Stress transmission inside three-dimensional granular packings is investigated using computer simulations. Localized force perturbation techniques are implemented for frictionless and frictional shallow, ordered, granular arrays confined by solid boundaries for a range of system sizes. Stress response profiles for frictional packings agree well with the predictions for the semi-infinite half-plane of classical isotropic elasticity theory down to boxes of linear dimensions of approximately forty particle diameters and over several orders of magnitude in the applied force. The response profiles for frictionless packings exhibit a regime of transition to strongly anisotropic features with increasing box size. The differences between the natures of the stress responses are shown to be characterized by very different particle displacement fields.