State-Space Model of Behavior

We utilized a state-space modeling approach to describe the dynamic features of behavioral data in trial-structured experiments. The model consists of a state equation and an observation equation. The state equation defines the temporal evolution of an unobserved process from one trial to the next, and the observation equation defines the influence of the unobserved process on the observed behavioral signals.

We assumed that the state variable is dynamic and changes from trial to trial through the course of the experiment; we call this variable the cognitive state. This state could relate to cognitive features such as attention or learning progression. We assume that the cognitive state is inaccessible to direct measurement and that its time evolution must be inferred from the observed behavioral signal(s). Let x_k be the value of the cognitive state at trial k, and assume that its evolution from trial to trial is defined by the conditional probability density function f(x_k|x_k−1). Here we assume that the state model has the Markov property; thus, the probability distribution of the cognitive state at time k depends only on the cognitive state at time k − 1. Although this assumption simplifies the notation in the derivation, it is possible to extend these methods to non-Markovian processes (Cox, 1955).

The observation process defines the influence of the state process on the observed behavioral signals. For example, a higher inattention state might lead to longer reaction times and more incorrect responses. We consider two classes of observations. The first, y_k, is a continuously valued signal with an upper bound T_k, where the observation y_k may differ from trial to trial. If the value of this process exceeds the upper bound, the trial is censored, and features of the data are missing. Therefore, y_k takes on values in the set

The second class of observations is defined by the random variable or vector z_k. These are signals have values that are observed whenever y_k ∈ [0, T_k], but missing whenever y_k = {“Censored”}. For example, y_kmight represent a reaction time, and z_k might be an indicator of a correct response in a binary choice trial. Trials with excessive reaction times are cut off, and neither the reaction time nor the response is observed. In experiments in which reaction time is the sole observed signal, z_k may be omitted. This class of censored data falls in the NMAR category; from this point onward, missing data are assumed to come from the NMAR class, except as otherwise mentioned (Little & Rubin, 2014).

We define f(y_k|x_k) to be the distribution of the observation process that can lead to censoring as a function of the state process. In cases in which additional signals may be missing during the censored trials, we define an additional observation model f(z_k|x_k, y_k), which characterizes the distribution of z_k as a function of both the state process and the censored observation process y_k. For each trial, we can compute the likelihood of any set of observations as a function of the state process:

The objective is to estimate the distribution of the cognitive process x_k for each trial k, on the basis of the observed signals up to trial k.