Summary: Let \(S_1\) and \(S_2\) be two signals of a random variable \(X\), where \(G_1(s_1\mid x)\) and \(G_2(s-2\mid x)\) are their conditional distributions given \(X = x\). If, for all \(s_1\) and \(s_2\), \(G_1(s_1\mid x)-G_2(s_2\mid x)\) changes sign at most once from negative to positive as \(x\) increases, then the conditional expectation of \(X\) given \(S_2\) is greater than the conditional expectation of \(X\) given \(S_2\) in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when \(S_1\) and \(S_2\) have the same marginal distribution and, when \(S_1\) and \(S_2\) are sums of \(X\) and independent noises, it is equivalent to the dispersive order of the noises.