# Analysed non-linear data with GAM regression, but reviewer has suggested fitting exponential or logarithmic curves instead. Which to use?

I am trying to publish my Masters thesis which looked at elephant impacts on vegetation, with a focus on the effect of artificial waterholes. I have made my initial submission and am now performing major revision, which involves some questioning of my statistical approaches.

Some of the elephant impact measures I looked at showed a non-linear relationship with distance to water which led me to use GAMs. However, a reviewer has said that they feel this is statistical overkill and achieves little more than "fitting complicated nonlinear functions that demonstrate that distance matters". They suggest I should rather describe non-linear relationships between elephant impacts and distance to water by fitting simple functions which others can use (e.g. exponential decay curves). I think the point being made is that these simpler functions would allow me to say something more generalisable about the elephant impacts (e.g. they resemble a logarithmic curve), while the GAMs are only useful for describing the impacts I found in my particular study area and is thus of less general usefulness.

I'm still a novice at statistics so I would appreciate some thoughts on which approach seems more sensible. I have included some examples from my data below.

The top graph shows canopy volume/ha in one vegetation type at different distances from water(the circles represent individual sampling plots). The line shows the predicted relationship from the GAM regression. This sort of relationship appears to resemble a logarithmic curve so the reviewer would perhaps recommend I fit that instead in a case like this.

The second graph shows canopy volume/ha in relation to distance to water, but in a second vegetation type. This vegetation type was more heavily impacted by elephants. Again the line on the graph is derived from the GAM prediction, but this could be one where I could look to fit an exponential curve?

I'd appreciate any thoughts on the merits of my use of GAMs versus the reviewer's recommendation to use simpler functions. As an aside, I also have not really explored fitting logarithmic or exponential curves to data in R, so I would also much appreciate any pointers in that regard.

• You have done well using GAMs, the reviewer's comment seems somewhat over-simplifying to me. Jul 10, 2021 at 15:33
• I have similar "distance to" ecological analysis and I am weighing the pros and cons of NLS vs. GAM. I have gone through your paper and find that you decided to stick with the GAMs for the non-linear relationships. I am perplexed however that your description of the relationship canopy volume vs distance to waterhole is "canopy volume [...] increased initially with distance from the nearest waterhole, before plateauing at around 1000 m (Fig. 3a and b)." However, plateauing behaviour is not what your GAM is showing but what an asymptotic regression would have showed. Details why? Mar 13 at 15:55

1. The use of GAM is well-established in the field of Ecology so I would add certain books/influential articles. Show you are not reinventing the wheel rather that you are abreast with modern modelling approaches.
2. You do not describe your sample size but you might want to try a validation schema to show that through the use of GAMs you get better goodness-of-fit. While hand-wavy if something like an AIC/BIC shows a clear preference for a particular model this can pacify some (not too sophisticated) criticism...
3. I would emphasise how the GAM fitting procedure looks into shrinkage. It is plausible that someone oversimplified GAMs in his/her head as "a polynomial basis of sorts" and therefore prone to overfit.
4. Take their view-point for a moment: are there any established studies suggesting logarithmic, or exponential decay curves already? The reviewer might be satisfied that you acknowledge them as a possibility. Maybe you can make a critical assessment of that prior work and show how your work is a step forward.
5. As Dimitri mentioned, specifying a functional form without prior knowledge can induce strong bias. You can politely double-down on the fact you are using a non-parametric approach. Maybe even try a different basis functions (e.g. cubic regression splines and thin-plate splines) and show how the results are (hopefully) very similar and thus not dependant on the choice of basis functions.

Just to be clear: In my opinion, using GAMs is the correct approach here; the criticism of "why not X-functional form" is weak. Such criticism might be warranted if prior research suggested robust evidence for a particular modelling assumption but even then it would not be a particularly strong position to take. That said, try to see where they are come from too, criticism can be helpful strength your manuscript and/or alleviate worries of future readers too.

• I recommend acceptance of this answer as this, in conjunction with my own, offers the best strategy for responding to the review. +1 Jul 10, 2021 at 16:16
• Thank you both for these really helpful responses, it definitely gives me more confidence in the merits of using GAMs and how I might defend it. Perhaps one final consideration I did not mention explicitly was that the reviewer feels the study's main value is to "contribute to efforts to parameterize spatial models of elephant herbivory". From what I've read, you can't extract a predictive equation with model coefficients from a GAM as you would from a GLM? If so, this is perhaps part of the reason the reviewer feels using simpler functions would be more useful to other researchers? Jul 11, 2021 at 15:30
• I am glad I could help. Hmm... This now moves to non-Stats issues: I think you and your co-authors should take ownership of what is the study's main value and be clear about that. More to the Stats: Aiming to fit a model A and get some coefficient X to use X into another model B is one approach, but it is a strongly parametric approach. From my limited view of your work you are also doing "spatial models of elephant herbivory" through the use a non-parametric approach (GAMs). You just cut off the middle man/model. :) Jul 11, 2021 at 22:38
• @LukeWilson re the predictive equation; yes, providing an "equation" is simpler for simpler models. But if you wanted users to be able to predict from your model, there are ways to achieve that. The counter argument is that often simpler models don't fit the data as well because they over simplify. So what is their utility? You could package up your model as an R object and ship that to anyone that needs to use it for prediction, for example. Jul 12, 2021 at 11:03

That GAMs are somehow “statistical overkill” and simpler functions are “more generalizable” is a contentious claims. A priori, if you had no idea about the functional relationship between your inputs and outputs, a GAM is a sensible approach which subsumes exponential and logarithmic shapes as a special case. To fit a model and decide post facto that “hey this looks like this other function, I’m going to fit that instead” is not a proper inferential procedure anyway. Besides, the first plot is clearly non-monotonic so I don’t see why a log would be suggested.

Assuming you fit the models appropriately (looks like these are count or ratio data e.g. something per unit something meaning you should use an offset of some sort) I would probably politely respond to the reviewer to say that GAMs would be more generalizable because they freely estimate the effect as compared to specifying the functional form to be exponential or logarithmic. Specifying the functional form is a higher bias model than estimating the functional form with penalized basis functions. This means however that GAMs are more variable, which may or may not be an issue. You can at least be honest with the uncertainty by bootstrapping.

• Re "not a proper inferential procedure": Just to clarify, you'd consider it inappropriate to replace a GAM by a linear model during the analysis if the investigation of the initial GAM shows a clear linear effect of the predictor? Jul 12, 2021 at 8:45
• @Schmuddi I would consider it inappropriate to look at the data to determine which shape would best fit the data, and then fit that model as if it were my original hypothesis. That would be a case of testing the hypothesis suggested by the data Jul 12, 2021 at 13:09