Monotonic regression is a nonparametric method for estimation ofmodels in which the expected value of a response variable y increases ordecreases in all coordinates of a vector of explanatory variables x = (x₁, …, x_p).Here, we examine statistical and computational aspects of our recentlyproposed generalization of the pool-adjacent-violators (PAV) algorithm fromone to several explanatory variables. In particular, we show how the goodnessof-fit and accuracy of obtained solutions can be enhanced by presortingobserved data with respect to their level in a Hasse diagram of the partial orderof the observed x-vectors, and we also demonstrate how these calculations canbe carried out to save computer memory and computational time. Monte Carlosimulations illustrate how rapidly the mean square difference between fittedand expected response values tends to zero, and how quickly the mean squareresidual approaches the true variance of the random error, as the number of observations increases up to 10⁴.

The International Council for the Exploration of the Sea (ICES) has longcompiled extensive data on contaminants in biota. We investigated how trendassessment of mercury in muscle tissue from fish (flounder and Atlantic cod)might be facilitated by using nonparametric regression to normalise observedlevels of this contaminant with respect to body length and weight. Specifically,we examined response surfaces and annual normalised means obtained byemploying purely additive models (AM), thin plate splines (TPS), andmonotonic regression (MR) to model mercury levels as functions of samplingyear and one or two covariates. Our analysis showed that TPS and MR modelscan be more satisfactory than purely additive models, because the formertechniques enable estimation of time-dependent relationships between themercury concentration and the covariates. However, the major obstacle fortrend assessment of the collected mercury data was a substantial interannualvariation that was related to factors other than body length and weight.Nevertheless, several time series of flounder data that started in the 1970s and1980s showed obvious downward trends, and these trends were particularly2strong in the Elbe estuary. When the analysis was limited to data collected after1990, an overall Mann-Kendall test for all sampling sites revealed astatistically significant downward trend for flounder, whereas it was notsignificant for cod.

The reunification of Germany led to dramatically reduced emissions of nitrogen (N) and phosphorus (P) to the environment. The aim of the present study was to examine how these exceptional decreases influenced the amounts of nutrients carried by the Elbe River to the North Sea. In particular, we attempted to extract anthropogenic signals from time series of riverine loads of nitrogen and phosphorus by developing a normalization technique that enabled removal of natural fluctuations caused by several weather-dependent variables. This analysis revealed several notable downward trends. The normalized loads of total-N and NO₃-N exhibited an almost linear trend, even though the nitrogen surplus in agriculture dropped dramatically in 1990 and then slowly increased. Furthermore, the decrease in total-P loads was found to be considerably smaller close to the mouth of the river than further upstream. Studying the predictive ability of different normalization models showed the following: (i) nutrient loads were influenced primarily by water discharge; (ii) models taking into account water temperature, load of suspended particulate matter, and salinity were superior for some combinations of sampling sites and nutrient species; semiparametric normalization models were almost invariably better than ordinary regression models.