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 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 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 4.1-4
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.