Signal Detection Theor y (SDT)Hervé Abdi11 OverviewSignal Detection Theory (often abridged as SDT) is used to analyzedata co ming from experiments where the task is to categorize am-biguous stimuli which can be generated either by a known process(called the signal) o r be obtained by chance (called the noise in theSDT framework). For example a radar operator must decide if whatshe sees on the radar screen indicates the presence of a plane (thesignal) or the presence of parasites (the noise). This type of appli-cations was the original framework ofSDT (see the founding workof Green & Swets, 1966) But the notion of signal and noise canbe so mewhat metaphorical is some experi mental contexts. Forexample, in a memory recognition experiment, participants haveto decide if the stimulus they currently see was presented before.Here the signal corresponds to a familiarity feeling generated by amemorized stimulus whereas the noise corresponds to a fa miliar-ity feeling generated by a new stimulus.The goal of detection theory is to estimate two main parame-ters from the experimental data. The fir s t parameter, ca lled d′, in-dicates the strength of the signal (relative to the noise). The secondparameter called C (a variant of it is called β), reflects the strategy1In: Neil Salkind (Ed.) (2007).Encyclopedia of Measurement and Statistics.Thousand Oaks ( CA): Sage.Address correspondence to: Hervé AbdiProgram in Cognition and Neurosciences, MS: Gr.4.1,The University of Texas at Dallas,Richardson, TX 75083–0688, USAE-mail:[email protected] http://www.utd.edu/∼herve1Herv é Abdi: Signal Detection TheoryTable 1: The four possible types of response in sdtDECISION: (PARTICIPANT’S RESPONSE)REALITY Yes NoSignal Present Hit MissSignal Absent False Alarm (FA) Correct Rejectionof response of the participant (e.g., saying easily yes rather thanno). SDT is used in very different domains from psychology (psy-chophysics, perception, memory), medical diagnostics (do the sy-mptoms match a known diagnostic or can they be dismissed areirrelevant), to statistical decision (do the data indicate that the ex-periment has an effect or not).2 The ModelIt is easier to introduce the model with an example, so supposethat we have designed a face memory experiment. In the first partof the experiment, a participant was asked to memorize a list offaces. At test, the pa rticipant is presented with a set of fa ces one ata time. Some faces in the test were seen before (these are old faces)and some were not seen before (these are new faces). The task isto decide for each face if this face was seen (responseYes) or not(responseNo) in the first part of the experiment.What are the different types of responses? A Yes response givento an old stimulus is a correct response, it is called a Hit; but a Yesresponse to a new stimulus is a mistake, it is called a False Alarm(abbreviated as FA). A No response given to a new stimulus is acorrect response, it is called a Correct Rejection; but a No responseto an old stimulus is a mistake, it is called a Miss (abbreviated asFA). These four types of response (and their frequency) can be or-ganized as shown in Table 1.2Herv é Abdi: Signal Detection TheoryThe relative frequency of these four types of response are notall independent. For example when the signal is present (first rowof Table 1) the proportion of Hits and the proportion o f Misses addup to one (because when the signal is present the subject can sayeither Yes or No). Likewise when th e s ig n al is absent, he propor-tion of FA and the proportion o f Correct Rejection add up to o ne.Therefore all the information in a Table such as Table 1) i s given bythe proportion of Hits and FAs.Even though the proportions of Hits and FAs provide all the in-formation in the data, these values are hard to interpret becausethey cruci a lly depend upon two parameters. The first parameteris the difficulty of the task: The easier the task the larger the pro-portion of Hits and the smaller the proportion of FAs. Wh en thetask is easy, we say that the signal and th e noise are well separated,or that there is a large distance between the signal and the noise(conversely, for a hard task, the signal and the noise are close andthe distance between them is small). The secon d parameter is thestrategy of the participant: A participant who always says No willnever commit a FA; on the other hand, a pa rticipant who alwayssays Yes is guaranteed all Hits. A participant who tends to give theresponse Yes is called liberal and a par tic i pa nt who tends to givethe response No is called conservative.3 The SDT modelSo, the proportions of Hits and FAs reflect the effect of two under-lying parameters: the first one reflects the separation between thesignal and the noise and the second one the strategy of the partic-ipant. The goal ofSDT is to estimate the value of these two para-meters from the experimental data. In order to do so,SDT createsa model of the participant’s response. Basically theSDT model as-sumes that the participant’s response depends upon the intensityof a hidden variable (e.g., familiarity of a face) and that the partici-pant responds Yes when the value o f this var ia ble for the stimulusis larger than a predefined threshold.3Herv é Abdi: Signal Detection TheorySDT also assumes that the stimuli generated by the noise con-dition var y naturally for that hidden variable. As is often the caseelsewhere, SDT, in addition, assumes that the hidden variable val-ues for the noise follow a normal distributio n. Recall at this point,that when a variable x follows a Gaussian (a.k.a Normal) distrib-ution, this distribution depends upon two parameters: the mean(denoted µ) and the variance (denoted σ2). It is defined as:G (x,µ,σ ) =1σp2πexp½−(x −µ)22σ2¾. (1)In general within theSDT framework the values of µ and σ are arbi-trary and therefore we ch o o s e the simpler values of µ = 0 and σ = 1(other values will g i ve the same results but with more cumbersomeprocedures). In this case, Equation 1 reduces toN (x) =1p2πexp½−12x2¾. (2)Figure 1: The model of sdt.Finally,SDT assumes that the signal is added to the noise. Inother words, the distribution of the values generated by the si g na l4Herv é Abdi: Signal Detection Theorycondition has th e same shape (and therefore the same variance) asthe noise distribution.Figure 1 illustrates the SDT model. The x-axis shows the i n-tensity of underlying hidden variable (e.g., familiarity