Are d prime (d') in signal detection theory and Cohen's d (mainly reported in the context of the general linear model) measures for the same thing (i.e., the difference of the means in SD-units), and just termed differently? Or is there any difference between those two measures?
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$\begingroup$ might as well add the z-score in there too. :) $\endgroup$– JohnJan 20, 2012 at 17:56
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$\begingroup$ @John, there are similarities, but there is one important distinction: Cohen's d (etc.) is calculated from the relationship between 2 distributions, whereas z-scores are calculated within 1. $\endgroup$– gung - Reinstate MonicaJan 21, 2012 at 4:57
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They are essentially the same thing: differences between means measured in units of standard deviations, as you say. There are some theoretical differences in the substance from which they arise. Cohen's d (and the closely related Hedges' g) are calculated on real observations, whereas the distributions underlying observed responses--and used to compute d'--are latent. In the signal detection world, there has been a good deal of work on the possibility that these latent distributions may not be Gaussian in some cases, with most researchers arguing that in that case d' is not an appropriate metric. Other measures, such as the area under the ROC curve, are advocated in that case. As far as I'm aware, in meta-analysis people are fine with using a scaled mean difference even if the distributions are not Gaussian. Nonetheless, they are fundamentally the same idea. You should realize that statistics is loaded with things that are the same, but have different names and historically developed in isolation from each other.
answered Jan 21, 2012 at 4:53
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$\begingroup$ +1 but this answer would improve if you include explicit formulas for Cohen's d and for d-prime. They both seem to be equal to $\mu_1-\mu_2$ divided by some measure of dispersion, it's just that they use slightly different measures of "pooled dispersion". $\endgroup$– amoebaSep 14, 2016 at 14:21
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$\begingroup$ @amoeba, Cohen's d uses the 'population' formula for the SD, whereas Hedges' g uses the 'sample' version, but this issue does not exist for d'. In that case, the assumed two Gaussian distributions are latent. I take the question to be primarily about d vs. d', not d vs. g. $\endgroup$ Sep 14, 2016 at 14:39
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$\begingroup$ Hmm, I googled "d prime" and here is the formula that I get from Wikipedia: en.wikipedia.org/wiki/Sensitivity_index - it is fully determined by the means and standard deviations of the two groups and is very similar to the formula for Cohen's d. I don't see anything "latent" here. Are you perhaps talking about some other definition of d prime? $\endgroup$– amoebaSep 14, 2016 at 14:44
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1$\begingroup$ @amoeba, nope, that's the one. The "formal definition" they list first is assumed for the latent variables. Note that the "estimate of d'... is calculated as: $d' = Z(\text{hit rate}) − Z(\text{false alarm rate})$". No SDs are actually computed using either the population or sample formulas. $\endgroup$ Sep 14, 2016 at 14:58
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$\begingroup$ Ah, thanks, I understand now. So d' is defined for any black-box detection procedure; it's just that if both stimulus and noise distributions are one-dimensional Gaussian and the detection procedure is optimal, then d' is equivalently given by the formula that is very close to the Cohen's d. Right? $\endgroup$– amoebaSep 14, 2016 at 15:10