Identifying and Explaining Intensification in Prehistoric Fishing Practices V: Quantifying the Relationship between Fish Size and Fish Bone Size

The previous post in this series established that a positive relationship exists between the live weight of fish and caudal vertebra height, providing support for the use of vertebra size as an index of overall fish size. I will attempt to quantify this relationship more precisely. Many different models could be chosen, but how should we select the most appropriate one?

The model needs to be appropriate for the structure of the data. The following graph shows a plot of the data with a linear model superimposed over the data points. The graph also depicts the deviation between data and the model with vertical lines. Notice that these deviations seem to get larger as the size of the fish gets larger. Linear models assume, among other things, that the variation remains constant. A linear model may not be appropriate.

Fish Live Weight and Vertebra Height Scatterplot with Linear Model

Fish Live Weight and Vertebra Height Scatterplot with Linear Model

A transformation of the data may help. Taking the log of both the live weight and the vertebra height produces more consistent variation. The next graph shows a linear model applied to the log of the data.

Log Transform of Live Fish Weight and Vertebra Height

Log Transform of Live Fish Weight and Vertebra Height

The deviations from the model are much more consistent. This model now seems reasonably appropriate to the structure of the transformed data in the sense that it doesn’t appear to violate the model assumptions. Those assumptions include normally-distributed variation and constant variation. Ideally, however, I’d like to fit a model that has an easier interpretation. Is there any theoretical basis for applying a particular type of model to fit to this data? As it turns out, the answer is “yes”, and I will discuss this model in the next post in the series.

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

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