Assessing the distortions introduced when calculating d’: A simulation approach

The discriminability measure \(d'\) is widely used in psychology to estimate sensitivity independently of response bias. The conventional approach to estimate \(d'\) involves a transformation from the hit rate and the false-alarm rate. When performance is perfect, correction methods must be applied to calculate \(d'\), but these corrections distort the estimate. In three simulation studies, we show that distortion in \(d'\) estimation can arise from other properties of the experimental design (number of trials, sample size, sample variance, task difficulty) that, when combined with application of the correction method, make \(d'\) distortion in any specific experiment design complex and can mislead statistical inference in the worst cases (Type I and Type II errors). To address this problem, we propose that researchers simulate \(d'\) estimation to explore the impact of design choices, given anticipated or observed data. An R Shiny application is introduced that estimates \(d'\) distortion, providing researchers the means to identify distortion and take steps to minimize its impact.

The supplemental materials for simulations can be found at https://github.com/Van-Zandt-Lab-at-OSU/Estimation-of-d-prime

This study does not include empirical data. The R code used to produce simulated data are available at https://github.com/Van-Zandt-Lab-at-OSU/Estimation-of-d-prime.

1. All values used in Simulation 3 were specified in Simulation 3.

2. See the Supplemental materials for negative values of c.

3. Recall that the effects of \(c \ne 0\) are symmetric (see Fig. 4), for any effects observed for \(c=.2\) or .4, we would also expect to see the same effects for \(c=-.2\) or \(-.4\)

Authors and Affiliations

1. Department of Psychology, University of Kansas, Lawrence, Kansas, USA

   Yiyang Chen

2. Department of Psychology, The Ohio State University, Columbus, Ohio, USA

   Heather R. Daly, Mark A. Pitt & Trisha Van Zandt

Corresponding author

Correspondence to Yiyang Chen.

This material is based upon work performed while Dr. Trisha Van Zandt was serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors have no conflicts of interest to disclose. The authors did not receive support for the submitted work. Drs. Mark Pitt and Trisha Van Zandt contributed equally.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1: Why increasing the sample size does not reduce \(d'\) distortion

In this appendix, we explain why increasing the sample size (more participants) would exacerbate the distortion problem in \(d'\) estimation when mathematical corrections are applied. Assume that a group of M participants each completed \(N_s\) signal trials and \(N_n\) noise trials. In the ideal situation, we hope that the average of their corresponding \(d'\) estimates (\(\hat{d'_1}\), \(\hat{d'_2}\), ..., \(\hat{d'_M}\)) converges to the true group mean \(d'\) as \(M \rightarrow \infty \), following the central limit theorem.

However, the distribution of \(\hat{d'}\), computed as a transformation of \(p_{\text {hit}}\) (hit rate) and \(p_{\text {false alarm}}\) (false-alarm rate) in Eq. 1, does not satisfy the conditions for the central limit theorem for independent, non-identical distributions to hold when the trial sizes are fixed at \(N_s\) and \(N_n\). For Participant i with true SDT parameters of \(d'_i\) and \(c_i\), the distribution of hit rate (\(p_{\text {hit},i}\)) is

where \(\Phi \) is the Gaussian CDF. The mean of \(p_{\text {hit},i}\), \(\Phi (\frac{d'_i}{2} - c_i)\), is the probability to correctly identify each signal trial. Similarly, for noise trials, the mean of correct rejection rate is \(\Phi (\frac{d'_i}{2} + c_i)\), thus the mean of false-alarm rate, \(p_{\text {false alarm},i}\) is \(1-\Phi (\frac{d'_i}{2} + c_i)\). Ideally, using Eq. 1 for \(d'\) estimation,

where z is the inverse of Gaussian CDF, we hope to achieve

However, with a finite trial number \(N_s\), \(P(p_{\text {hit}} =1)>0\), thus there is always a non-zero probability of perfect performance when identifying signals. Similarly, \(P(p_{\text {false alarm}}=0)>0\) when \(N_n\) is finite, thus there is always a non-zero probability of perfect performance when identifying noise. Because the inverse of the Gaussian CDF z(p) in Eq. 1 has a support of (0, 1) and is not defined when \(p=0\) or \(p=1\), the distributions of \(\hat{d'}_i\) (\(i=1,2,...,M\)) is ill-defined when \(p_{\text {hit}}\) or \(p_{\text {false alarm}}\) are 0 or 1. As a result, Eq. 3 does not hold, and there is no finite expected value for \(\hat{d'}_i\). Consequently, \(\hat{d'_1}\), \(\hat{d'_2}\), ..., \(\hat{d'_M}\) do not satisfy the conditions for the central limit theorem for independent, non-identical distributions to hold because they do not have well-defined, finite expected values or variances.

After applying the mathematical corrections, the distributions of \(\hat{d'}_r\) or \(\hat{d'}_{ll}\) do have finite expected values and finite variances. Thus they satisfy the conditions for the central limit theorem to hold. However, the mathematical correction distorts the distributions of \(\hat{d'}_r\) or \(\hat{d'}_{ll}\). For example, using replacement with a correction value of a, the distribution of hit rate for Participant i is corrected to

and

Thus the mean of \(p_{\text {hit},r,i}\) is no longer \(\Phi (\frac{d'_i}{2} - c_i)\), but is corrected to

Similarly, the mean of \(p_{\text {false alarm},r,i}\) is no longer \(1-\Phi (\frac{d'_i}{2} + c_i)\), but is corrected to

Therefore,

As a result, the average of \(\hat{d'_{r,1}}\), \(\hat{d'_{r,2}}\), ..., \(\hat{d'_{r,3}}\) does not converge to the true group mean \(d'\) as \(M \rightarrow \infty \): the mean is asymptotically biased. Instead, it converges to a different value different from the true mean \(d'\), where the amount of bias is determined by both the correction method and other experimental parameters shown in Eq. 4. The same issue applies to \(\hat{d'}_{ll}\). Examples of values that they converge to are shown in Fig. 5, demonstrating that these values can have a large difference from the true mean \(d'\) in many cases. As a result, increasing the sample size would exacerbate distortion problems in \(d'\) estimation caused by mathematical correction.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions