A complex matrix is called doubly quasistochastic if all its row sums and column sums are 1. Matrices \(A\) and \(B\) are doubly quasistochastically similar if \(B= SAS^{-1}\), where \(S\) is double quasistochastic. The author obtains a necessary and sufficient condition for a given matrix \(A\) to be doubly stochastically similar to a matrix with equal diagonal elements, and a necessary and sufficient condition for \(A\) to be doubly stochastically similar to a matrix with any diagonal elements the sum of which equals the trace of \(A\). In addition, an inverse elementary divisor result for doubly quasistochastic matrices is obtained.