Signal Detection Theory: d' & β

Criterion Position
Distribution Separation

Background:

In signal detection theory, there are two distrubtions of events -- the distribution of events when only noise is present (often assumed to have a mean of 0, but this is not necessary) and the distribution of events when both the signal and noise are present. Because the signal is considered a constant, the signal plus noise distribution has the same shape as the noise distribution, but is displaced on the X axis by an amount that represents the strength of the constant signal. Thus, in the diagram above (assumming that you haven't moved the sliders), the noise distribution is on the right with a mean of 0 and the signal plus noise distribution is on the left with a mean of 3. To simply the math, signal detection theory assumes that both distributions are normal in shape with a standard deviation of 1. This is not a necessary assumption, but will make life a little easier in the calculations.

d' (dee prime), which is signal detection theory's measure of sensitivity, is simply the distance between the means of the two distributions. With the default values, d' equals 3 in the above diagram because the signal plus noise distribution has a mean of 3 and the noise distribution has a mean of 0 (3 = 3 - 0).

Signal detection theory assumes that there is a criterion. Whenever the perception is greater than or equal to the value of the criterion, the observer (signal detection theory's name for a participant) will always respond that the signal is present. Whenever the perception is less than the criterion, the observer will always respond that the signal is absent. The position of the criterion is under the control of the observer. If the signal occurs on most trials, the observer may shift the criterion to the left -- making its value smaller so that the observer will respond that the signal is present more of the time. Conversely, if the signal rarely occurs, the observer may shift the criterion to the right -- making its value larger so that the observer will be less likely to respond that the signal was present. If the cost of making a false alarm (saying that the signal was present when in reality it was not) is high, the observer might move the criterion to the right -- the observer requires more evidence of the signal before being willing to state that the signal is present.

β is the signal dection theory's measure of response bias -- how willing the observer is to say that the signal was present. β is defined as the ratio of the height of the signal plus noise distribution at the criterion to the height of the noise distribution at the criterion. As the criterion gets larger, β gets larger and the observer is said to be more conservative. As the criterion gets smaller, β gets closer to 0 and the observer is said to be liberal. Values of β from 0 to, but not including 1, are said to be liberal, while values of β greater than 1 are said to be conservative. Because of the asymmetry of β the base 10 logarithm of β is often used instead. If the base 10 logarithm of β is less than 0, the observer is liberal. If the base 10 logarithm of β is greater than 0, the observer is conservative. When β equals 1 or the base 10 logarithm of β equals 0, the observer is unbiased -- neither liberal nor conservative.

The Activity:

In the activity, you can manipulate the value of the criterion by sliding the "Criterion Position" slider left or right. As you do so, the values of the probability of a false alarm (the red hashing in the distribution) and the probability of a hit (the green hashing in the distribution) will change. As you would expect, the value of β will also change. The value of d' remains constant. What should happen as you move the criterion to the right? What should happen as you move the criterion to the left?

You can also manipulate how strong the signal is by sliding the "Distribution Separation" slider. Moving the slider to the right makes the signal stronger relative to the noise while moving the slider to the left makes the signal weaker relative to the noise. What should happen to d', p(hit) and p(false alarm) as you increase the separation of the distributions? What should happen to d', p(hit) and p(false alarm) as you decrease the separation of the distributions?

One of the most important features of signal detection theory is that the values of d' and β are independent of each other. That is, any (reasonable) value of d' can be paired with any (reasonable) value of β. Thus, bottom-up processes (as measured by d') and top-down processes (as measured by β) are independent of each other as they should be.

This is not something that you can do in the activity. One would normally want to have a large probability of a hit and a small probability of a false alarm. But manipulating the separation of the distributions does not achieve both and moving the position of the criterion tends to increase or decrease both together. Yet, it is possible to simultaneous increase the probabilty of a hit while decreasing the probability of a false alarm -- how?

If you are really astute, you will have noticed the similarity of signal detection theory and null hypothesis statistical testing. In particular, the probability of a false alarm corresponds to p -- the probability of making a Type I error. The probability of a hit corresponds to 1 - β (where this β is the probability of making a Type II error) which is otherwise known as statistical power. d' is the value z in a z-score test. You can also think of d' as a measure of effect size.

When one does a signal detection study, one presents hundreds of trials in which the signal is sometimes added to the noise. On each trial, the observer must state whether the signal is present or not. From the responses, the researcher calculates the probability of a hit (the number of times that the observer responded that the signal was present and it really was present divided by the total number of times that the signal was presented with noise) and the probability of a false alarm (the number of times that the observer responded that the signal was present but it was not actually present divided by the total number of times that the signal was not present). From these two values, d' and β can be calculated. Step by step instructions for calculating d' and β from p(hit) and p(fa) are available.