On Jun 9, 2011, at 11:35 AM, Simon Wood wrote: > I think that the main problem here is that smooths are not constrained to > pass through the origin, so the covariate taking the value zero doesn't > correspond to no effect in the way that you would like it to. Another way of > putting this is that smooths are translation invariant, you get essentially > the same inference from the model y_i = f(x_i) + e_i as from y_i = f(x_i + k) > + e_i (which implies that x_i=0 can have no special status).
OK, I understand the translation invariance, I think, and why x_i=0 has no special status. I don't understand the consequence that you are saying follows from that, though. Onward... > All mgcv does in the case of te(a) + te(b) + te(d) + te(a, b) + > te(a, d) is to remove the bases for te(a), te(b) and te(d) from the basis of > te(a,b) and te(a,d). Further constraining te(a,b) and te(a,d) so that > te(0,b) = te(a,0) = 0 etc wouldn't make much sense (in general 0 might not > even be in the range of a and b). Hmm. Perhaps my question would be better phrased as a question of model interpretation. Can I think of the smooth found for te(a) as the "main effect" of a? If so, should I just not be bothered by the fact that te(a,b) at b=0 has a different shape from te(a)? Or is the "main effect" of a really, here, te(a) + te(a,b | b=0) + te(a,d | d=0) (if my notation makes any sense, I'm not much at math)? Or is the whole concept of "main effect" meaningless for these kinds of models -- in which case, how do I interpret what te(a) means? Or perhaps I should not be trying to interpret a by itself; perhaps I can only interpret the interactions, not the main effects. In that case, do I interpret te(a,b) by itself, or do I need to conceptually "add in" te(a) to understand what te(a,b) is telling me? My head is spinning. Clearly I just don't understand what these GAM models even mean, on a fundamental conceptual level. > In general I find functional ANOVA not entirely intuitive to think about, but > there is a very good book on it by Chong Gu (Smoothing spline ANOVA, 2002, > Springer), and the associated package gss is on CRAN. Is what I'm doing functional ANOVA? I realize ANOVA and regression are fundamentally related; but I think of ANOVA as involving discrete levels (factors) in the independent variables, like treatment groups. My independent variables are all continuous, so I would not have thought of this as ANOVA. Anyhow, OK. I will go get that book today, and see if I can figure all this out. Thanks for your help! Ben Haller McGill University http://biology.mcgill.ca/grad/ben/ Begin forwarded message: > From: Simon Wood <s.w...@bath.ac.uk> > Date: June 9, 2011 11:35:11 AM EDT > To: r-help@r-project.org, rh...@sticksoftware.com > Subject: Re: [R] gam() (in mgcv) with multiple interactions > > I think that the main problem here is that smooths are not constrained to > pass through the origin, so the covariate taking the value zero doesn't > correspond to no effect in the way that you would like it to. Another way of > putting this is that smooths are translation invariant, you get essentially > the same inference from the model y_i = f(x_i) + e_i as from y_i = f(x_i + k) > + e_i (which implies that x_i=0 can have no special status). > > All mgcv does in the case of te(a) + te(b) + te(d) + te(a, b) + > te(a, d) is to remove the bases for te(a), te(b) and te(d) from the basis of > te(a,b) and te(a,d). Further constraining te(a,b) and te(a,d) so that > te(0,b) = te(a,0) = 0 etc wouldn't make much sense (in general 0 might not > even be in the range of a and b). > > In general I find functional ANOVA not entirely intuitive to think about, but > there is a very good book on it by Chong Gu (Smoothing spline ANOVA, 2002, > Springer), and the associated package gss is on CRAN. > > best, > Simon > > > > On 07/06/11 17:00, Ben Haller wrote: >> Hi! I'm learning mgcv, and reading Simon Wood's book on GAMs, as >> recommended to me earlier by some folks on this list. I've run into >> a question to which I can't find the answer in his book, so I'm >> hoping somebody here knows. >> >> My outcome variable is binary, so I'm doing a binomial fit with >> gam(). I have five independent variables, all continuous, all >> uniformly distributed in [0, 1]. (This dataset is the result of a >> simulation model.) Let's call them a,b,c,d,e for simplicity. I'm >> interested in interactions such as a*b, so I'm using tensor product >> smooths such as te(a,b). So far so good. But I'm also interested >> in, let's say, a*d. So ok, I put te(a,d) in as well. Both of these >> have a as a marginal basis (if I'm using the right terminology; all I >> mean is, both interactions involve a), and I would have expected them >> to share that basis; I would have expected them to be constrained >> such that the effect of a when b=0, for one, would be the same as the >> effect of a when d=0, for the other. This would be just as, in a GLM >> with formula a*b + a*d, that formula would expand to a + b + d + a:b >> + a:d, and there is only one "a"; a doesn't get to be different for >> the a*b interaction than it is for the! a*d interaction. But with >> tensor product smooths in gam(), that does not seem to be the case. >> I'm still just getting to know mgcv and experimenting with things, so >> I may be doing something wrong; but the plots I have done of fits of >> this type appear to show different marginal effects. >> >> I tried explicitly including terms for the marginal basis; in my >> example, I tried a formula like te(a) + te(b) + te(d) + te(a, b) + >> te(a, d). No dice; in this case, the main effect of a is different >> between all three places where it occurs in the model. I.e. te(a) >> shows a different effect of a than te(a, b) shows at b=0, which is >> again different from the effect shown by te(a, d) at d=0. I don't >> even know what that could possibly mean; it seems wrong to me that >> this could even be the case, but what do I know. :-> >> >> I could move up to a higher-order tensor like te(a,b,d), but there >> are three problems with that. One, the b:d interaction (in my >> simplified example) is then also part of the model, and I'm not >> interested in it. Two, given the set of interactions that I *am* >> interested in, I would actually be forced to do the full five-way >> te(a,b,c,d,e), and with a 300,000 row dataset, I shudder to think how >> long that will take to run, since it would have something like 5^5 >> free parameters to fit; that doesn't seem worth pursuing. And three, >> interpretation of a five-way interaction would be unpleasant, to say >> the least; I'd much rather be able to stay with just the two-way (and >> one three-way) interactions that I know are of interest (I know this >> from previous logistic regression modelling of the dataset). >> >> For those who like to see the actual R code, here are two fits I've >> tried: >> >> gam(outcome ~ te(acl, dispersal) + te(amplitude, dispersal) + >> te(slope, curvature, amplitude), family=binomial, data=rla, >> method="REML") >> >> gam(outcome ~ te(slope) + te(curvature) + te(amplitude) + te(acl) + >> te(dispersal) + te(slope, curvature) + te(slope, amplitude) + >> te(curvature, amplitude) + te(acl, dispersal) + te(amplitude, >> dispersal) + te(slope, curvature, amplitude), family=binomial, >> data=rla, method="REML") >> >> So. Any advice? How can I correctly do a gam() fit involving >> multiple interactions that involve the same independent variable? >> >> Thanks! >> >> Ben Haller McGill University >> >> http://biology.mcgill.ca/grad/ben/ >> >> ______________________________________________ R-help@r-project.org >> mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do >> read the posting guide http://www.R-project.org/posting-guide.html >> and provide commented, minimal, self-contained, reproducible code. >> > > > -- > Simon Wood, Mathematical Science, University of Bath BA2 7AY UK > +44 (0)1225 386603 http://people.bath.ac.uk/sw283 [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
