In [H. Suzuki, Imprimitive Q-polynomial association schemes, J. Algebraic Combin. 7 (2) (1998) 165-180], it was shown that an imprimitive Q-polynomial scheme X = (X, {R(i)}(0 <= i <= d)) is either dual bipartite, dual antipodal or of class 4 or 6. In this paper, it will be shown that the scheme of class 4 does not occur using the integrality conditions of the entries of the first eigenmatrix of X. These integrality conditions arise from the fact that X has exactly one Q-polynomial ordering [H. Suzuki, Association schemes with multiple Q-polynomial structures,J. Algebraic Combin. 7 (2) (1998) 181-196]. (C) 2008 Elsevier Ltd. All rights reserved.