nestfs: Cross-validated (nested) forward selection

nestfs provides an implementation of forward selection based on linear and logistic regression which adopts cross-validation as a core component of the selection procedure.

Forward selection is an inherently slow approach, as for each variable a model needs to be fitted. In our implementation, this issue is further aggravated by the fact that an inner cross-validation happens at each iteration, with the aim of guiding the selection towards variables that have better generalization properties.

The code is parallelized over the inner folds, thanks to the parallel package. User time therefore depends on the number of available cores, but there is no advantage in using more cores than inner folds. The number of cores assigned to computations must be registerd before starting by setting the "mc.cores" option.

The main advantage of forward selection is that it provides an immediately interpretable model, and the panel of variables obtained is in some sense the least redundant one, particularly if the number of variables to choose from is not too large (in our experience, up to about 30-40 variables).

However, when the number of variables is much larger than that, forward selection, besides being unbearably slow, may be more subject to overfitting, which is in the nature of its greedy-like design.

A precompiled package is available on CRAN.

Usage

First load the package and register the number of cores to use by setting the mc.cores option. If you are lucky enough to work on a large multicore machine, best performance is achieved by registering as many cores as the number of inner folds being used (the default is 30).

r library(nestfs) options(mc.cores=10)

To run forward selection from a baseline model that contains only age and sex, the following is enough:

```r data(diabetes) fs.res <- fs(Y ~ age + sex, diabetes, family=gaussian()) summary(fs.res)

vars fdr llks diffs iter

1 age NA NA NA NA

2 sex NA -2136.854 NA NA

3 ltg 7.008928e-10 -2058.831 78.022766 1

4 bmi 1.850715e-05 -2009.568 49.263387 2

5 map 2.020038e-03 -1999.253 10.314799 3

6 age.sex 1.508210e-02 -1992.544 6.709064 4

7 hdl 4.039276e-02 -1985.208 7.336623 5

8 bmi.map 7.474167e-02 -1980.913 4.294736 6

```

By default, selection happens over all variables present in the data.frame that are not part of the initial model. This can be controlled through the choose.from option, which accepts variable names or indices.

It is possible to promote sparser selection by requesting a larger improvement in log-likelihood (option min.llk.diff): this is advisable especially when the number of variables to choose from exceeds 10-15, as it's our experience that even the default setting of 2 (already stricter than what adopted by other packages) may lead to some overfitting. In any case, it's possible to set a maximum size of the panel selected by reducing the number of iterations (option max.iters, by default set to 10).

Nested forward selection is helpful to assess the stability of the selection process by performing it on each training split of the cross-validation folds:

```r folds <- create.folds(10, nrow(diabetes), seed=1) nest.res <- nested.fs(Y ~ age + sex, diabetes, family=gaussian(), folds=folds) summary(nest.res)

vars percent coef coefIQR rank rankIQR diffLogLik diffLogLikIQR

1 bmi 100 24.547 (23.61, 25.48) 2 (1.00, 2.00) 61.021 (44.49, 76.85)

2 ltg 100 23.729 (22.39, 24.41) 2 (1.00, 2.00) 52.868 (36.09, 69.36)

3 map 100 15.147 (14.45, 15.88) 3 (3.00, 3.75) 8.366 (8.04, 9.61)

4 hdl 100 -13.297 (-13.65, -12.55) 4 (4.00, 4.00) 6.728 (6.35, 7.83)

5 age.sex 80 8.825 (8.72, 9.24) 5 (5.00, 6.00) 4.625 (4.45, 5.37)

6 bmi.map 70 8.165 (7.55, 8.27) 6 (5.50, 7.00) 3.604 (2.66, 4.15)

7 bmi.glu 20 4.460 (4.07, 4.85) 5 (5.00, 5.00) 3.535 (3.09, 3.98)

8 glu.2 20 6.477 (6.47, 6.49) 6 (6.25, 6.75) 2.984 (2.56, 3.41)

9 sex.map 20 6.862 (6.71, 7.01) 6 (5.25, 5.75) 2.936 (2.89, 2.98)

10 age.glu 10 7.469 (7.47, 7.47) 3 (3.00, 3.00) 4.826 (4.83, 4.83)

11 age.map 10 7.365 (7.36, 7.36) 6 (6.00, 6.00) 2.679 (2.68, 2.68)

12 bmi.2 10 7.987 (7.99, 7.99) 6 (6.00, 6.00) 2.466 (2.47, 2.47)

```

The output above shows that bmi, ltg, map and hdl are chosen in all folds, and most of the improvement in fit is provided by the first two variables, which agrees with what was found when running forward selection on all data.

Most importantly, nested forward selection produces a cross-validated measure of performance of the selection process, which is an unbiased estimate of the predictive performance of the selected panels on withdrawn data: ```r nested.performance(nest.res)

Correlation coefficient: 0.7097

```

This can be compared with what is obtained by the baseline model on the same set of cross-validation folds:

```r base.res <- nested.glm(Y ~ age + sex, diabetes, family=gaussian(), folds=folds) nested.performance(base.res)

Correlation coefficient: 0.1551

```

References

• M. Colombo, H.C. Looker, B. Farran et al., Serum kidney injury molecule 1 and beta-2-microglobulin perform as well as larger panels for prediction of rapid decline in renal function in type 2 diabetes, Diabetologia (2019) 62 (1): 156-168. https://doi.org/10.1007/s00125-018-4741-9

• H.C. Looker, M. Colombo, S. Hess et al., Biomarkers of rapid chronic kidney disease progression in type 2 diabetes, Kidney International (2015), 88 (4): 888-896. https://doi.org/10.1038/ki.2015.199

• H.C. Looker, M. Colombo, F. Agakov et al., Protein biomarkers for the prediction of cardiovascular disease in type 2 diabetes, Diabetologia (2015) 58 (6): 1363-1371. https://doi.org/10.1007/s00125-015-3535-6