install.packages("devtools")
devtools::install_github("garthtarr/mplot")
require(mplot)

… or get it on CRAN

install.packages("mplot")

Main functions

• vis() for variable inclusion and model stability plots
• af() for the adaptive fence
• bglmnet() for bootstrapping glmnet
• mplot() for an interactive shiny interface

Google

• 2 million pages with "model selection"
• 1/2 million pages with "variable selection"

Google Scholar

• 860,000 articles with "model selection"
• 200,000 articles with "variable selection"

Do we really need more?

Aim: To provide scientists/researchers/analysts with tools that give them more information about the model selection choices that they are making.

Concept of model stability independently introduced by Meinshausen and Bühlmann (2010) and Müller and Welsh (2010) for different linear regression situations.

Key idea: small changes should have small effects

Information Criterion

• Generalised IC: \(\text{GIC}(\alpha;\lambda) = -2\times \text{LogLik}(\alpha)+\lambda p_\alpha\)

With important special cases:

• AIC: \(\lambda=2\)
• BIC: \(\lambda = \log(n)\)

Regularisation routines

• Lasso: minimises \(-\text{LogLik}(\alpha) +\lambda\ ||\beta_\alpha||_1\)
• Many variants of the lasso, SCAD,…

Aim: To visualise inclusion probabilities as a function of the penalty multiplier \(\lambda\in [0,2\log(n)]\).

Procedure

1. Calculate (weighted) bootstrap samples \(b=1,\ldots,B\).
2. For each bootstrap sample, at each \(\lambda\) value, find \(\hat{\alpha}_\lambda^{(b)}\in\mathcal{A}\) as the model with smallest \(\text{GIC}(\alpha;\lambda) = -2\times \text{LogLik}(\alpha)+\lambda p_\alpha\).
3. The inclusion probability for variable \(x_j\) is estimated as \(\frac{1}{B}\sum_{b=1}^B 1\{j\in \hat{\alpha}_\lambda^{(b)}\}\).

References

• Müller and Welsh (2010) for linear regression models
• Murray, Heritier, and Müller (2013) for generalised linear models

Aim: To add value to the loss against size plots using a symbol size proportional to a measure of stability.

Procedure

1. Calculate (weighted) bootstrap samples \(b=1,\ldots,B\).
2. For each bootstrap sample, identify the best model at each dimension.
3. Add this information to the loss against size plot using model identifiers that are proportional to the frequency with which a model was identified as being best at each model size.

References

• Murray, Heritier, and Müller (2013) for generalised linear models

• Let \(Q(\alpha)\) be a measure of lack of fit
• Specifically we consider \(Q(\alpha)=-2\text{LogLik}(\alpha)\)

Main idea

The fence is based aroung the inequality: \[Q(\alpha) \leq Q(\alpha_f) + c .\]

• Model \(\alpha\) is inside the fence if the inequality holds.
• For any \(c\geq 0\), the full model \(\alpha_f\) is always inside the fence.
• Among the set of models that are inside the fence, model(s) with smallest dimension are preferred.

Solution: Bootstrap over a range of values of \(c\).

Procedure

1. For each value of \(c\):
   • Perform parametric bootstrap under \(\alpha_f\).
   • For each bootstrap sample, identify the smallest model that is inside the fence, \(\hat{\alpha}(c)\). Jiang, Nguyen, and Rao (2009) suggest that if there is more than one model, choose the one with the smallest \(Q(\alpha)\).
   • Let \(p^∗(\alpha) = P^∗\{\hat{\alpha}(c) = \alpha\}\) be the empirical probability of selecting model \(\alpha\) at a given value of \(c\).
   • Calculate \(p^* = \max_{\alpha\in\mathcal{A}} p^*(\alpha)\)
2. Plot values of \(p^*\) against \(c\) and find first peak.
3. Use this value of \(c\) with the original data.

• Tibshirani (1996) did regression with an \(L_1\) norm penalty and called it the lasso (least absolute shrinkage and selection operator).

• The lasso parameter estimates are obtained by minimising the residual sum of squares subject to the constraint that \[\sum_j |\beta_j| \leq t.\]

bgn.g = bglmnet(lm.d)
plot(bgn.g, which = "boot", highlight = "ltg")

bgn.d = bglmnet(lm.d)
plot(bgn.d, which = "vip")

• Increase the speed of GLM (approximations)?
• Mixed models (speed is an issue here too)
• Robust alternatives (other than simple screening)
• Cox regression has been requested

Find out more

• Tarr G, Mueller S and Welsh AH (2015). “mplot: An R package for graphical model stability and variable selection.” arXiv:1509.07583 [stat.ME], http://arxiv.org/abs/1509.07583.

Slides: garthtarr.com/pres/hobart2015