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.
nestedcv adds functionality to enable
cross-validation of the elastic net alpha parameter when fitting
nestedcv partitions the dataset into outer and inner
folds (default 10x10 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
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.
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
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 4.1-4 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) #> AUC Accuracy Balanced accuracy #> 0.9642857 0.8800000 0.8782468 ## 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 #> 0.0001439 0.7000000 #> #> Final coefficients: #> (Intercept) V9730 V7701 V7149 V5121 V181 #> 1.01051 -1.22493 1.06727 -0.89861 0.80456 0.79643 #> V6311 V19946 V14177 V4971 V896 V710 #> 0.78827 0.78478 -0.76635 0.71868 -0.69013 0.68093 #> V133 V483 V5704 V11573 V14628 V8243 #> -0.64858 -0.64803 -0.63564 -0.62821 -0.62014 0.61855 #> V13522 V188 V9073 V11944 V19581 V17645 #> 0.61581 0.60526 0.58630 -0.57317 0.56912 0.56899 #> V5803 V6518 V4087 V15337 V3712 V6394 #> 0.51300 -0.50912 0.50208 -0.49684 0.48826 0.48627 #> V49 V17860 V7294 V15135 V13768 V4590 #> 0.48236 0.47561 0.45865 -0.45787 -0.43488 0.43103 #> V12441 V4537 V5183 V3223 V7322 V13266 #> 0.42777 0.42535 0.42160 -0.42047 -0.40287 0.38810 #> V17816 V7255 V4315 V15384 V2054 V12899 #> -0.37643 -0.37280 0.36663 0.33959 0.31741 0.30857 #> V6506 V4234 V16698 V1486 V6171 V4603 #> -0.30799 0.30782 0.30250 0.30121 -0.28270 0.28131 #> V3108 V10938 V3269 V19345 V775 V17135 #> 0.25656 -0.23655 -0.22815 0.22607 0.21699 -0.21528 #> V6124 V1064 V11411 V8974 V8420 V15671 #> 0.21320 -0.20876 0.20405 -0.20381 -0.19711 -0.19432 #> V4364 V2402 V126 V3302 V1583 V658 #> -0.19142 -0.19114 -0.18329 0.17972 -0.15390 0.14211 #> V6404 V15804 V8317 V38 V11324 V10000 #> -0.12674 -0.11861 0.07605 0.07143 -0.07112 0.06776 #> V5497 V6540 V5925 V2548 V12427 #> 0.06608 -0.05820 0.04333 -0.03802 -0.03797 #> #> Result: #> AUC Accuracy Balanced accuracy #> 0.5786 0.5733 0.5639 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.
Figures C & D below show two valid methods for fitting a model with CV for tuning parameters as well as unbiased estimates of model performance. Figure C is a traditional hold-out test set, with the dataset partitioned 2/3 training, 1/3 test. Notably the critical difference between scheme B above, is that the filtering is only done on the training set and not on the whole dataset.
Figure D shows the scheme for fully nested cross-validation. Note that filtering is applied to each outer CV training fold. The key advantage of nested CV is that outer CV test folds are collated to give an improved estimate of performance compared to scheme C since the numbers for total testing are larger.