This study clarifies the causes of inter-model differences in the global-average temperature response to doubled CO2, commonly known as equilibrium climate sensitivity (ECS). We begin by noting several issues with the standard approach for decomposing ECS into a sum of forcing and feedback terms. This leads us to derive an alternative method based on linearizing the effect of the net feedback. Consistent with previous studies, the new method identifies shortwave cloud feedback as the dominant source of inter-model spread in ECS. Our new approach also reveals that covariances between cloud feedback and forcing, between lapse rate and longwave cloud feedbacks, and between albedo and shortwave cloud feedbacks play an important and previously underappreciated role in determining model differences in ECS. Defining feedbacks based on fixed relative rather than specific humidity (as suggested by Held and Shell 2012) reduces the covariances between processes and leads to more straightforward interpretations of results.