LOOCV in Caret package ( randomForest example) - not unique results I pose you my doubts:
For what I know there is only a single way to perform a LOOCV for a model (i.e. testing each one of the N elements vs the model trained with the other N-1 elements).
Namely, this should be a LOOCV AUC:
library('randomForest')
library('pROC') #for ROC curve

irisData <- iris[1:60,]
irisData$Species <- as.factor(as.character(irisData$Species))

predictions <- 1:60

for (k in 1:60){ predictions[k] <- predict(randomForest(Species ~ Sepal.Length , data = irisData[-k,], mtry=1),type="prob", newdata = irisData[k,,drop=F])[2] }
auc(irisData$Species,predictions,direction="<", levels = levels(irisData$Species))


Area under the curve: 0.776

Repeating the code i always obtain the same value.
By using caret, i obtain
library('caret')

fitControl <- trainControl(
  method = 'LOOCV',                # k-fold cross validation 'cv'
  number = 1,                     # number of folds
  savePredictions = 'final',       # saves predictions for optimal tuning parameter
  classProbs = T ,                 # should class probabilities be returned
  summaryFunction=twoClassSummary  # results summary function
) 

train(Species ~ Sepal.Length, data=irisData ,method='rf',   tuneGrid=data.frame(mtry=1)  ,trControl = fitControl)

With AUC values between 0.770 and 0.780. 
I tried to change number to 60 but the result is the same.
Where is the issue?
Best.
 A: Yes, for randomForest you need to set the seed, you can see below:
library('randomForest')
library('pROC') 

irisData <- iris[1:60,]
irisData$Species <- as.factor(as.character(irisData$Species))

predictions <- 1:60

for (k in 1:60){
set.seed(1) 
predictions[k] <- predict(randomForest(Species ~ Sepal.Length , 
data = irisData[-k,], mtry=1),type="prob", 
newdata = irisData[k,,drop=F])[2] 
}
auc(irisData$Species,predictions,direction="<", 
levels = levels(irisData$Species))

Area under the curve: 0.776

For caret, you need a list of seed integers the length of your resampling, and the last one is the seed used for prediction on final model. So that makes a list of 61, all 1s, similar to above:
library('caret')

fitControl <- trainControl(
  method = 'LOOCV',                
  number = 1,                     
  savePredictions = 'final',        
  classProbs = T ,
  seed = as.list(rep(1,61)),                
  summaryFunction=twoClassSummary 
) 

train(Species ~ Sepal.Length, data=irisData ,method='rf',   
tuneGrid=data.frame(mtry=1)  ,trControl = fitControl)

60 samples
 1 predictor
 2 classes: 'setosa', 'versicolor' 

No pre-processing
Resampling: Leave-One-Out Cross-Validation 
Summary of sample sizes: 59, 59, 59, 59, 59, 59, ... 
Resampling results:

  ROC    Sens  Spec
  0.776  1     0.6 

