Making decisions based on a linear combination L of features is of course very common in pattern recognition. For distinguishing between two hypotheses or classes, the test is of the form sign(L - τ ) for some threshold τ. Due mainly to fixing τ , such tests are sensitive to changes in illumination and other variations in imaging conditions. We propose a special case, a "self-normalized linear test" (SNLT), hard-wired to be of the form sign(L_1 - L_2) with unit weights. The basic idea is to "normalize" L_1, which involves the usual discriminating features, by L_2, which is composed of non-discriminating features. For a rich variety of features (e.g., based directly on intensity differences), SNLTs are largely invariant to illumination and robust to unexpected background variations. Experiments in face detection are promising: they confirm the expected invariances and out-perform some previous results in a hierarchical framework.