6.1 Simulation scenario 1 We use an observed data structure ( W, A, Y ), where W ≜ ( W$_{1}$, W$_{2}$, . . . , W$_{5}$ ) is drawn from a standard 5-dimensional normal distribution, A is drawn according to a Bernoulli(0 . 5) distribution, and Y = µ ( A, W ) + 5 ξ ( A, W ), where the different forms of the conditional mean function µ ( a, w ) are given in Table 1, and ξ ( a, w ) is a random variate following a Beta distribution with shape parameters α = 3 expit( aw$_{2}$ ) and β = 2 expit[(1 − a ) w$_{1}$ ] shifted to have mean zero, where expit( x ) = 1 / (1 + exp( − x )). We performed tests of the null in which µ (1 , W ) is equal to µ (0 , W ) almost surely and in distribution, as presented in Examples 1 and 2, respectively. Our estimate ˆ P$_{n}$ of P$_{0}$ was constructed using the knowledge that P$_{0}$ ( A = 1 | W ) = 1 / 2, as would be available, for example, in the context of a randomized trial. The conditional mean function µ ( a, w ) was estimated using the ensemble learning algorithm Super Learner (van der Laan et al., 2007), as implemented in the SuperLearner package (Polley and van der Laan, 2013). This algorithm was implemented using 10-fold cross-validation to determine the best convex combination of regression function candidates minimizing mean-squared error using a candidate library consisting of SL.rpart , SL.glm.interaction , SL.glm , SL.earth , and SL.nnet . We used the results of 2 to evaluate significance for Example 1, and the eigenvalue approach presented in Section 3.2.2 to evaluate significance for Example 2, where we used all of the positive eigenvalues for n = 125 and the largest 200 positive eigenvalues for n > 125 using the rARPACK package (Qiu et al., 2014). We ran 1,000 Monte Carlo simulations with samples of size 125, 250, 500, 1000, and 2000, except for the np package, which we only ran for 500 Monte Carlo simulations due to its burdensome computation time. For Example 1 we compared our approach with that of Racine et al. (2006) using the npsigtest function from the np package. This requires first selecting a bandwidth, which we did using the npregbw function, specifying that we wanted a local linear estimator and the bandwidth to be selected using the cv.aic method (Hayfield and Racine, 2008). 27