Each stage is described in more detail below for the case when all the concentration measurements are above the detection limit. Other help files describe how the methodology is adapted when there are 'less-than' measurements.
The concentrations are first normalised to account for changes in the bulk physical composition of the sediment such as particle size distribution or organic carbon content. Normalisation requires pivot values, estimates of the concentrations of contaminants and normalisers in pure sand. A normalised concentration is given by:
`c_(ss)=c_x + \frac{(c_m-c_x)(n_(ss)-n_x)}{n_m-n_x}`
where:
The analytical variance of the normalised concentrations is given by:
`\text{Var}(c_(ss))= A^2 (\text{Var}(c_m)+B^2 \text{Var}(n_m))`
where:
Normalised concentrations that are environmentally inadmissible (i.e. negative), or with an analytical coefficient of variation of more than 100%, are excluded from the assessment. The procedures for normalisation are described in more detail in Technical Annex 5 of OSPAR (2015) and the derivation of pivot values is described in ICES (2002).
OSPAR, 2015. Agreement 2002-16, JAMP Guidelines for Monitoring Contaminants in Sediments. Updated 2015
ICES, 2002. Annexes 8 and 9 to the 2002 report of the ICES Working Group on Marine Sediments in Relation to Pollution.
The annual contaminant index is a weighted average of the log concentrations, where the weights are a suitable combination of the analytical variation of each measurement and, where possible, an estimate of the within-year environmental (field) variation.
Let cti, i = 1 ... nt be the (normalised) concentrations measured in year t, t = 1 ... T, and let σti be the analytical standard deviation of these concentrations. The analytical standard deviation of log(cti) is then approximately σti / cti. Let τ be the within-year environmental standard deviation, assumed constant throughout the time series (and known for now). The joint analytical and within-year environmental variance of log(cti) is then:
`v_(ti)=\tau^2 + \frac{\sigma_(ti)^2}{c_(ti)^2}`
The annual contaminant index in year t is the weighted average of the log(cti)
`y_t=\frac{1}{u_t.}\sum_i u_(ti) \log(c_(ti))`
where uti = 1 / vti and `u_t.=\sum_i u_(ti)`. The joint analytical and within-year environmental variance of yt is
`\text{Var}(y_t)=\frac{1}{u_t.}`
When there are multiple samples in at least one year of the time series, the within-year environmental standard deviation τ is estimated by restricted maximum likelihood with the analytical standard deviations assumed known and equal to σti. When there is only one sample each year, τ is taken to be zero, and the variance of the annual contaminant index is a measure of the analytical variability only. Thus, the variance of the index incorporates all the information available about within-year variability.
The scaled weights are the precisions of the indices (the reciprocal of the variances) scaled so that the most precise index has weight 1:
`w_t=\frac{u_t.}{max{u_t.,t=1...T}}`
The contaminant time series are assessed for temporal trends by fitting a weighted regression model to the annual contaminant indices. Doing so is straightforward if the statistical weights are known beforehand. The statistical weights should be inversely related to the total environmental and analytical variance each year. However, until recently, the analytical variances of the contaminant and normaliser measurements were not routinely available, so optimal weights could not be calculated. Sorting this out is an area of current activity. In the meantime, the pragmatic approach described below is used to convert the scaled weights to statistical weights that account for the relative magnitudes of the environmental and analytical variances.
Assume that the contaminant time series can be described by the model:
`y_t=f(t)+\varepsilon_t`
where yt is the annual contaminant index in year t, f(t) is a smooth function of time (possibly linear) describing the underlying trend in contaminant levels, and εt is the ‘noise’ in year t from both environmental and analytical variation. Further, assume that the noise can be decomposed into two terms:
`\varepsilon_t=\tau_t + \delta_t`
where τt is the noise due to analytical variation (and possibly within-year environmental variation) and δt is the noise due to all remaining sources of environmental variation. Finally, assume that the noise terms are mutually independent and normally distributed:
`\tau_t ~ \text{N}(0,\frac{\sigma_t^2}{w_t})`
`\delta ~ \text{N}(0,\sigma_\delta^2)`
where wt is the scaled weight for year t. Given this model, the appropriate statistical weights are:
`W_t=(\sigma_\delta^2+\sigma_\tau^2)(\sigma_\sigma^2+\frac{\sigma_\tau^2}{w_t})^{-1}`
The statistical weights provide an appropriate balance between the two variance components and satisfy 0 ≤ wt ≤ Wt ≤ 1.
The variance components `\sigma_\delta^2` and `\sigma_\tau^2` are unknown, so must be estimated to give the statistical weights. The approach used relies on the fact that the residuals rt from an unweighted fit to the data should become more variable as the scaled weights decrease (for example as analytical quality degrades). To a first approximation, the squared residuals `r_t^2` have mean `\sigma_\delta^2+\sigma_\tau^2\text{/}w_t`. Thus, if the squared residuals `r_t^2` are regressed against 1 / wt the intercept and slope should provide estimates of `\sigma_\delta^2` and `\sigma_\tau^2` respectively. Formally, the regression is done using a generalised linear model with gamma errors and identity link. The estimates of `\sigma_\delta^2` and `\sigma_\tau^2` are then plugged into the formula for the statistical weights. Sometimes, the approach will give negative variance estimates, in which case the relevant estimates are taken to be zero.
The annual contaminant indices are modelled as:
`y_t=f(t)+\epsilon_t`
where yt is the annual contaminant index in year t, f(t) is a smooth function of time (possibly linear) describing the underlying trend in contaminant levels, and εt is an error term assumed to be independent and normally distributed with variance σ2 / Wt, where Wt are appropriate statistical weights.
The form of f(t) depends on the number of years of data:
A loess smoother is used to estimate the smooth function of time. The amount of smoothing is controlled by the neighbourhood of contaminant indices that is used to estimate each f(t) as t runs from 1 to T. A neighbourhood of 9, for example, uses the 9 indices that are closest to t to estimate f(t). A sequence of neighbourhoods are considered (7, 9, 11 up to T, if T is odd, or T + 1, if T is even) with the final choice based on Akaike's Information Criterion corrected for small sample size (AICc). However, if there is no evidence of nonlinearity in the data (i.e. if the AICc of the linear model is lower than that of the best smoother) then the linear model f(t) = µ + βt is used instead.
Weighted linear regression is described by e.g. Draper & Smith (1998). Loess smoothers were developed by Cleveland (1979). The application of loess smoothers to contaminant time series is described by Fryer & Nicholson (1999).
Cleveland WS, 1979. Robust locally-weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74: 829-836.
Draper NR & Smith H, 1998. Applied regression analysis, 3rd edition. Wiley
Fryer RJ & Nicholson MD, 1999. Using smoothers for comprehensive assessments of contaminant time series in marine biota. ICES Journal of Marine Science 56: 779-790.
Environmental status and temporal trends are assessed using the model fitted to the annual contaminant indices.
Environmental status is assessed by comparing the upper one-sided 95% confidence limit on the fitted value in the most recent monitoring year to the available assessment criteria. For example, if the upper confidence limit is below the Background Assessment Concentration (BAC), then the mean contaminant index in the most recent monitoring year is significantly below the BAC and concentrations are said to be 'at background'.
No formal assessment of status is made when there are only 1 or 2 years of data. However, an ad-hoc assessment is made by comparing the contaminant index (1 year) or the larger of the two contaminant indices (2 years) to the assessment criteria.
Temporal trends are assessed for all time series with at least five years of data. When a linear model has been fitted (i.e. when there are 5-6 years of data, or if there are 7+ years of data and no evidence of nonlinearity), there is evidence of a temporal trend if the slope β of the linear regression of yt on t is significant at the 5% level. When a smooth model has been fitted, the fitted smoother is used to test for evidence of any systematic change in contaminant levels over time; this test is also decomposed into both a nonlinear and linear component. The results for each time series can be found in the statistical analysis output on the right hand side of the map under Graphics. Details of the methodology are in Fryer & Nicholson (1999). However, the summary maps focus on changes in contaminant levels between the start and the end of the time series. The fitted value of the smoother at the start of the time series is compared to the fitted value at the end of the time series using a t-test, with significance assessed at the 5% level. The correlation between the two fitted values is accounted for by the t-test. In the statistical analysis output, this is referred to as the change in the last fifteen years, since the longest time series is fifteen years.
Fryer RJ & Nicholson MD, 1999. Using smoothers for comprehensive assessments of contaminant time series in marine biota. ICES Journal of Marine Science 56: 779-790.