A recent work showed that it is possible to transform a single-cycle test for stuck-at faults into a launch-on-shift (LOS) test that is guaranteed to detect the same stuck-at faults without any logic or fault simulation. The LOS test also detects transition faults. This was used for obtaining a compact LOS test set that detects both types of faults. In the scenario where LOS tests are used for both stuck-at and transition faults, this article observes that, under certain conditions, the detection of a stuck-at fault guarantees the detection of a corresponding transition fault. This implies that the two faults are equivalent under LOS tests. Equivalence can be used for reducing the set of target faults for test generation and test compaction. The article develops this notion of equivalence under LOS tests with equal primary input vectors and provides an efficient procedure for identifying it. It presents experimental results to demonstrate that such equivalences exist in benchmark circuits, and shows an unexpected effect on a test compaction procedure.