A note on the effect on power of score tests via dimension reduction by penalized regression under the null

We consider the problem of score testing for certain low dimensional parameters of interest in a model that could include finite but high dimensional secondary covariates and associated nuisance parameters. We investigate the possibility of the potential gain in power by reducing the dimensionality of the secondary variables via oracle estimators such as the Adaptive Lasso. As an application, we use a recently developed framework for score tests of association of a disease outcome with an exposure of interest in the presence of a possible interaction of the exposure with other co-factors of the model. We derive the local power of such tests and show that if the primary and secondary predictors are independent, then having an oracle estimator does not improve the local power of the score test. Conversely, if they are dependent, there is the potential for power gain. Simulations are used to validate the theoretical results and explore the extent of correlation needed between the primary and secondary covariates to observe an improvement of the power of the test by using the oracle estimator. Our conclusions are likely to hold more generally beyond the model of interactions considered here.