When species simultaneously compete with two or more species of competitor, higher order interactions (HOIs) can lead to emergent properties not present when species interact in isolated pairs. In order to extend ecological theory to multi-competitor communities, ecologists must develop a practical and general definition for HOIs that can be applied to a wide range of competition models. In this paper we propose a definition for HOIs and outline a set of criteria for testing whether a model has or does not have HOIs. These criteria are valuable for empirical ecologists in need of clarity when discussing HOIs in empirical data. We also provide thorough discussion of how our definition compares with previous definitions of HOIs and interaction modification in the literature. In the second part of the paper we demonstrate the steps required for a rigorous test of HOIs in empirical data. To do this we simulate resource competition between three annual plant species which differ in phenology. We then fit phenomenological competition models to the outcome of simulated competition and use these to test for the presence of HOIs. In our simulations, we find the strength of HOIs varies with phenology: species that grow later experience stronger HOIs than earlier growing species. Our simulation shows how HOIs could emerge in ecosystems where resource availability and individual size change rapidly throughout the course of the growing season and where there are differences in the timing of resource acquisition between competitors.