Recently Liu and Wang derived the likelihood ratio test (LRT) statistic and its asymptotic distribution for testing equality of two multinomial distributions vs. the alternative that the second distribution is larger in terms of increasing convex order (ICX). ICX is less restrictive than stochastic order and is a notion that has found applications in insurance and actuarial science. In this paper we propose a new test for ICX. The new test has several advantages over the LRT and over any test procedure that depends on asymptotic theory for implementation. The advantages include the following: (i) The test is exact (non-asymptotic). (ii) The test is performed by conditioning on marginal column totals (and row totals in a full multinomial model for a $2\times C$ table). (iii) The test has desirable monotonicity properties. That is, the test is monotone in all practical directions (to be formally defined). (iv) The test can be carried out computationally with the aid of a computer program. (v) The test has good power properties among a wide variety of possible alternatives. (vi) The test is admissible. The basis of the new test is the directed chi-square methodology developed by Cohen, Madigan, and Sackrowitz.