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Consider the data dput here.

I can predict y from x using a cubic spline:

> m <- mgcv::gam(y ~ s(x, bs = "cr"), data = d)
> summary(m)

Family: gaussian 
Link function: identity 

Formula:
y ~ s(x, bs = "cr")

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  6.01174    0.03043   197.6   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
       edf Ref.df    F p-value    
s(x) 6.421  7.538 8.34 1.6e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.0119   Deviance explained = 1.32%
GCV = 4.7363  Scale est. = 4.7294    n = 5109

The p-value for the significance of the smooth term is very small. But what does this mean? When it's a linear regression, for example, we know that the coefficient is likely not zero (if we accept the frequentist assumptions of the model). But it isn't a coefficient—it's an estimated degrees of freedom.

We can also look at a number of coefficients:

> coef(m)
 (Intercept)       s(x).1       s(x).2       s(x).3       s(x).4       s(x).5       s(x).6       s(x).7 
 6.011743981  0.183409981  0.619214052  0.127993730 -0.101947068  0.103454193 -0.005406753 -0.002020089 
      s(x).8       s(x).9 
 0.267600842  2.180296427 

But they don't have their own significance values:

> summary(m)$coef
NULL

We can see the relationship between x and y here:

enter image description here

We see more certainty (less distance from the line) in certain parts than other parts of the spline—how is this interpreted, since each cubic part doesn't have its own significance?

Are significance values here even meaningful, since mgcv chooses the knots based on generalized cross validation?

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You don't say how the data are generated (the response is an integer, and there doesn't look to be much relationship between x and y), and the model diagnostics are terrible for a Gaussian GAM so take what I say below with a good pinch of salt and assume that we fitted the right model for these data.

In the output from a GLM fitted by glm() you'd get a Wald test of the null hypothesis that $\hat{\beta}_j = 0$. The output from summary.gam() is conceptually the same thing, except now, because we are thinking about the null hypothesis for a function, the null hypothesis is that the function $\hat{f}_j(x_{ij}) = 0$.

Essentially, this is a test of deviation from a flat or null function that is constant at 0 over all observed $x_{ij}$.

We're not testing the effective degrees of freedom (EDF) here; thy're just reported as a summary of the complexity of the estimated smooth function. It often doesn't make much sense to test the individual coefficients here either, as we're typically interested in the estimated function, not individual coefficients.

Yes, the significance values are meaningful. They are approximate, in the same sense that p-values for a GLM are approximate (relying on asymptotic behaviour for example), but also they do not account for the selection of the smoothness parameter(s). The theory behind and behaviour of the test statistic used here is given in Wood (2013), and their properties are pretty good in most instances. That said, the p-values have better behaviour when the model is estimated using ML smootheness selection (method = 'ML") or ReML smoothness selection (method = 'REML'), in that order, than where GCV is used. Indeed, the default to use GCV is known to undersmooth in many practical applications, and will likely be not be the default in a future version of mgcv.

Wood, S.N. (2013) On p-values for smooth components of an extended generalized additive model. Biometrika 100:221-228

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  • $\begingroup$ Data generation: x-axis is age, y-axis is how people rated a movie on a 1-to-10 scale. It came out when the 25-35 year-olds were kids, so that nostalgic bump is what I'm interested in. Why do you say that there doesn't look to be much relationship between x and y, given this? Model diagnostics: how do you know they are terrible from this output? The null hypothesis that the function is zero makes sense to me, and I'll definitely fit with maximum-likelihood, considering that citation, which I'll read now. $\endgroup$ – Mark White May 27 at 21:39
  • $\begingroup$ Because I actually went and fitted your model to the data you provided and the diagnostics (mgcv::gam.check() or gratia::appraise() were terrible; if the response is a score like this, then a Gaussian model doesn't make much sense - score can only be an integer, not a continuous value, it is also bounded at 1 and 10. An ordered categorical model may make sense but I am not that familiar with models for ratings like this. $\endgroup$ – Gavin Simpson May 28 at 15:19

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