All of the measures used to describe performance in SDT are derived from the relationships between the Signal Present and Signal Absent distributions.
In the SDT applet, the Signal Present distribution is shown in green and the Signal Absent distribution is shown in red. If a person is more sensitive to the signal, the difference between the two distributions is greater. We can simulate this by dragging the “d’ =” box to the right.
d’
The most commonly used SDT measure of sensitivity is d’ (d prime), which is the standardized difference between the means of the Signal Present and Signal Absent distributions. To calculate d’, we need to know only a person’s hit rate and false alarm rate.
The formula for d’ is as follows: d’ = z(FA) – z(H)
where FA and H are the False Alarm and Hit rates, respectively, that correspond to right-tail probabilities on the normal distribution. Thus, z(FA) and z(H) are the z-scores that correspond to these right-tail p-values represented by FA and H. Larger absolute values of d’ mean that a person is more sensitive to the difference between the Signal Present and Signal Absent distributions. d’ values near zero indicate chance performance.
Some statisticians use z(FA) and z(H) to indicate the z-scores for the left-tail p-values from the normal distribution. However, in signal detection applications, hit rate and false alarm rates refer to the right-tail of the normal distribution. Thus, we follow Swets (1996) and use z(FA) and z(H) to indicate z-scores for the right-tail probabilities. If we were to use the left-tail p-values, then the correct formula for d’ would be z(H) – z(FA).
What if p = 0 or p = 1?
It is possible that a respondent detects every signal, giving a hit rate of H = 1.00, or a respondent might make no false alarms giving FA = 0. A right-tail p-value of 0 corresponds to z = infinity while a p-value of 1 corresponds to z =negative infinity. This poses a problem for computing d‘ !
A conventional adjustment is to set the minimum p = 1/N where N is the number of trials used in the calculation of p. Similarly, we can set the maximum value for p = (N-1)/N.
Ideally, we design the study to avoid a large portion of p-values at the extremes.
Exercise 3: A subject has a false alarm rate of .30 and a hit rate of .90. Calculate d’ for this subject using the formula: d’ = z(FA) – z(H).
Ask the Expert
The False Alarm rate of .30 indicates that when the signal is absent the subject says ‘Yes’ 30% of the time. Thus, the right-tail p-value on the red ‘Signal Absent’ distribution is .30. Use the p-z converter to find z(FA), the z value that corresponds to this p-value.
The Hit rate of .90 indicates that when the signal is present the subject says “Yes” 90% of the time. Thus, the right-tail p-value on the green ‘Signal Present’ distribution is .90. Use the p-z converter to find z(H), the z-value that corresponds to this p-value.
We can now compute d’ = z(FA) – z(H).
(Now, input the hit and false alarm rates into the Signal Detection Theory applet above to have it calculate d’ for you (Remember to press the ‘Set Hits and False Alarms’ button to update the values in the text boxes). Did the d’ value that you calculated match the d’ value calculated by the applet?
Check Your Answers
An observer has a hit rate of .90 and a false alarm rate of .30. Calculate d’, z(Hits), and z(False Alarms) for this person.
z(False Alarms) = z(.30) = .524
z(Hits) = z(.90) = -1.282
d’ = z(False Alarms) – z(Hits)
= .524 – (-1.282) = 1.806 (rounded to 1.81 by the applet)
Good! Now that you’ve got the basics down, we can move on to an interactive example of an application of Signal Detection Theory and calculation of d’.
Exercise 4 (Sensitivity): The value of d‘ is a measure of the sensitivity of a respondent to the presence of the signal. In this exercise we will use the WISE Signal Detection Theory applet to compare the sensitivity of two respondents to the presence of the signal. Remember to press the ‘Return’ key or the ‘Set Hits and False Alarms’ button after changing a hit or false alarm value
a. Find d’ for Anita who had a hit rate of .90 and a false alarm rate of .30.
Check Your Answers
Anita’s d’ = 1.81
b. Next, calculate d’ for Bob who had a hit rate of .68 and false alarm rate of .09.
Check Your Answers
Bob’s d’ = 1.81
c. How would you describe the relative sensitivity of Alice and Bob? How would you explain this finding?
Check Your Answers
Anita and Bob are virtually identical in their sensitivity. Anita was more willing than Bob to say ‘Old,’ which resulted in her higher hit rate and also a higher false alarm rate.
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