Cox模型和寿险退保研究中的比例危险性

利用生存分析研究寿险退保问题是一个很好的工具，因为可以将寿险保单的持续期（persistency duration）视为生存期长，而将保单的退保或失效看作一个“保单生命”的结束，这其中的保单退保或失效就成为生存研究的目标事件。而导致保单失效的因素会有很多，只有通过利用Cox比例危险模型拟合寿险退保数据以分析影响客户退保的原因，并在对Cox模型的比例危险假设进行检验时，发现部分影响因素并不遵守此前提条件，从而推理得到这些影响因素在不同的时间段对客户退保的影响方式不同。也就是说，其影响有短期效应和长期效应之分。     

On quantile residuals in beta regression

Beta regression is often used to model the relationship between a dependent variable that assumes values on the open interval (0, 1) and a set of predictor variables. An important challenge in beta regression is to find residuals whose distribution is well approximated by the standard normal distribution. Two previous works compared residuals in beta regression, but the authors did not include the quantile residual. Using Monte Carlo simulation techniques, this article studies the behavior of certain residuals in beta regression in several scenarios. Overall, the results suggest that the distribution of the quantile residual is better approximated by the standard normal distribution than that of the other residuals in most scenarios. Three applications illustrate the effectiveness of the quantile residual.     

Identifying Nonproportional Covariates in the Cox Model

This paper addresses problem of testing whether an individual covariate in the Cox model has a proportional (i.e., time-constant) effect on the hazard. Two existing methods are considered: one is based on the component of the score process, and the other is a Neyman type smooth test. Simulations show that, when the model contains both proportional and nonproportional covariates, these methods are not reliable tools for discrimination. A simple yet effective solution is proposed based on smooth modeling of the effects of the covariates not in focus.     

The extreme residuals in logistic regression models

Goodness-of-fit tests for logistic regression models using extreme residuals are considered. Approximations to the moments of the Pearson residuals are given for model fits made by maximum likelihood, minimum chi-square and weighted least squares and used to define modified residuals. Approximations to the critical values of the extreme statistics based on the ordinary and modified Pearson residuals are developed and assessed for the case of a single explanatory variable.     

Automatic bandwidth selection in robust nonparametric regression

Nonparametric regression techniques have been studied extensively in the literature in recent years due to their flexibility.In addition robust versions of these techniques have become popular and have been incorporated into some of the standard statistical analysis packages.With new techniques available comes the responsibility of using them properly and in appropriate situations. Often, as in the case presented here, model-fitting diagnostics, such as cross-validation statistics,are not available as tools to determine if the smoothing parameter value being used is preferable to some other arbitrarily chosen value.     

Heterogeneity and varying effect in hazards regression

In the analysis of survival data, when nonproportional hazards are encountered, the Cox model is often extended to allow for a time-dependent effect by accommodating a varying coefficient. This extension, however, cannot resolve the nonproportionality caused by heterogeneity. In contrast, the heteroscedastic hazards regression (HHR) model is capable of modeling heterogeneity and thus can be applied when dealing with nonproportional hazards. In this paper, we study the application of the HHR model possibly equipped with varying coefficients. An LRR (logarithm of relative risk) plot is suggested when investigating the need to impose varying coefficients. Constancy and degeneration in the plot are used as diagnostic criteria. For the HHR model, a ‘piecewise effect’ (PE) analysis and an ‘average effect’ (AE) analysis are introduced. For the PE setting, we propose a score-type test for covariate-specific varying coefficients. The Stanford Heart Transplant data are analyzed for illustration. In the case of degeneration being destroyed by a polynomial covariate, piecewise constancy and/or monotonicity of the LRRs is considered as an alternative criterion based on the PE analysis. Finally, under the framework of the varying-coefficient HHR model, the meanings of the PE and AE analyses, along with their dynamic interpretation, are discussed.     

Examination and analysis of residuals: A test for detecting a monotonic relation between mean and variance in regression through the origin

This paper presents a simple and exact test for detecting a monotonic relation between the mean and variance in linear regression through the origin. This test resulted from utilizing uncorrelated Theil-residuals and the Goldfeld-Quandt peak test. A numerical example is provided to elucidate the method. A simulation experiment was performed to compare the empirical power of this test with those of the existing tests.     

Auxiliary covariate in additive hazards regression for survival data

We consider the additive hazards regression analysis by utilising auxiliary covariate information to improve the efficiency of the statistical inference when the primary covariate is ascertained only for a randomly selected subsample. We construct a martingale-based estimating equation for the regression parameter and establish the asymptotic consistency and normality of the resultant estimator. Simulation study shows that our proposed method can improve the efficiency compared with the estimator which discards the auxiliary covariate information. A real example is also analysed as an illustration.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

Summary.  We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates t  ∈  R ^d ,  d =1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for d =1 and d =2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension d this becomes more drastic. If d 4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when d 4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.     

A class of residuals for outlier identification in zero adjusted regression models

Zero adjusted regression models are used to fit variables that are discrete at zero and continuous at some interval of the positive real numbers. Diagnostic analysis in these models is usually performed using the randomized quantile residual, which is useful for checking the overall adequacy of a zero adjusted regression model. However, it may fail to identify some outliers. In this work, we introduce a class of residuals for outlier identification in zero adjusted regression models. Monte Carlo simulation studies and two applications suggest that one of the residuals of the class introduced here has good properties and detects outliers that are not identified by the randomized quantile residual.     

Cox regression analysis in presence of collinearity: an application to assessment of health risks associated with occupational radiation exposure

This paper considers the analysis of time to event data in the presence of collinearity between covariates. In linear and logistic regression models, the ridge regression estimator has been applied as an alternative to the maximum likelihood estimator in the presence of collinearity. The advantage of the ridge regression estimator over the usual maximum likelihood estimator is that the former often has a smaller total mean square error and is thus more precise. In this paper, we generalized this approach for addressing collinearity to the Cox proportional hazards model. Simulation studies were conducted to evaluate the performance of the ridge regression estimator. Our approach was motivated by an occupational radiation study conducted at Oak Ridge National Laboratory to evaluate health risks associated with occupational radiation exposure in which the exposure tends to be correlated with possible confounders such as years of exposure and attained age. We applied the proposed methods to this study to evaluate the association of radiation exposure with all-cause mortality.     

Estimation in the Cox proportional hazards model with doubly censored and truncated data

In longitudinal studies, the proportional hazard model is often used to analyse covariate effects on the duration time, defined as the elapsed time between the first and second event. In this article, we consider the situation when the first event suffers partly interval-censoring and the second event suffers left-truncation and right-censoring. We proposed a two-step estimation procedure for estimating the regression coefficients of the proportional model. A simulation study is conducted to investigate the performance of the proposed estimator.     

Leverage,influence and residuals in regression models when observations are correlated

This paper considers the concepts of leverage and influence in the linear regression model with correlated errors when the error covariance structure is completely specified. Generalizations of the usual measures are given. Extensions of residuals also naturally arise. The theory is illustrated using two examples     

How Cox models react to a study-specific confounder in a patient-level pooled dataset: random effects better cope with an imbalanced covariate across trials unless baseline hazards differ

Combining patient-level data from clinical trials can connect rare phenomena with clinical endpoints, but statistical techniques applied to a single trial may become problematical when trials are pooled. Estimating the hazard of a binary variable unevenly distributed across trials showcases a common pooled database issue. We studied how an unevenly distributed binary variable can compromise the integrity of fixed and random effects Cox proportional hazards (cph) models. We compared fixed effect and random effects cph models on a set of simulated datasets inspired by a 17-trial pooled database of patients presenting with ST segment elevation myocardial infarction (STEMI) and non-STEMI undergoing percutaneous coronary intervention. An unevenly distributed covariate can bias hazard ratio estimates, inflate standard errors, raise type I error, and reduce power. While uneveness causes problems for all cph models, random effects suffer least. Compared to fixed effect models, random effects suffer lower bias and trade inflated type I errors for improved power. Contrasting hazard rates between trials prevent accurate estimates from both fixed and random effects models.     

Low order approximations in deconvolution and regression with errors in variables

Summary.  We suggest two new methods, which are applicable to both deconvolution and regression with errors in explanatory variables, for nonparametric inference. The two approaches involve kernel or orthogonal series methods. They are based on defining a low order approximation to the problem at hand, and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently. Of course, both techniques could be employed to construct consistent estimators, but in many contexts of importance (e.g. those where the errors are Gaussian) consistency is, from a practical viewpoint, an unattainable goal. We rephrase the problem in a form where an explicit, interpretable, low order approximation is available. The information that we require about the error distribution (the error-in-variables distribution, in the case of regression) is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors, e.g. through repeated measurements. In particular, we do not need to estimate a function, such as a characteristic function, which expresses detailed properties of the error distribution. This feature of our methods, coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages, means that the methods are particularly economical in computing time.     

Detecting discontinuities in nonparametric regression curves and surfaces

The existence of a discontinuity in a regression function can be inferred by comparing regression estimates based on the data lying on different sides of a point of interest. This idea has been used in earlier research by Hall and Titterington (1992), Müller (1992) and later authors. The use of nonparametric regression allows this to be done without assuming linear or other parametric forms for the continuous part of the underlying regression function. The focus of the present paper is on assessing the evidence for the presence of a discontinuity within a regression function through examination of the standardised differences of ‘left’ and ‘right’ estimators at a variety of covariate values. The calculations for the test are carried out through distributional results on quadratic forms. A graphical method in the form of a reference band to highlight the sources of the evidence for discontinuities is proposed. The methods are also developed for the two covariate case where there are additional issues associated with the presence of a jump location curve. Methods for estimating this curve are also developed. All the techniques, for the one and two covariate situations, are illustrated through applications.     

On the correct regression function (in L2) and its applications when the dimension of the covariate vector is random

We derive the optimal regression function (i.e., the best approximation in the L₂ sense) when the vector of covariates has a random dimension. Furthermore, we consider applications of these results to problems in statistical regression and classification with missing covariates. It will be seen, perhaps surprisingly, that the correct regression function for the case with missing covariates can sometimes perform better than the usual regression function corresponding to the case with no missing covariates. This is because even if some of the covariates are missing, an indicator random variable δ$\delta$, which is always observable, and is equal to 1 if there are no missing values (and 0 otherwise), may have far more information and predictive power about the response variable Y than the missing covariates do. We also propose kernel-based procedures for estimating the correct regression function nonparametrically. As an alternative estimation procedure, we also consider the least-squares method.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.