We study the directed maximum common edge subgraph problem (DMCES) for the class of directed graphs that are finite, weakly connected, oriented, and simple. We use DMCES to define a metric on partially ordered sets that can be represented as weakly connected directed acyclic graphs. While most existing metrics assume that the underlying sets of the partial order are identical, and only the relationships between elements can differ, the metric defined here allows the partially ordered sets to be different. The proof that there is a metric based on DMCES involves the extension of the concept of line digraphs. Although this extension can be used to compute the metric by a reduction to the maximum clique problem, it is computationally feasible only for sparse graphs. We provide an alternative techniques for computing the metric for directed graphs that have the additional property of being transitively closed.