Appendix B. A description of how data were simulated for the evaluation of the different analysis techniques for Example 2.
Coefficients from simple linear regressions of increment width vs. temperature in individual fish were selected from a bivariate normal distribution with: mean intercept, -0.327; mean slope, 0.0263; variance of intercept, 0.646; variance of slope, 0.00202; and covariance of intercept and slope, -0.0357. These values are empirical estimates based on the 53 pairs of regression coefficients from fits to the original data. No weighting scheme was used in obtaining these estimates, which is why the mean slope here (0.0263) is higher than the slope estimates reported in the text.
For each of 53 fish, a pair of regression coefficients was selected from this bivariate distribution and applied to one of the time series of temperatures in the original data set to generate a trajectory of mean increment width vs. time for that fish.
A time series of 4996 random errors was generated by stringing together the 53 time series of residuals from the original data in such a way as to minimize the discontinuities between the end of one series and the beginning of the next. Specifically, starting with the time series of errors for the first fish, I searched the remaining 52 series for the one having an initial value as close as possible to the final value of the first series. The two series were then joined, with the "joining" value replaced by the average of the last residual of the first series and the first residual of the new series. This process was repeated until all 53 time series of errors were incorporated into the large string.
As shown in Fig. B1, the correlation structure of the resulting time series of 4996 errors is quite complicated, with evidence of irregular, high-order serial correlation that would be hard to capture with the models usually available in time-series software.
For each simulated fish, a sub-sequence of the string of 4996 errors was randomly chosen and added to the time series of mean increment width generated as described above. The result was a set of 53 time series of increment width, corresponding to 53 time series of temperatures, with each series consisting of a "signal" generated by regression coefficients applied to a particular time series of temperatures and a random component intended to mimic as closely as possible the error structure observed in the original data.
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| FIG. B1. Partial autocorrelation coefficients for the time series of 4996 random errors. The dashed lines enclose coefficients that are not significantly different from zero. |