In the previous post in this series, I showed that mixture models of two lognormal distributions provide a reasonable fit to the size-frequency data for fish caudal vertebrae from my midden assemblages. I have interpreted these distributions as result of the use of nets and other gear types. Nets should take smaller fish than other gear types, and both nets and other gear types may also take large fish. During the course of my model-fitting, I determined that a few vertebrae in each assemblage were too large to fit the mixture model. Such fish were excluded from the data used to fit the mixture models. These very large fish may be attributable to some other gear type than was used to take the other fish. The mixture models show that most fish, quantified in terms of minimum number of individuals, were caught by nets in each assemblage. Archeological analysis should not just end, however, with an identification of the number of fish in an assemblage that were caught by nets or by other gear.

The overall contribution to the diet of fish caught by nets in comparison to fish caught by other gear is of particular interest. Smaller fish produce a lower return on the work invested in fishing, all other things being equal. Many small fish may have to be caught to provide the contribution to the diet that a single, large fish would provide. The proportion of “net-caught” or “hook/spear-caught” fish bone in an assemblage thus does not by itself accurately reflect that return.

This contribution can be determined by calculating the total live weight of fish represented by the modeled distributions of net-caught fish and fish caught with other gear. The total live weight of fish in an assemblage has a more obvious relationship to the potential dietary contribution of the fish than the count of those fish. The positive correlation between caudal vertebra height and fish live weight allows these amounts to be inferred, using a simple transformation of the data.

The total live weight of “net-caught” and “hook/spear-caught” fish can be calculated from the mixture model results. The mixture model provides parameters that can be employed to create an idealized size-frequency distribution for each population. These distributions are scaled using the inferred number of fish from each population and the live weight equation. Remember that the relationship between live weight of fish and caudal vertebrae height can be represented by the following equation:

where the parameters were estimated from modern data.

The next equation illustrates the calculation of total live fish weight from one of the modeled distributions represented in an assemblage, where N is the inferred number of fish in the assemblage that belongs to that population and* f(x, µ, σ)* represents the lognormal probability density function:

The parameters N, *µ, *and* σ *are estimated from the maximum likelihood analysis of the mixture models.

The scaled distributions are integrated over the range of observed vertebra heights to obtain the total weight of fish. For this study, the equation was integrated from zero to the maximum observed caudal vertebra height among all of the assemblages, which was 14.2 mm. The scaled distributions are then integrated over the range of observed vertebra heights to obtain the total weight of fish. Remember that some fish vertebrae were so large that they were considered outliers and possibly part of a third mode, caught using a different technique than was used to catch fish from the other two distributions. The weight represented by these very large fish was calculated directly from the live weight (power law) equation. Once these mathematical operations have been completed for both of the populations that comprise the mixture distribution, assemblages can be compared for patterns in the amount of fish caught by net and by other gear. The following table provides the results.

The table shows the weight of net-caught fish from the distribution of smaller fish, the weight of larger fish from the second distribution, and the weight of very large fish. The weight of net-caught fish was compared to the combined weight of other fish to calculate the proportion of net-caught fish in each assemblage. These results contrast with my earlier results based on the number of fish from these distributions. Net-caught fish are much less important by weight in all of the assemblages.

At this point, sufficient middle-level theory has been developed to consider the variation among these assemblages. High-level theory provides possible explanations for the patterns observed through the application middle-theory. In subsequent posts in this series, I will present and apply some formal high-level theories that may explain variation in the intensification of fishing.

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

Tags: archaeology, faunal analysis, intensification of subsistence, mathematical modeling in archaeology, middle-level theory, middle-range theory

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