  In some scientific fields, it is common to have certain variables of interest
that are of particular importance and for which there are many studies
indicating a relationship with different explanatory variables. In such cases,
particularly those where no relationships are known among the explanatory
variables, it is worth asking under what conditions it is possible for all such
claimed effects to exist simultaneously. This paper addresses this question by
reviewing some theorems from multivariate analysis showing that, unless the
explanatory variables also have sizable dependencies with each other, it is
impossible to have many such large effects. We discuss implications for the
replication crisis in social science.
