By far the most fruitful technique for showing lower bounds for the CONGEST model is reductions to two-party communication complexity. This technique has yielded nearly tight results for various fundamental problems such as distance computations, minimum spanning tree, minimum vertex cover, and more. In this work, we take this technique a step further, and we introduce a framework of reductions to $t$-party communication complexity, for every $t\geq 2$. Our framework enables us to show improved hardness results for maximum independent set. Recently, Bachrach et al.[PODC 2019] used the two-party framework to show hardness of approximation for maximum independent set. They show that finding a $(5/6+ε)$-approximation requires $Ω(n/\log^6 n)$ rounds, and finding a $(7/8+ε)$-approximation requires $Ω(n^2/\log^7 n)$ rounds, in the CONGEST model where $n$ in the number of nodes in the network. We improve the results of Bachrach et al. by using reductions to multi-party communication complexity. Our results: (1) Any algorithm that finds a $(1/2+ε)$-approximation for maximum independent set in the CONGEST model requires $Ω(n/\log^3 n)$ rounds. (2) Any algorithm that finds a $(3/4+ε)$-approximation for maximum independent set in the CONGEST model requires $Ω(n^2/\log^3 n)$ rounds.