Carp and smallmouth bass fillets were collected at Plainwell Impoundment and Lake Allegan from 1983 to 1999 and weight, length, lipid content, and total PCB concentration were measured. PCB concentration in fish tissue is often associated with lipid content and length or weight, so we investigated the appropriateness of adjusting tissue PCB concentrations for covariation with lipid, length, and/or weight. Weight was highly correlated with length and added almost nothing to the fit when included in the presence of length, so we excluded it from models to avoid problems with multicolinearity. For this reason, we will discuss only lipid and length.
We assumed a model of the form:
is a linear model representing the relationship of log-PCB with log-length and log-lipid, is the mixed-order model, and is a lognormal error distribution. We fit this model using a two-stage process. In order to account for the effect of lipid and length, we fit a linear model with log-PCB, log-lipid and log-length treating year as a categorical variable. The factors and interactions included in the model were chosen separately for each species in order to simplify the model fitting procedure and the comparison of results within species across sites. Years with fewer than four data points were insufficient for modeling the interaction terms and so were collapsed into the nearest year (if this occurred, it was in the first year of sample data for a given site and species). Using the residuals from the linear model, we calculated the adjusted concentrations as:
is a vector of the measurements for a representative fish, which we chose to be the overall average lipid and length within each species. We fit the MO temporal trend model to the adjusted data.
Due to differences between the two species and two locations, we modeled each species-location combination separately. We fit the model to the data using the maximum likelihood estimators for the parameters.
The choice of tP can be made to coincide with some initial time t0 (as in Stow et al., 1999), but the choice is arbitrary as long as it is within a certain (possibly infinite) interval. This interval is dependent on the parameters. The details of this relationship are provided in the appendix. We chose tP to be 1990 because it seemed to result in better convergence for the optimization routines (see appendix). We constrained the parameters such that the model had positive concavity and was real-valued between 1975 and 2030. A more detailed account of these choices is provided in the appendix. As noted in Stow, et al (1999), as θ approaches 1, the mixed-order model approaches first order decay. We assumed the errors were lognormally distributed. Rather than transforming the data and working in log-scale with normal error, we modeled the mean directly using lognormal error. The second derivative matrix of the likelihood function was ill-conditioned so we used a derivative free algorithm called the downhill simplex method developed by Nelder and Mead (1965) to maximize the likelihood function in Matlabę. We checked our assumption of lognormal error by performing a test for normality given by Looney and Gulledge (1985) on the log of the standardized residuals.
Profile likelihood approaches have been developed as a way to make inferences about a particular parameter of interest when there are a number of other parameters that are necessary for the model, but uninteresting apart from that (i.e. nuisance parameters). We were interested in predicting mean PCB concentration in fish at 2010. To estimate approximate confidence intervals for the future mean concentration, the other parameters were treated as nuisance parameters. The likelihood was calculated for fixed values of the mean parameter by maximizing over the other parameters. In order to accomplish this, we re-parameterized the model with μ2010 (C2010) as a parameter rather than C1990 (consider that C1990= μ1990). Using the asymptotic χ2 distribution of the generalized likelihood ratio test (Bain and Engelhardt, 1992), we generated profile-likelihood based confidence intervals for the mean in 2010.