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 4.1-8fit <- 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 21 3
## 1 3 23
##
## AUC Accuracy Balanced accuracy
## 0.9327 0.8800 0.8798
## 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
## 0.000134 0.700000
##
## Final coefficients:
## (Intercept) V19583 V10235 V18767 V15040 V2810
## 0.54417 -1.12955 1.07773 1.04470 -0.93285 -0.91779
## V15563 V16357 V8713 V17561 V7235 V6842
## -0.91047 -0.87128 0.83231 0.80264 -0.80175 -0.78071
## V2583 V2389 V6657 V18942 V17139 V13049
## -0.75805 -0.72744 0.71658 0.70005 -0.69589 0.67341
## V17506 V9349 V14745 V10960 V15981 V11725
## 0.66658 -0.66383 0.66304 -0.66170 -0.66160 0.63766
## V14211 V9122 V8130 V15046 V8766 V12289
## -0.63109 -0.61695 -0.58749 0.55648 -0.55578 -0.54262
## V7175 V13311 V18904 V159 V18451 V8567
## -0.53899 -0.53711 -0.52282 -0.50498 0.50383 -0.50110
## V11255 V12980 V17085 V6872 V11489 V19010
## 0.48472 -0.47895 0.44595 0.44090 -0.38280 -0.37894
## V17213 V12162 V12199 V2953 V8288 V9036
## -0.37002 0.34370 -0.33035 0.32582 -0.31367 0.30640
## V11937 V19259 V11406 V16629 V16858 V9860
## 0.29887 -0.29701 -0.27376 -0.26593 -0.26362 -0.25490
## V2823 V8726 V11667 V13770 V5851 V1101
## -0.24259 0.23690 -0.22467 -0.21823 0.21608 0.21401
## V8236 V1524 V133 V14987 V6530 V2854
## -0.19900 0.18288 0.17236 0.16616 -0.16322 -0.14418
## V17283 V8217 V3435 V15048 V5348 V14800
## -0.11862 -0.11170 0.10529 0.10172 0.09143 0.08681
## V5354 V19564 V7438 V9789 V13970 V9505
## -0.08265 -0.07764 0.07651 -0.07013 -0.06825 0.05871
## V15315 V17455 V17145
## -0.05273 0.02718 -0.02508
##
## Result:
## Reference
## Predicted 0 1
## 0 31 46
## 1 42 31
##
## AUC Accuracy Balanced accuracy
## 0.4095 0.4133 0.4136
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.