In the previous post in this series, I demonstrated that caudal vertebra height can be used to estimate live fish weight. The use of vertebrae, however, introduces an additional issue of quantification that requires resolution. Bony fish have two main types of vertebra: abdominal and caudal. Predictable variation in form occurs among the abdominal and caudal vertebrae in the vertebral column of an individual fish. Despite this predictability, an undifferentiated pile of fish vertebrae in an archaeological collection is usually just separated into these main types because the variation can be subtle. Multiple specimens of a single vertebra type (like caudal vertebrae) from a particular species thus could be attributable to a single individual or to multiple individuals. Many statistical tests, however, require that each observation be independent of the others. This assumption is particularly critical for the analysis of size-frequency distributions. The assumption of independent observations would be violated if multiple bone specimens derived from the same individual. This violation could dramatically affect inferences regarding the shape of that distribution. Some method must be used to eliminate potentially redundant specimens.

Two criteria can be used to identify vertebrae from separate individuals. First, the size of vertebrae within an individual bony fish (excluding the length of the centrum) typically varies only a little. The vertebral centra of sharks, skates, and rays (elasmobranchs) seem to vary to a much greater extent within an individual. Subsequent analysis focused on bony fish for this reason. Second, each species of bony fish has a characteristic number of abdominal and caudal vertebrae, and this number varies modestly among individuals. A small number of caudal vertebrae within a narrow size range from a particular taxon, for example, may well have come from the same individual. Vertebrae from a particular taxon that span a large size range or that occur in large number within a small size range are likely to have derived from multiple fish.

In the sample of fish specimens, vertebra height typically varied less than 0.3 mm when comparing abdominal and caudal vertebrae. The size difference within individuals was not strongly correlated with the overall size of the caudal vertebra.

A simple linear regression returned estimates of -0.19 for the y-intercept and 0.13 for the slope of the line, while r^{2}=0.49 and p < 0.01. Two of the large vertebrae appear to be outliers, however, and may be unduly influencing these results. With these two cases removed, the simple linear regression estimates the y-intercept to be 0.05 and the slope to be 0.06, while r^{2}=0.22 and p =0.01. This rule of thumb may therefore be generally applicable.

When the number of caudal vertebrae within a 0.3-mm size interval exceeds the typical number for that taxon, more than one individual from that size range may be represented. Additional work should be undertaken, using a larger sample of fish, to confirm and refine this observation. In the interim, the foregoing principles and observations can be used to calculate the minimum number of individuals represented in an archaeological assemblage and to estimate the size of each fish.

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

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

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