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Incorporating Functional Response Time Effects into a Signal Detection Theory Model

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Signal detection theory (SDT; Tanner & Swets in Psychological Review 61:401–409, 1954) is a dominant modeling framework used for evaluating the accuracy of diagnostic systems that seek to distinguish signal from noise in psychology. Although the use of response time data in psychometric models has increased in recent years, the incorporation of response time data into SDT models remains a relatively underexplored approach to distinguishing signal from noise. Functional response time effects are hypothesized in SDT models, based on findings from other related psychometric models with response time data. In this study, an SDT model is extended to incorporate functional response time effects using smooth functions and to include all sources of variability in SDT model parameters across trials, participants, and items in the experimental data. The extended SDT model with smooth functions is formulated as a generalized linear mixed-effects model and implemented in the gamm4 R package. The extended model is illustrated using recognition memory data to understand how conversational language is remembered. Accuracy of parameter estimates and the importance of modeling variability in detecting the experimental condition effects and functional response time effects are shown in conditions similar to the empirical data set via a simulation study. In addition, the type 1 error rate of the test for a smooth function of response time is evaluated.

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  1. We use the term functional to refer to the intrinsic structure of the data rather than their explicit form (Ramsay & Silverman, 2005, p. 38).

  2. The 28 image groups are: baby, backpack, banana, belt, bird, boot, box, chair, desk, dog, flag, grapes, hair, hat, jacket, juice, pants, paper, pie, pig, ring, shirt, shoe, skirt, sock, swords, tree, watch.

  3. There are three identifiable parameterizations of smooth functions of \(RT_{lji}\): (a) the mean level of \(\texttt{isold}_{lji}\) and a smooth function of \(RT_{lji}\) for each level of \(\texttt{isold}_{lji}\), (b) a smooth function of \(RT_{lji}\) and a smooth function for the differences in the effect of response time by \(\texttt{isold}_{lji}\), and (c) the mean level of \(\texttt{isold}_{lji}\), a smooth function of \(RT_{lji}\), and a smooth function for the differences in the effect of response time by \(\texttt{isold}_{lji}\). In this current study, we chose the third parameterization to estimate the fixed d parameter (\(\mu ^{d}\)), a smooth function of \(RT_{lji}\) for the \(c_{lji}\) parameter, and a smooth function of \(RT_{lji}\) for the \(d_{lji}\) parameter.

  4. “0” in the superscripts of \(\mu ^{c0}\) and \(\mu ^{d0}\) indicates “null,” which means that they are estimated without random effects and without experimental condition effects.

  5. With ordered factors in R (e.g., ordered.disOLD= as.ordered(factor.disOLD) for a isold covariate), the interactions are not strictly interactions as in the analysis of variance (ANOVA) but as regression coefficients of the covariates.

  6. Summations start with 2 because of the identification constraints.


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Correspondence to Sun-Joo Cho.

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Edward Ip was the handling ARCS Editor for this paper. This research was supported in part by Grant No. SES-1851690 from the National Science Foundation (NSF) to Sun-Joo Cho, Sarah Brown-Schmidt, and Paul De Boeck. Nothing in this article necessarily reflects the positions or policies of the agency.

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Cho, SJ., Brown-Schmidt, S., Boeck, P.D. et al. Incorporating Functional Response Time Effects into a Signal Detection Theory Model. Psychometrika 88, 1056–1086 (2023).

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