We generalize results on the monotonicity of equilibria for network games with incomplete information. In those games players know the stochastic process of network formation and their own degree in the realized network, and decide an action whose payoff depends on the strategic interaction in the network between their own action and a statistic (as, for example, the mean, the maximum or the minimum) of neighbors’ actions. We show that, even under degree independence, not only the distinction between strategic complements and strategic substitutes is important in determining the nature of Bayesian Nash equilibria, but also the nature itself of the statistic.