Events correlated with respect to every subposet of a fixed poset

Posets are said to be (positively) correlated with respect to a third posetR onX (we writeA _R B) ifP(AR) P(AR B). HereP(CR) is the probability that a randomly chosen linear extension ofR is also a linear extension ofC. We classify posetsR onX such that(x, y) _s (u, v) holds for all posetsS onX which are subposets ofR, wherex, y, u, v are distinct elements ofX. On the way to proving this result, we show when a correlation inequality due to Shepp holds strictly.