# d-prime as dependent variable in mixed effects model

d-prime refers to the sensitivity index in the signal detection theory, calculated as the z(probability of hit)-z(probability of false alarm).

If I have to build a linear mixed-effects model with d-prime as the dependent variable, should I use the generalized linear mixed effect model with binomial distribution with a link to probit? family = binomial(link='probit') Should the difference between z-transformed probabilities be treated as probability?

I have read a highly relevant question here: How do I compare d-prime scores for 2 different conditions for the same individual?

However, I do not have access to the original trial data and cannot put the trial-by-trial response as the dependent variable and the reality (the correct answer) as fixed effect. All I have is the d-prime data.

• Do you know the number of trials on which each d-prime score is calculated? Regarding your questions "Should the difference between z-transformed probabilities be treated as probability?", the answer is no, d-prime do not follow a binomial distribution. Sep 19, 2019 at 9:33
• Hi matteo, I do know about how many trials the d-prime is based on. I am also thinking whether I should treat d' as contiNuous and just use the lme. However, I have tried fitting a lme, but the residuals were not normally distributed. Sep 19, 2019 at 10:04
• d-prime is, in principle, unbounded so you should treat it as such. Perhaps if you added a plot of the residuals you could get some more useful answer. However, I asked about the number of trials because I was trying to figure out if there was a way to calculate the proportions of hits and false alarm given what you have. Do you also know the values of the criterion $c$? Sep 19, 2019 at 16:47
• I realize that I am 4 years too late, but for the record, you can not recover the proportions of hits and false alarms from d'. For example, both A, B, and C result in a d' of 1.35: A) hits = 75%, FA = 25% B) hits = 95%, FA = 61.5% C) hits = 50%, FA = 8.8% Sep 12, 2023 at 19:16

To follow up on my comment, it may be possible to recover the proportions of hits and false alarms, allowing modelling of the data via a GLM as indicated in the other answer that you linked. To do so however you'd need to know the number of trials run in each condition, and the value of the other signal detection theory parameter, that is the criterion $$c$$. Additionally, I also assumed that trials with signal present and signal absent were presented in equal proportion. Say the total number of trials was $$2n$$ for each condition ($$n$$ trials with signal absent and $$n$$ trials with signal present). Recall the formula for $$d'$$ and $$c$$ $$d'=\Phi^{-1}\left(p_{\text{HITS}}\right) - \Phi^{-1}\left(p_{\text{FA}}\right)$$ $$c= - \Phi^{-1}\left(p_{\text{FA}}\right)$$ where $$p_{\text{HITS}}=\frac{x}{n}$$ and $$p_{\text{FA}}=\frac{y}{n}$$ are the proportions of hits and false alarms. Thus the observer responded $$x+y$$ times "signal present", out of the total $$2n$$ trials, of which $$x$$ were hits and $$y$$ false alarms. The formula for the $$d'$$ can be written as $$d'= \sqrt{2} \left[ \text{erf}^{-1}\left(\frac{2x}{n} -1\right) -\text{erf}^{-1}\left(\frac{2y}{n}-1 \right) \right]$$ By solving the the equations for $$y$$ and $$x$$ it should be possible to recover the values and use them in a probit GLM $$y = \frac{1}{2}n\left[1-\text{erf}\left(\frac{c}{\sqrt{2}}\right) \right]$$ and $$x = \frac{1}{2}n\left[\text{erf}\left(\frac{1}{2}\left(\sqrt{2}d'+\text{erf}^{-1}\left(\frac{2y}{n}-1\right)\right)\right)+1 \right]$$