# Compare linear vs. nonlinear model fit

Hello everyone,

some colleagues and I were discussing how to decide whether a linear or a sigmoidal curve fits the data best. E.g., does the probability of catching fish increase in a linear or sigmoidal fashion depending on the number of worms used?
We thought about comparing a model with a gaussian link to a model with a logit link, but as far as I understand it, you cannot compare the fit of models with a different link function in a meaningful way? Plus, a sigmoidal function might look linear depending on the parameters of the function.
The other option would be polynomials/spline regression, but that seems pretty â€śheavyâ€ť considering that we know we want to compare linear vs. sigmoidal?

This sounds like a straightforward problem, but I couldnâ€™t find an answer so far. Sorry if I overlooked something.
Iâ€™m grateful for any resources you can point me towards!

Juli

Iâ€™ve never heard this, but that might be my inexperience with model comparison. It strikes me though that the approach taken by loo for comparison shouldnâ€™t care about the internal structures of the model as being different, just that there be a log_prob for each of the same set of observables. @avehtari , true?

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Oh, and as you say, there is a range of values for the sigmoid model that make it indistinguishable from a linear model, so one approach is to fit the sigmoid model and simply look at how much probability mass is in that linear range. No model comparison necessary.

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True.

True.

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Thank you both! I especially like the approach to see hoch much probability mass is in the linear range - this will probably give the most meaningful answer to the question.

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BTW, while you expressed that you have domain expertise that led to the parametric form of the non-linear variant, if you didnâ€™t have such prior knowledge and wanted to more flexibly accommodate possibly-non-lienear effects, then you could use a Gaussian Process, or, if the data are too dense and GPs are too slow, use a generalized additive model as an approximation. With both GPs and GAMs, the models will revert to linearity if the data donâ€™t support higher complexity.