Previous analyses of the presence of non-partisans in cabinets consider the relative power of presidents as the explanatory factor. However, their analysis either uses indices of presidential power or is in terms of regime type – semi-presidential, parliamentary, or monarchical. Using a novel dataset on non-partisan appointments in 30 European democracies, we deploy an innovative two-step fractional response regression. This enables us to disentangle different determinants of the presence of non-partisans and how many (their magnitude). We show that these determinants have partly different effects on whether any non-partisans are appointed to cabinets and on their magnitude. Direct presidential elections increase the likelihood, but not the magnitude, of non-partisan appointments, and a president’s power to dissolve parliament increases both likelihood and magnitude. Furthermore, we discover that a prime minister’s power to dissolve parliament decreases the magnitude of such appointments but does not affect their likelihood. Our analysis fine-tunes the institutional details that affect the likelihood and magnitude of non-partisan appointments. In so doing, we show that regime types are concealing important within-type differences.