Time‐Dependent Predictive Accuracy in the Presence of Competing Risks

Summary Competing risks arise naturally in time‐to‐event studies. In this article, we propose time‐dependent accuracy measures for a marker when we have censored survival times and competing risks. Time‐dependent versions of sensitivity or true positive (TP) fraction naturally correspond to consideration of either cumulative (or prevalent) cases that accrue over a fixed time period, or alternatively to incident cases that are observed among event‐free subjects at any select time. Time‐dependent (dynamic) specificity (1–false positive (FP)) can be based on the marker distribution among event‐free subjects. We extend these definitions to incorporate cause of failure for competing risks outcomes. The proposed estimation for cause‐specific cumulative TP/dynamic FP is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time. On the other hand, incident TP/dynamic FP can be estimated using a possibly nonproportional hazards Cox model for the cause‐specific hazards and riskset reweighting of the marker distribution. The proposed methods extend the time‐dependent predictive accuracy measures of Heagerty, Lumley, and Pepe (2000, Biometrics 56, 337–344) and Heagerty and Zheng (2005, Biometrics 61, 92–105).     

Impact of climatic variability on parameters used in typology and ecological quality assessment of surface waters—implications on the Water Framework Directive

In most cases the negative impacts of climate change to aquatic ecosystems cannot be mitigated by measures in the river basin management. Ignoring climate change by the Water Framework Directive may have strong implications for the typology and quality assessment systems used for water bodies. As a result of climate change, water bodies, especially those located near the type boundaries may change their type. Compared to typology characteristics, water quality parameters are even more labile and may be easily affected by climate change. The paper exemplifies that the anticipated deterioration of water quality within the time frame relevant for WFD implementation may be large enough to endanger the fulfillment of the set water quality objectives. The review of the river basin characterization every six years, as required by the WFD, might also include re-evaluation of reference conditions according to the changes observed at pristine reference sites. As a consequence, the restoration targets (i.e., the good ecological status) would also need to be evaluated periodically.     

Assessing eutrophication in the Portuguese continental Exclusive Economic Zone within the European Marine Strategy Framework Directive

This study reports the state and causes of eutrophication in the Portuguese continental Exclusive Economic Zone (EEZ), during a 14-year period (1995–2008), following the European Marine Strategy Framework Directive (MSFD) and using the trophic index TRIX for an integrated evaluation of indicators of eutrophication, and identifies areas where monitoring is needed to improve the eutrophication assessment. A non-continuous dataset for the 8 indicators specified by the MSFD for eutrophication assessment was used, including published and grey data. Eutrophication indicators were validated and thresholds reviewed, considering regional differences. The diatom:flagellate ratio was found a poor indicator of eutrophication as shifts in the diatom:flagellate ratio naturally occur associated with alternating water column turbulence and upwelling, and stratification, and therefore, could not be associated with anthropogenic nutrient enrichment effects. Assessment areas were, as a whole, classified as non-problem areas concerning eutrophication. Although nutrient enrichment was observed in coastal waters, related to river plume influence, nutrient enrichment direct and indirect effects were generally not detectable, possibly due to water column dispersion and mixing processes. Only occasionally, mild eutrophication was found in specific areas under the influence of major river (Douro, Vouga and Guadiana) plumes, associated with high nutrient and phytoplankton biomass levels and seagrass decline, which indicates the need for directed monitoring on eutrophication in those areas.     

Dilemma of Non‐Steady State in Lakes – Development and Predictability of In‐Lake P Concentration in Dimictic Lake Scharmützelsee (Germany) after Abrupt Load Reduction

Based on measured and calculated long‐term data on external phosphorus (P) load (1920–2009), hypolimnetic P accumulation and trophic parameters for the dimictic Lake Scharmützelsee, we aimed to identify factors which cause variable P net sedimentation and the importance of internal P loading for different time periods especially focusing on non‐steady state after abrupt external load reduction in 1988. P retention (R) decreased from 0.85 during the high external P loading (0.37 g m^–2 a^–1) phase (1950–1988) to 0.71 during the following transient phase, and increased to 0.81 for the present recovery phase (0.17 g m^–2 a^–1) beginning in 2003. Mean net sedimentation coefficients for the same periods were 0.47, 0.22 and 0.30. Our results show that a) empirical models overestimate R during the high loading phase and underestimate R during the transient phase after load reduction, and b) the application of simple one‐box models which assume that a portion of in‐lake P stock is retained requires the consideration of the variability of the net sedimentation coefficient. We identified multiple reasons for variable gross sedimentation (e.g., particle trapping in the elongated lake; efficient accumulation of sewage P) as well as release of P (delayed release of mobile P from sediment; changes in plant colonization and food webs), so that their relation (net sedimentation) varies. Despite a new equilibrium reached in 2003, it is still unclear when the lake will reach mesotrophic reference conditions and a good ecological status. Historical data and elevated Cl^– concentration (22 mg L^–1) indicate that P import from sewage contaminated groundwater still continues, so that the mean in‐lake P concentration is still too high (53 µg L^–1), and biological structures have not fully recovered yet. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluating prognostic accuracy of biomarkers under competing risk

To develop more targeted intervention strategies, an important research goal is to identify markers predictive of clinical events. A crucial step toward this goal is to characterize the clinical performance of a marker for predicting different types of events. In this article, we present statistical methods for evaluating the performance of a prognostic marker in predicting multiple competing events. To capture the potential time-varying predictive performance of the marker and incorporate competing risks, we define time- and cause-specific accuracy summaries by stratifying cases based on causes of failure. Such definition would allow one to evaluate the predictive accuracy of a marker for each type of event and compare its predictiveness across event types. Extending the nonparametric crude cause-specific receiver operating characteristics curve estimators by Saha and Heagerty (2010), we develop inference procedures for a range of cause-specific accuracy summaries. To estimate the accuracy measures and assess how covariates may affect the accuracy of a marker under the competing risk setting, we consider two forms of semiparametric models through the cause-specific hazard framework. These approaches enable a flexible modeling of the relationships between the marker and failure times for each cause, while efficiently accommodating additional covariates. We investigate the asymptotic property of the proposed accuracy estimators and demonstrate the finite sample performance of these estimators through simulation studies. The proposed procedures are illustrated with data from a prostate cancer prognostic study.     

Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Concomitant Information in Competing Risk Analysis

A competing risk model, accommodating both Type I censoring and random withdrawals, is expanded to incorporate concomitant information by allowing the parameters of the underlying distributions to be a linear function of two covariates. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution. A method is developed for testing the equality of the coefficients for a given covariate for each of the competing risks using MLE'.     

Depth diversity gradients of macrophytes: Shape,drivers, and recent shifts

Investigating diversity gradients helps to understand biodiversity drivers and threats. However, one diversity gradient is rarely assessed, namely how plant species distribute along the depth gradient of lakes. Here, we provide the first comprehensive characterization of depth diversity gradient (DDG) of alpha, beta, and gamma species richness of submerged macrophytes across multiple lakes. We characterize the DDG for additive richness components (alpha, beta, gamma), assess environmental drivers, and address temporal change over recent years. We take advantage of yet the largest dataset of macrophyte occurrence along lake depth (274 depth transects across 28 deep lakes) as well as of physiochemical measurements (12 deep lakes from 2006 to 2017 across Bavaria), provided publicly online by the Bavarian State Office for the Environment. We found a high variability in DDG shapes across the study lakes. The DDGs for alpha and gamma richness are predominantly hump‐shaped, while beta richness shows a decreasing DDG. Generalized additive mixed‐effect models indicate that the depth of the maximum richness (D _max) is influenced by light quality, light quantity, and layering depth, whereas the respective maximum alpha richness within the depth gradient (R _max) is significantly influenced by lake area only. Most observed DDGs seem generally stable over recent years. However, for single lakes we found significant linear trends for R _max and D _max going into different directions. The observed hump‐shaped DDGs agree with three competing hypotheses: the mid‐domain effect, the mean–disturbance hypothesis, and the mean–productivity hypothesis. The DDG amplitude seems driven by lake area (thus following known species–area relationships), whereas skewness depends on physiochemical factors, mainly water transparency and layering depth. Our results provide insights for conservation strategies and for mechanistic frameworks to disentangle competing explanatory hypotheses for the DDG.