Estimating multilevel signal detection theory models using maximum likelihood

Signal detection theory (SDT) is one of the most influential modeling frameworks in psychological research. One of its main contributions is the possibility to disentangle two central components in decisions under uncertainty: sensitivity, the ability to differentiate between signal and noise, and response bias, a tendency to favor one decision over the other. When applying such models to common psychological data comprising multiple trials of multiple participants, multilevel modeling is considered the state-of-the-art in psychological research. While the estimation of non-linear multilevel models such as SDT models is usually done in a Bayesian framework, this is not necessary to benefit from the advantages of this modeling approach: Multilevel SDT models can, in principle, also be fitted using maximum likelihood (ML) – although this is rarely done in practice, presumably due to the lack of appropriate software for doing so. We present our work on an R package that is aimed at supporting the straightforward application of this approach for researchers applying SDT. To fit multilevel SDT models using ML, we exploit the equivalence of SDT models and a subclass of generalized linear models (GLMs; DeCarlo, 1998). GLMs can easily be extended to multilevel models by including random effects in the model, yielding generalized linear mixed models (GLMMs). Thereby, multilevel SDT models can be fitted with ML by using commonly-known software packages for fitting GLMMs. Our R package allows one to fit different variants of multilevel SDT models with sensitivity and response bias parameters that can vary according to user-specified predictor variables and different sources of random variation. It "translates" the given SDT model to a GLMM, selects an appropriate random-effects structure, estimates the parameters, and transforms the parameter estimates for both population and subject level back to the SDT framework. In addition, likelihood ratio tests for given predictors can be calculated. We demonstrate the validity of our implementation through simulation studies.