ROC curve as predictive tool in human performance--relationship between $\beta$, $X_C$, and signal probability I am taking an engineering psychology course in which we are using ROC curves to evaluate human performance in signal detection. Many of the questions on this site about ROC curves are using them as a data visualization tool, to plot the performance of a binary classifier.
But I am struggling to find information about ROC isosensitivity curves and their use as a predictive tool in human performance. (I've looked around, and it seems like this SE site is the best place to ask, but I am happy to be redirected elsewhere.)
In particular, I would like to understand the relationship between

*

*signal probability;

*the optimal response bias $\beta$ and response threshold $X_C$;

*the experimental response bias $\beta$ and response threshold $X_C$;

*sensitivity (as measured by area under the ROC curve, distance between the S and N distributions, etc.); and

*the payoff/cost associated with hits, false alarms, misses, and correct rejections.

The textbook we are using (Wickens et al., Engineering Psychology and Human Performance, 4e) indicates that all these things are related, but doesn't make it clear which items are positively and negatively correlated in which situations. In particular, it doesn't clearly differentiate between the optimal and experimental cases, so I am having to connect the dots myself. I have divided my questions into three parts. I will not post them as separate questions because I think that a complete answer to just one or two of the parts will suffice to understand the others.
Definition of $\beta$
The text defines (experimental) $\beta$ as $\beta = \frac{P(X |S)}{P(X|N)}$, where $S$ denotes a signal trial, $N$ a noise trial. But this definition is of little use because $X$ is left undefined (see p. 29). What does $X$ mean here?
The text also defines $$\beta_\text{opt} = \frac{P(N)}{P(S)}$$ for the case where payoffs are unspecified, and then provides a more specific formula for optimal $\beta$ given a payoff matrix. What assumptions are made about the payoffs in the definition of $\beta_\text{opt}$ given above?
Sluggish beta
This explanation of "sluggish beta" is also perplexing:

Laboratory experiments have shown that beta is not adjusted as much as
it should be. That is, subjects demonstrate a sluggish beta .... They are less risky than they should be if the optimal
beta is high, and less conservative than they should be if the optimal
beta is low (p. 30).

If the optimal beta is high, then so is the optimal $X_C$, which corresponds to a conservative policy (i.e. averse to false alarms, per p. 29). So, I think the words in boldface should be switched. Am I wrong?
ROC curve and changes in signal probability
Here is the figure I am referencing:

We are given three isosensitivity curves, one where the signal and noise distributions are well distributed, one where there is no differentiation (the straight line), and one in the middle.
One practice question I have asks, what happens to $\beta$ when signal probability increases? (The question doesn't specify if we are talking about optimal $\beta$ or experimental $\beta$; instead it says "according to the ROC curve.")
The three matrices on the left of the figure above associate a decrease in the signal probability (from $20/30$ to $15/30$ to $10/30$) with motion down the ROC curve and an increase in $\beta$, which corresponds to a more conservative threshold, i.e. lower $X_C$. So when signal probability increases, we would expect a lower beta, or a less conservative response.
But in these three matrices, the connection between $\beta$ and the signal probability seems totally contrived to me. Since (as you can clearly see from the matrices) the probabilities appearing in each cell are already conditioned on the signal (or noise) probability, a low setting for $\beta$ will always yield a response matrix like
$$\begin{bmatrix}0.95 & 0.80 \\ 0.05 & 0.20\end{bmatrix}$$
regardless of whether 1%, 5%, or 99% of the trials were true signals.
So, what point are the authors trying to make by including the signal probabilities $\begin{bmatrix}20 & 10\end{bmatrix}$ above this matrix?

*

*Are they claiming that in a typical human observer, increasing the signal probability yields a decrease in experimental $\beta$? This, if I understand correctly, is the opposite of what the "sluggish beta" phenomenon would predict.

*Or are they claiming that the optimal $\beta$ is negatively correlated with signal probability? This seems like an untenable claim in general, because we have no information about the costs associated with false alarms, etc. If the cost of a false alarm is extremely high and the number of noise trials is greater than zero, then the optimal $X_c$, and hence optimal $\beta$, will also be extremely high, right?

What do the ROC curves shown above represent? Are they meant to represent idealized experimental data? Or some sort of generalization of the optimal $\beta$-value?
 A: This topic, or at least this formulation of it, has proven surprisingly obscure, and the inconsistencies in the textbook haven't been much help. However, I think I have figured out the answer to most of the questions posed in the OP.
Definition of $\beta$

The text defines (experimental) $\beta$ as $\beta = \frac{P(X |S)}{P(X|N)}$, where $S$ denotes a signal trial, $N$ a noise trial. But this definition is of little use because $X$ is left undefined (see p. 29). What does $X$ mean here?

$X$ is the decision threshold that the operator (perhaps unknowingly) is using to categorize trials as signal or noise. Let $f_N$ denote the pdf of the stimulus during a noise trial, and $f_S$ denote the pdf of the stimulus during a signal trial.  Then the formula is better rendered
$$\beta = \frac{f_S(X)}{f_N(X)}$$
which is sometimes described as "the ratio of the ordinate of the signal and noise distributions at the decision threshold." Here ordinate is an old-fashioned way of referring to the y-value or density of the pdf.
This definition assumes that both $f_N$ and $f_S$ describe normal variables of equal variance (but different means).

The text also defines $$\beta_\text{opt} = \frac{P(N)}{P(S)}$$ for the case where payoffs are unspecified, and then provides a more specific formula for optimal $\beta$ given a payoff matrix. What assumptions are made about the payoffs in the definition of $\beta_\text{opt}$ given above?

The longer formula is
$$\beta_\text{opt} = \frac{P(N)}{P(S)} \frac{v_{CR} + v_{FA}}{v_H + v_M}$$
(where $v_{FA}$ and $v_{M}$ are typically negative). The two formulas are equivalent when $v_{CR} + v_{FA} = v_H + v_M$, e.g. when the payoff matrix is symmetrical.
Sluggish beta

If the optimal beta is high, then so is the optimal $X_C$, which corresponds to a conservative policy (i.e. averse to false alarms, per p. 29). So, I think the words in boldface should be switched.

I stand by this.
ROC curve and changes in signal probability

One practice question I have asks, what happens to $\beta$ when signal probability increases? (The question doesn't specify if we are talking about optimal $\beta$ or experimental $\beta$; instead it says "according to the ROC curve.")

I think this is a poorly formulated question.

*

*If the question is about optimal beta, then increasing the signal probability decreases $\beta_\text{opt}$, per the formula above.

*OTOH, suppose the question is about observed beta.

*

*If the system is automated, then an increase in signal probability has no effect on the decision criterion, and $\beta$ is unchanged per the formula above.

*If a human is making a decision, then in practical terms, humans tend to try to equalize the number of false alarms and misses, so increasing the signal probability will generate more misses, causing the user to adopt a more liberal policy, i.e. lowering beta.




(figure)
What point are the authors trying to make by including the signal probabilities $\begin{bmatrix}20 & 10\end{bmatrix}$ above this matrix?

I believe this is an attempt to show how optimal beta shifts in response to changing signal probabilities under the assumption that the payoff matrix is symmetrical.

What do the ROC curves shown above represent? Are they meant to represent idealized experimental data? Or some sort of generalization of the optimal $\beta$-value?

Basically, the latter. These are isosensitivity curves. In theory, a signal detection system (like a Bayesian classifier) with a fixed sensitivity—that is, one whose "internal" representation of the signal and noise variables creates two normal distributions of equal variance—can modulate its decision freely between a totally risky criterion ($\beta \to 0$) and a totally conservative one ($\beta \to \infty$), and doing so will create a curve with the form above, whose shape is determined entirely by the underlying sensitivity (the difference in the means of the normal distributions, called $d'$) and the assumption of normality and equal variance.
There is a nice interactive demonstration of this on this website. It provides the formula
$$d'= \Phi^{−1}(p_H)−\Phi^{−1}(p_{FA})$$
from which the formula for the shape of the isosensitivity curve itself, which gives $p_H$ as a function of $p_{FA}$, easily follows:
$$p_H(p_{FA}) = \Phi\left( d' + \Phi^{−1}(p_{FA})\right)$$
where $\Phi$ is the cdf of a standard normal variable.
We can also get humans to adjust their internal criterion decision (albeit suboptimally, because of sluggish beta) in a signal detection task, e.g. by varying the signal probability or payoff matrix. However, because humans don't have a nice, normal internal representation of the stimulus to work with, doing so often results in a change in both observed beta and sensitivity. The empirical sensitivity can be calculated by using the confusion matrix and applying the assumption of normal stimulus variables with equal variances.
(Formulas for this are given in the appendix to the chapter; the gist is that you figure out how far apart the means of $f_N$ and $f_S$ would have to be from $X_C$ in order to generate the confusion matrix in question.)
