Logistic回归模型中连续变量交互作用的分析

Rothman提出生物学交互作用的评价应该基于相加尺度即是否有相加交互作用,而logistic回归模型的乘积项反映的是相乘交互作用.目前国内外文献讨论logistic回归模型中两因素的相加交互作用以两分类变量为主,本文介绍两连续变量或连续变量与分类变量相加交互作用可信区间估计的Bootstrap方法,文中以香港男性肺癌病例对照研究资料为例,辅以免费软件R的实现程序,为研究人员分析交互作用提供参考.     

Logistic回归模型中连续变量交互作用的分析

Rothman提出生物学交互作用的评价应该基于相加尺度即是否有相加交互作用,而logistic回归模型的乘积项反映的是相乘交互作用.目前国内外文献讨论logistic回归模型中两因素的相加交互作用以两分类变量为主,本文介绍两连续变量或连续变量与分类变量相加交互作用可信区间估计的Bootstrap方法,文中以香港男性肺癌病例对照研究资料为例,辅以免费软件R的实现程序,为研究人员分析交互作用提供参考.     

Logistic回归模型中连续变量交互作用的分析

Rothman提出生物学交互作用的评价应该基于相加尺度即是否有相加交互作用,而logistic回归模型的乘积项反映的是相乘交互作用.目前国内外文献讨论logistic回归模型中两因素的相加交互作用以两分类变量为主,本文介绍两连续变量或连续变量与分类变量相加交互作用可信区间估计的Bootstrap方法,文中以香港男性肺癌病例对照研究资料为例,辅以免费软件R的实现程序,为研究人员分析交互作用提供参考.     

Estimating measures of interaction on an additive scale for preventive exposures

Measures of interaction on an additive scale (relative excess risk due to interaction [RERI], attributable proportion [AP], synergy index [S]), were developed for risk factors rather than preventive factors. It has been suggested that preventive factors should be recoded to risk factors before calculating these measures. We aimed to show that these measures are problematic with preventive factors prior to recoding, and to clarify the recoding method to be used to circumvent these problems. Recoding of preventive factors should be done such that the stratum with the lowest risk becomes the reference category when both factors are considered jointly (rather than one at a time). We used data from a case-control study on the interaction between ACE inhibitors and the ACE gene on incident diabetes. Use of ACE inhibitors was a preventive factor and DD ACE genotype was a risk factor. Before recoding, the RERI, AP and S showed inconsistent results (RERI = 0.26 [95%CI: −0.30; 0.82], AP = 0.30 [95%CI: −0.28; 0.88], S = 0.35 [95%CI: 0.02; 7.38]), with the first two measures suggesting positive interaction and the third negative interaction. After recoding the use of ACE inhibitors, they showed consistent results (RERI = −0.37 [95%CI: −1.23; 0.49], AP = −0.29 [95%CI: −0.98; 0.40], S = 0.43 [95%CI: 0.07; 2.60]), all indicating negative interaction. Preventive factors should not be used to calculate measures of interaction on an additive scale without recoding.     

Estimating the difference between differences: measurement of additive scale interaction for proportions

A simple, effective closed-form method to calculate a confidence interval for the difference between two differences of proportions is proposed. The method is based on the Wilson score interval for the single proportion, and may be used to compare either unpaired or paired differences. It is equally applicable whether cell frequencies are large or small; and coverage properties are favourable. It is easily adapted to give a confidence interval for the treatment effect for a binary outcome in a two-period cross-over trial.     

Estimating adjusted NNT measures in logistic regression analysis

The number needed to treat (NNT) is a popular measure to describe the absolute effect of a new treatment compared with a standard treatment or placebo in clinical trials with binary outcome. For use of NNT measures in epidemiology to compare exposed and unexposed subjects, the terms 'number needed to be exposed' (NNE) and 'exposure impact number' (EIN) have been proposed. Additionally, in the framework of logistic regression a method was derived to perform point and interval estimation of NNT measures with adjustment for confounding by using the adjusted odds ratio (OR approach). In this paper, a new method is proposed which is based upon the average risk difference over the observed confounder values (ARD approach). A decision has to be made, whether the effect of allocating an exposure to unexposed persons or the effect of removing an exposure from exposed persons should be described. We use the term NNE for the first and the term EIN for the second situation. NNE is the average number of unexposed persons needed to be exposed to observe one extra case; EIN is the average number of exposed persons among one case can be attributed to the exposure. By means of simulations it is shown that the ARD approach is better than the OR approach in terms of bias and coverage probability, especially if the confounder distribution is wide. The proposed method is illustrated by application to data of a cohort study investigating the effect of smoking on coronary heart disease.     

Estimating treatment efficacy over time: a logistic regression model for binary longitudinal outcomes

This paper presents a case study in longitudinal data analysis where the goal is to estimate the efficacy of a new drug for treatment of a severe chronic constipation. Data consist of long sequences of binary outcomes (relief/no relief) on each of a large number of patients randomized to treatment (low and high dose) or placebo. Data characteristics indicate: (1) the treatment effects vary non-linearly with time; (2) there is substantial heterogeneity across subjects in their responses to treatment; and (3) there is a high proportion of subjects who never experience any relief (the non-responders).To overcome these challenges, we develop a hierarchical model for binary longitudinal data with a mixture distribution on the probability of response to account for the high frequency of non-responders. While the model is specified conditionally on subject-specific latent variables, we also draw inferences on key population-average parameters for the assessment of the treatments' efficacy in a population. In addition we employ a model-checking method to compare the goodness-of-fit for our model against simpler modelling approaches for aggregated counts, such as the zero-inflated Poisson and zero-inflated negative binomial models.We estimate subject-specific and population-average rate ratios of relief for the treatment with respect to the placebo as functions of time (RR(t)), and compare them with the rate ratios estimated from the models for aggregated counts. We find that: (1) the treatment is effective with respect to the placebo with higher efficacy at the beginning of the study; (2) the estimated rate ratios from the models for aggregated counts appear to be similar to the average across time of the population-average rate ratios estimated under our model; and (3) model-checking suggests that the hierarchical and zero-inflated negative binomial model fit the data best.If we are mainly interested to establish the overall efficacy (or safety) of a new drug, it is appropriate to aggregate the longitudinal data over time and analyse the count data by use of standard statistical methods. However, the models for aggregated counts cannot capture time trend of treatment such as the initial treatment benefit or the development of tolerance during the early stage of the treatment which may be important information to physicians to predict the treatment effects for their patients.     

logistic回归模型中交互作用的分析及评价

流行病学病因学研究常运用logistic回归模型分析影响因素的作用,并利用纳入乘积项的方法分析因素间交互作用,如有统计学意义表示两因素间存在相乘交互作用,但乘积项若无统计学意义并不表示两因素问相加交互作用或生物学交互作用的有无.文中介绍Rothman提出的针对logistic或Cox回归模型的三个评价相加交互作用的指标及其可信区间的计算,并以SPSS 15.0软件应用实例分析得出logistic回归模型的参数估计值和协方差矩阵,引入Andersson等编制的Excel计算表,计算相加交瓦作用指标及其可信区间,用于评价因素间的相加交互作用,为研究人员分析生物学交互作用提供依据.该方法方便快捷,且Excel计算表可在线免费下载.     

Estimating spatial accessibility to facilities on the regional scale: an extended commuting-based interaction potential model

Background  

There is growing interest in the study of the relationships between individual health-related behaviours (e.g. food intake and physical activity) and measurements of spatial accessibility to the associated facilities (e.g. food outlets and sport facilities). The aim of this study is to propose measurements of spatial accessibility to facilities on the regional scale, using aggregated data. We first used a potential accessibility model that partly makes it possible to overcome the limitations of the most frequently used indices such as the count of opportunities within a given neighbourhood. We then propose an extended model in order to take into account both home and work-based accessibility for a commuting population.     

实现logistic与Cox回归相乘相加交互作用的临床实践宏程序

病例对照研究常采用条件或非条件logistic分析,生存资料分析常采用Cox比例模型,但多数文献仅纳入主效应模型,然而广义线性模型不同于一般线性模型,其交互作用分为相乘交互与相加交互作用,前者只有统计学意义而后者更符合生物学意义。笔者以SAS 9.4软件编写宏,在计算logistic与Cox相乘交互项同时计算交互对比度、归因比、交互作用指数指标及利用Wald、Delta、PL(profile likelihood) 3种方法的可信区间评价相加交互作用,便于临床流行病学与遗传学大数据分析相乘相加交互作用时参考。     

Change point testing in logistic regression models with interaction term

A threshold effect takes place in situations where the relationship between an outcome variable and a predictor variable changes as the predictor value crosses a certain threshold/change point. Threshold effects are often plausible in a complex biological system, especially in defining immune responses that are protective against infections such as HIV‐1, which motivates the current work. We study two hypothesis testing problems in change point models. We first compare three different approaches to obtaining a p‐value for the maximum of scores test in a logistic regression model with change point variable as a main effect. Next, we study the testing problem in a logistic regression model with the change point variable both as a main effect and as part of an interaction term. We propose a test based on the maximum of likelihood ratios test statistic and obtain its reference distribution through a Monte Carlo method. We also propose a maximum of weighted scores test that can be more powerful than the maximum of likelihood ratios test when we know the direction of the interaction effect. In simulation studies, we show that the proposed tests have a correct type I error and higher power than several existing methods. We illustrate the application of change point model‐based testing methods in a recent study of immune responses that are associated with the risk of mother to child transmission of HIV‐1. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluating the discriminatory power of a multiple logistic regression model

Various measures for estimating the goodness-of-fit of the multiple logistic regression (MLR) model have been suggested, although there is no clear consensus as to which measure is most suitable. In this paper, a simple measure of the discriminatory power of the fitted MLR model, based on maximization of Youden's J index (J*), is proposed and compared with several goodness-of-fit statistics described previously. The relative effectiveness of the measure is illustrated using data from the Lipid Research Clinics Prevalence Study. It is suggested that J* may be a useful alternative index of goodness-of-fit of an MLR model, with the added advantage of having a simple practical interpretation.     

A bivariate logistic regression model based on latent variables

Bivariate observations of binary and ordinal data arise frequently and require a bivariate modeling approach in cases where one is interested in aspects of the marginal distributions as separate outcomes along with the association between the two. We consider methods for constructing such bivariate models based on latent variables with logistic marginals and propose a model based on the Ali-Mikhail-Haq bivariate logistic distribution. We motivate the model as an extension of that based on the Gumbel type 2 distribution as considered by other authors and as a bivariate extension of the logistic distribution, which preserves certain natural characteristics. Basic properties of the obtained model are studied and the proposed methods are illustrated through analysis of two data sets: a basic science cognitive experiment of visual recognition and awareness and a clinical data set describing assessments of walking disability among multiple sclerosis patients.     

Visualizing and assessing discrimination in the logistic regression model

Logistic regression models are widely used in medicine for predicting patient outcome (prognosis) and constructing diagnostic tests (diagnosis). Multivariable logistic models yield an (approximately) continuous risk score, a transformation of which gives the estimated event probability for an individual. A key aspect of model performance is discrimination, that is, the model's ability to distinguish between patients who have (or will have) an event of interest and those who do not (or will not). Graphical aids are important in understanding a logistic model. The receiver‐operating characteristic (ROC) curve is familiar, but not necessarily easy to interpret. We advocate a simple graphic that provides further insight into discrimination, namely a histogram or dot plot of the risk score in the outcome groups. The most popular performance measure for the logistic model is the c‐index, numerically equivalent to the area under the ROC curve. We discuss the comparative merits of the c‐index and the (standardized) mean difference in risk score between the outcome groups. The latter statistic, sometimes known generically as the effect size, has been computed in slightly different ways by several different authors, including Glass, Cohen and Hedges. An alternative measure is the overlap between the distributions in the outcome groups, defined as the area under the minimum of the two density functions. The larger the overlap, the weaker the discrimination. Under certain assumptions about the distribution of the risk score, the c‐index, effect size and overlap are functionally related. We illustrate the ideas with simulated and real data sets. Copyright © 2010 John Wiley & Sons, Ltd.     

Global tests in the additive hazards regression model

In this article, we discuss testing for the effect of several covariates in the additive hazards regression model. Bhattacharyya and Klein (Statist. Med. 2005; 24(14):2235-2240) note that an ad hoc weight function suggested by Aalen (Statist. Med. 1989; 8:907-925) is inconsistent when used as a global test for comparing groups since the test statistic depends on which group is used as the baseline group. We will suggest a simple alternative test that does not exhibit this problem. This test is a natural extension of the logrank test. We shall also discuss an alternative covariance estimator. The tests are applied to a data set and a simulation study is performed.     

基于Logistic回归的早期可疑异位妊娠诊断模型的研究

目的 建立由超声及血清指标组成的综合诊断模型,以获得早期诊断异位妊娠(EP)患者的较佳指标组合,提高早期不明位置妊娠患者(PUL)的综合诊断水平.方法 早期PUL患者随机分成建模组(184例)和测试组(90例),按最终妊娠结果分为EP组与非EP组.以拟合累积方式得出建模组每个病例EP的诊断概率,并利用受试者工作特性曲线(ROC)评价诊断指标,最后在独立的测试组中检验模型的诊断效率、正确性与稳定性.结果 采用logistic逐步回归分析显示,血清孕酮、内膜厚度和对称性的意义较大,对PUL的妊娠结局有影响.受试者工作特征曲线(ROC)证实,logistic模型较单一变量的诊断性能高(P＜0.05).当模型预测概率临界值取0.25时,对EP鉴别的敏感度和特异度分别为98.4%、92.6%,正确指数为0.91,曲线下面积为0.992.以同样标准应用于测试组,2组的诊断效率相仿.结论 综合考虑子宫内膜形态学变化与相应的生化指标,将有助于提高阴道超声诊断EP的正确率,能较为准确地判断早期PUL的妊娠状态.     

Using logistic regression in perinatal epidemiology: an introduction for clinical researchers. Part 2: The logistic regression equation

In Part 1 basic concepts were introduced as a preparation for an introductory explanation of logistic regression. Logistic regression is a statistical modelling technique, designed for the estimation of the simultaneous effects of predictors on the risk of a certain dichotomous outcome variable where each effect is estimated while adjusting for the effect of the other factors considered. The basic concepts--odds, odds ratio, confounding and interaction--were introduced in such a way that they naturally lead to the concept of logistic regression. In Part 2 the concepts are 'translated' into simple equations. By studying these equations the equivalence between such mathematical expressions and the underlying clinical assessment of risk will become clear.     

Using logistic regression in perinatal epidemiology: an introduction for clinical researchers. Part 2: the logistic regression equation

Summary. In Part 1 basic concepts were introduced as a preparation for an introductory explanation of logistic regression. Logistic regression is a statistical modelling technique, designed for the estimation of the simultaneous effects of predictors on the risk of a certain dichotomous outcome variable where each effect is estimated while adjusting for the effect of the other factors considered. The basic concepts - odds, odds ratio, confounding and interaction - were introduced in such a way that they naturally lead to the concept of logistic regression. In Part 2 the concepts are translated into simple equations. By studying these equations the equivalence between such mathematical expressions and the underlying clinical assessment of risk will become clear.