Re: [R] Understanding why a GAM can't suppress an intercept
hi Andrew. gam does suppress the intercept, it's just that this doesn't force the smooth through the intercept in the way that you would like. Basically for the parameteric component of the model '-1' behaves exactly like it does in 'lm' (it's using the same code). The smooths are 'added on' to the parametric component of the model, with sum to zero constraints to force identifiability. There is a solution to forcing a spline through a particular point at http://r.789695.n4.nabble.com/Use-pcls-in-quot-mgcv-quot-package-to-achieve-constrained-cubic-spline-td4660966.html (i.e. the R help thread Re: [R] Use pcls in mgcv package to achieve constrained cubic spline) best, Simon On 16/04/13 22:36, Andrew Crane-Droesch wrote: Dear List, I've just tried to specify a GAM without an intercept -- I've got one of the (rare) cases where it is appropriate for E(y) - 0 as X -0. Naively running a GAM with the -1 appended to the formula and the calling predict.gam, I see that the model isn't behaving as expected. I don't understand why this would be. Google turns up this old R help thread: http://r.789695.n4.nabble.com/GAM-without-intercept-td4645786.html Simon writes: *Smooth terms are constrained to sum to zero over the covariate values. ** **This is an identifiability constraint designed to avoid confounding with ** **the intercept (particularly important if you have more than one smooth). * If you remove the intercept from you model altogether (m2) then the smooth will still sum to zero over the covariate values, which in your case will mean that the smooth is quite a long way from the data. When you include the intercept (m1) then the intercept is effectively shifting the constrained curve up towards the data, and you get a nice fit. Why? I haven't read Simon's book in great detail, though I have read Ruppert et al.'s Semiparametric Regression. I don't see a reason why a penalized spline model shouldn't equal the intercept (or zero) when all of the regressors equals zero. Is anyone able to help with a bit of intuition? Or relevant passages from a good description of why this would be the case? Furthermore, why does the -1 formula specification work if it doesn't work as intended by for example lm? Many thanks, Andrew [[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. -- Simon Wood, Mathematical Science, University of Bath BA2 7AY UK +44 (0)1225 386603 http://people.bath.ac.uk/sw283 __ 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.
Re: [R] Understanding why a GAM can't suppress an intercept
Simon, Many thanks as always for your help. I see and appreciate the example that you cited, but I'm having a hard time generalizing it to a multivariate case. A bit about my context -- my dataset is response ratios; the log of a treatment over a control. One of my explanatory variables is treatment intensity. When intensity goes to zero, the expectation of the response ratio should also go to zero. Here is the model that I would like to fit: model = (ResponseRatio~ +s(as.factor(study),bs=re,by=intensity) +s(intensity) +s(x1),by=intensity) +s(x2),by=intensity) +te(x1,x2),by=intensity) +te(x1,intensity) ) Here is the example that you gave: library(mgcv) set.seed(0) n - 100 x - runif(n)*4-1;x - sort(x); f - exp(4*x)/(1+exp(4*x));y - f+rnorm(100)*0.1;plot(x,y) dat - data.frame(x=x,y=y) ## Create a spline basis and penalty, making sure there is a knot ## at the constraint point, (0 here, but could be anywhere) knots - data.frame(x=seq(-1,3,length=9)) ## create knots ## set up smoother... sm - smoothCon(s(x,k=9,bs=cr),dat,knots=knots)[[1]] ## 3rd parameter is value of spline at knot location 0, ## set it to 0 by dropping... X - sm$X[,-3]## spline basis S - sm$S[[1]][-3,-3] ## spline penalty off - y*0 + .6 ## offset term to force curve through (0, .6) ## fit spline constrained through (0, .6)... b - gam(y ~ X - 1 + offset(off),paraPen=list(X=list(S))) lines(x,predict(b)) ## compare to unconstrained fit... b.u - gam(y ~ s(x,k=9),data=dat,knots=knots) lines(x,predict(b.u)) *My question*: how can I extend your example to more than one smooth terms, and several smooth interactions? In the context of a model where E[ResponseRatio] must equal 0 when intensity equals zero? I am not sure that I understand exactly what is going on when you call smoothCon to specify the `sm`. I've written my own penalized spline code from scratch, but it is far less sophisticated than mgcv, and is basically a ridge regression where I optimize to get a lambda after specifying a model with a lot of knots. mgcv clearly has a lot more going on, and is far preferable to my rudimentary code for its handling of tensors and random effects. (Also, for prediction, how can I do by=dum AND by=intensity at the same time?) Many thanks, Andrew PS: I am aware that interacting my model with an intensity variable makes my model quite heteroskedastic. I am thinking of using a cluster wild bootstrap to construct confidence intervals. If a better way forward immediately comes to your mind -- especially if its computationally cheaper -- I'd greatly appreciate if you could share it. On 04/17/2013 02:16 AM, Simon Wood wrote: hi Andrew. gam does suppress the intercept, it's just that this doesn't force the smooth through the intercept in the way that you would like. Basically for the parameteric component of the model '-1' behaves exactly like it does in 'lm' (it's using the same code). The smooths are 'added on' to the parametric component of the model, with sum to zero constraints to force identifiability. There is a solution to forcing a spline through a particular point at http://r.789695.n4.nabble.com/Use-pcls-in-quot-mgcv-quot-package-to-achieve-constrained-cubic-spline-td4660966.html (i.e. the R help thread Re: [R] Use pcls in mgcv package to achieve constrained cubic spline) best, Simon On 16/04/13 22:36, Andrew Crane-Droesch wrote: Dear List, I've just tried to specify a GAM without an intercept -- I've got one of the (rare) cases where it is appropriate for E(y) - 0 as X -0. Naively running a GAM with the -1 appended to the formula and the calling predict.gam, I see that the model isn't behaving as expected. I don't understand why this would be. Google turns up this old R help thread: http://r.789695.n4.nabble.com/GAM-without-intercept-td4645786.html Simon writes: *Smooth terms are constrained to sum to zero over the covariate values. ** **This is an identifiability constraint designed to avoid confounding with ** **the intercept (particularly important if you have more than one smooth). * If you remove the intercept from you model altogether (m2) then the smooth will still sum to zero over the covariate values, which in your case will mean that the smooth is quite a long way from the data. When you include the intercept (m1) then the intercept is effectively shifting the constrained curve up towards the data, and you get a nice fit. Why? I haven't read Simon's book in great detail, though I have read Ruppert et al.'s Semiparametric Regression. I don't see a reason why a penalized spline model shouldn't equal the intercept (or zero) when all of the regressors equals zero. Is anyone able to help with a bit of intuition? Or relevant passages from a good description of why this would be the case? Furthermore, why does the -1
[R] Understanding why a GAM can't suppress an intercept
Dear List, I've just tried to specify a GAM without an intercept -- I've got one of the (rare) cases where it is appropriate for E(y) - 0 as X -0. Naively running a GAM with the -1 appended to the formula and the calling predict.gam, I see that the model isn't behaving as expected. I don't understand why this would be. Google turns up this old R help thread: http://r.789695.n4.nabble.com/GAM-without-intercept-td4645786.html Simon writes: *Smooth terms are constrained to sum to zero over the covariate values. ** **This is an identifiability constraint designed to avoid confounding with ** **the intercept (particularly important if you have more than one smooth). * If you remove the intercept from you model altogether (m2) then the smooth will still sum to zero over the covariate values, which in your case will mean that the smooth is quite a long way from the data. When you include the intercept (m1) then the intercept is effectively shifting the constrained curve up towards the data, and you get a nice fit. Why? I haven't read Simon's book in great detail, though I have read Ruppert et al.'s Semiparametric Regression. I don't see a reason why a penalized spline model shouldn't equal the intercept (or zero) when all of the regressors equals zero. Is anyone able to help with a bit of intuition? Or relevant passages from a good description of why this would be the case? Furthermore, why does the -1 formula specification work if it doesn't work as intended by for example lm? Many thanks, Andrew [[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.
