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I realize this may be a potentially broad question, but I was wondering whether there are 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 time-series
  • 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 its use is indicated over other regression analyses.

Marco Plebani
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mluerig
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    GAM's are used when the linear predictor depends linearly on unknown smooth functions of some predictor variables. – user2974951 Dec 05 '18 at 11:09
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    The distinction is blurry as you can represent numeric covariables e.g. by a spline also in a GLM. – Michael M Dec 05 '18 at 11:31
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    While the distinction is blurry, gam's can represent interactions also the smae way as glm's so strict additivity is not needed, the big difference is in inference: gam's need special methods, since estimation is not done via projection, but via smoothing. What that does imply in practice, I don' t understand. – kjetil b halvorsen Dec 05 '18 at 11:37
  • GLM $\subset$ GAM. – usεr11852 Dec 06 '18 at 00:31

3 Answers3

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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').

matteo
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    Well, but as said in comments, all of that can be done with glm's also ... I suspect the main difference is pragmatic. The R implementation in `mgcv` does a lot of things you cannot do with `glm`, but could have been done in that framework also ... – kjetil b halvorsen Dec 05 '18 at 12:30
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    Yes, I agree with you, GAMs are an extension of GLMs. However the question was about when to use GAM and when to use GLM, and it seemed to me that the op meant "classical" forms of GLMs, which do not usually include a set of basis function as predictors and are not used to reveal/approximate unknown nonlinear relationship. – matteo Dec 05 '18 at 13:14
  • thanks - this is helpful. and yes, I was talking about classic GLMs – mluerig Dec 05 '18 at 18:41
  • @ matteo just two more things: i) what exactly do you mean by "true relationship is likely to be smooth, rather than wiggly"? and ii) "particularly useful for uncovering nonlinear effects of numerical covariates" - how would one describe / quantify nonlinearity (e.g. with `mgcv`)? – mluerig Dec 05 '18 at 18:49
  • The true relationship might not actually be smooth, however GAMs typically control for model complexity by adding a "wiggliness" penalty during the process of likelihood maximization (usually implemented as a proportion of the integrated square of the second derivative of the estimated function). Nonlinear effects of numerical covariates means that the influence of a particular numerical variable on the dependent variable might, for example, not increases/decreases monotonically with the variable value, but have an unknown shape, e.g. with local maxima, minima, inflection points,... – matteo Dec 05 '18 at 21:10
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    If you want to know more about the details, I warmly recommend Simon Wood's book: https://www.crcpress.com/Generalized-Additive-Models-An-Introduction-with-R-Second-Edition/Wood/p/book/9781498728331 – matteo Dec 05 '18 at 21:11
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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.

Wayne
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    I imagine that I most often do what those grad students did: throw my data in a gam and be dazzled by how well `mgcv` handles my data. I try to be parsimonious with my parameters, and I check how well the predicted values match my data. your comments are a good reminder to be a bit more rigorous - and maybe finally get simon woods book! – mluerig Dec 05 '18 at 18:59
  • Heck, I'll go so far as to use a smoother to explore a variable, and then either fix the degrees of freedom at a low value or eliminate the smooth and use, say, a squared term if the smoother was basically quadratic. A quadratic makes sense for an age effect, for example. – Wayne Dec 05 '18 at 19:39
  • @Wayne, I came here exactly for an answer on data exploration in relation to GAMs, and saw you pointed it out. How do you use GAMs for data exploration? And how would you decide whether a GAM is needed, or if a GLM would suffice. Would it make sense to simply run a simple GAM in which you run the response and each of the potential predictors in turn, plot that, and see if the relationship warrants a GAM (i.e. non-linear and non-monotonic relationship)? – Tilen Jan 19 '19 at 19:46
  • This answer is fundamentally confused about how GAM works. There is nothing necessarily dangerous about using very high-order smoothers in GAM. You're supposed to rely on the smoothing to keep them from getting too wiggly. You may need to use the REML criterion instead of CV, as Wood now recommends. – Paul Dec 07 '20 at 15:16
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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).

From 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.

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

kjetil b halvorsen
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Rafa_Mas
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