Table 1: Squared error, R detailed values and corresponding rank on Jaguar trial data for 10-fold, 4-fold, 3-fold, and 2-fold. 3 6 2 4 5 1 3 6 2 4 5 1 SqE without Logic Reg. Rank 4-fold Rank 3-fold 1 − R without Logic Reg. Rank 4-fold Rank 3-fold Mean 0.246 1.267 0.241 0.264 0.267 0.238 Mean 0.554 0.724 0.552 0.623 0.644 0.539 3 6 2 4 5 1 3 6 2 4 5 1 Mean 0.238 1.650 0.235 0.255 0.258 0.228 Mean 0.534 0.686 0.532 0.607 0.611 0.508 Rank 3 6 2 4 5 1 Rank 1.5 6 3 5 4 1.5 2-fold 2-fold Mean 0.293 1.117 0.275 0.295 0.298 0.273 Mean 0.651 0.754 0.656 0.746 0.743 0.651 Mean rank 2.625 6 2.25 4.25 4.75 1.125 Mean rank 2.25 6 2.5 4.5 4.5 1.25 3 AIDS Research and Treatment Method LM(1) LM(2) Random Forest D/S/A CART Super Learner-5 Method LM(1) LM(2) Random Forest D/S/A CART Super Learner-5 Rank 1.5 6 3 5 4 1.5 Rank 1.5 6 3 5 4 1.5 10-fold 10-fold Mean 0.216 1.218 0.258 0.283 0.264 0.216 Mean 0.464 0.808 0.609 0.712 0.632 0.464 learners to those providing poor performance. We investigate four splits: 10-fold, 4-fold, 3-fold, and 2-fold that correspond to 90%, 75%, 66%, and 50% of data use as training samples and 10%, 25%, 33%, and 50% as validation sample respectively. Learners were evaluated using two distinct func- tions usually used as loss functions: squared error (SqE) and ﬁrst-order coeﬃcient (R). The SqE is (Y − E(Y X))2, that is, the squared diﬀerence between observed and predicted outcome. R is the ﬁrst-order correlation coeﬃcient between Y and E(Y |X), which has been recently used in this context [29]. It is important to note that SqE is unbounded while −1 ≤ R ≤ 1. For all full models, R2 estimates and accuracy were also computed in addition to SqE and R. We deﬁned two threshold values to deﬁne patients having a virologic response: −0.6log10 copies/mL and −0.5log10 copies/mL. For example, a patient with an HIV-1 RNA reduction larger than 0.6log10 copies/mL was classiﬁed as responder, otherwise as nonresponder. Patients may also be classiﬁed responders or not according to the predicted reduc- tion by a given method. 3. Candidate Learners We investigate the following learners: Logic Regression, Dele- tion/Substitution/Addition, Least squares regression, Ran- dom Forest, Classiﬁcation and Regression Trees. All algo- rithms are available as free packages of R software. Logic Regression (package named LogicReg) is an adap- tive regression methodology that attempts to construct pre- dictors as Boolean combinations of covariables [30]. Dele- tion/Substitution/Addition (package named DSA) is polyno- mial regression dataadaptive that generates candidate pre- dictors as polynomial combinations of binary covariables [31]. Classiﬁcation and Regression Trees (CARTs) build a re- gression tree in continuous outcome setting (package rpart) [32]. Random Forest (package RandomForest) is a “bagging predictor” (Bootstrap Aggregating), this method build a model from a combination of high number of regression trees resulting in the so-called Forest [33]. Least squares reg- ression was set up on two datasets: one consisted of all main terms and the second consisted of all main terms plus all two- way interactions (resp. denoted as LM(1) and LM(2)). From those learners, we set up two Super Learners: Super Learner using ﬁve learners, built with D/S/A, LM(1), LM(2), random forest and CART (noted Super Learner-5 in the fol- lowing), and Super Learner with six learners, the same as Super Learner-5 plus Logic Regression (denoted as Super Learner-6 in the following). Internal ﬁne-tuning procedure by internal cross-valida- tion was used to obtain the best performance for Logic Regression and D/S/A. The tuning parameters of D/S/A were maxsize = 20 (two times the number of co-variables), max- orderint = 2 and maxsumofpow = 2. All three steps were allowed (Deletion, Substitution, and Addition). CART has complexity parameter (cp) equal to 0.01. For Random Forest the number of trees was 1,000 and the number of variables to randomly consider at each node of each tree was ﬁxed at three (mtry = 3). That corresponds to the number of co-vari- ables divided by 3 which is usually used in regres- sion setting. Simple linear regression was used as refer- ence (without variable selection procedure). Methods were ranked; if two or more methods produced the same risk value, the mean rank was assigned (e.g., if Super Learner-5 and LM (1) gave the same SqE, in spite of assigning rank 1 and 2, resp., we noted 1.5 for both). 4. Results Results of the Discrete Super Learner and Super Learner-5 are given in Table 1. For example, based on the SqE as loss