PACKAGE |STAT Data Manipulation and Analysis, by Gary Perlman
NAME dprime - compute d' and beta for signal detection data
SYNOPSIS dprime [-p] [hit-rate false-alarm-rate] [hits false-alarms misses correct-rejections]
DESCRIPTION dprime works from data collected in a signal detection task in which a signal is either present or not in a stimulus, and an observer either indicates that a signal was detected or not. 
                         Signal Presented?
                         Yes                 No
     Signal Detected?
     Yes                 hit                 fa
     No                  miss                cr
In the table above, a hit is a signal that was presented and detected, a miss is a signal that was presented but not detected, a fa (for false alarm) is a signal that was not presented but mistakenly detected, and a cr (for correct rejection) is a signal that was not presented and not reported as detected by the observer. The hit rate, sometimes called hr, is the percentage of signals that were detected and is based on the number of hits and misses: 
hr = #hit / (#hit + #miss)
The false alarm rate, sometimes called far, is the percentage of non-signals that were reported as signals and is based on the number of false-alarms and correct-rejections: 
far = #fa / (#fa + #cr)
dprime reads its input in several formats. dprime can be given two arguments: the hit-rate and the false-alarm-rate, for which dprime will compute d', beta, and the criterion. dprime can also be give four arguments: the number of hits, the number of false-alarms, the number of misses and the number of correct-rejections, for which dprime will compute the hit-rate and false-alarm-rate and then compute d', beta, and the criterion. Otherwise, if no values are provided on the command line, dprime reads raw data from the standard input. If raw data are input, dprime assumes a two column input in which the first column tells whether signal+noise or just noise were presented, and the second column tells how the observer responded. The following strings can be used to indicate affirmative answers 
signal, yes, 1
while the following can be used to indicate negative: 
noise, no, 0
Upper case forms for the above are allowed. When reading the standard input, dprime allows an optional third column that gives a repetition count for the case.
OPTIONS
-p
print the `precision' of the observer, calculated as: 
#hit / (#hit + #false-alarm)
In the information retrieval field, it is common to evaluate the effectiveness of a retrieval with two values:
R =
recall = proportion of relevant items retrieved (the hit-rate)
P =
precision = proportion of retrieved items that are relevant
The correspondence to signal detection is that `relevance' is the signal to be detected and has been discussed in Swets, J. A. (1969) Effectiveness of Information Retrieval Methods. American Documentation, 20, 72-89.
The following standard help options are supported. The program exits after displaying the help.
-L
Display limits
-O
Display options and values
-V
Display version number and date
ALGORITHM The value for d' is the Z value of the hit-rate minus that of the false-alarm-rate. 
d' = Z(hr) - Z(far)
This reflects the distance between the two distributions: signal, and signal+noise. Though Z values can have any real value, normally distributed ones are between -2 and 2 about 95% of the time, so differences of twice that would be rare.

The value for beta is the ratio of the normal density functions at the criterion of the Z values used in the computation of d'. This reflects an observer's bias to say `yes' or `no' with the unbiased observer having a value around 1.0. As the bias to say `yes' increases, resulting in a higher hit-rate and false-alarm-rate, beta approaches 0.0. As the bias to say `no' increases, resulting in a lower hit-rate and false-alarm-rate, beta increases over 1.0 on an open-ended scale. A major reason for doing a signal detection analysis is to get a measure of discrimination that is constant over observer biases, but the invariance of d' over different beta is often not certain. The `criterion' is the distance from the mean of the noise (non-signal) distribution to the point above which the observer says `yes' and is in the same units as d'.

If results indicate a perfect observer (e.g., all hits and no misses), then then d' would be infinite. In realistic conditions, this would never happen. In such cases, d' is reported as 0.0, and beta is reported as 1.0.

EXAMPLE 
dprime .7 .4     # will print
hr     far    dprime beta   criterion
0.700  0.400  0.778  0.900  0.253

dprime 70 400 30 600 # will print the same, but with a table
REFERENCE Coombs, C. H., Dawes, R. M., and Tversky, A. (1970) Mathematical Psychology, Englewood Cliffs, NJ: Prentice-Hall. See Chapter 6 on The Theory of Signal Detectability, pp. 165-201.

Salton, G, & McGill, M. J. (1983) Introduction to Modern Information Retrieval, New York: McGraw-Hill. See Chapter 5 on Retrieval Evaluation, especially pp. 164-184.

BUGS The program has not been tested extensively.

In 2016, a bug was reported by a user. In dprime, the 'criterion' is not calculated correctly. In line 167 of dprime.c, it should be: 

criterion = (zhr+zfar)/-2.000;
to compute the criterion as recommeded in: http://people.brandeis.edu/~sekuler/stanislawTodorov1999.pdf
UPDATED 2016-05-16