knitr::opts_chunk$set( collapse = TRUE, warning = FALSE ) library(nestedcv) library(pROC)

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.

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.

install.packages("nestedcv") library(nestedcv)

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) 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) ## Nested CV fit2 <- nestcv.glmnet(y, x, family = "binomial", alphaSet = 7:10 / 10, filterFUN = ttest_filter, filter_options = list(nfilter = 100)) fit2 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.

knitr::include_graphics("fig1.svg")

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.

knitr::include_graphics("fig2.svg")

In the real life example below, RNA-Sequencing gene expression data from synovial biopsies from patients with rheumatoid arthritis in the R4RA randomised clinical trial (Humby et al, 2021) is used to predict clinical response to the biologic drug rituximab. Treatment response is determined by a clinical measure, namely Clinical Disease Activity Index (CDAI) 50% response, which has a binary outcome: treatment success or failure (response or non-response). This dataset contains gene expression on over 50,000 genes in arthritic synovial tissue from 133 individuals, who were randomised to two drugs (rituximab and tocilizumab). First, we remove genes of low expression using a median cut-off (this still leaves >16,000 genes), and we subset the dataset to the rituximab treated individuals (n=68).

# Raw RNA-Seq data for this example is located at: # https://www.ebi.ac.uk/arrayexpress/experiments/E-MTAB-11611/ # set up data load("/../R4RA_270821.RData") index <- r4ra.meta$Outliers_Detected_On_PCA != "outlier" & r4ra.meta$Visit == 3 & !is.na(r4ra.meta$Visit) metadata <- r4ra.meta[index, ] dim(metadata) # 133 individuals medians <- Rfast::rowMedians(as.matrix(r4ra.vst)) data <- t(as.matrix(r4ra.vst)) # remove low expressed genes data <- data[index, medians > 6] dim(data) # 16254 genes # Rituximab cohort only yrtx <- metadata$CDAI.response.status.V7[metadata$Randomised.medication == "Rituximab"] yrtx <- factor(yrtx) data.rtx <- data[metadata$Randomised.medication == "Rituximab", ] # no filter res.rtx <- nestcv.glmnet(y = yrtx, x = data.rtx, family = "binomial", cv.cores = 8, alphaSet = seq(0.7, 1, 0.05)) res.rtx

## Nested cross-validation with glmnet ## No filter ## ## Final parameters: ## lambda alpha ## 0.1511 0.7950 ## ## Final coefficients: ## (Intercept) AC016582.3 PCBP3 TMEM170B EIF4E3 SEC14L6 CEP85 APLF ## 0.8898659 -0.2676580 -0.2667770 0.2456329 0.2042326 -0.1992225 0.1076051 -0.1072684 ## EARS2 PTK7 EFNA5 MEST IQANK1 MTATP6P1 GSK3B STK40 ## -0.1036846 -0.0919594 -0.0882686 0.0769173 -0.0708992 0.0545392 0.0469272 0.0316988 ## SUV39H2 AC005670.2 ZNF773 XIST STAU2 DIRAS3 ## 0.0297370 0.0184851 -0.0170861 -0.0100934 0.0016182 -0.0009975 ## ## Result: ## AUC Accuracy Balanced accuracy ## 0.7648 0.7059 0.6773

Use `summary()`

to see full information from the nested model fitting. `coef()`

can be used to show the coefficients of the final fitted model. For comparison,
performance metrics from the left-out inner CV test folds can be viewed using
`innercv_summary()`

. Performance metrics on the outer training folds can be
viewed with `train_summary()`

, provided the argument `outer_train_predict`

was
set to `TRUE`

in the original call to either `nestcv.glmnet()`

, `nestcv.train()`

or `outercv()`

.

Filters can be used by setting the `filterFUN`

argument. Options for the filter
function are passed as a list through `filter_options`

.

# t-test filter res.rtx <- nestcv.glmnet(y = yrtx, x = data.rtx, filterFUN = ttest_filter, filter_options = list(nfilter = 300, p_cutoff = NULL), family = "binomial", cv.cores = 8, alphaSet = seq(0.7, 1, 0.05)) summary(res.rtx)

Output from the nested CV with glmnet can be plotted to show how deviance is affected by alpha and lambda.

plot_alphas(res.rtx) plot_lambdas(res.rtx)

knitr::include_graphics("plot_alpha_lam.png")

The tuning of alpha for each outer fold can be plotted.

# Fold 1 line plot plot(res.rtx$outer_result[[1]]$cvafit) # Scatter plot plot(res.rtx$outer_result[[1]]$cvafit, type = 'p') # Number of non-zero coefficients plot(res.rtx$outer_result[[1]]$cvafit, xaxis = 'nvar')

knitr::include_graphics("plot_cva.png")

ROC curves from left-out folds from both outer and inner CV can be plotted. Note that the AUC based on the left-out outer folds is the unbiased estimate of accuracy, while the left-out inner folds demonstrate bias due to the optimisation of the model's hyperparameters on the inner fold data.

# Outer CV ROC plot(res.rtx$roc, main = "Outer fold ROC", font.main = 1, col = 'blue') legend("bottomright", legend = paste0("AUC = ", signif(pROC::auc(res.rtx$roc), 3)), bty = 'n') # Inner CV ROC rtx.inroc <- innercv_roc(res.rtx) plot(rtx.inroc, main = "Inner fold ROC", font.main = 1, col = 'red') legend("bottomright", legend = paste0("AUC = ", signif(pROC::auc(rtx.inroc), 3)), bty = 'n')

knitr::include_graphics("roc.png")

The overall expression level of each gene selected in the final model can be compared with a boxplot.

boxplot_model(res.rtx, ylab = "VST")

knitr::include_graphics("boxplot.png")

Leave-one-out cross-validation (LOOCV) can be performed on the outer folds.

# Outer LOOCV res.rtx <- nestcv.glmnet(y = yrtx, x = data.rtx, min_1se = 0, filterFUN = ttest_filter, filter_options = list(nfilter = 300, p_cutoff = NULL), outer_method = "LOOCV", family = "binomial", cv.cores = 8, alphaSet = seq(0.7, 1, 0.05)) summary(res.rtx)

`glmnet`

can only accept a numeric matrix of predictors. Dataframes of
predictors with factors or character variables can be converted to a matrix
using one-hot encoding to create dummy variables using the function `one_hot()`

.
For `nestcv.glmnet`

models, `one_hot()`

should be used with the argument
`all_levels = FALSE`

to avoid the predictor matrix not being full rank.

Multiple filters for variable reduction are available including:

`ttest_filter`

t-test

`wilcoxon_filter`

Wilcoxon (Mann-Whitney) test

`anova_filter`

one-way ANOVA

`correl_filter`

Pearson or Spearman correlation for regression modelling

`lm_filter`

linear model with covariates

`rf_filter`

random forest variable importance using randomForest package

`ranger_filter`

random forest variable importance using ranger package

`relieff_filter`

ReliefF and other methods available via CORElearn

`glmnet_filter`

uses sparsity of elastic net regression using glmnet to
restrict variables

`boruta_filter`

Boruta

`stat_filter`

A swiss army knife univariate statistical filter for mixed
data with combined continuous & categorical data

# Random forest filter res.rtx <- nestcv.glmnet(y = yrtx, x = data.rtx, min_1se = 0.5, filterFUN = rf_filter, filter_options = list(nfilter = 300), family = "binomial", cv.cores = 8, alphaSet = seq(0.7, 1, 0.05)) summary(res.rtx) # ReliefF algorithm filter res.rtx <- nestcv.glmnet(y = yrtx, x = data.rtx, min_1se = 0, filterFUN = relieff_filter, filter_options = list(nfilter = 300), family = "binomial", cv.cores = 8, alphaSet = seq(0.7, 1, 0.05)) summary(res.rtx)

Bootstrapped versions of the univariate filters are available [see
`boot_ttest()`

]. These use repeated random sampling to try to improve stability
of ranking of predictors based on univariate statistics.

`stat_filter()`

is a swiss army knife filter which can handle dataframes with
mixed data with combined continuous & categorical predictors. It uses different
univariate statistics dependent on data type. For example, t-test is used for
binary outcome with continuous predictors; correlation or linear regression for
continuous outcome with continuous predictors etc. It can also be used to
quickly generate summary statistics relating predictors to the response
variable.

library(mlbench) data(BostonHousing2) dat <- BostonHousing2 y <- dat$cmedv ## continuous outcome x <- subset(dat, select = -c(cmedv, medv, town)) stat_filter(y, x, type = "full")

It is fairly straightforward to create your own custom filter, which can be
embedded within the outer CV loops via `nestcv.glmnet`

, `nestcv.train`

or
`outercv`

. The function simply must be of the form

filter <- function(y, x, ...) {}

Other arguments can be passed in to the filter function as a named list via the
`filter_options`

argument. The function must return a vector of indices of those
predictors in `x`

which are to be retained for downstream model fitting as well
as prediction on left-out outer folds. Importantly the filter function is
applied independently to each outer CV fold and not run on the whole data.

Finally once the model performance has been calculated by nested CV. The filter is applied to the whole dataset when refitting the final model to the full dataset.

Class imbalance is known to impact on model fitting for certain model types, e.g. random forest, SVM. Models may tend to aim to predict the majority class and ignore the minority class since selecting the majority class can give high accuracy purely by chance. While performance measures such as balanced accuracy can give improved estimates of model performance, techniques for rebalancing data have been developed. These include:

- Random oversampling of the minority class
- Random undersampling of the majority class
- Combination of oversampling and undersampling
- Synthesising new data in the minority class, e.g. SMOTE (Chawla et al, 2002)

These are available within `nestedcv`

using the `balance`

argument to specify a
balancing function. Other arguments to control the balancing process are passed
to the function as a list via `balance_options`

.

`randomsample`

Random oversampling of the minority class and/or undersampling
of the majority class

`smote`

Synthetic minority oversampling technique (SMOTE)

Note that in `nestedcv`

balancing is performed only on the outer training folds,
immediately prior to filtering of features. This is important as balancing the
whole dataset prior to the outer CV leads to data leakage of outer CV hold-out
samples into the outer training folds.

The number of samples in each class in the outer CV folds can be checked on
`nestedcv`

objects using the function `class_balance()`

.

For logistic regression methods like glmnet, balancing may actually be deleterious. Using weights to apply increased weight to minority samples is typically preferred as a method for improving model fitting.

The following example simulates an imbalanced dataset with 150 samples and 20,000 predictors of which only the first 30 are weak predictors.

## Imbalanced dataset set.seed(1, "L'Ecuyer-CMRG") x <- matrix(rnorm(150 * 2e+04), 150, 2e+04) # predictors y <- factor(rbinom(150, 1, 0.2)) # imbalanced binary response table(y) ## first 30 parameters are weak predictors x[, 1:30] <- rnorm(150 * 30, 0, 1) + as.numeric(y)*0.7

Here we illustrate the use of `randomsample()`

to balance x & y *outside* of the
CV loop by random oversampling minority group. Then we fit a nested CV glmnet
model on the balanced data.

out <- randomsample(y, x) y2 <- out$y x2 <- out$x table(y2) ## Nested CV glmnet with unnested balancing by random oversampling on ## whole dataset fit1 <- nestcv.glmnet(y2, x2, family = "binomial", alphaSet = 1, n_outer_folds = 4, cv.cores=2, filterFUN = ttest_filter) fit1$summary

Alternatively choices for dealing with imbalance include balancing x & y outside of CV loop by random oversampling minority group, or by SMOTE.

out <- randomsample(y, x, minor=1, major=0.4) y2 <- out$y x2 <- out$x table(y2) ## Nested CV glmnet with unnested balancing by random undersampling on ## whole dataset fit1b <- nestcv.glmnet(y2, x2, family = "binomial", alphaSet = 1, n_outer_folds = 4, cv.cores=2, filterFUN = ttest_filter) fit1b$summary ## Balance x & y outside of CV loop by SMOTE out <- smote(y, x) y2 <- out$y x2 <- out$x table(y2) ## Nested CV glmnet with unnested balancing by SMOTE on whole dataset fit2 <- nestcv.glmnet(y2, x2, family = "binomial", alphaSet = 1, n_outer_folds = 4, cv.cores=2, filterFUN = ttest_filter) fit2$summary

Finally, we show full nesting of both sample balancing and feature filtering
within the outer CV loop, using `nestcv.glmnet()`

.

## Nested CV glmnet with nested balancing by random oversampling fit3 <- nestcv.glmnet(y, x, family = "binomial", alphaSet = 1, n_outer_folds = 4, cv.cores=2, balance = "randomsample", filterFUN = ttest_filter) fit3$summary

For regression, an alternative method is to use weights. Increasing the weight
of the minority class can improve performance. A simple function `weight()`

is
provided which increases weight for minority samples to give equal balance
across classes according to their overall proportions. This function works for
both binary and multiclass classification.

## Nested CV glmnet with weights w <- weight(y) table(w) fit4 <- nestcv.glmnet(y, x, family = "binomial", alphaSet = 1, n_outer_folds = 4, cv.cores=2, weights = w, filterFUN = ttest_filter) fit4$summary

Finally we plot ROC curves to illustrate the differences between these approaches.

plot(fit1$roc, col='green') lines(fit1b$roc, col='red') lines(fit2$roc, col='blue') lines(fit3$roc) lines(fit4$roc, col='purple') legend('bottomright', legend = c("Unnested random oversampling", "Unnested SMOTE", "Unnested random undersampling", "Nested random oversampling", "Nested glmnet with weights"), col = c("green", "blue", "red", "black", "purple"), lty = 1, lwd = 2, bty = "n", cex=0.8)

This shows that unnested oversampling and unnested SMOTE leads to a data leak resulting in upward bias in apparent performance. The correct unbiased estimate of performance is 'nested random oversampling'. Interestingly unnested random undersampling does not lead to any leakage of samples into left-out test folds, but the reduction in data size means that training is adversely affected and performance of the trained models is poor.

A custom balancing function can be provided. The function must be of the form:

balance <- function(y, x, ...) { return(list(y = y, x = x)) }

Other arguments can be passed in to the balance function as a named list via the
`balance_options`

argument. The function must return a list containing `y`

an
expanded/altered response vector and `x`

the matrix or dataframe of predictors
with increased/decreased samples in rows and predictors in columns.

Nested CV can also be performed using the `caret`

package framework written by
Max Kuhn (https://topepo.github.io/caret/index.html). This enables access to the
large library of machine learning models available within caret.

Here we use `caret`

for tuning the alpha and lambda parameters of `glmnet`

.

# nested CV using caret tg <- expand.grid(lambda = exp(seq(log(2e-3), log(1e0), length.out = 100)), alpha = seq(0.8, 1, 0.1)) ncv <- nestcv.train(y = yrtx, x = data.rtx, method = "glmnet", savePredictions = "final", filterFUN = ttest_filter, filter_options = list(nfilter = 300), tuneGrid = tg, cv.cores = 8) ncv$summary # Plot ROC on outer folds plot(ncv$roc) # Plot ROC on inner LO folds inroc <- innercv_roc(ncv) plot(inroc) pROC::auc(inroc) # Extract coefficients of final fitted model glmnet_coefs(ncv$final_fit$finalModel, s = ncv$finalTune$lambda)

It is important to try calls to `nestcv.train`

with `cv.cores=1`

first. With
`caret`

this may flag up that specific packages are not installed or that there
are problems with input variables `y`

and `x`

which may have to be corrected for
the call to run in multicore mode. Once it is clear that the call to
`nestcv.train`

is working ok, you can quit single core execution and restart in
multicore mode.

It is important to realise that the `train()`

function in `caret`

sets a
parameter known as `tuneLength`

to 3 by default, so the default tuning grid is
minimal. `tuneLength`

can easily be increased to give a tuning grid of greater
granularity. Tuneable parameters for individual models can be inspected using
`modelLookup()`

, while `getModelInfo()`

gives further information.

When fitting classification models, the usual default metric for tuning model
hyperparameters in `caret`

is Accuracy. However, with small datasets, accuracy
is disproportionately affected by changes in a single individual's prediction
outcome from incorrect to correctly classified or vice versa. For this reason,
we suggest using logLoss with smaller datasets as it provides more stable
measures of model tuning behaviour. In `nestedcv`

, when fitting classification
models with `caret`

, the default metric is changed to use logLoss.

We recommend that the results of tuning are plotted to understand whether parameters have a systematic effect on model accuracy. With small datasets tuning may pick parameters partially at random because of random fluctuations in measured accuracy during tuning, which may worsen noise surrounding performance than if they were fixed.

# Example tuning plot for outer fold 1 plot(ncv$outer_result[[1]]$fit, xTrans = log) # ggplot2 version ggplot(ncv$outer_result[[1]]$fit) + scale_x_log10()

By default `nestedcv`

stratifies outer & inner CV folds based on the outcome
variable using caret's `createFolds()`

function. Users can control CV folds
directly by providing their own lists of fold indices. Outer CV fold indices can
simply be provided via the argument `outer_folds`

to `nestcv.glmnet()`

or
`nestcv.train()`

. This is a list of vectors with one vector of indices used for
each test CV fold.

The inner CV fold indices can also be controlled via the `inner_folds`

argument.
This is a list of a list of vectors. This can be useful if control of
stratification is desired. The toy example below demonstrates how the
`outer_folds`

and `inner_folds`

objects have to be structured.

data(iris) y <- iris$Species x <- iris[, -5] out_folds <- caret::createFolds(y, k = 8) in_folds <- lapply(out_folds, function(i) { train_y <- y[-i] caret::createFolds(train_y, k = 8) }) res <- nestcv.train(y, x, method = "rf", cv.cores = 8, inner_folds = in_folds, outer_folds = out_folds) summary(res) res$outer_folds # show which outer fold indices were used

In this example the outcome variable `y`

has been used to stratify the folds for
both outer and inner CV for 8 x 8 nested CV. But it could easily be replaced
with a variable from the predictor dataframe if stratification by an imbalanced
group is desired. For grouped data where members of a group must be kept
together try using caret's `groupKFold()`

function. For stratification with
multiple columns (e.g. response + one or more grouping variables), try
`multi_strata()`

from the `splitTools`

package.

For all of the `nestedcv`

model training functions described above, predictions
on new data can be made by referencing the final fitted object which is fitted
to the whole dataset.

# for nestcv.glmnet object preds <- predict(res.rtx, newdata = data.rtx, type = 'response') # for nestcv.train object preds <- predict(ncv, newdata = data.rtx)

See the accompanying vignette "Using *outercv* with Bayesian shrinkage models".

Currently `nestcv.glmnet`

, `nestcv.train`

and `outercv`

all allow
parallelisation of the outer CV loop using `mclapply`

from the `parallel`

package on unix/mac and `parLapply`

on windows. This is enabled by specifying
the number of cores using the argument `cv.cores`

. Since in some cases the
filtering process can be time consuming (e.g. `rf_filter`

, `relieff_filter`

), we
tend to recommend parallelisation of the outer CV loop over parallelisation of
the inner CV loop.

To check the number of physical cores, use:

parallel::detectCores(logical = FALSE)

To maintain consistency with random numbers during parallelisation set the random number seed in the following way:

set.seed(123, "L'Ecuyer-CMRG")

The `caret`

package is set up for parallelisation using `foreach`

(https://topepo.github.io/caret/parallel-processing.html). We generally do not
recommend nested parallelisation of both outer and inner loops, although this is
theoretically feasible if you have enough cores. If *P* processors are
registered with the parallel backend, some `caret`

functionality leads to *P*^2^
processes being generated. We generally find this does not lead to much speed up
once the number of processes reaches the number of physical cores, as all
processors are saturated and there is both time and memory overheads for
duplicating data and packages for each process.

A key problem with parallelisation in R is that errors, warnings and user input
have to be suppressed during multicore processing. If a `nestedcv`

call is not
working, we recommend that you try it with `cv.cores=1`

first to check it starts
up without error messages.

If you use this package, please cite as:

Lewis MJ, Spiliopoulou A, Goldmann K, Pitzalis C, McKeigue P, Barnes MR (2023).
nestedcv: an R package for fast implementation of nested cross-validation with
embedded feature selection designed for transcriptomics and high dimensional
data. *Bioinformatics Advances*. https://doi.org/10.1093/bioadv/vbad048

Chawla, NV, Bowyer KW, Hall LO, Kegelmeyer WP. Smote: Synthetic minority
over-sampling technique. *Journal of Artificial Intelligence Research*
2002; 16:321-357

Humby F, Durez P, Buch MH, Lewis MJ et al. Rituximab versus tocilizumab in
anti-TNF inadequate responder patients with rheumatoid arthritis (R4RA): 16-week
outcomes of a stratified, biopsy-driven, multicentre, open-label, phase 4
randomised controlled trial. *Lancet* 2021; 397(10271):305-317.
https://doi.org/10.1016/s0140-6736(20)32341-2

Kuhn, M. Building Predictive Models in R Using the caret Package.
*Journal of Statistical Software* 2008; 28(5):1-26.

Rivellese F, Surace AEA, Goldmann K, et al, Lewis MJ, Pitzalis C and the R4RA
collaborative group. Rituximab versus tocilizumab in rheumatoid arthritis:
Synovial biopsy-based biomarker analysis of the phase 4 R4RA randomized trial.
*Nature medicine* 2022; 28(6):1256-1268.
https://doi.org/10.1038/s41591-022-01789-0

Vabalas A, Gowen E, Poliakoff E, Casson AJ. Machine learning algorithm
validation with a limited sample size. *PloS one* 2019;14(11):e0224365.

Zou, H, Hastie, T. Regularization and variable selection via the elastic net.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*
2005; 67(2): 301-320.

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