Simulations

Using Lake Allegan carp and Plainwell Impoundment carp as the base datasets, we performed three pairs of simulations to evaluate three different aspects of our procedure. These were the coverage of the likelihood methods (lognormal error), robustness to faulty error assumptions (empirical distribution), and the failure to account for variability from the initial linear model (full procedure empirical). In each simulation we needed a ′true′ model to use as the basis for the simulation. We chose to use the fitted models for the base datasets as the true model of the mean. These simulations differed only in how the data for the non-linear regression was generated.

The data for the lognormal error simulations was model based. Each
dataset was generated having the same within year sample sizes as the base
dataset. The generated data was lognormal with the mean in a given year
equal to the model and shape parameter equal to the fitted value of the shape
parameter for the base dataset. In the empirical distribution simulations,
we used a re-sampling approach and sampled with replacement by year from the
length and lipid *adjusted values *of the base dataset. Yearly sample
sizes in the generated data were kept consistent with those found in the base
dataset. The data for the full procedure empirical simulations was also
re-sampled, except this time it was from the *unadjusted values* of the
base dataset. We sampled with replacement by year from the unadjusted values
of the base dataset and then adjusted these data for length and lipid with
two-way interactions as described above. Yearly sample sizes in
the generated data were kept consistent with those found in the base
dataset.

Once the data had been generated, we followed the profile likelihood procedure described above to generate a confidence interval for the predicted mean PCB concentration in 2010. If the optimization routines did not converge for a given dataset, it was noted. After generating 1000 samples and their corresponding confidence intervals, we tabulated the percentage of generated confidence intervals that contained the true value. We used this Monte Carlo estimate of our coverage probability to evaluate how well our procedure performed under the assumptions of the simulation.