In the previous post in this series, I suggested that existing theory could provide a model appropriate to the observed relationship between live fish weight and vertebra size. This relationship appears to be an instance of allometric scaling. Many animal species exhibit a power-law relationship between the scale of particular traits and overall body size. Such relationships take the following form. Let *y* = body size and *x* = the size of a particular trait. Then:

where *a* and *b* are constants and the parameters to be estimated from the data.

Note that a log transform of this relationship would result in the following equation:

.

This equation produces a straight line with the y-intercept at log(*a*) and a slope of *b*. The log-log transform of the data illustrated in the previous post showed this kind of line, which is a necessary (but not sufficient) condition for demonstrating a power-law relationship.

For my sample of fish, I evaluated the relationship between fish live weight and caudal vertebra size using a linear regression analysis. This technique identifies the values of *a* and *b* that minimize the deviation between the observed values and the predicted values. The estimates for the sample are *a*=4.54 and *b*=2.77. The 95 percent confidence interval for *a* ranges from 3.05 to 6.76, while the 95 percent confidence interval for *b* ranges from 2.49 to 3.05. For this model, r^{2}=0.93, and the p-values for both parameters are less than 0.001. The low p-values indicate that the sample size was sufficiently large.

The following plot illustrates the fit of this model to the data. In the plot, the dashed lines show the prediction interval around the model. This interval depicts the range within which 95 percent of live weights for new samples would be expected to fall for a given value of caudal vertebra height.

The sample of fish for this analysis should be expanded. Nevertheless, it supports the common-sense notion that fish bone size reflects the overall size of fish. The data that corroborate this middle-level theory are not comprehensive, but they provide sufficient justification to proceed. Caudal vertebra height will be used as a measure of fish size.

© Scott Pletka and *Mathematical Tools, Archaeological Problems*, 2009.