Quantification of Treatment Effect Modification on Both an Additive and Multiplicative Scale

BackgroundIn both observational and randomized studies, associations with overall survival are by and large assessed on a multiplicative scale using the Cox model. However, clinicians and clinical researchers have an ardent interest in assessing absolute benefit associated with treatments. In older patients, some studies have reported lower relative treatment effect, which might translate into similar or even greater absolute treatment effect given their high baseline hazard for clinical events.MethodsThe effect of treatment and the effect modification of treatment were respectively assessed using a multiplicative and an additive hazard model in an analysis adjusted for propensity score in the context of coronary surgery.ResultsThe multiplicative model yielded a lower relative hazard reduction with bilateral internal thoracic artery grafting in older patients (Hazard ratio for interaction/year = 1.03, 95%CI: 1.00 to 1.06, p = 0.05) whereas the additive model reported a similar absolute hazard reduction with increasing age (Delta for interaction/year = 0.10, 95%CI: -0.27 to 0.46, p = 0.61). The number needed to treat derived from the propensity score-adjusted multiplicative model was remarkably similar at the end of the follow-up in patients aged < = 60 and in patients >70.ConclusionsThe present example demonstrates that a lower treatment effect in older patients on a relative scale can conversely translate into a similar treatment effect on an additive scale due to large baseline hazard differences. Importantly, absolute risk reduction, either crude or adjusted, can be calculated from multiplicative survival models. We advocate for a wider use of the absolute scale, especially using additive hazard models, to assess treatment effect and treatment effect modification.     

An extension of the proportional-hazards model for grouped data

A family of transformations for probabilities is considered for the analysis of grouped survival data. Additive and multiplicative models for the hazard function are compared. A method is suggested for estimating the scale on which an additive representation of the hazard, in terms of explanatory variables, is appropriate. Computations are performed using GLIM. A test for departures from the grouped proportional-hazards model is introduced. An example is analyzed by the methods put forward.     

Analyses of two-way chronic studies

This paper proposes a method for analyzing tumor data from chronic studies when the experimental design includes combinations of two factors, for example, sex and dose. Both main effects and combined-effect (interaction) hypotheses are considered. A stratified log-rank statistic is presented for tests of no column or row (main) effects. The paper shows that when the numbers of animals in the cells are unequal and disproportional, the null distribution of the unstratified log-rank statistic does not have a chi-square distribution. Two simple models, additive and multiplicative, for representing the combined effect of row and column are considered under the proportional hazards model. A simple conservative statistic is proposed for testing the additivity of the row and column effects. A simulation experiment to examine the behavior of the null distribution of the combined-effect test statistic under the additive model and the power of the test against the multiplicative model is reported. The procedure is illustrated by analyzing mammary tumors induced by 7,12-dimethylbenz[a]anthracene (DMBA) in yellow and agouti F1 female mice from a laboratory experiment.     

An additive‐multiplicative mean residual life model for right‐censored data

Many studies have focused on determining the effect of the body mass index (BMI) on the mortality in different cohorts. In this article, we propose an additive‐multiplicative mean residual life (MRL) model to assess the effects of BMI and other risk factors on the MRL function of survival time in a cohort of Chinese type 2 diabetic patients. The proposed model can simultaneously manage additive and multiplicative risk factors and provide a comprehensible interpretation of their effects on the MRL function of interest. We develop an estimation procedure through pseudo partial score equations to obtain parameter estimates. We establish the asymptotic properties of the proposed estimators and conduct simulations to demonstrate the performance of the proposed method. The application of the procedure to a study on the life expectancy of type 2 diabetic patients reveals new insights into the extension of the life expectancy of such patients.     

Analysis of failure time data with multilevel clustering, with application to the child vitamin a intervention trial in Nepal

We consider the problem of estimating covariate effects in the marginal Cox proportional hazard model and multilevel associations for child mortality data collected from a vitamin A supplementation trial in Nepal, where the data are clustered within households and villages. For this purpose, a class of multivariate survival models that can be represented by a functional of marginal survival functions and accounts for hierarchical structure of clustering is exploited. Based on this class of models, an estimation strategy involving a within-cluster resampling procedure is proposed, and a model assessment approach is presented. The asymptotic theory for the proposed estimators and lack-of-fit test is established. The simulation study shows that the estimates are approximately unbiased, and the proposed test statistic is conservative under extremely heavy censoring but approaches the size otherwise. The analysis of the Nepal study data shows that the association of mortality is much greater within households than within villages.     

A statistical model for red blood cell survival

A statistical model for the survival time of red blood cells (RBCs) with a continuous distribution of cell lifespans is presented. The underlying distribution of RBC lifespans is derived from a probability density function with a bathtub-shaped hazard curve, and accounts for death of RBCs due to senescence (age-dependent increasing hazard rate) and random destruction (constant hazard), as well as for death due to initial or delayed failures and neocytolysis (equivalent to early red cell mortality). The model yields survival times similar to those of previously published studies of RBC survival and is easily amenable to inclusion of drug effects and haemolytic disorders.     

Accelerated hazards regression model and its adequacy for censored survival data

The accelerated hazards regression model is introduced to study the relationship between survival times and covariates through a scale change between hazard functions. The model is also compared with several other popular classes of regression models for censored survival data in statistical literature. Test statistics are proposed and studied to assess the model's adequacy. Actual data from a randomized clinical trial of biodegradable carmustine polymer for treatment of brain cancer are analyzed to demonstrate the potential application of the regression model and the proposed test statistics.     

大兴安岭东部主要林分类型乔木层生物量估算模型

大尺度森林生物量的估算方法是人们目前关注的焦点,建立林分生物量模型成为一种趋势.本研究以大兴安岭东部6个主要林分类型为研究对象,构建了其总量及各分项一元、二元可加性林分生物量模型.采用似然分析法判断总量及各分项生物量异速生长模型的误差结构(可加型或相乘型),采用非线性似乎不相关回归模型方法估计模型参数.结果表明: 经似然分析法判断,大兴安岭东部6个主要林分类型总量及各分项生物量异速生长模型的误差结构都是相乘型的,对数转换的可加性生物量可以被选用.各林分类型可加性生物量模型的调整后确定系数为0.78～0.99,平均相对误差为-2.3%～6.9%,平均相对误差绝对值6.3%～43.3%.增加林分平均高可以提高绝大多数生物量模型的拟合效果和预测能力,而且总量、地上和树干生物量模型效果较好,树根、树枝、树叶和树冠生物量模型效果较差.为了使模型参数估计更有效,所建立的生物量模型应当考虑林分总生物量及各分项生物量的可加性.本研究建立的林分总量与各分项生物量模型都能对大兴安岭东部6个主要林分类型生物量进行较好的估计.     

Unemployment and mortality in Denmark, 1970-80.

Relative mortality in the period 1970-80 was studied among Danish men and women who were unemployed and employed on the day of the 1970 census. The study population consisted of the total labour force in the age range 20-64 on 9 November 1970--that is, about 2 million employed and 22,000 unemployed people. Relative mortality was analysed by a multiplicative hazard regression model (as a natural extension of the standardised mortality ratio) and a multiplicative regression model with extra-Poisson variation. A significantly increased death rate (40-50%) was found among the unemployed after adjusting for occupation, housing category, geographical region, and marital state. Analysis of five main causes of death showed increased mortality from all causes, but especially from suicide or accidents. In areas where the local unemployment rate was comparatively high the relative mortality among the unemployed was lower. The increased mortality among the unemployed was interpreted as a consequence of health related selection as well as increased susceptibility associated with the psychosocial stress of unemployment.     

Sperm competition games: a comparison of loaded raffle models and their biological implications

Our main aim is to compare the additive model, due to Mesterton-Gibbons, and the multiplicative model, due to Parker, of sperm allocation under sperm competition, when other influences are treated in the same way. We first review these (and other) models and their foundations, leading to a generalization of the multiplicative model. Sperm is assumed to cost energy, and this constraint is incorporated differently in the two models. These give the same results in the random-roles situation when the males occupy roles (of first and second to mate) randomly: the number of sperm ejaculated in the favoured role is greater than that in the disfavoured role by an amount that depends on the effect of sperm limitation (i.e. the probability that there is insufficient sperm to ensure full fertility). If the latter is negligible, or the fertilization raffle fair, this difference is zero, as Parker found originally. In the constant roles situation (where males of a particular type always occupy the same role) the predictions differ: the additive model has the same predictions as in the random roles case, but the multiplicative model predicts that males of the type occupying the favoured role ejaculate less than males of the type occupying the disfavoured role, in accord with Parker's original conclusion. The fitnesses of the two types of male can be calculated in the multiplicative model: the fitness of the favoured male is usually higher, even if he has to expend more energy in "finding" a female, e.g. through fighting, etc. These conclusions relate to inter-male behaviour (i.e. of different male types), as distinct from intra-male behaviour (i.e. of a given male when in different roles). We analyse situations in which one male type has some probability of acting in its less usual role: calculations with varying amounts of sperm limitation are presented. It is found that the presence of a male of a different type has an effect on intra-male ejaculate behaviour, which also depends critically on the role usually occupied. We conclude that the multiplicative model is the more accurate model and provides more information. Some experimental data on sperm numbers are used to find the effects of sperm limitation. For species which conform to the loaded raffle model, sperm limitation typically has small or negligible effects: in this case, we argue that empiricists should look for equal ejaculates in the two roles when studying random role situations; when roles are occupied non-randomly average sperm expenditure should be greater by male types typically occupying the disfavoured role, but within a male type, expenditure should be greater in the role it typically occupies.     

Exact percentage points of the likelihood-ratio test for a change-point hazard-rate model

We wish to test that the hazard rate of survival or failure-time data is constant against the alternative of a change in hazard after an unspecified time. The likelihood ratio is unbounded but the exact null distribution of a restricted likelihood-ratio test statistic is found. This distribution is not affected by Type II censoring but it does depend very strongly on the interval in which the unknown change-point is assumed to lie. Some exact percentage points are given which are much larger than simulated points that have been reported in the literature.     

Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

Choice of Scores in Trend Tests for Case‐Control Studies of Candidate‐Gene Associations

When applying the Cochran‐Armitage (CA) trend test for an association between a candidate allele and a disease in a case‐control study, a set of scores must be assigned to the genotypes. Sasieni (1997, Biometrics 53 , 1253–1261) suggested scores for the recessive, additive, and dominant models but did not examine their statistical properties. Using the criteria of minimizing the required sample size of the CA trend test to achieve prespecified type I and type II errors, we show that the scores given by Sasieni (1997) are optimal for the recessive and dominant models and locally optimal for the additive one. Moreover, the additive scores are shown to be locally optimal for the multiplicative model. The tests are applied to a real dataset.     

Goodness-of-fit in generalized nonlinear mixed-effects models

In recent years, generalized linear and nonlinear mixed-effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data. To date, much of the work has focused on various methods for estimating and comparing the parameters of mixed-effects models. Very little work has been done in the area of model selection and goodness-of-fit, particularly with respect to the assumed variance-covariance structure. In this paper, we present a goodness-of-fit statistic which can be used in a manner similar to the R2 criterion in linear regression for assessing the adequacy of an assumed mean and variance-covariance structure. In addition, we introduce an approximate pseudo-likelihood ratio test for testing the adequacy of the hypothesized convariance structure. These methods are illustrated and compared to the usual normal theory likelihood methods (Akaike's information criterion and the likelihood ratio test) using three examples. Simulation results indicate the pseudo-likelihood ratio test compares favorably with the standard normal theory likelihood ratio test, but both procedures are sensitive to departures from normality.     

Density compensation,isocline shape and single-level competition models

Criticism of the Lotka-Volterra competition model implies that the theory of competition should be based upon more general concepts. It is suggested that the shape of the competitor isoclines can provide this basis.The relationship between the total density of a competitive community and species number depends crucially upon isocline shape. This has immediate relevance to the interpretation of the excess density compensation seen in some island communities, since if isoclines are sufficiently concave (curved towards the origin) then this phenomenon is expected to be the rule rather than the exception.These observations do not depend upon any specific model, but in order to determine the shape of isoclines in natural communities a link must be made between the biological processes of competition and isocline shape. To this end three types of single-level competition model are distinguished (additive, multiplicative and temporal resource models) depending upon how gains from resources interact in determining individual fitness.The models are based upon resource availability functions (RAFs), which are decreasing functions of the level of competition and determine the availability of each resource to each species. Provided that the argument of these functions is always a weighted sum of the number of competitors then in the case of the additive resource model the shape of the RAFs determines directly the shape of the isoclines. For the multiplicative model, the shape of the logarithm of the RAFs adopts this role.Analysis of a special case of the additive resource model suggests that concave isoclines are likely to predominate, and that the degree of concavity is of an order which minimizes the tendency of total numbers to increase with species number. In some circumstances, involving “scramble”-type competition and habitat selection, the expected concavity is sufficient to cause a decrease. In any event the expected occurrence of concave competition isoclines predicts a much higher incidence of excess density compensation (due to carrying capacity differences) than expected from any model having linear isoclines.The effect of shifting from an additive to a multiplicative resource model is to make the existence of purely concave isoclines less probable and to raise the possibility of purely convex isoclines. On the other hand, shifting from an additive resource model to a temporal resource model apparently has no such simple interpretation and specific predictions must await further analysis.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

梵净山自然保护区珙桐天然种群生命表与生存分析

以种群生命表和生存分析理论为基础,采用空间代替时间法和分段匀滑技术,编制梵净山自然保护区珙桐天然种群特定时间生命表,绘制其死亡率曲线、消失率曲线、存活曲线和生存函数曲线,分析种群数量动态变化。结果表明:珙桐种群结构存在波动性,趋于DeeveyⅢ型,其幼年阶段的个体较丰富;珙桐种群死亡率和消失率曲线变化趋势基本一致,在这一发育过程中有两个死亡高峰,一个出现在幼苗向幼树的过渡期(Ⅰ龄级→Ⅱ龄级),另一个出现在从中龄向老龄过渡的阶段(Ⅷ龄级→Ⅸ龄级);种群生存分析表明,珙桐种群的生存率单调下降,累计死亡率单调上升,Ⅷ龄级后,种群生存率小于8%,累计死亡率大于92%,危险率超过生存率;4个生存函数曲线表明,梵净山珙桐有前期锐减、中期稳定和后期衰退的特点。幼苗和中龄级个体的不足是导致珙桐濒危的重要原因。     

湖北赛武当国家级自然保护区金钱槭(Dipteronia sinensis)种群的数量动态

金钱槭(Dipteronia sinensis)是我国特有的珍稀植物。为探究其种群的数量动态变化规律, 以湖北赛武当自然保护区金钱槭种群为研究对象, 通过样地调查, 编制了金钱槭种群的静态生命表并绘制其存活曲线、死亡率曲线、消失率曲线以及4个生存函数曲线, 同时结合种群年龄结构的数量变化动态和时间序列分析方法来探讨金钱槭种群的数量动态。研究表明: 1)金钱槭种群年龄结构存在一定的波动性, 其幼龄阶段的个体较多, 中龄次之, 成龄的个体数量最少; 种群的生存曲线更接近于DeeveyⅢ型。2)金钱槭种群的生存率呈现单调下降的趋势, 而累积死亡率呈现单调上升的趋势。3)金钱槭种群的死亡密度曲线和危险率曲线的走向相一致, 但是危险率的值都比死亡密度的值略大一些。4)时间序列分析显示金钱槭种群在3、6、9年后的数量预测中整体上呈现出小幅度增长的趋势, 并且种群动态指数>0, 表明金钱槭种群为增长型种群。研究结果为进一步保护金钱槭提供依据。     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.     

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

For randomized clinical trials where the endpoint of interest is a time-to-event subject to censoring, estimating the treatment effect has mostly focused on the hazard ratio from the Cox proportional hazards model. Since the model’s proportional hazards assumption is not always satisfied, a useful alternative, the so-called additive hazards model, may instead be used to estimate a treatment effect on the difference of hazard functions. Still, the hazards difference may be difficult to grasp intuitively, particularly in a clinical setting of, e.g., patient counseling, or resource planning. In this paper, we study the quantiles of a covariate’s conditional survival function in the additive hazards model. Specifically, we estimate the residual time quantiles, i.e., the quantiles of survival times remaining at a given time t, conditional on the survival times greater than t, for a specific covariate in the additive hazards model. We use the estimates to translate the hazards difference into the difference in residual time quantiles, which allows a more direct clinical interpretation. We determine the asymptotic properties, assess the performance via Monte-Carlo simulations, and demonstrate the use of residual time quantiles in two real randomized clinical trials.