Assessing covariate-dependent contaminant time-series in the marine environment

R. J. Fryer 0 M. D. Nicholson 0 0 R. J. Fryer: FRS Marine Laboratory , PO Box 101, Victoria Road, Aberdeen, AB11 9DB, Scotland , UK; M. D. Nicholson: CEFAS, Lowestoft Laboratory , Pakefield Road This paper describes a method for assessing contaminant time-series when contaminant levels vary with some covariate and the contaminant-covariate relationship changes over time. At each time point, the contaminant data are split into two groups according to whether the corresponding values of the covariate are smaller or larger than some specified value, leading to two contaminant time-series, for small and large values of the covariate respectively. Smoothers are then used to model the small and large time-series, and to construct analyses of variance that test for any change over time, any covariate effect, and whether the pattern of temporal change in the small time-series is the same as in the large time-series. The smoothers can also be displayed graphically, allowing easy interpretation of the results. The method is applied to four time-series of mercury concentrations in fish muscle from monitoring sites around the UK. - Introduction Contaminant levels in marine time-series have often been found to vary with some covariate. For example, heavy metal concentrations in fish muscle are often related to total body length (Phillips, 1980; MacCrimmon et al., 1983; ICES, 1989; Dixon and Jones, 1994). Assessing temporal trends in contaminant levels can then be problematic, because it is necessary to disentangle changes over time from changes due to the covariate. A common solution is to adjust for the covariate using analysis of covariance (Fisher, 1932). For example, trend assessments of heavy metals in fish muscle have been based on length-adjusted mean (log-) concentrations (ICES, 1989; Jensen and Cheng, 1987; Evans et al., 1993; Jorgensen and Pedersen, 1994). Similar analyses have been made adjusting for tissue, shell, and body weight (ICES, 1990; Allard and Stokes, 1989; Rees and Nicholson, 1989; Nicholson et al., 1991). Although analysis of covariance can be very effective, the results can be difficult to interpret if the slope of the contaminant-covariate relationship changes over time. In this case, any temporal trend will depend on the value of the covariate to which the contaminant levels have been adjusted. Various biological (Braune, 1987) and genetic (Misra et al., 1990) explanations for variation in slope have been suggested. Indeed, statistical arguments suggest that such variation should always be present (Nester, 1996), although whether it can be detected depends on how much the slopes vary, the volume of available data, and so on. Methods for dealing with variation in slope include adjusting contaminant levels to an average covariate-value, using either an estimate of some average slope (Misra et al., 1990), or using the individual slopes themselves (McMurtry et al., 1989; Jorgensen and Pedersen, 1994). However, trends constructed in this way have been criticised as misleading or meaningless unless the average covariate-value has some a priori significance, since choosing some other value would reveal a different trend (Magnusson, 1989). Other methods adopt a stochastic model for the variation in slope. For example, linear mixed models (Hocking, 1996) can be used to model the slopes as random realisations from some statistical distribution. An alternative is to place some constraints on the slopes, for example by using a state-space framework (Jones, 1993) that allows the slope to evolve stochastically over time. This approach was used by Warren (1995) and naturally leads to covariate-dependent temporal trends, which can be displayed and assessed graphically. Unfortunately, our experience from analysing the many time-series generated in regional monitoring programmes is that the methods described above are not particularly suited to routine assessments, tending to get bogged down by questions of data quality and model assumptions. For example, what should be done with concentrations reported as below the limit of detection of the analytical method, or with a few unexplained, very high concentrations? And what happens when a linear contaminant-covariate relationship is adequate at most time points, but not at others? Addressing these questions is always time consuming, and sometimes fails to lead to a satisfactory resolution. But more importantly, a disproportionate amount of time is spent investigating what is going on within each time point, when the important management questions are generally concerned what is going on between time points. These considerations are particularly relevant to assessment meetings where many time-series are assessed in a short space of time, and statistical resources are often limited. Here, we present a simpler approach that circumvents many problems of data quality and model assumptions, accounts for contaminant-covariate relationships that vary over time, and correctly describes and tests for temporal trends. The method is based on smoothers (e.g. Cleveland, 1979), and extends the work described in Fryer and Nicholson (1999), where smoothers are used to assess contaminant time-series in the absence of a covariate. At each time point, the contaminant data are split into two groups according to whether the corresponding values of the covariate are smaller or larger than some specified value. This leads to two contaminant time-series, for small and large values of the covariate respectively. Smoothers are then used to model the small and large time-series, and to construct analyses of variance that test for any change over time, any covariate effect, and whether the pattern of temporal change in the small time-series is the same as in the large time-series. The smoothers can also be displayed graphically, allowing easy interpretation of the results. In essence, detailed modelling of the contaminant-covariate relationships is replaced by the simpler comparison of contaminant levels in two groups. The main technical difficulty remaining is to model appropriately any correlations between the small and large time-series. The next section describes the theory behind the construction and statistical comparison of the small and large time-series. We present the theory in terms of an annual monitoring programme of contaminant concentrations in a marine organism. However, the methodology is more generally applicable and might be used for assessing other quantities (e.g. contaminant loads, biological effects measurements, fish catches-at-age) measured in other matrices (e.g. sediments, fish stocks) that satisfy the statistical assumptions below. The following section uses the methodology to assess four time-series of mercury concentrations in fish muscle from monitoring sites around the UK. Finally, we discuss various issues related to model assumptions and performance. This section describes the theory for using smoothers to asses (...truncated)