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Deep neural networks (DNNs) show promise in image-based medical diagnosis, but cannot be fully trusted since their performance can be severely degraded by dataset shifts to which human perception remains invariant. If we can better understand the differences between human and machine perception, we can potentially characterize and mitigate this effect. We therefore propose a framework for comparing human and machine perception in medical diagnosis. The two are compared with respect to their sensitivity to the removal of clinically meaningful information, and to the regions of an image deemed most suspicious. Drawing inspiration from the natural image domain, we frame both comparisons in terms of perturbation robustness. The novelty of our framework is that separate analyses are performed for subgroups with clinically meaningful differences. We argue that this is necessary in order to avert Simpson's paradox and draw correct conclusions. We demonstrate our framework with a case study in breast cancer screening, and reveal significant differences between radiologists and DNNs. We compare the two with respect to their robustness to Gaussian low-pass filtering, performing a subgroup analysis on microcalcifications and soft tissue lesions. For microcalcifications, DNNs use a separate set of high frequency components than radiologists, some of which lie outside the image regions considered most suspicious by radiologists. These features run the risk of being spurious, but if not, could represent potential new biomarkers. For soft tissue lesions, the divergence between radiologists and DNNs is even starker, with DNNs relying heavily on spurious high frequency components ignored by radiologists. Importantly, this deviation in soft tissue lesions was only observable through subgroup analysis, which highlights the importance of incorporating medical domain knowledge into our comparison framework.