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# Short Notes on Theory of Signal Detection

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```									     Short Notes on Theory of Signal
Detection
Walter Schneider
calculator http://wise.cgu.edu/sdtmod/index.asp

handout
http://www.cns.nyu.edu/~david/handouts/sdt/sdt.html
Normal Distribution
Hit/Miss & Criterion

Turor             No Tumor

Say “Tumor”             Hit            False Alarm
Say “No              Miss               Correct
Turmor"                                Rejection

Turor             No Tumor

Say “Tumor”             40                   5
Say “No                10                  45
Turmor"
TOTAL                 50                  50

The hit rate is 40/50 or as a proportion .80.
The false alarm rate is 5/50 or .10.
Criterion Shift
d’ and ROC
Basic Curves
d’ - Sensitivity

• The formula for d' is:
d’ = z(p(False Alarms))- z(p(Hits))
• where z (H) and z (FA) represent the
transformation of the hit and false alarm
rates to z-scores.
• Example
False Alarms = 0.10, Hits = 0.70,
d’= Z (0.10) – Z (0.70) = [1.28] – [-0.52] = 1.8
Beta - Criterion

• Criterion bias ordinate of hit/ ordinate FA
• The formula for d' is:
Beta = [Ordinate p(Hit) ] / [Ordinate p(FalseAlarm) ]
• Example False Alarms = 0.10, Hits = 0.70,
Beta = [Ordinate   (0.70)   ] / [Ordinate (0.10) ]
= [0.349]              / [0.176] = 1.98
Example of Mean 3 subjects
d’ = z(p(False Alarms))- z(p(Hits))

Beta = [Ordinate p(Hit) ] / [Ordinate p(FalseAlarm) ]

d' Calcuation
Sub          Hit            FA            Z(hit)       Z(FA)      d'     Ordinate Hit Ordinate FA Beta
1            0.7           0.1          0.52      -1.282 1.806          0.348       0.175     1.981
2            0.5           0.4          0.00      -0.253 0.253          0.399       0.386     1.033
3          0.85          0.05           1.04      -1.645 2.681          0.233       0.103     2.261
Aver               0.683         0.183          0.520      -1.060 1.5802         0.327       0.222     1.758
NOT                0.683         0.183           0.48      -0.903 1.380          0.356       0.265     1.341

Note d’ and Beta must be calculated for each run of an expected
sensitivity and criterion separately
Excel Hints
Z(Hit)=NORMINV(Hit,0,1)
Ordinate(Hit) = =1/SQRT((2*PI()))*EXP(-POWER(Hit,2)/2)
Practical Considerations
• Need significant FAs and misses (>10%)
– (NOTE A’ less sensitive to low FAs)
• Data must be done with consistent bias
and sensitivity
– Calculations must be done within subject and
if need be within run
– To determine average d’ and beta calculate
the individual estimates (DO NOT AVERAGE
THE RAW HITS AND FALSE ALARMS)

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