Refining Gamson: the isometric log-ratio transformation and portfolio proportionality in multiparty governments

One of the most robust empirical findings in political science is that in multiparty democracies cabinet ministries are distributed in rough proportion to parties' legislative seat shares, a pattern known as Gamson's Law. Yet existing research often overlooks the fact that portfolio and seat shares are compositional---mutually dependent parts of a whole. Standard methods treat them as unconstrained, risking bias, misleading uncertainty estimates, and flawed inference. Unfortunately, the most common strategy for handling compositions---the additive log-ratio (ALR) transformation combined with seemingly unrelated regression (SUR)---fails when the number and identity of compositional elements (like parties) vary across cases. We propose the isometric log-ratio (ILR) transformation---new to political science---as an alternative that both respects compositional geometry and adapts to differing compositional structures. Monte Carlo simulations show that ILR sharply outperforms standard approaches, reducing bias, improving coverage probability, and increasing statistical power. While we apply it to portfolio allocation, ILR provides a general solution for modeling compositional outcomes with other potential uses, including in electoral competition, where ALR+SUR has required strong assumptions or ad hoc adjustments. Using this improved methodology, we find that seat--portfolio proportionality is weaker overall than conventionally reported and varies substantially across governments.