How to summarize and compare non-linear relationships?

Question

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:

2 non-linear examples:

An example with no obvious relationship:

Answer 1

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.

Answer 2

For comparison sake, it will be helpful to parametrize the relationship between OM (organic matter) and SED (sediment) similarly across lakes -- so that you are estimating the same model for each lake. That way, you can directly compare coefficient estimates.

If you limit potential nonlinear relationships to an order two polynomial (quadratic), then it would be as simple as adding a second term to a linear model:

OM = beta_0 + beta_1 * SED + beta_2 * (SED^2)

You could then do a t-test to see if the coefficients of two lakes are equal... to each other, or to zero depending on the questions you are trying to answer.

You stated your question as: "I am interested in comparing how lakes differ in the relationship between percent organic matter and sediment depth (i.e., slope)."

If you word your question more specifically, this will aid in selecting the right approach. Why would the relationship between OM and SED differ across lakes? Is there some other observable that would explain the differing relationship?

If so, you might want to include this explanatory variable in your model, via an interaction term or elsewhere. Without more information on the specific question you are trying to answer -- other than "is the relationship between OM and SED the same across lakes?" -- it is difficult to suggest a more specific approach.