Functional Approaches to Modeling Response Data

Statistical applications often involve the estimation of smooth curves, and the item response function (IRF) P_i (θ), where i = 1, …, n, the probability of success on item i that is conditional on value θ, is an iconic example. Functional data analysis as articulated in Ramsay and Silverman (2005) and Ramsay et al. (2009) aims to allow the data analyst as much flexibility as is required for this task, while at the same time providing tools to ensure that the estimated curves satisfy whatever smoothness and shape constraints the application requires. Chapter 9 of Ramsay and Silverman (2002) illustrated these principles in a test analysis situation.