Short Notes on Theory of Signal Detection

Walter Schneider

Links

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 Turmor"

Miss Correct Rejection

  Turor No Tumor

Say “Tumor” 40 5

Say “No Turmor"

10 45

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.

• ExampleFalse 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

Sub Hit FA Z(hit) Z(FA) d' Ordinate Hit Ordinate FA Beta1 0.7 0.1 0.52 -1.282 1.806 0.348 0.175 1.9812 0.5 0.4 0.00 -0.253 0.253 0.399 0.386 1.0333 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 1.758NOT 0.683 0.183 0.48 -0.903 1.380 0.356 0.265 1.341

d' CalcuationSub 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 1.9812 0.5 0.4 0.00 -0.253 0.253 0.399 0.386 1.0333 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 1.758NOT 0.683 0.183 0.48 -0.903 1.380 0.356 0.265 1.341

d’ = z(p(False Alarms))- z(p(Hits))

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

Note d’ and Beta must be calculated for each run of an expected sensitivity and criterion separately

Excel HintsZ(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)