GAMs: Model Selection

• Model selection
• Shrinkage smooths
• Shrinkage via double penalty (select = TRUE)
• Confidence intervals for smooths
• p values
• anova()
• AIC

Model (or variable) selection — and important area of theoretical and applied interest

• In statistics we aim for a balance between fit and parsimony
• In applied research we seek the set of covariates with strongest effects on \( y \)

We seek a subset of covariates that improves interpretability and prediction accuracy

Smoothing parameter estimation allows selection of a wide range of potentially complex functions for smooths…

But, cannot remove a term entirely from the model because the penalties used act only on the range space of a spline basis. The null space of the basis is unpenalised.

• Null space — the basis functions that are smooth (constant, linear)
• Range space — the basis functions that are wiggly

mgcv has two ways to penalize the null space, i.e. to do selection

• double penalty approach via select = TRUE
• shrinkage approach via special bases for thin plate and cubic splines

Other shrinkage/selection approaches are available

\( \mathbf{S}_j \) is the smoothing penalty matrix & can be decomposed as

\[ \mathbf{S}_j = \mathbf{U}_j\mathbf{\Lambda}_j\mathbf{U}_j^{T} \]

where \( \mathbf{U}_j \) is a matrix of eigenvectors and \( \mathbf{\Lambda}_j \) a diagonal matrix of eigenvalues (i.e. this is an eigen decomposition of \( \mathbf{S}_j \)).

\( \mathbf{\Lambda}_j \) contains some 0s due to the spline basis null space — no matter how large the penalty \( \lambda_j \) might get no guarantee a smooth term will be suppressed completely.

To solve this we need an extra penalty…

Create a second penalty matrix from \( \mathbf{U}_j \), considering only the matrix of eigenvectors associated with the zero eigenvalues

\[ \mathbf{S}_j^{*} = \mathbf{U}_j^{*}\mathbf{U}_j^{*T} \]

Now we can fit a GAM with two penalties of the form

\[ \lambda_j \mathbf{\beta}^T \mathbf{S}_j \mathbf{\beta} + \lambda_j^{*} \mathbf{\beta}^T \mathbf{S}_j^{*} \mathbf{\beta} \]

Which implies two sets of penalties need to be estimated.

In practice, add select = TRUE to your gam() call

The double penalty approach requires twice as many smoothness parameters to be estimated. An alternative is the shrinkage approach, where \( \mathbf{S}_j \) is replaced by

\[ \tilde{\mathbf{S}}_j = \mathbf{U}_j\tilde{\mathbf{\Lambda}}_j\mathbf{U}_j^{T} \]

where \( \tilde{\mathbf{\Lambda}}_j \) is as before except the zero eigenvalues are set to some small value \( \epsilon \).

This allows the null space terms to be shrunk by the standard smoothing parameters.

Use s(..., bs = "ts") or s(..., bs = "cs") in mgcv

\( \mathbf{S}_j \) can be viewed as prior precision matrices and \( \lambda_j \) as improper Gaussian priors on the spline coefficients.

The impropriety derives from \( \mathbf{S}_j \) not being of full rank (zeroes in \( \mathbf{\Lambda}_j \)).

Both the double penalty and shrinkage smooths remove the impropriety from the Gaussian prior

• Double penalty — makes no assumption as to how much to shrink the null space. This is determined from the data via estimation of \( \lambda_j^{*} \)
• Shrinkage smooths — assumes null space should be shrunk less than the wiggly part

Marra & Wood (2011) show that the double penalty and the shrinkage smooth approaches

• performed significantly better than alternatives in terms of predictive ability, and
• performed as well as alternatives in terms of variable selection

Family: poisson 
Link function: log 

Formula:
y ~ s(x0) + s(x1) + s(x2) + s(x3) + s(x4) + s(x5)

Parametric coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  1.21758    0.04082   29.83   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Approximate significance of smooth terms:
            edf Ref.df  Chi.sq p-value    
s(x0) 1.7655119      9   5.264  0.0397 *  
s(x1) 1.9271039      9  65.356  <2e-16 ***
s(x2) 6.1351372      9 156.204  <2e-16 ***
s(x3) 0.0002618      9   0.000  0.4088    
s(x4) 0.0002766      9   0.000  1.0000    
s(x5) 0.1757146      9   0.195  0.2963    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.545   Deviance explained = 51.6%
-REML = 430.78  Scale est. = 1         n = 200

plot.gam() produces approximate 95% intervals (at +/- 2 SEs)

What do these intervals represent?

Nychka (1988) showed that standard Wahba/Silverman type Bayesian confidence intervals on smooths had good across-the-function frequentist coverage properties.

Marra & Wood (2012) extended this theory to the generalised case and explain where the coverage properties failed:

Musn't over-smooth too much, which happens when \( \lambda_j \) are over-estimated

Two situations where this might occur

1. where true effect is almost in the penalty null space, \( \hat{\lambda}_j \rightarrow \infty \)
2. where \( \hat{\lambda}_j \) difficult to estimate due to highly correlated covariates
   • if 2 correlated covariates have different amounts of wiggliness, estimated effects can have degree of smoothness reversed

In summary, we have shown that Bayesian componentwise variable width intervals… for the smooth components of an additive model should achieve close to nominal across-the-function coverage probability, provided only that we do not over-smooth so heavily… Beyond this requirement not to oversmooth too heavily, the results appear to have rather weak dependence on smoothing parameter values, suggesting that the neglect of smoothing parameter variability should not significantly degrade interval performance.

Marra & Wood (2012) suggested a solution to situation 1., namely true functions close to the penalty null space.

Smooths are normally subject to identifiability constraints (centred), which leads to zero variance where the estimated function crosses the zero line.

Instead, compute intervals for \( j \) th smooth as if it alone had the intercept; identifiability constraints go on the other smooth terms.

Use seWithMean = TRUE in call to plot.gam()

…are approximate:

1. they don't really account for the estimation of \( \lambda_j \) — treated as known
2. rely on asymptotic behaviour — they tend towards being right as sample size tends to \( \infty \)

Also, p values in summary.gam() have changed a lot over time — all options except current default are deprecated as of v1.18-13.

The approach described in Wood (2006) is “no longer recommended”!

…are a test of zero-effect of a smooth term

Default p values rely on theory of Nychka (1988) and Marra & Wood (2012) for confidence interval coverage.

If the Bayesian CI have good across-the-function properties, Wood (2013a) showed that the p values have

• almost the correct null distribution
• reasonable power

Test statistic is a form of \( \chi^2 \) statistic, but with complicated degrees of freedom.

The results of Nychka (1988) and Marra & Wood (2012) break down if smooth terms are unpenalized.

This include i.i.d. Gaussian random effects, (e.g. bs = "re".)

Wood (2013b) proposed instead a test based on a likelihood ratio statistic:

• the reference distribution used is appropriate for testing a \( \mathrm{H}_0 \) on the boundary of the allowed parameter space…
• …in other words, it corrects for a \( \mathrm{H}_0 \) that a variance term is zero.

have the best behaviour when smoothness selection is done using ML, then REML.

Neither of these are the default, so remember to use method = "ML" or method = "REML" as appropriate

…are based on Wald statistics using the Bayesian covariance matrix for the coefficients.

This is the “right thing to do” when there are random effects terms present and doesn't really affect performance if there aren't.

Hence in most instances you won't need to change the default freq = FALSE in summary.gam()

mgcv provides an anova() method for "gam" objects:

1. Single model form: anova(m1)
2. Multi model form: anova(m1, m2, m3)

This differs from anova() methods for "lm" or "glm" objects:

• the tests are Wald-like tests as described for summary.gam() of a \( \mathrm{H}_0 \) of zero-effect of a smooth term
• these are not sequential tests!

b1 <- gam(y ~ x0 + s(x1) + s(x2) + s(x3), method = "REML")
anova(b1)

Family: gaussian 
Link function: identity 

Formula:
y ~ x0 + s(x1) + s(x2) + s(x3)

Parametric Terms:
   df     F  p-value
x0  3 26.94 1.57e-14

Approximate significance of smooth terms:
        edf Ref.df      F  p-value
s(x1) 1.000  1.001 26.677 5.83e-07
s(x2) 6.694  7.807 18.755  < 2e-16
s(x3) 1.000  1.000  0.068    0.795

The multi-model form should really be used with care — the p values are really approximate

b1 <- gam(y ~ s(x0) + s(x1) + s(x2) + s(x3) + s(x4) + s(x5), data = dat,
         family=poisson, method = "ML")
b2 <- update(b1, . ~ . - s(x3) - s(x4) - s(x5))
anova(b2, b1, test = "LRT")

Analysis of Deviance Table

Model 1: y ~ s(x0) + s(x1) + s(x2)
Model 2: y ~ s(x0) + s(x1) + s(x2) + s(x3) + s(x4) + s(x5)
  Resid. Df Resid. Dev     Df Deviance Pr(>Chi)
1    186.23     248.97                         
2    183.34     248.01 2.8959  0.96184    0.795

For general smooths deviance is replaced by \( -2\mathcal{L}(\hat{\beta}) \)

• Comparison of GAMs by a form of AIC is an alternative frequentist approach to model selection
• Rather than using the marginal likelihood, the likelihood of the \( \mathbf{\beta}_j \) conditional upon \( \lambda_j \) is used, with the EDF replacing \( k \), the number of model parameters
• This conditional AIC tends to select complex models, especially those with random effects, as the EDF ignores that \( \lambda_j \) are estimated
• Wood et al (2015) suggests a correction that accounts for uncertainty in \( \lambda_j \)

\[ AIC = -2l(\hat{\beta}) + 2\mathrm{tr}(\widehat{\mathcal{I}}V^{'}_{\beta}) \]

In this example, \( x_3 \), \( x_4 \), and \( x_5 \) have no effects on \( y \)

AIC(b1, b2)

         df      AIC
b1 15.03493 847.7961
b2 12.12435 842.9368

• Marra & Wood (2011) Computational Statistics and Data Analysis 55 2372–2387.
• Marra & Wood (2012) Scandinavian journal of statistics, theory and applications 39(1), 53–74.
• Nychka (1988) Journal of the American Statistical Association 83(404) 1134–1143.
• Wood (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC.
• Wood (2013a) Biometrika 100(1) 221–228.
• Wood (2013b) Biometrika 100(4) 1005–1010.