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I realize this may be a potentially broad question, but I was wondering whether there are generalizable assumptions that indicate the use of a GAM (Generalized additive model) over a GLM (Generalized linear model)?

Someone recently told me that GAMs should only be used when I assume the data structure to be "additive", i.e. I expect additions of x to predict y.
Another person pointed out that a GAM does a different type of regression analysis than a GLM, and that a GLM is preferred when linearity can be assumed.

In the past I have been using a GAM for ecological data, e.g.:

• continuous timeseries


• when the data did not have a linear shape


• I had multiple x to predict my y that I thought to have some nonlinear interaction that I could visualize using "surface plots" together with a statistical test

I obviously don't have a great understanding of what a GAM does different than a GLM. I believe it's a valid statistical test, (and I see an increase in the use GAMs, at least in ecological journals), but I need to know better when it's use is indicated over other regression analyses.

The main difference imho is that while "classical" forms of linear, or generalized linear, models assume a fixed linear or some other parametric form of the relationship between the dependent variable and the covariates, GAM do not assume a priori any specific form of this relationship, and can be used to reveal and estimate non-linear effects of the covariate on the dependent variable.
More in detail, while in (generalized) linear models the linear predictor is a weighted sum of the $n$ covariates, $sum_i=1^n beta_i x_i$, in GAMs this term is replaced by a sum of smooth function, e.g. $sum_i=1^n sum_j=1^q beta_i , s_j left( x_i right)$, where the $s_1(cdot),dots,s_q(cdot)$ are smooth basis functions (e.g. cubic splines) and $q$ is the basis dimension. By combining the basis functions GAMs can represent a large number of functional relationship (to do so they rely on the assumption that the true relationship is likely to be smooth, rather than wiggly). They are essentially an extension of GLMs, however they are designed in a way that makes them particularly useful for uncovering nonlinear effects of numerical covariates, and for doing so in an "automatic" fashion (from Hastie and Tibshirani original article, they have 'the advantage of being completely automatic, i.e. no "detective" work is needed on the part of the statistician').

I'd emphasize that GAMs are much more flexible than GLMs, and hence need more care in their use. With greater power comes greater responsibility.

You mention their use in ecology, which I have also noticed. I was in Costa Rica and saw some kind of study in a rainforest where some grad students had thrown some data into a GAM and accepted its crazy-complex smoothers because the software said so. It was pretty depressing, except for the humorous/admirable fact that they rigorously included a footnote that documented the fact that they'd used a GAM and the high-order smoothers that resulted.

You don't have to understand exactly how GAMs work to use them, but you really need to think about your data, the problem at hand, your software's automated selection of parameters like smoother orders, your choices (what smoothers you specify, interactions, if a smoother is justified, etc), and the plausibility of your results.

Do lots of plots and look at your smoothing curves. Do they go crazy in areas with little data? What happens when you specify a low-order smoother or remove smoothing entirely? Is a degree 7 smoother realistic for that variable, is it overfitting despite assurances that it's cross-validating its choices? Do you have enough data? Is it high-quality or noisy?

I like GAMS and think they're under-appreciated for data exploration. They're just super-flexible and if you allow yourself to science without rigor, they will take you farther into the statistical wilderness than simpler models like GLMs.

I have no reputation to simply add a comment. I do totally agree with Wayne’s comment: With greater power comes greater responsibility. GAMs can be very flexible and often we get/see crazy-complex smoothers. Then, I strongly recommend researchers to restrict degrees of freedom (number of knots) of the smooth functions and to test different model structures (interactions/no interactions etc.). GAMs can be considered in between of model-driven approaches (although the border is fuzzy I would include GLM in that group) and data-driven approaches (e.g. Artificial Neural Networks or Random Forests who assume fully interacting non-linear variables’ effects). In accordance, I do not totally agree with Hastie and Tibshirani because GAMs still need some detective work (Hope no one kills me for saying so). Form an ecological perspective, I would recommend using the R package scam to avoid these unreliable variable crazy-complex smoothers. It was developed by Natalya Pya and Simon Wood and it allows constraining the smooth curves to desired shapes (e.g. unimodal or monotonic), even for two-way interactions. I think GLM becomes a minor alternative after constraining the shape of the smooth functions but this is only my personal opinion.

HTH

Rafael

Pya, N., Wood, S.N., 2015. Shape constrained additive models. Stat. Comput. 25 (3), 543–559. 10.1007/s11222-013-9448-7