The intermediate result cardinality - the number of objects satisfying a condition given in a query - is an important factor for estimating the cost of the query in query optimization. In this paper we show that an object-oriented query often involves partial participation of classes in a relationship. We then present a new technique for estimating the intermediate result cardinality in such a query. Partial participation has not been considered seriously in existing techniques. Since the proposed technique uses detailed statistics to accommodate partial participation, it estimates the intermediate result cardinality more accurately than existing ones. We also show that these statistics are easily obtained by using inherent properties of object-oriented databases.