gam smoother vs parametric term (concurvity difference)

Question

I have a gam model that is:

 gam=gam(sv~s(day,bs="tp")+s(range,bs="tp")+s(time,bs="cc"),data=train.all,gamma=1.4,method="REML")

the s(range) produces an e.d.f of 1, so I made the model:

gam1=gam(sv~s(day,bs="tp")+range+s(time,bs="cc"),data=train.all,gamma=1.4,method="REML")

There is very high concurvity (~0.85) between day and range in the first model (gam), but that goes away in the gam1 model. I am wondering why that is if s(range) is essentially the same as the parametric form of range. Is the concurvity/collinearity (not sure what to call it between a smoother and parametric term) still there, but simply not calculated by mgcv when it is a parametric term? Or are any co-dependence effects truly removed by simply changing "range" to its parametric form?

Answer 1

The concurvity moves from the stated smooth terms to the parametric terms, which concurvity groups in total under the para column of the matrix or matrices returned.

Here's a modified example from ?concurvity

library("mgcv")
## simulate data with concurvity...
set.seed(8)
n<- 200
f2 <- function(x) 0.2 * x^11 * (10 * (1 - x))^6 + 10 *
            (10 * x)^3 * (1 - x)^10
t <- sort(runif(n)) ## first covariate
## make covariate x a smooth function of t + noise...
x <- f2(t) + rnorm(n)*3
## simulate response dependent on t and x...
y <- sin(4*pi*t) + exp(x/20) + rnorm(n)*.3

## fit model...
b <- gam(y ~ s(t,k=15) + s(x,k=15), method="REML")

Now add a linear term and refit

x2 <- seq_len(n) + rnorm(n)*3
b2 <- update(b, . ~ . + x2)

Now look at the concurvity of the two models

## assess concurvity between each term and `rest of model'...
concurvity(b)
concurvity(b2)

These produce

> concurvity(b)
                para       s(t)      s(x)
worst    1.06587e-24 0.60269087 0.6026909
observed 1.06587e-24 0.09576829 0.5728602
estimate 1.06587e-24 0.24513981 0.4659564
> concurvity(b2)
              para      s(t)      s(x)
worst    0.9990068 0.9970541 0.6042295
observed 0.9990068 0.7866776 0.5733337
estimate 0.9990068 0.9111690 0.4668871

Note that x2 is essentially a noisy version of t:

> cor(t, x2)
[1] 0.9975977

and hence the concurvity is gone up from essentially 0 in b to almost 1 in b2.

Now if we add x2 as a smooth function instead...

concurvity(update(b, . ~ . + s(x2)))

we see that the para entries return to being very small and we get a measure for the spline term s(x2) directly

> concurvity(update(b, . ~ . + s(x2)))
                 para      s(t)      s(x)     s(x2)
worst    1.506201e-24 0.9977153 0.6264654 0.9976988
observed 1.506201e-24 0.9838018 0.5893737 0.9963857
estimate 1.506201e-24 0.9909506 0.4921592 0.9943990

This is just how the function works in terms of the parametric terms; the focus is on the smooth terms.

Note: you are specifying gamma but fitting using REML. gamma only affects GCV and UBRE/AIC methods of smoothness selection, so you can remove this argument as it is having zero effect on the model fits. From version 1.8-23 of mgcv, the gamma argument no also affects models fitted using REML/ML, where smoothness parameters are selected BY REML/ML as if the sample size was $n/\gamma$ instead of $n$.