Two different values for criterion in signal detection theory?

Question

I'm a signal detection theory newbie. I have data with a classical SDT design: the participants can answer YES or NO to a question where the correct answer is YES or NO, depending on the presence or absence of a stimulus feature. I looked online for a script to calculate d', and found one which is confusing me.

What's confusing is that the criterion is calculated in two different ways. Does anybody have an idea why this is the case? These are the relevant lines of (matlab) code:

% Convert to Z scores zHit = norminv(proportion_of_hits) ; zFA = norminv(proportion_of_false_alarms) ; % Calculate d-prime dPrime = zHit - zFA ; % calculate BETA beta = exp((zFA^2 - zHit^2)/2); % calculate C C = -.5 * (zHit + zFA);

I'm interested in beta and C - how one interprets them, how they differ from each other, what a high/low score in each of the two means. And also, which is the one that is more likely to be understandable to a reader who knows SDT.

Answer 1

For these types of questions I really like Detection Theory: A User's Guide by Macmillan and Creelman. They consider 3 types of bias. The criterion location $c$ is calculated relative to the zero-bias point and expressed in units of standard deviations, such that a $c$ of 1 means the criterion is 1 standard deviation to the right of the zero-bias location and a $c$ of -2 means the criterion is 2 standard deviations to the left of the zero-bias location. Since $c$ is relative to the zero-bias location, if $c$ is held constant, then as $d'$ changes the distance of $c$ to the mean of the distribution changes. Arguably, this means that there is a change in response bias, but the measure of $c$ does not reflect that (after all, the distance to the zero-bias location, which is $c$, does not change with changes in $d'$). The response bias metric $c'=c/d'$ avoids this issue by taking sensitivity ($d'$) into account. The third measure of bias $\beta$ is defined as $\beta = e^{cd'}$ (which if you do the math is what your code calculates). This measure of bias is based on the likelihood ratio of the two distributions. Because of the mathematical relationship between the measures, they tell you the same thing. There isn't a correct measure and it is easy to switch between the measures, so it does not matter which you report.

For further reading, see also Stanislaw & Todorov (1999), Calculation of Signal Detection Theory Measures