- The concentration measurements each year are summarised by an annual contaminant index. A weight is assigned to each index, scaled to lie between 0 and 1, that incorporates information about the analytical quality of the index.
- The scaled weights are converted into statistical weights that account for the relative magnitudes of the analytical and environmental variability in the data.
- A weighted regression model is fitted to the annual contaminant indices.
The type of model depends on the number of years of data:
- 1-2 years: no model
- 3-4 years: mean
- 5-6 years: linear trend
- 7+ years: smoother

- The fitted models are used to assess environmental status against available assessment criteria and evidence of temporal change in contaminant levels in the last ten years

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:

*c*is the normalised concentration of the contaminant_{ss}*c*is the measured concentration of the contaminant_{m}*c*is the pivot concentration for the contaminant_{x}*n*is the reference concentration of the normaliser_{ss}*n*is the measured concentration of the normaliser_{m}*n*is the pivot concentration for the normaliser_{x}

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:

- `A = \frac{n_(ss)-n_x}{n_m-n_x}`
- `B = \frac{c_m-c_x}{n_m-n_x}`
- Var(
*c*) and Var(_{m}*n*) are the analytical variances of the contaminant and normaliser measurements, as calculated by the MERMAN database._{m}

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 OSPAR (2002) and the derivation of pivot values is described in ICES (2002).

OSPAR, 2002. Technical Annex 5 to the JAMP Guidelines for Monitoring Contaminants in Sediment

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 *c _{ti}, i* = 1 ...

`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(*c _{ti}*)

`y_t=\frac{1}{u_t.}\sum_i u_(ti) \log(c_(ti))`

where *u _{ti}* = 1 /

`\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 σ* _{ti}*. 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 *y _{t}* is the annual contaminant index in year

`\varepsilon_t=\tau_t + \delta_t`

where τ* _{t}* is the noise due to analytical variation (and possibly within-year
environmental variation) and δ

`\tau_t ~ \text{N}(0,\frac{\sigma_t^2}{w_t})`

`\delta ~ \text{N}(0,\sigma_\delta^2)`

where *w _{t}* is the scaled weight for year

`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 ≤ *w _{t}* ≤

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 *r _{t}* 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

The annual contaminant indices are modelled as:

`y_t=f(t)+\epsilon_t`

where *y _{t}* is the annual contaminant index in year

The form of *f*(*t*) depends on the number of years of data:

- 1-2 years
- no model is fitted as there are too few years for formal statistical analysis
- 3-4 years
- mean model
*f*(*t*) = ¦ - there are too few years for a formal trend assessment, but the mean level is summarised by ¦ and is used to assess status
- 5-6 years
- linear trend
*f*(*t*) = ¦ + β*t* - the contaminant indices vary linearly with time; the fitted model is used to assess status and evidence of temporal change
- 7+ years
- smoother
function of time*f*(*t*) = smooth - a loess smoother is fitted to the contaminant indices with a fixed window width (Fryer & Nicholson, 1999) of either 7, 9 or 11 years; the choice of window width is based on Akaike's Information Criteron (AIC); the fitted model is used to assess status and evidence of temporal change

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 time series with at least five years of data. When there are
5-6 years, there is evidence of a temporal trend if the slope β of the linear regression of *y _{t}* on

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.