between the two bootstrap methods, and both clearly outperform Korn's confidence intervals. The non-negative least squares estimator yields biased results for moderate sample sizes. The bias adjustment for this estimator works well, even for small and moderate sample sizes, and surprisingly outperforms the isotonic regression method in certain situations.     

Order-restricted dose-related trend tests

Methods of isotonic regression are proposed to increase the power of common trend tests in situations where a monotonicity constraint is imposed upon the dose-response function. Isotonic versions of Cochran-Armitage type trend tests for binary response data are developed, and the bootstrap method is used in finding the empirical distributions of the test statistics and their critical values. The isotonic likelihood ratio test with a survival adjustment is also proposed. This survival adjustment can be applied to the likelihood ratio test for either the order-restricted or unrestricted parameter cases. To achieve the isotonic modifications, an amalgamation algorithm is applied when the observed dose-response is non-monotonic. A Monte Carlo simulation study comparing these trend tests shows the advantages of the isotonic modifications and survival adjustment. By applying the proposed methods to data from a toxicology and carcinogenesis study conducted as part of the National Toxicology Program, the effect of CI Pigment Red 23 is investigated.     

Additive isotonic regression models in epidemiology

Stone's method for assessing disease risk around a point source through isotonic regression is routinely used in spatial epidemiology. It is useful in situations where the relationship of risk with exposure (distance being commonly used as a surrogate variable) is assumed monotonic but otherwise of unknown form. This paper extends this method to non-spatial epidemiology, where typically two or more risk factors are present. The methodology described is based on the additive isotonic model approach of Bacchetti; versions appropriate to count (Poisson) data and case-control (binomial) data are described. In both cases, adjustment for covariates is incorporated, and a Monte Carlo method of hypothesis testing and interval estimation is presented. The methodology is illustrated through a case-control example concerning the analysis of the possible effect of preconceptional external ionizing radiation doses on the sex ratio at birth among children of fathers working at the Sellafield nuclear installation, Cumbria, U.K.     

A comparison of regression trees, logistic regression, generalized additive models, and multivariate adaptive regression splines for predicting AMI mortality

Clinicians and health service researchers are frequently interested in predicting patient-specific probabilities of adverse events (e.g. death, disease recurrence, post-operative complications, hospital readmission). There is an increasing interest in the use of classification and regression trees (CART) for predicting outcomes in clinical studies. We compared the predictive accuracy of logistic regression with that of regression trees for predicting mortality after hospitalization with an acute myocardial infarction (AMI). We also examined the predictive ability of two other types of data-driven models: generalized additive models (GAMs) and multivariate adaptive regression splines (MARS). We used data on 9484 patients admitted to hospital with an AMI in Ontario. We used repeated split-sample validation: the data were randomly divided into derivation and validation samples. Predictive models were estimated using the derivation sample and the predictive accuracy of the resultant model was assessed using the area under the receiver operating characteristic (ROC) curve in the validation sample. This process was repeated 1000 times-the initial data set was randomly divided into derivation and validation samples 1000 times, and the predictive accuracy of each method was assessed each time. The mean ROC curve area for the regression tree models in the 1000 derivation samples was 0.762, while the mean ROC curve area of a simple logistic regression model was 0.845. The mean ROC curve areas for the other methods ranged from a low of 0.831 to a high of 0.851. Our study shows that regression trees do not perform as well as logistic regression for predicting mortality following AMI. However, the logistic regression model had performance comparable to that of more flexible, data-driven models such as GAMs and MARS.     

Analysis on binary responses with ordered covariates and missing data

We consider the situation of two ordered categorical variables and a binary outcome variable, where one or both of the categorical variables may have missing values. The goal is to estimate the probability of response of the outcome variable for each cell of the contingency table of categorical variables while incorporating the fact that the categorical variables are ordered. The probability of response is assumed to change monotonically as each of the categorical variables changes level. A probability model is used in which the response is binomial with parameters p(ij) for each cell (i, j) and the number of observations in each cell is multinomial. Estimation approaches that incorporate Gibbs sampling with order restrictions on p(ij) induced via a prior distribution, two-dimensional isotonic regression and multiple imputation to handle missing values are considered. The methods are compared in a simulation study. Using a fully Bayesian approach with a strong prior distribution to induce ordering can lead to large gains in efficiency, but can also induce bias. Utilizing isotonic regression can lead to modest gains in efficiency, while minimizing bias and guaranteeing that the order constraints are satisfied. A hybrid of isotonic regression and Gibbs sampling appears to work well across a variety of scenarios. The methods are applied to a pancreatic cancer case-control study with two biomarkers.     

Discriminant analysis when all variables are ordered.

Determination of the equation that relates an ordered dependent variable to ordered independent variables is sought. One solution, non-parametric discriminant analysis (NPD), involves obtaining the best monotonic step function by means of a computer search procedure. Although one can use alternative selection criteria in obtaining the equation, the illustrative examples use absolute distance. This paper compares the prediction procedures obtained from NPD with those from linear discriminant analysis, linear regression (with and without transformed variables), and logistic regression. We show that NPD is analogous to regression tree analysis with incorporation of ordered variables and monotonicity. We use various prediction functions to predict the example data, the data using the leave-one-out technique, and a verification set. Consistently, non-parametric discriminant analysis performs as good as or better than the tested alternatives.     

Result variation and efficiency kinetics in real-time PCR

Fluorescent monitoring of DNA amplification is the basis of real-time PCR. Absolute quantification can be achieved using a standard curve method. The standard curve is constructed by amplifying known amounts of standards under identical conditions to that of the samples.The objective of the current study is to propose a mathematical model to assess the acceptability of PCR results. Four commercial standards for HCV-RNA (hepatitis C virus RNA) along with 6 patient samples were measured by real-time PCR, using two different RT-PCR reagents. The standard deviation of regression (Sy,x) was calculated for each group of standard and compared by F-Test. The efficiency kinetics was computed by logistic regression, c2 goodness of fit test was preformed to assess the appropriateness of the efficiency curves.Calculated efficiencies were not significantly different from the value predicted by logistic regression model. Reactions with more variation showed less stable efficiency curves, with wider range of amplification efficiencies.Amplification efficiency kinetics can be computed by fitting a logistic regression curve to the gathered fluorescent data of each reaction. This model can be employed to assess the acceptability of PCR results calculated by standard curve method.     

A practical guide to dose-response analyses and risk assessment in occupational epidemiology

Dose-response modeling in occupational epidemiology is usually motivated by questions of causal inference (eg, is there a monotonic increase of risk with increasing exposure?) or risk assessment (eg, how much excess risk exists at any given level of exposure?). We focus on several approaches to dose-response in occupational cohort studies. Categorical analyses are useful for detecting the shape of dose-response. However, they depend on the number and location of cutpoints and result in step functions rather than smooth curves. Restricted cubic splines and penalized splines are useful parametric techniques that provide smooth curves. Although splines can complement categorical analyses, they do not provide interpretable parameters. The shapes of these curves will depend on the degree of "smoothing" chosen by the analyst. We recommend combining categorical analyses and some type of smoother, with the goal of developing a reasonably simple parametric model. A simple parametric model should serve as the goal of dose-response analyses because (1) most "true" exposure response curves in nature may be reasonably simple, (2) a simple parametric model is easily communicated and used by others, and (3) a simple parametric model is the best tool for risk assessors and regulators seeking to estimate individual excess risks per unit of exposure. We discuss these issues and others, including whether the best model is always the one that fits the best, reasons to prefer a linear model for risk in the low-exposure region when conducting risk assessment, and common methods of calculating excess lifetime risk at a given exposure from epidemiologic results (eg, from rate ratios). Points are illustrated using data from a study of dioxin and cancer.     

An EM-based semi-parametric mixture model approach to the regression analysis of competing-risks data

We consider a mixture model approach to the regression analysis of competing-risks data. Attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types. These two quantities are specified in the mixture model using the logistic model and the proportional hazards model, respectively. We propose a semi-parametric mixture method to estimate the logistic and regression coefficients jointly, whereby the component-baseline hazard functions are completely unspecified. Estimation is based on maximum likelihood on the basis of the full likelihood, implemented via an expectation-conditional maximization (ECM) algorithm. Simulation studies are performed to compare the performance of the proposed semi-parametric method with a fully parametric mixture approach. The results show that when the component-baseline hazard is monotonic increasing, the semi-parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples. When the component-baseline hazard is not monotonic increasing, the semi-parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring. The methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol.     

Estimation of the time of a linear trend in monitoring survival time

Change point estimation is recognized as an essential tool of root cause analyses within quality control programs as it enables clinical experts to search for potential causes of change in hospital outcomes more effectively. In this paper, we consider estimation of the time when a linear trend disturbance has occurred in survival time following an in-control clinical intervention in the presence of variable patient mix. To model the process and change point, a linear trend in the survival time of patients who underwent cardiac surgery is formulated using hierarchical models in a Bayesian framework. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. We use Markov Chain Monte Carlo to obtain posterior distributions of the change point parameters including the location and the slope size of the trend and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time cumulative sum control chart (CUSUM) control charts for different trend scenarios. In comparison with the alternatives, step change point model and built-in CUSUM estimator, more accurate and precise estimates are obtained by the proposed Bayesian estimator over linear trends. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.     

A sequential testing approach to detecting multiple change points in the proportional hazards model

The semi‐parametric proportional hazards model has been widely adopted in clinical trials with time‐to‐event outcomes. A key assumption in the model is that the hazard ratio function is a constant over time, which is frequently violated as there is often a lag period before an experimental treatment reaches its full effect. One existing approach uses maximal score tests and Monte Carlo sampling to identify multiple change points in the hazard ratio function, which requires the number of change points that exist in the model to be known. We propose a sequential testing approach to detecting multiple change points in the hazard ratio function using likelihood ratio tests, and the distributions of the likelihood ratio statistics under the null hypothesis are evaluated via resampling. An important feature of the proposed approach is that the number of change points in the model is inferred from the data and does not need to be specified. Numerical results based on simulated clinical trials and a real time‐to‐event study show that the proposed approach can accurately detect the change points in the hazard ratio function. Copyright © 2012 John Wiley & Sons, Ltd.     

The mixed model for the analysis of a repeated-measurement multivariate count data

Clustered overdispersed multivariate count data are challenging to model due to the presence of correlation within and between samples. Typically, the first source of correlation needs to be addressed but its quantification is of less interest. Here, we focus on the correlation between time points. In addition, the effects of covariates on the multivariate counts distribution need to be assessed. To fulfill these requirements, a regression model based on the Dirichlet-multinomial distribution for association between covariates and the categorical counts is extended by using random effects to deal with the additional clustering. This model is the Dirichlet-multinomial mixed regression model. Alternatively, a negative binomial regression mixed model can be deployed where the corresponding likelihood is conditioned on the total count. It appears that these two approaches are equivalent when the total count is fixed and independent of the random effects. We consider both subject-specific and categorical-specific random effects. However, the latter has a larger computational burden when the number of categories increases. Our work is motivated by microbiome data sets obtained by sequencing of the amplicon of the bacterial 16S rRNA gene. These data have a compositional structure and are typically overdispersed. The microbiome data set is from an epidemiological study carried out in a helminth-endemic area in Indonesia. The conclusions are as follows: time has no statistically significant effect on microbiome composition, the correlation between subjects is statistically significant, and treatment has a significant effect on the microbiome composition only in infected subjects who remained infected.     

Constructing longitudinal disease progression curves using sparse,short‐term individual data with an application to Alzheimer's disease

In epidemiology, cohort studies utilised to monitor and assess disease status and progression often result in short‐term and sparse follow‐up data. Thus, gaining an understanding of the full‐term disease pathogenesis can be difficult, requiring shorter‐term data from many individuals to be collated. We investigate and evaluate methods to construct and quantify the underlying long‐term longitudinal trajectories for disease markers using short‐term follow‐up data, specifically applied to Alzheimer's disease. We generate individuals' follow‐up data to investigate approaches to this problem adopting a four‐step modelling approach that (i) determines individual slopes and anchor points for their short‐term trajectory, (ii) fits polynomials to these slopes and anchor points, (iii) integrates the reciprocated polynomials and (iv) inverts the resulting curve providing an estimate of the underlying longitudinal trajectory. To alleviate the potential problem of roots of polynomials falling into the region over which we integrate, we propose the use of non‐negative polynomials in Step 2. We demonstrate that our approach can construct underlying sigmoidal trajectories from individuals' sparse, short‐term follow‐up data. Furthermore, to determine an optimal methodology, we consider variations to our modelling approach including contrasting linear mixed effects regression to linear regression in Step 1 and investigating different orders of polynomials in Step 2. Cubic order polynomials provided more accurate results, and there were negligible differences between regression methodologies. We use bootstrap confidence intervals to quantify the variability in our estimates of the underlying longitudinal trajectory and apply these methods to data from the Alzheimer's Disease Neuroimaging Initiative to demonstrate their practical use. Copyright © 2017 John Wiley & Sons, Ltd.     

The trend odds model for ordinal data

Ordinal data appear in a wide variety of scientific fields. These data are often analyzed using ordinal logistic regression models that assume proportional odds. When this assumption is not met, it may be possible to capture the lack of proportionality using a constrained structural relationship between the odds and the cut‐points of the ordinal values. We consider a trend odds version of this constrained model, wherein the odds parameter increases or decreases in a monotonic manner across the cut‐points. We demonstrate algebraically and graphically how this model is related to latent logistic, normal, and exponential distributions. In particular, we find that scale changes in these potential latent distributions are consistent with the trend odds assumption, with the logistic and exponential distributions having odds that increase in a linear or nearly linear fashion. We show how to fit this model using SAS Proc NLMIXED and perform simulations under proportional odds and trend odds processes. We find that the added complexity of the trend odds model gives improved power over the proportional odds model when there are moderate to severe departures from proportionality. A hypothetical data set is used to illustrate the interpretation of the trend odds model, and we apply this model to a swine influenza example wherein the proportional odds assumption appears to be violated. Copyright © 2012 John Wiley & Sons, Ltd.     

A semiparametric approach to analysing dose-response data

In the analysis of a quantal dose-response experiment with grouped data, the most commonly used parametric procedure is logistic regression, commonly referred to as 'logit analysis'. The adequacy of the fit by the logistic regression curve is tested using the chi-square lack-of-fit test. If the lack-of-fit test is not significant, then the logistic model is assumed to be adequate and estimation of effective doses and confidence intervals on the effective doses can be made. When the tolerance distribution of the dose-response data is not known and cannot be assumed by the user, one can use non-parametric methods, such as kernel regression or local linear regression, to estimate the dose-response curve, effective doses and confidence intervals. This research proposes another alternative based on semi-parametric regression to analysing quantal dose-response data called model-robust quantal regression (MRQR). MRQR linearly combines the parametric and non-parametric predictions with the use of a mixing parameter. MRQR uses logistic regression as the parametric portion of the model and local linear regression as the non-parametric portion of the model. Our research has shown that the MRQR procedure can improve the fit of the dose-response curve by producing narrower confidence intervals for predictions while providing improved precision of estimates of the effective doses with respect to either logistic or local linear regression results.     

多元线性回归方程中共线影响点的诊断

目的介绍在多元线性回归方程中共线影响点的来源及其诊断方法.方法通过利用三种不同的方法:①删除行变量观测条件数的变化②利用迹对行列式的比值和③杠杆成分来确定共线影响点.结果在有异常点存在时可以利用三种诊断方法来确定该点是否对共线性产生了影响.结论在建立多元线性回归方程时可以利用诊断共线影响点的方法来建立更符合实际规律的方程.