nestedcv
Myles Lewis, Athina Spiliopoulou, Katriona Goldmann
Introduction
The motivation for this package is to provide functions which help with the development and tuning of machine learning models in biomedical data where the sample size is frequently limited, but the number of predictors may be significantly larger (P >> n). While most machine learning pipelines involve splitting data into training and testing cohorts, typically 2/3 and 1/3 respectively, medical datasets may be too small for this, and so determination of accuracy in the left-out test set suffers because the test set is small. Nested cross-validation (CV) provides a way to get round this, by maximising use of the whole dataset for testing overall accuracy, while maintaining the split between training and testing.
In addition typical biomedical datasets often have many 10,000s of possible predictors, so filtering of predictors is commonly needed. However, it has been demonstrated that filtering on the whole dataset creates a bias when determining accuracy of models (Vabalas et al, 2019). Feature selection of predictors should be considered an integral part of a model, with feature selection performed only on training data. Then the selected features and accompanying model can be tested on hold-out test data without bias. Thus, it is recommended that any filtering of predictors is performed within the CV loops, to prevent test data information leakage.
This package enables nested cross-validation (CV) to be performed
using the commonly used glmnet package, which fits elastic
net regression models, and the caret package, which is a
general framework for fitting a large number of machine learning models.
In addition, nestedcv adds functionality to enable
cross-validation of the elastic net alpha parameter when fitting
glmnet models.
nestedcv partitions the dataset into outer and inner
folds (default 10 x 10 folds). The inner fold CV, (default is 10-fold),
is used to tune optimal hyperparameters for models. Then the model is
fitted on the whole inner fold and tested on the left-out data from the
outer fold. This is repeated across all outer folds (default 10 outer
folds), and the unseen test predictions from the outer folds are
compared against the true results for the outer test folds and the
results concatenated, to give measures of accuracy (e.g. AUC and
accuracy for classification, or RMSE for regression) across the whole
dataset.
A final round of CV is performed on the whole dataset to determine hyperparameters to fit the final model to the whole data, which can be used for prediction with external data.
Variable selection
While some models such as glmnet allow for sparsity and
have variable selection built-in, many models fail to fit when given
massive numbers of predictors, or perform poorly due to overfitting
without variable selection. In addition, in medicine one of the goals of
predictive modelling is commonly the development of diagnostic or
biomarker tests, for which reducing the number of predictors is
typically a practical necessity.
Several filter functions (t-test, Wilcoxon test, anova,
Pearson/Spearman correlation, random forest variable importance, and
ReliefF from the CORElearn package) for feature selection
are provided, and can be embedded within the outer loop of the nested
CV.
Installation
install.packages("nestedcv")
library(nestedcv)Examples
Importance of nested CV
The following simulated example demonstrates the bias intrinsic to datasets where P >> n when applying filtering of predictors to the whole dataset rather than to training folds.
## Example binary classification problem with P >> n
x <- matrix(rnorm(150 * 2e+04), 150, 2e+04) # predictors
y <- factor(rbinom(150, 1, 0.5)) # binary response
## Partition data into 2/3 training set, 1/3 test set
trainSet <- caret::createDataPartition(y, p = 0.66, list = FALSE)
## t-test filter using whole test set
filt <- ttest_filter(y, x, nfilter = 100)
filx <- x[, filt]
## Train glmnet on training set only using filtered predictor matrix
library(glmnet)
## Loading required package: Matrix
## Loaded glmnet 4.1-8
fit <- cv.glmnet(filx[trainSet, ], y[trainSet], family = "binomial")
## Predict response on test set
predy <- predict(fit, newx = filx[-trainSet, ], s = "lambda.min", type = "class")
predy <- as.vector(predy)
predyp <- predict(fit, newx = filx[-trainSet, ], s = "lambda.min", type = "response")
predyp <- as.vector(predyp)
output <- data.frame(testy = y[-trainSet], predy = predy, predyp = predyp)
## Results on test set
## shows bias since univariate filtering was applied to whole dataset
predSummary(output)
## Reference
## Predicted 0 1
## 0 22 5
## 1 4 19
##
## AUC Accuracy Balanced accuracy
## 0.9167 0.8200 0.8189
## Nested CV
fit2 <- nestcv.glmnet(y, x, family = "binomial", alphaSet = 7:10 / 10,
filterFUN = ttest_filter,
filter_options = list(nfilter = 100))
fit2
## Nested cross-validation with glmnet
## Filter: ttest_filter
##
## Final parameters:
## lambda alpha
## 0.0002116 0.7000000
##
## Final coefficients:
## (Intercept) V99 V3223 V9456 V12948 V17633
## 0.01672 1.19589 -1.10712 0.91087 0.82747 0.82666
## V19857 V11425 V16734 V3137 V7611 V17451
## 0.81021 -0.79535 -0.79168 0.76315 -0.75806 -0.74216
## V9407 V10165 V10692 V18079 V15195 V18124
## -0.73168 0.72503 -0.72442 -0.70206 -0.69217 0.64493
## V10061 V11338 V3297 V4779 V19993 V10399
## 0.63476 0.59524 -0.58129 -0.57781 0.57085 -0.56759
## V11311 V8527 V11238 V16738 V7527 V19599
## -0.56660 0.56580 0.55922 -0.55844 0.54260 -0.53113
## V16504 V7447 V9240 V2769 V18423 V14809
## -0.52702 0.51212 0.50956 -0.49778 0.48351 -0.47591
## V10193 V2707 V8645 V14476 V271 V6045
## -0.46911 0.46586 0.45852 0.44108 -0.43848 -0.41254
## V7681 V2891 V18605 V11883 V18563 V2200
## 0.40246 0.35620 0.35218 -0.34953 0.33033 -0.31492
## V19651 V18198 V14620 V4521 V7219 V7257
## -0.30946 0.28250 -0.28178 -0.27187 0.26827 -0.26542
## V8512 V19174 V1976 V429 V8086 V14961
## -0.24532 0.23876 0.23275 -0.23151 0.22628 0.22058
## V9038 V3845 V9950 V19945 V6101 V16942
## 0.20790 0.18985 -0.17692 0.17144 -0.15608 0.14826
## V14138 V14439 V16317 V1020 V14435 V4989
## 0.14416 0.12220 -0.11686 -0.11430 -0.09818 -0.09571
## V15043 V293 V11170 V18509 V14340 V2239
## 0.09477 0.09271 0.08407 0.07047 0.05934 -0.01955
##
## Result:
## Reference
## Predicted 0 1
## 0 51 42
## 1 27 30
##
## AUC Accuracy Balanced accuracy
## 0.5235 0.5400 0.5353
testroc <- pROC::roc(output$testy, output$predyp, direction = "<", quiet = TRUE)
inroc <- innercv_roc(fit2)
plot(fit2$roc)
lines(inroc, col = 'blue')
lines(testroc, col = 'red')
legend('bottomright', legend = c("Nested CV", "Left-out inner CV folds",
"Test partition, non-nested filtering"),
col = c("black", "blue", "red"), lty = 1, lwd = 2, bty = "n")
In this example the dataset is pure noise. Filtering of predictors on the whole dataset is a source of leakage of information about the test set, leading to substantially overoptimistic performance on the test set as measured by ROC AUC.
Figures A & B below show two commonly used, but biased methods in which cross-validation is used to fit models, but the result is a biased estimate of model performance. In scheme A, there is no hold-out test set at all, so there are two sources of bias/ data leakage: first, the filtering on the whole dataset, and second, the use of left-out CV folds for measuring performance. Left-out CV folds are known to lead to biased estimates of performance as the tuning parameters are ‘learnt’ from optimising the result on the left-out CV fold.
In scheme B, the CV is used to tune parameters and a hold-out set is used to measure performance, but information leakage occurs when filtering is applied to the whole dataset. Unfortunately this is commonly observed in many studies which apply differential expression analysis on the whole dataset to select predictors which are then passed to machine learning algorithms.