Copula modeling for the estimation of measures of marker classification and predictiveness performance with survival outcomes.

The discriminative and predictive power of a continuous-valued marker for survival outcomes can be summarized using the receiver operating characteristic and predictiveness curves, respectively. In this paper, fully parametric and semi-parametric copula-based constructions of the joint model of the marker and the survival time are developed for characterizing, plotting, and analyzing both curves along with other underlying performance measures. The formulations require a copula function, a parametric specification for the margin of the marker, and either a parametric distribution or a non-parametric estimator for the margin of the time to event, to respectively characterize the fully parametric and semi-parametric joint models. Estimation is carried out using maximum likelihood and a two-stage procedure for the parametric and semi-parametric models, respectively. Resampling-based methods are used for computing standard errors and confidence bounds for the various parameters, curves, and associated measures. Graphical inspection of residuals from each conditional distribution is employed as a guide for choosing a copula from a set of candidates. The performance of the estimators of various classification and predictiveness measures is assessed in simulation studies, assuming different copula and censoring scenarios. The methods are illustrated with the analysis of two markers using the familiar primary biliary cirrhosis data set.

Copula modeling of receiver operating characteristic and predictiveness curves.

Receiver operating characteristic (ROC) and predictiveness curves are graphical tools to study the discriminative and predictive power of a continuous-valued marker in a binary outcome. In this paper, a copula-based construction of the joint density of the marker and the outcome is developed for plotting and analyzing both curves. The methodology only requires a copula function, the marginal distribution of the marker, and the prevalence rate for the model to be characterized. The adoption of the Gaussian copula and the customization of the margin for the marker are proposed for such characterization. The computation of both curves is numerically more feasible than methods that attempt to obtain one curve in terms of the other. Estimation is carried out using maximum likelihood and resampling-based methods. Randomized quantile residuals from each conditional distribution are employed for both assessing the adequacy of the model and identifying outliers. The performance of the estimators of both curves and their underlying quantities is evaluated in simulation studies that assume different dependence structures and sample sizes. The methods are illustrated with an analysis of the level of progesterone receptor gene expression for the diagnosis and prediction of estrogen receptor-positive breast cancer.

Long-Term Cause-Specific Mortality After Surgery for Women With Breast Cancer: A 20-Year Follow-Up Study From Surveillance, Epidemiology, and End Results Cancer Registries.

BACKGROUND: Research into long-term cause-specific mortality of women diagnosed with breast cancer is important because it allows for the splitting of the population into patients who eventually die from breast cancer and from other causes. The adoption of this approach helps to identify patients with an elevated risk of eventual death from breast cancer. OBJECTIVE: The primary aim of this study was to examine the associations between both sociodemographic and clinicopathologic characteristics and the underlying risks of death from breast cancer and from other causes for women diagnosed with breast cancer. A second aim was to propose a predictive biomarker of cause-specific mortality in terms of treatment and several important characteristics of a patient. METHODS: A cohort of 16 511 female patients diagnosed with breast cancer in 1990 was obtained from the Surveillance, Epidemiology, and End Results cancer registries and followed for 20 years. A mixture model for the regression analysis of competing risks was used to identify factors and confounders that affected either the eventual cause-specific mortality or conditional cause-specific hazard rates, or both. Missing data were handled with multiple imputation. RESULTS: Curvilinear relationships of age at diagnosis along with race, marital status, breast cancer type, tumor size, estrogen receptor status, extension, lymph node status, type of surgery, and radiotherapy status were significant risk factors for the cause-specific mortality, with extension and lymph node status appearing to be confounded with the effects of both type of surgery and radiotherapy status. The score obtained from combining a set of predictors showed to be an accurate predictive biomarker. CONCLUSIONS: In cause-specific mortality of women diagnosed breast cancer, prognosis appears to depend on both sociodemographic and clinicopathologic factors. The predictive biomarker proposed in this study may help identifying the level of seriousness of the disease earlier than traditional methods, potentially guiding future allocation of resources for better patient care and management strategies.

Addressing missing covariates for the regression analysis of competing risks: Prognostic modelling for triaging patients diagnosed with prostate cancer.

Competing risks arise in medical research when subjects are exposed to various types or causes of death. Data from large cohort studies usually exhibit subsets of regressors that are missing for some study subjects. Furthermore, such studies often give rise to censored data. In this article, a carefully formulated likelihood-based technique for the regression analysis of right-censored competing risks data when two of the covariates are discrete and partially missing is developed. The approach envisaged here comprises two models: one describes the covariate effects on both long-term incidence and conditional latencies for each cause of death, whilst the other deals with the observation process by which the covariates are missing. The former is formulated with a well-established mixture model and the latter is characterised by copula-based bivariate probability functions for both the missing covariates and the missing data mechanism. The resulting formulation lends itself to the empirical assessment of non-ignorability by performing sensitivity analyses using models with and without a non-ignorable component. The methods are illustrated on a 20-year follow-up involving a prostate cancer cohort from the National Cancer Institutes Surveillance, Epidemiology, and End Results program.

Temporal trend, clinicopathologic and sociodemographic characterization of age at diagnosis of breast cancer among US women diagnosed from 1990 to 2009.

ABSTRACT: This paper investigates the distribution of age at diagnosis of female breast cancer and its association with temporal trend, clinicopathologic and sociodemographic variables in the presence of two latent clusters that are directly unobservable. Such clusters help to identify two subpopulations of either young or old patients whose etiologies are thought to be different. A large sample drawn from registry data from the National Cancer Institute's Surveillance, Epidemiology, and End Results program from 1990 to 2009 was analyzed using a two-component Gaussian mixture model. Evidence of a steady delay of age at diagnosis and an increasing proportion of young patients being diagnosed during the 20-year period was found. Histopathologic effects indicate that duct and lobular carcinomas differ significantly in regard to subpopulation membership, which confirms that they represent different etiologies. While the presence of estrogen receptor status in the model overlaps the effects of other important variables it is highly correlated with, it is found that the grade, extension and size of the tumor along with lymph node involvement status, race and marital status are important predictors of age at diagnosis. The results highlight the significant impacts that such features can have on breast cancer control efforts, and point to the importance of ensuring that medical decision making should use them along with an indicator of the age subpopulation a patient may belong to.

Extreme value modeling for the analysis and prediction of time series of extreme tropospheric ozone levels: a case study.

The occurrence of high concentrations of tropospheric ozone is considered as one of the most important issues of air management programs. The prediction of dangerous ozone levels for the public health and the environment, along with the assessment of air quality control programs aimed at reducing their severity, is of considerable interest to the scientific community and to policy makers. The chemical mechanisms of tropospheric ozone formation are complex, and highly variable meteorological conditions contribute additionally to difficulties in accurate study and prediction of high levels of ozone. Statistical methods offer an effective approach to understand the problem and eventually improve the ability to predict maximum levels of ozone. In this paper an extreme value model is developed to study data sets that consist of periodically collected maxima of tropospheric ozone concentrations and meteorological variables. The methods are applied to daily tropospheric ozone maxima in Guadalajara City, Mexico, for the period January 1997 to December 2006. The model adjusts the daily rate of change in ozone for concurrent impacts of seasonality and present and past meteorological conditions, which include surface temperature, wind speed, wind direction, relative humidity, and ozone. The results indicate that trend, annual effects, and key meteorological variables along with some interactions explain the variation in daily ozone maxima. Prediction performance assessments yield reasonably good results.

A flexible class of parametric transition regression models based on copulas: application to poliomyelitis incidence.

This paper presents an extension of a general parametric class of transitional models of order p. In these models, the conditional distribution of the current observation, given the present and past history, is a mixture of conditional distributions, each of them corresponding to the current observation, given each one of the p-lagged observations. Such conditional distributions are constructed using bivariate copula models which allow for a rich range of dependence suitable to model non-Gaussian time series. Fixed and time varying covariates can be included in the models. These models have the advantage of straightforward construction and estimation for the analysis of time series and more general longitudinal data. A poliomyelitis incidence data set is used to illustrate the proposed methods, contrary to other researches' conclusions whose methods are mainly based on linear models, we find significant evidence of a decreasing trend in polio infection after accounting for seasonality.

Fitting competing risks with an assumed copula.

We propose a fully parametric model for the analysis of competing risks data where the types of failure may not be independent. We show how the dependence between the cause-specific survival times can be modelled with a copula function. Features include: identifiability of the problem; accessible understanding of the dependence structures; and flexibility in choosing marginal survival functions. The model is constructed in such a way that it allows us to adjust for concomitant variables and for a dependence parameter to assess the effects of these on each marginal survival model and on the relationship between the causes of death. The methods are applied to a prostate cancer data set. We find that, with the copula model, more accurate inferences are obtained than with the use of a simpler model such as the independent competing risks approach.