Solved – How to summarize and compare non-linear relationships

I have data on the percent of organic matter in lake sediments from 0 cm (i.e., the sediment – water interface) down to 9 cm for approximately 25 lakes. In each lake 2 cores were taken from each location so I have 2 replicate measures of organic matter percentage at each sediment depth for each lake.

I am interested in comparing how lakes differ in the relationship between percent organic matter and sediment depth (i.e., slope). In some lakes the relationship between percent organic matter and sediment depth appears linear but in other cases the relationship is more complex (see examples below).

My initial thoughts were to fit linear relationships where appropriate either to the whole curve or to a subset of the curve if it was "mainly" linear and only compare those lakes where a significant linear relationship was found. However I am unhappy with this approach in that it requires eliminating data for no other reason than they do not fit the linear model and it ignores potentially interesting information about the relationship between percent organic matter and sediment depth.

What would be a good way to summarize and compare the curves from different lakes?

Thank you

Example curves: In all cases the y-axis is the percent organic matter in the sediment and the x-axis is the sediment depth where 0 = the sedi
ment-water interface.

A nice linear example:

A nice linear example

2 non-linear examples:

nonlinear 1

enonlinear 2

An example with no obvious relationship:

no relationship

Check out Generalized Additive Models, which permit fitting non-linear functions without a priori specification of the non-linear form. I'm not sure how one would go about comparing the subsequent fits however. Another similar (in that I believe they both employ cubic splines) approach is achieved by Functional Data Analysis, where I understand there are methods for characterizing differences between fitted functions.

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