To catch a thief with a recognition test: The model and some empirical results

COGNITIVE

PSYCHOLOGY

21, 423-468 (1989)

To Catch a Thief with a Recognition Test: The Model and Some Empirical Results SAM S.RAKOVER Haifa

University

AND BARUCH CAHLON Oakland

University

The purpose of the present paper is to describe a new technique and a mathematical model-called the “Catch model”-for identifying a face previously seen (i.e., the target face). Both the technique and the model were developed on the basis of the general approach of information processing used with respect to human memory. Subjects were presented with a pair of test faces on each trial. Neither of the test faces was the target face. Their task was to choose from the two test faces the one most similar to the target face. The data furnished by the subjects were used to reconstruct the target face in the following way: At each trial the differentiating values, such as a long nose and blue eyes, of the test face chosen by the subject were recorded. These values were the ones that accounted for the difference between the two test faces. Over the whole run of the test trials, the differentiating values were associated with various frequencies of occurrence. The target face was reconstructed by selecting the differentiating values having the highest frequency of occurrence. Only one differentiating value per facial dimension such as a nose and eyes could be selected. Thus, given that the facial dimension of the nose has three different values consisting of the long, short, and wide varieties of nose, the value chosen would be the one associated with the highest frequency of occurrence. Mathematical derivations show that, given different variations of the proposed technique, the target face will be detected. These Q 1989 Academic derivations were supported by the results of three experiments. Press, Inc.

Experiments 1 and 2 reported in this paper were conducted at Oakland University, Rochester, MI, when the first author was on sabbatical in 1979. Experiment 3 was conducted at Haifa University and funded by the University’s Faculty of the Social Sciences and Research Authority. The authors are very grateful to Simcha F. Landua, Director of the Institute of Criminology of the Hebrew University at Jerusalem, for making available to us Penry’s PhotoFit Kit. Special thanks go to David Budescu, Itamar Gati, David Navon, and Asher Koriat, who read an earlier version of the manuscript and made helpful suggestions. Requests for reprints should be addressed to Sam S. Rakover, Department of Psychology, Haifa University, Mt. Carmel, Haifa 31 999, Israel. 423 OOlO-0285/89$7.50 Copyright All rights

0 1989 by Academic Press, Inc. of reproduction in any form reserved.

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INTRODUCTION The development of the model for the recognition of faces, and the experiments reported in this paper, were first undertaken in 1975 in response to a practical problem presented to the first author by the Israel Police Force. Israeli police had found that the customary identikit technique fell far short of being a successful means of identifying criminal suspects. In this technique, a police expert attempts to reconstruct the face previously seen, the target face (F,), with the help of an eyewitness. After a number of preliminary questions to the witness concerning the target person’s more obvious attributes, such as age and sex, the police expert proceeds to reconstruct Ft by using transparencies, each of which contains a line drawing of selected facial dimensions of the human face, such as a nose, a mouth, eyes, ears, and hair. These he superimposes one over the other, each time asking the witness whether the reconstruction resembles Ft. If the answer is in the negative, the police expert changes the transparency in question. This procedure is repeated until Ft is reconstructed. The attempts to solve this practical police problem have led us to propose a completely new solution, rather than a suggestion for improving the identikit technique. A survey of the literature has confirmed our suspicion that the reliance of identikit technique on reconstruction memory puts many obstacles in the way of identifying Ft. Some of these difficulties can be traced to such factors as the difference of context between the circumstances in which Ft was seen and the test situation, the insensitivity of the witness, and the existence of many alternatives to each facial feature (for a review see Davies, 1981; Ellis, 1984; Yarmey, 1979). One of the major obstacles to reconstructing Ft in this procedure is that the witness has to compare isolated arbitrary features contained in the identikit with the features of Ft stored in his memory. This operation gives rise to at least two problems: First, it is not known whether Ft is stored in the witness’ memory as a gestalt or as a set of facial dimensions. Second, it is not known if the dimensions of the identikit are the same as the “facial units” stored in the witness’ memory. Given these considerations, our idea was to develop a technique for identifying Ft by using the whole face in a procedure like the one followed in recognition experiments of the yes-no or two-alternative type. In experiments of this type involving recognition memory of faces, subjects are asked to judge whether they have seen a given face-that is, if the test face is the same as Ft. A special advantage in using the whole face in this kind of test of recognition memory is that recognition of faces is known to be very accurate even after a lapse of many years; by contrast, verbal information about faces is only very poorly recalled and has little or no effect on the recognition of faces

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(for a review, see Ellis, 1981, 1984; Goldstein & Chance, 1981; Yarmey, 1979). The trouble with recognition procedure is that it cannot be applied straightforwardly to the situation in which the identification of Ft is required, simply because Ft is unknown. It was to deal with this problem that the particular technique and model set forth in this paper were developed. The paper presents a new experimental paradigm and a mathematical model that combines certain components of recognition memory and similarity judgments. In the experimental situations, subjects (as eyewitnesses) view Ft first and then are asked to view a series of pairs of faces, the test faces, and to choose from each pair the face that is most similar to the remembered Ft. Given the choice data and the model’s postulates, the values in Ft (e.g., long nose and blue eyes) will be present more often in the faces chosen than values not in Ft. This will lead to the possibility of reconstructing Ft from the values that were present most often in the chosen test faces. The results of three experiments supported the predictions of the present model. More specifically we shall describe first the proposed technique which is based on two subroutines which can be computerized-the subject or the witness’ task and the policeman’s task or the procedure for analyzing the data. Second, we shall present the theoretical basis of the model which we have named the “Catch model.” The theoretical basis consists of the delineation of the general assumptions or the framework within which the specific Catch model is developed. Finally we shall present the Catch model as an idealized set of postulates. From the Catch model several consequences will be deduced mathematically and they will also be supported empirically. Note that our general strategy is to start with an idealized and simple model and modify its postulates to cope with the requirements of reality, so that the gap between the model and real-life situations will be narrowed (see also footnote 1). I. The subject’s task: choosing the most similar face. At each trial the subject in our experiment (i.e. the witness) is presented with a pair of test faces, one face on the left side (FL) and the other on the right (FR). From these two test faces the subject is required to choose the one which most resembles Ft. Neither of the test faces was Ft. This is a relatively easy task to accomplish. For example, if one were shown the faces of, say, President Washington and President Reagan, and were asked which most resembled the face of President Lincoln, the answer would be the face of President Reagan. At each successive trial, the subject is presented with a fresh combination of two test faces from which he must select the one that seems best to resemble Ft. Thus in contrast to the feature-by-feature

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routine of the identikit technique, the procedure described here calls for a holistic comparison of faces. 2. The policeman’s

procedure:

The procedure for analyzing

the data.

Before setting out to describe the policeman’s routine in reconstructing F,, we have to distinguish betweenfacial dimensions and values. By facial dimensions we mean the general categories of physical features that make up the human face, such as the nose, the mouth, the eyes, the ears, and hair. By values along each of these dimensions we mean the particular types of noses, mouths, eyes, ears, and hair, that belong to the general categories of facial dimensions. ’ A face is an organized composition of values in which only one value is chosen for any dimension. For example, we might describe Ft by specifying the following values: completely bald head, blue eyes, long nose, bushy eyebrows, thick lips, and narrow chin. In this case the italicized words stand for the facial dimensions, and in conjunction with their modifying adjectives they represent the values. According to our procedure, the subject would be asked to choose at each trial one of the two test faces that most resembles Ft. Let us therefore assume a trial using the following test-F=: completely bald, blue eyes, long nose, bushy eyebrows, thick lips, and broad chin; and FR: completely bald, brown eyes, short nose, bushy eyebrows, thin lips, and broad chin. Let us assume further that the subject has chosen FL. The difference between the two test faces is in the italicized values; the unitalicized values are the same in both faces. We shall call the former the differentiating values. Note that the differentiating values of FL are not those of FR. The major task of the policeman or the procedure for analyzing the data is (a) to record the differentiating values of the test faces chosen by the subject, (b) to select for each facial dimension one differentiating value associated with the highest frequency of occurrence, and (c) to reconstruct Ft by using the values in (b). For each pair of test faces at each trial, the policeman records the differentiating values of the test face selected by the subject. In the course of all of the trials, the differentiating values r The description of a face probably needs a much more elaborate system of representation than the utilization of the two simplistic concepts of dimensions and of values along these dimensions. Although these concepts are sufficient for the present stage of the research program, hierarchy of dimensions, subdimensions, and values might be required for future development of the model. Here are two examples which the present model does not deal with. Long wavy hair can be conceived of as a composite of two values belonging to two diierent facial subdimensions: Length of the hair (short, long) and texture ofrhe hair (straight, wavy, curly). The distance between the eyes or the distance between the nose and the upper lip can also be conceived of as two more dimensions or subdimensions that the mode1 does not deal with. These delineations are based on empirical work concerning the saliency of facial dimensions (e.g., Davies, 1981; Harmon, 1976).

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of the chosen test faces will be associated with different frequencies of occurrence. With the expectation that the test faces chosen by the subject will contain a greater number of differentiating values belonging to Ft than those not chosen, the policeman can proceed to reconstruct Ft by selecting for each facial dimension the differentiating value associated with the highest frequency of occurrence. Thus, in the case of the facial dimension of the nose, if the three differentiating values consisting of long, medium, and short nose are associated respectively with a frequency of occurrence of 12,4, and 7, then the long nose is selected as the expected value of Ft. The expectation that the differentiating values having the highest frequency are those that constitute Ft will be proved under the assumptions of the proposed model. THE THEORETICAL

BASIS OF THE CATCH MODEL

The Catch model is developed within the approach to human memory known as information processing. However, since this is a general approach which includes many different theories and models, it is necessary to narrow the scope and delineate the theoretical domain within which our specific model has been developed. This we do by setting out the five basic assumptions which constitute the theoretical basis of the present model: 1. We assume that Ft is stored in the memory as a gestalt, together with background stimulation and the knowledge of when and where Ft was perceived. These we shall call collectively raw-memory information. 2. We assume that the raw-memory information is processed in various ways, depending on the subject’s intentions and goals, and the requirements of the memory task. 3. We assume that in a yes-no or two-alternative recognition task the subject decides whether a given test face is identical to Ft by following the general procedure known as feature analysis. Briefly, this means that all faces are analyzed in the subject’s mind in terms of a given set of particular values, and that the subject compares the set of values of the test face with the values of Ft stored in his long-term memory. If there is a match between the two sets of values, then the test face is identified by the subject as Ft (for more details, see Rakover, 1983). 4. We assume that what the subject does in choosing the most similar face can be understood in terms of the theory of the yes-no recognitionmemory task that has been described above. Our basic assumption is that the subject makes a match between FL and Ft and between FR and F,, and decides on the basis of these matchings which of the two test faces is more similar to Ft.

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5. Finally, we assume that the policeman’s procedure has to be such that, in conjunction with certain assumptions concerning human recognition memory, it will lead to the reconstruction of Ft. As can be seen, our solution to the problem of identifying Ft falls within the general approach to human recognition memory-namely, the feature-analysis paradigm. In other words, we do not regard the memory for faces to be a unique faculty, in a class apart from the recognition of other visual stimuli. Rather, working within the general information-processing approach, we believe that a schema consisting of a set of facial dimensions together with their appropriate values, by which faces are analyzed and compared, is developed over a lifetime (for a similar view, see Ellis, 1981; Goldstein & Chance, 1981). THE CATCH MODEL As we have mentioned, the subject’s task involves two stages. In the first of these, Ft is projected on a screen; in the second, different pairs of test faces which do not include Ft are presented to the subject in a sequence of trials. At each trial the subject is requested to select from the two test faces the one which most resembles Ft. Figure 1, containing a male Ft and a pair of test faces, will serve to illustrate the procedure. The Ft in Fig. 1 includes a finite number of organized values, with only one value selected from the relevant facial dimension. Of the set of all possible facial dimensions, only four are important for the purpose of identifying Ft in this example: the head, mouth, chin, and eyes. These four facial dimensions are varied from one test face to the other-the remaining facial dimensions, such as the nose, remain constant. Put another way, these four facial dimensions are variable facial dimensions, their values being used to compose the different test faces. By contrast, the constantfacial dimensions are invariable, since only a single value of each such dimension appears in all possible test faces. For the sake of convenience we shall use the expression “facial dimensions with their values” to denote variable facial dimension. In Fig. 1, each of the four facial dimensions comprise two values. These facial dimensions and their values may be listed as follows: (A) Head

(a~) bald

(B) iah

(~2) with hair (C) Chin

(cl) with beard (4 without beard

@d smiling

@2) open (0) Eyes

(dJ with eyeglasses (d2) without eyeglasses

There exist 24 = 16 possible faces. Each face is defined as a profile, or as a set of four values a,bjc&,, where i, j, k, 1 = 1 or 2. Only one of the

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Face

Ft : a2 b, c, d,

The FI : a2 b, c2 d2

Test

Facas Fr: a, b, c2 d,

FIG. 1. The target face and a test pair used in Experiment 1.

16 possible profiles is F,, and the rest are test faces. In Fig. 1 the profde of Ft is defined as F,:a,b,c,d,--or head with hair, smiling mouth, bearded chin, and eyes without eyeglasses. Stated in more general terms, a profile is defined as a set of organized values based on m independent facial dimension and 12independent values per facial dimension. The set of all possible profiles is called the facial space and is denoted by R. The number of profiles in fi is denoted by k(a), which is to say p(n) = P. The facial space excluding the target face is denoted by U. The number of profiles in U is denoted by l.~(U) = nm - 1. The set of all possible pairs of the test faces excluding the target face is denoted by 2. The number of pairs in Z is denoted by p(Z) = t~Ju)[(l~(u) - l)%]. (Note that the Catch model will be developed when Ft is not included in the pairs in Z, since in actual practice Ft may not appear in the pairs of the test faces. Further-

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more, although the model’s predictions can be derived without the inclusion of Ft in the pairs in Z, the inclusion of Ft in these pairs would only enhance the effectiveness of the procedure.) Our model is based on five postulates. It will be proven that, given these postulates, Ft will be identified. POSTULATE 1. Given the raw-memory information of a face, it is possible to represent a face as analyzed by a finite set of independent facial dimensions with a finite set of independent values per dimension. POSTULATE 2. The facial dimensions together with their values in the memory are identical with those constituting Ft and the test faces. POSTULATE

3. The representation of a face and its recollection are

without errors. POSTULATE 4. In deciding which of the two test faces resembles Ft most, the subject goes through the following steps:

(a) He compares each of the test faces with Ft to see if it has the same values as F,; (b) He determines for each test face the number of “matches,” that is, the number of values common to both Ft and a test face. We denote the number of matches by p,(w); (c) He selects the test face with the highest number of matches, or PO0 For example, in Fig. 1 it is predicted that the subject will choose FL, since the number of matches between FL and Ft is three (the matched values being u2, b,, and d,). By comparison, the number of matches between FR and Ft is only one (the sole common value being b1).2 5. In attempting to identify F,, the policeman will go through the following steps: POSTULATE

(a) For each pair of test faces presented to the subject, he will determine the differentiating values of the chosen test face; (b) On the basis of the summation of the subject’s choices over all of the test faces, he will determine for each facial dimension the differentiating value with the highest frequency of occurrence; (c) Finally, he will reconstruct Ft determined in step b. * While in essence Postulate 4 is a simple definition of similarity that resembles Tversky’s (1977) model, the present idea for constructing a scale of psychological distance was created before we became aware of the latter model (see Discussion: Similarity or psychological distance).

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As can be seen, the model’s postulates are in many respects a particular application of the theoretical basis to the problem of identifying Ft. Postulate 1 through Postulate 3 are developed on the basis of Assumptions 1 and 2. Postulate 4 is based on Assumptions 3 and 4, and Postulate 5, which describes the policeman’s procedure, is anchored in Assumption 5. Hence, our general strategy in developing the Catch model is to start with the information-processing paradigm as our theoretical approach to the area of human memory, then to narrow the theoretical domain by delineating our theoretical basis, which allows us to describe in detail the postulates of the Catch model. The model’s postulates are based on a simple and idealized situation which, in many cases, is not in accordance with what is known from the literature. For example, the model has assumed a perfect memory and has attributed equal weight to all values (e.g., Ellis, 1984; Shepherd, Davies, & Ellis, 1981). Nevertheless we have adopted this approach for the following reasons. First, in order to facilitate the mathematical analysis we decided to start with a simple model. Attributing different weights to different dimensions and values, or assuming certain interactions between dimensions and values, will complicate the mathematics to a great extent. Second, starting with a simple model provides us with the information of how far this model could take us. If the reconstructed face does not deviate too much from F,, then it is possible to trade precision for simplicity. And finally, starting with a simple model paves the way for the development of a better approximation of reality. Although we started with the unrealistic assumption of a perfect memory, as the model developed we indeed changed this assumption. But the alteration was not made at random. Rather it was dictated by the model’s framework. Hence, although we started with an idealized model, through its development we ended with a modified model which is much closer to real-life situations. Given the postulates, we derive three propositions and two corollaries. The proofs of these derivations are based on two general considerations. First, since we assumed independence among facial dimensions as well as among values, and since there is a symmetry among all values in Z, all we need to do in order to show success in the identification of Ft is to demonstrate mathematically the following: Given all the chosen test faces in Z, the frequency of occurrence of any given arbitrary differentiating value belonging to Ft is greater than the frequency of any differentiating value not belonging to Ft. Take for example Fig. 1: given F,:a,b,c,d,, all possible test faces such as F,:a,b,c,d, and F,:a,b,c,d, and all chosen test faces such as FL, we need only show that the frequency of the differentiating value a2 (i.e., head with hair) is greater than that of a, (i.e., bald). Assuming the general case in which the value per facial dimension is

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equal or greater than two (n 2 2), we shall make use of the following notations: Given a particular facial dimension, a differentiating value belonging to Ft will be denoted by t and its frequency of occurrence by (I. A differentiating value which does not belong to Ft will be denoted by nr. Since there are many M’S, it is important to discriminate between two kinds of nt. The first is an arbitrary particular nt which is to be compared with t in terms of frequency of occurrence, and will be designated as nt*; its frequency of occurrence will be designated by l&t*). The second kind of nt consists of all nt’s which are not nt*. For example, let us assume the case in which m = 4 and n = 4, with F,:u,b,c,d,. Given the facial dimensions A, B, C, and D, the r’s will be u2, b,, ci, and d,, respectively. A facial featur-say A-contains four values a1a2&a& Since u2 is t, then a,, u3, and u4 are nt’s. Out of these three, a value-say at-is arbitrarily chosen as an nt* to be compared in terms of frequency of occurrence with u2, which is the t. That is, while a, is nt*, the other two values (a, and u4) are nt’s. Accordingly, Ft will be identified if p(t) > p,(ni*). The difference p(t) - p,(nt*) expresses the target’s advantage in frequency of occurrence of t over nt *. This advantage will be denoted by ci. Given the assumptions of the model, it follows that when the policeman summates all the differentiating values, he determines the (Yfor all of the target’s dimensions simultaneously. Since (Yfor facial dimension A is the same as for B, C, D, etc., we define the identification of Ft as the identification of t for any given facial dimension. The second general consideration underlying our proof is that we have discriminated among several types of pairs of test faces: contributory pairs, noncontributory pairs, and equal pairs. Contributory

and Noncontributory

Pairs

Given a facial dimension, a pair of test faces has the important function of contributing one point to the frequency of either t or nt*. In the example above, if the subject chooses FL, then u2 (i.e., t) would receive one point: but if he chose FR, then a, (i.e., nt*) would receive one point. Note that in this example the other values receive no points since they play no part in differentiating between FL and FR. Hence, given a facial dimension, the pairs that can contribute points to either t or nt* are those for which there is a contributory difference between (1) r and nt*, (2) t and nt, or (3) nt* and nt. Other pairs are noncontributory pairs. For example, let us assume F,:u,b,c,d2 and three pairs of test faces: (1) F,:a,b,c,& (2) F,:a,b,c,dz,

(3) F~:~,b,c,d2,

F,:u,b,c,d, FR:u3b,c2d2 F,:u,b,c,d,

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In the first pairs, the model prefers FL, with the result that two t’s (i.e., a2 and d,) receive points. In the second pair, the model prefers FL, with the result that f (i.e., az) receives a point. In the third pair, the model prefers FR, with the result that nt* (i.e., ai) receives a point. (Note that in this case another t (i.e., c,) also receives a point .) Equal Pairs There are pairs for which the model cannot make a choice. These are pairs that receive an equal number of matches. For example, given F,:a,b,c,d, and the test pair F,:u,b,c,d, and FR:ulb,c,d,, the p(W) calculated by the model is 2 for both FL and FR. Hence, in terms of the Catch model these two profiles resemble Ft equally. The difftculty with equal pairs is that they may be either contributory or noncontributory pairs-that is, some of them may be among those pairs that contribute points to t and/or to nt*. Put another way, only some of the equal pairs constitute a subgroup of those pairs for which there is a contributory difference. This fact may interfere with the calculation of (Y in a real-life situation. The reason is that as a consequence of not knowing F,, we do not know which of the test pairs are equal pairs, nor do we know whether equal pairs will contribute to t or nt*. For instance, in the example above, if the subject chooses FL, then t (i.e., u2) receives a point; but if he chooses FR, then nt* (i.e., al) receives a point. Given this situation, we have decided to view equal pairs as contributing points only to nt*-that is, to treat them as though the choice were being made to t’s disadvantage. Our reason for doing so is as follows: If we can prove that Ft will be identified under the condition of t’s disadvantage, then clearly Ft will be identified in an actual situation, where there is very little chance that all equal pairs will contribute points solely to nt*. We denote p the number of points contributed to nt* by all possible contributory equal pairs. Therefore when considering equal pairs, we shall denote the selection advantage by y: that is, y = p(t) - p(nt*) - p = cx - f3. PROPOSITION 1. Excluding equal pairs, the number of differentiating values in (FJ and the chosen test face is greater than the number of differentiating values in Ft and the unchosen test face.

Proof. The full mathematical proof is given in Appendix I. The proof demonstrates the crucial condition for identifying Ft. Accordingly, given a pair of test faces, the number of differentiating values in Ft and FL (the chosen test face) is greater than the number of differentiating values in Ft and FR (the unchosen test-face).

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As an illustration consider the following example: F,:a,b,c,d, F,:a,b,c,d,: FR:a2b2c,d, Differentiating values b,cl bzcz 3 (FL is chosen) 1 P.(w) Ft II (differentiating 0 (an empty set) values) 01 p[F, fl (differentiating 2 0 values)] Clearly, the number of differentiating values in Ft and FL, which is 2, is greater than the number of differentiating values in Ft and FR, which is 0. Hence, it is possible to propose that the greater the number of pairs of test faces, the higher the chances of identifying Ft. Although this is a fundamental idea underlying the catch model, Proposition 1 does not specify how to detect the relevant differentiating value (i.e., t) from the other differentiating values. Moreover, Proposition 1 does not take equal pairs into account. These problems are solved by Proposition 2. PROPOSITION 2. If a subject is presented with all possible pairs of the test faces (i.e., Z), then Ft will be identified.

Proof. The full mathematical proof is given in Appendix II. The proof shows that, summating all the differentiating values over the chosen testfaces, the frequency of occurrence of t of a given facial dimension is always greater than the frequency of occurrence of nt*. Hence, t can be distinguished from nt* by its higher frequency of occurrence. Mathematical formulas have been derived from the Catch model postulates by which the frequency of t as well as of nt* may be calculated. Ft is identified by reconstructing its profile in the following way: For any given facial dimension, one and only one differentiating value associated with the highest frequency of occurrence is selected which, according to Proposition 2, is the target’s value. Given a finite number of facial dimensions for F,, the target face can be reconstructed and identitied. The proof of Proposition 2 is in two parts: the first part shows the above for all possible pairs of test faces excluding those pairs we call equal pairs; the second shows the above for all pairs of test faces including the equal pairs. The second part of the proof shows that even in the worst case, when the equal pairs are chosen to t’s disadvantage, the frequency of f will nevertheless be higher than nt* and Ft will therefore be identified.

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Table 1 presents a few numerical examples of the expected frequency of the differentiating values of target and nontarget faces for different m’s and n’s. As we can see from the table, Ft is identified even in the worst case, when p is subtracted from (Y.Hence, given all possible pairs of test faces, Ft is identified even in an actual situation in which equal pairs cannot be distinguished. PROPOSITION 3. If a subject is presented with a random sample of pairs of test faces (i.e., S of Z), then the probability of identifying Ft is a function of the number of S [i.e., t.~(S)l.Thus, if p,(S) 2 (k(Z) - (a - 1)) or k(S) b (p.(Z) - (y - I)), then the probability that Ft will be identified is 1. If k(S) < (t&Z) - CX),or if p(S) < (k(Z) - r), then the probability that Ft will be identified is less than 1. In this case, the probability of identifying Ft increases as a function of the increase of t.Q).

Proof. The full mathematical computations are given in Appendix III. The basic idea of the proof can be summarized as follows. First, 2 is divided into two sets: the set of S, which includes all the pairs of test faces that will be presented to the subject; and the set of K, which includes all pairs that will not be presented to the subject (where p(k) is the number of K-pairs). Second, a K-pair can be classified as a contributory, noncontributory, or equal pair. In regard to the identification of F,, the worst case for calculating the advantage of t over nt* would be if all K-pairs from among all of the test pairs that had been presented to the subject were to contribute points to t. For example, given Ft:a,b,c,d2 and the K-pair FL:a2b2c2d2and FR:albzczdl, it is clear that k(t) would be reduced by one point (since t is u2) had the subject chosen FL as the model predicts. TABLE 1 Frequencies of Differentiating Values of the Target and the Nontarget Face for Different m’s and n’s m=3 Frequencies CLQ t nt* P Target’s advantage a Y

n=2

n=3

m=4 n=4

n=2

n=3

n=4

21 I 1 4

325 96 28 53

1953 513 147 296

105 34 I 15

3160 920 310 419

32,385 8,262 2,853 4,061

6 2

68 15

366 70

21 12

610 191

5,409 1,348

Note. For the definitions of symbols see text or Appendix VI.

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To be on the safe side we have let the proof assume the worst case for K-pairs contributing to t, since in real-life situations, we cannot know whether K-pairs will contribute points to t, nt*, or nt. Hence, our calculations were made with the assumption that K-pairs reduce p(t) by JL(~). The first part of the proof of Proposition 3 shows that for Ft to be identified with a probability of 1, p(k) cannot be greater than cxor y. Thus p,(k) must be equal to or less than (CX- 1) or (y - 1). Since p.(S) = p(Z) - p,(k), it follows that u(S) 2 (p(Z) - (o - 1)) or @) 3 (~0 - (y 1)). The second part of the proof of Proposition 3 shows that the probability of identifying t (i.e., p(t) = Q(t) - p(nt*) > 0)) increases as a function of l.@) when &Xc)is greater than OLor y. In this regard there are two points to be kept in mind. First, in accordance with Postulate 1, since p(t) for identifying the value belonging to facial dimension A is the same as p(t) for identifying the value belonging to facial dimensions B, C, D, etc., thenp(t) is the probability for identifying Ft. Second, the reason p(t) increases as a function of p.(S) is established by Proposition 2. Accordingly, an increase in u.(S) augments the chances of presenting the subject with a greater number of test pairs that contribute to t rather than to nt*, since it has already been shown that there are a greater number of t contributory pairs than nt* contributory pairs. Figure 2 depicts p(t) as a function p,(S) when m = 3, n = 2, and F,:a,b,cl. As can be seen, p(t) increases monotonically with l&S). The function is calculated when considering equal pairs. Note that p(t) equals unity for p(S) = 20 and 21, since p(k) < (y - 1) for y = 1, 2 (i.e., according to Table 1, y can have the maximal value of 2, and therefore p.(k) = 0, 1 or p(S) = 21, 20, respectively). Until now we have assumed a subject with a perfect memory (see Postulate 3). However, this assumption cannot be maintained in real-life situations. To deal with the problem of imperfect memory we propose two corollaries. Let us begin by considering the circumstances to which the first of these applies. An obvious consequence of imperfect memory is error. For example, assume that the target face is F,:a,b,c,d, and the test pair is FL:a2b,c,d, and F,:a,b,c,d,. According to the model, FL should be chosen since FL contains three matches, whereas FR has only two matches. Nevertheless, due to imperfect memory the subject makes an error and chooses FR. As a result t (i.e., az) loses a point, while nt* (i.e., a,) earns a point. Of all possible conditions arising from imperfect memory, this condition interferes most with the attempt to identify Ft. There are in addition two other conditions that are less prejudicial to identifying F,: when only k(t) is reduced by a point, and when only k(nt*) is increased by a point. We shall call the worst of these cases the most error-prone condition. Under this

CATCH

A THIEF

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Ft : a, b, c,

9%

M*3 N=2 9Q-

&I-

55-

50-

45-

+

0

I

,

5

10

fibI = Nudw FIG. 2. p(t)

‘---x--s

of Test Focm Rmdomly sunp*d as a function

of (L(S).

TEST

437

438

RAKOVERANDCAHLON

condition the chances that Ft will be wrongly identified are the highest, since p(t) is reduced by e number of points at the same time that p.(nt*) is increased by e number of points (where e denotes the number of errors). Since (Y and y are large numbers (see Table 1) we must consider the question of how many errors a subject can make under the most errorprone condition and still identify Ft. It is to Corollary 1 that we must turn for the answer. Corollary 2 shows that as a general rule even under the conditions of imperfect memory, information aggregated from many subjects increases the probability of identifying Ft in comparison to information gathered from only one subject. COROLLARY 1. Given a subject with an imperfect memory, if he is presented with all possible test pairs (i.e., Z), then the target will be identified under the following conditions:

(1) when e < v2 (Y - 1 or e < 1/2y - 1, and OLor y are even; and (2) when e < [Y’z(w]or e < [9’2-y], and cxor y are odd. (Square brackets denote the integer part of the number.) Proof. The full mathematical proof is given in Appendix IV. What the proof shows is illustrated by the following example. Table 2 presents the percentages of errors (pe) made under the most error-prone condition. These percentages can be viewed as estimations of the upper limit of failures of memory that still allows for the identification of Ft. COROLLARY 2. The probability of identifying Ft increases as the number of subjects increases. Proof. The full mathematical proof is given in Appendix V. Numerical examples illustrating what this proof shows will be given in the analysis of the following three experiments (see under Effect of Number of Subjects).

TABLE2 Number of Errors (e) and Their Percentages (pe) under the Most Error-Prone Condition for Different m’s and n’s

m=3 n=2 Without equal pairs e pe With equal pairs e pe

n=3

m=4 n=4

n=2

n=3

n=4

2 25

33 26.6

182 27.5

13 31.7

304 24.7

2704 24.3

0 0

7 3.6

34 3.6

5 8.9

95 5.8

674 4.4

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Experiments 1 and 2 Having described the Catch model, we shall now report three experiments exemplifying and supporting some of the model’s major implications. Since the first two experiments used a set of stimuli different from the one employed in the third experiment, they will be presented jointly. Method Procedure. Subjects were students in the Department of Psychology at Oakland University in Michigan. The subjects were informed that the experiment would consist of two parts and were instructed about their tasks before each part of the experiment began. In the first part they were shown a single picture consisting of a line drawing of the face of a man or a woman projected with an overhead projector on a screen for about 15 s. This was the target face. Subjects were asked to examine all of the target’s features. In the second part of the experiment-immediately following the presentation of the target face-subjects were shown a test pair that consisted of line drawings representing the faces of two men or women who were not Ft. Subjects had to decide which of two faces projected on the left and right sides of the screen best resembled the target picture. They had 15 s to make up their minds and mark their answers on the answer sheet. Twenty-one test pairs were presented randomly. The experimental session lasted for 1 h of class time. Design and subjects. Two experiments are reported. In Experiment 1, both F1 and the test pairs were male; in Experiment 2, both F, and the test pairs were female. In Experiment 1 there were 23 subjects averaging 20.74 years of age, of whom 6 were male and 17 were female. In Experiment 2 there were 15 subjects averaging 23.47 years of age, of whom 3 were male and 12 were female. Experiment 1 strictly followed the procedure described above. Experiment 2, however, included additional instructions. After the target was presented, subjects were told that in addition to marking their choices between the two test faces, they should also check off the values upon which they had based their decision. The answer sheet contained the following values of F,: Hair (curly or long); Mouth (closed or open); Earring and necklace (with or without); and Eyeglasses (with or without). Subjects were also informed that if they had no particular reason for their choice, they could check the “general impression” category; and that if they identified features which were not included on the answer sheet, they could write these in under the category of “other.” They were told as well that they would be given an extra 45 s following the test pair presentation in order to perform the additional task. Stimuli. The target faces and test pairs were taken from a deck of cards of a game called Mr. X--Who are you Mr. X? invented by the fvst author. The faces on the cards exactly fitted our experimental requirements, since the rules of this game are very similar to the postulates of our model. The game was manufactured by Litho-Offset “ZIV.” Jerusalem, May 1977. A total of 16 male faces and 16 female faces were used. These were distinguished in terms of their values. The male and female faces were made up of four facial dimensions consisting of two values per dimension (i.e., m = 4 and n = 2). These facial dimensions and values are listed below. Male Faces (A) Head (ai = bald, az = with hair) (B) Mouth (b, = smiling, bZ = open) (C) Beard (c, = with, c2 = without)

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(0) Eyeglasses (d, = with, dz = without) Female Faces (A) Hair (a, = curly, a2 = long) (B) Mouth (b, = closed, b, = opened) (C’) Earrings and necklace (cr = with, c2 = without) (0) Eyeglasses (d, = with, d2 = without) These dimensions and values were selected on the basis of the following considerations. First, some of the dimensions-head, mouth, and chin-were chosen because they have been found to be the most salient dimensions in a face (e.g., Ellis, 1984; Shepherd, Davies, & Ellis, 1981). Second, some of the features-hair, beard, and eyeglasses-were chosen because their addition or removal results in a drastic reduction of face recognition (e.g., Patterson & Baddeley, 1977; Shepherd, Davies & Ellis, 1981). And finally, we decided to include in Ft and in the testing faces some transient and accessory dimensions-eyeglasses, earrings, and necklace. Since our model attributes equal weight to all facial dimensions and values, we wanted to see how well these dimensions would be detected by the present method. As we can see from Table 1, the total number of test pairs is 10.5; and according to Proposition 3 it is possible to present as many as 94 test pairs and still detect Ft (i.e., p(S) = p(Z) - (y - 1)). Practically speaking however, 94 test pairs could not have been used experimentally. We therefore chose 7 faces out of the 15 faces in order to generate 21 test pairs, so that Ft could be detected even when equal pairs were involved. Table 3 presents the model’s predictions for the identification of a male target face by a single subject with a perfect memory. For the sake of convenience, Table 3 has been so designed that p,(W) should be greater for FL than for FR (except for equal pairs). The cells in the table show the differentiating values of the chosen test face-that is, of FL. There are TABLE 3 Left and Right Test Pairs, with Their Differentiating Test profiles: FL-Fa:

(1) Gzcz4 (1)

aAh (1)

a2b2c2dl (1)

aJwA

(2)

a&,c,

a2W2

4

a,b2cld2

bw%

a&d2

&c,d, (3) %b,w-b (3)

WI

0,

F L..abcd 22 2 2 F,:a,b,c,d, F,: bzd, F,:b,d, a,

ad2

W,

w,dz

(2) (2)

Values Frequency a Y

4w,d,

(2)

02d2

(2)

Gw,d, (3)

%azb, F,:a,b, Cl

aAcA (2) I~, rq,b&

Values when F,: a,b,c,d,

a2 a1 I

2 5 2

F,:a,c, F,:a,c, b,c,d, b,

bl

bz

I

2 5 2

a2blc2

Cl G I

2 5 2

aWl

dz 4 -I 2 5 2

Note. Numerals in parentheses represent the number of matches (p,(W)).

FL:c2d2 F,:c,d,

a2hcA (3)

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five cells with equal pairs. These show the differentiating values of both FL and Fa. On the basis of our model, we computed the contributions made by equal pairs to r’s disadvantage. As can be seen, this assumption results in equal pairs contributing 3 points to each nt*. The frequencies for t’s, nl*‘s, o, and y are shown at the bottom of Table 3. A similar table of test pairs has been constructed for a female target face, F,:a,b,c,d,.

Results: Experiment 1 Ft detection. Table 4 presents the main results of Experiment 1. As can be seen from the table, three out of the four values of Ft were identifiedthat is, their frequencies of occurrence were greater than that of nontarget values. (Underlined frequencies in the table indicate the correct values.) This partial identification of Ft was achieved when data were summated over the 21 test pairs and 23 subjects. Given Ft:a2blcld2, the only value not identified was d2 (i.e., “without eyeglasses”). In the present case, equal pairs did not interfere with the identification of F,, although there was a slight tendency for a better Ft detection when equal pairs were not involved than when they were. This is demonstrated by the slightly higher percentage of observed (t) in the absence of equal pairs (see below). Effect ofnumber ofsubjects. The expectation that increasing the number of subjects would facilitate Ft detection is confirmed. This is shown by comparing the percentage of the average correct identification by individuals (1Z) with the percentage of average correct identifications by the group (GI). The II index is computed by the formula

n=xv.f 4.n 100, where v is the number of values identified; f is the number of subjects identifying a particular value; and 4 * n is the maximal number of values that can be identified by n subjects. The GI index is given by dividing the number of detected values by the maximal number of values that can be detected. While GI = 75% with or without equal pairs, II = 58% with equal pairs and II = 72% without equal pairs. Hence, group detection of Ft is better than individual detection of Ft. Note that without equal pairs, detection of Ft is improved on the individual level but not on the group level: the difference between II when equal pairs are present and II when equal pairs are absent is 14%. Fast method. As can be seen from Table 3 there are four test pairs, each containing a single differentiating value of F,, namely, u2, b,, ci, or d2. If we assume perfect memory, then all we need to do in order to identify Ft is to present these four test pairs. The group data have confirmed this expectation: GI = 100% (with the following frequencies: a, = 4; a2 = 19; b, = 19; b, = 4; ci = 21; c2 = 2; d, = 11; d2 = 12). On the other hand,

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TABLE 4 Frequencies oft and nt* Summated over 21 Test Pairs and 23 Subjects in Experiment 1 Values: Conditions: With equal pairs Frequencies Percentage expected 100 . 7/12 Percentage observed loo . t/(t + nr*) Without equal pairs Frequencies Percentage expected 108 . 719 Percentage observed 100 . t/(t + nt*)

(t)

A a, a,

b,

B

102 174 58

208 68 58

167 109 58

6444

63

75

61

42

65 -142

-162 45

-125 82

109 98

78

78

78

78

69

78

60

47

b

C Cl c2

D 4

4

58

(t)

(t) (t)

Note. See text for further explanations.

analysis of individual data reveals a lower level of success in the detection of F,: II = 77%. Once again, the data support Corollary 2, according to which increasing the number of subjects enhances Ft identification. Correct choices and errors. According to the Catch model, in the case of nonequal pairs, when a subject has to decide which of the two test faces resembles Ft most, he will choose the face with the higher p(w). If (for convenience) FL in a test pair is associated with the higher p,(w) (e.g., p,(W),+Il.@V)a= 3/l), then the correct choice is FL. From this two predictions concerning correct choices can be derived. First, the number of correct choices should be above the chance level. Second, the greater (~.L(W)~- p(w),) difference, the greater the tendency for choosing FL over FR. The latter prediction is based on the additional assumption that choosing the test face which resembles Ft most is a monotonical function Of(i-dW)L - t.@%). Table 5 presents percentage correct choices as a function of p,(W)s of the test pairs. As can be seen from Table 5, Experiment 1, the first prediction is supported. Percentage correct choices are significantly above the chance level for all the reported p(W)L/l.@V)a rations (i.e., 2/l, 3/l, 3/2) and for all choices summated over all the test pairs. The second prediction is supported partially. Percentage correct choices tends to increase as a function of the increase of l.r(W’), in the test pairs 2/l and 3/l (x’(l) = 7.02 p < .Ol). There was no significant difference in percentage correct choices between test pairs 3/l and 3/2, and there was no significant difference between the proportions associated with the Q.L(W)~ l.~(W),) difference giving 1 or 2, when these differences were calculated by the summation of correct choices over all the relevant test pairs.

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Percentage Correct Choices as a Function of p(W)‘s of a Test Pair (plier) p(W)s of a test pair:

2/o

Experiment 1

2/l

3/o

3/l

77

59 * 71

2

**

3

60 pJS

69 **

4/o

4/l

412

413

All choices

a6

74 **

**

a5 **

** loo

** 90

69 *

62 **

**

81 **

312

93 **

82 *

74 *

62 *

69 **

Nore. No data are available for empty cells. *p < .05 and **p < .Ol as calculated by the proportion test.

According to Corollary 1, in the case where m = 4 and it = 2, a maximum of two errors can be made and Ft still be detected in the absence of equal pairs, and zero errors when equal pairs are involved. However, these calculations are made under the most error-prone condition. When a less error-prone condition obtains-as for example, when an error subtracts a point from a2 but adds a point to b,--a precise prediction concerning Ft detection is hard to make. Nevertheless, one would think that the greater the number of errors, the smaller the number of detected values of Ft. This hunch is confirmed both when equal pairs are present and absent: R = -0.522 p < .05 and R = - .768 p < .Ol, respectively. Results: Experiment

2

Ft detection and the effect of number of subjects. Table 6 presents the main results of Experiment 2. As can be seen from the table, Ft was TABLE 6 Frequencies oft and nt* Summated over 21 Test Pairs and 15 Subjects in Experiment 2 Value: Conditions: With equal pairs Frequencies Percentage expected 100 . 7/12 Percentage observed 100 . t/(r + nr*) Without equal pairs Frequencies Percentage expected 100 . 719 Percentage observed 100 . t/(t + nr*)

(t)

A a, a2

b,

B

38 -142

C Cl cz

4

-128 52

-104 76

79 -101

58

58

58

58

79

71

58

56

26 -108

-106 29

-87 48

52 -83

78

78

78

78

81

79

64

61

b

D 4

(t)

(t) (t)

Note. See text for further explanations.

444

RAKOVERANDCAHLON

detected both when equal pairs were present and when they were absent. While the group data allow the complete identification of Ft (i.e., GI = lOO%), individual data do not permit the reconstruction of the whole face of Ft (i.e., II = 72% with equal pairs, and II = 93% without equal pairs). Once again, the data support the corollary stating that increasing the number of subjects enhances Ft detection. As in Experiment 1, here, too, the data illustrate the difficulties created by equal pairs for the identification of Ft. Fast method. Each of the four test pairs yields one differentiating value: a2, bi, cl, or d2. Data based on these four test pairs show the group index for Ft identification as GI = 100% (al = 3; a2 = 12; b, = 14; b, = 1, ci = 15; c2 = 0; d, = 2; d2 = 13). In comparison, the index of individual identification of Ft is less than 100%; II = 90%. Correct choices and errors. As in Experiment 1, the results tend to support our two predictions concerning correct choices (see Table 5, Experiment 2). First, percentage correct choices is significantly above the cance level. Second, percentage correct choices tends to increase as a function of the increase of Pi in test pairs 2/l and 3/l (x2(1) = 19.00 p < .Ol), and to decrease as a function of the increase of p,(w), in test pairs 3/l and 3/2 (x2(1) = 4.73 p -=c.05). Percentage correct choices tends to increase as a function of the increase in (Pi - p( W)n) difference giving 1 or 2 (x2(1) = 12.30~ < .Ol). The number of detected values of Ft is reduced with an increase in number of errors: in the case of equal pairs R = - .419 p > .05; and without equal pairs R = - .799 p < .Ol. Postdecision analysis. Experiment 2 allows us to examine the broader issue of how and why subjects decide between FL and FR. The analysis is based on the postdecision task of subjects in this experiment. Before proceeding with our analysis, however, we should consider the implications of asking subjects to mark the relevant values of Ft as the reason for their choice between test faces. First, our instructions may have heightened the subject’s awareness of the value of F,, with the result that their choice performance in the experiment may have been better than it would have been were they not requested to carry out the post-decision task (e.g., in Experiment 2 all facial components of Ft were identified, whereas in Experiment 1 only three out of four were identified). Second, quite probably a complex interaction was created between the subjects’ choices and the reasons for their decisions, so that the order of performance choice followed by reason-may sometimes have been reversed. Although we were aware of these and other complications arising from the method of probing awareness by means of postexperimental questionnaires (e.g., Erikson, 1962; Levin, 1964) we nevertheless had to employ the method because it offered a number of interesting answers to our

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question concerning the decision-making process involved in making a choice between test faces. (2) Reasons for making the choice. There appear to be four possible explanations of the reasons for the choice between FL and FR: (1) (2) (3) (4)

The differentiating values Either (Ft fl FL) or (Ft fl Either (F, II FL) or (Ft II p(F, fl FL) versus p(Ft rl

between FL and FR. FR) differentiates between FL and FR. FR). FR).

For example, given F,:a,b,c,d, and the test pair FL:u2b,c2d2 and given that FL is chosen, then the reasons for making a choice may be as follows:

FR:a,bzcld,,

According to (1) the choice can be explained in terms of bl, c2, and d2. According to (2) the choice can be explained in terms of b, and d2. According to (3) the choice can be explained in terms of u2b, and d,. According to (4) the choice can be explained in terms of the number of values common to Ft and FL versus the values common to Ft and FR that is, FL is chosen because the subject correctly thinks that p(F, n FL) > t-P, n 4th

Analysis of the data shows that reasons (1) and (2) were not used by the subjects. Had this been the case, then subjects would not have checked those values that appeared in both FL and FR. Thus, in the example above, subjects would not have checked a2 as their reason for choosing FL. However, our analysis shows that out of the total of 270 test pairs in which the same value appeared in both FL and FR (summated over subjects and facial dimension), in 208 such cases (i.e., 77%) subjects indicated the same value as reasons for their choice. This result leaves (3) and (4) as possible reasons for choosing between FL and Fn. But it would be difficult to establish which of the two remaining possibilities was the reason for the subjects’ choice. Part of the difficulty is that (3) and (4) are conceptually associated, and that the computation suggested by (4) could be carried out automatically, that is to say without the subject’s awareness of the value of Ft. Nevertheless, since (4) is an essential part of Postulate 4, the preceding analysis provides at least some support for the Catch model. (2) The problem of awareness. As an alternative explanation of the experimental results we can propose the subject’s awareness of the values of Ft. Accordingly, if the subject is aware of all the values of the target then it is clear that Ft will be detected. But what happens when the subject is aware of one or two, or even three, of the values of F,?

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RAKOVERANDCAHLON

For one thing, if the subject is not aware of all the values of F,, then in many cases the awareness hypothesis would generate predictions completely opposed to the predictions derived from the Catch model. For example, let us assume that the subject is aware that a2 is a value belonging to Ft. Hence, if he is presented with the test pair FL:a,b,c2d, and FR:a,b,~,dz, then he should choose FL. This result, while in agreement with the awareness hypothesis, is contradictory to the predictions of the Catch model. The Catch model predicts that the subject will select FR and not FL, since l.~(W,) = 3 while l~(W,) = 1. (It is noteworthy that while the Catch model attempts to describe the cognitive mechanism which stands behind the choice between the two test faces, the awareness hypothesis does not specify how a subject becomes aware, for example, of a2 and not of b,.) But how many values were the subjects actually aware of? To answer this question we have to compare the number of correct checks in the postdecision task with the number that can be made correctly by chance. A finding that the number of observed correct checks is significantly greater than the number of correct checks that can be made by chance would support the hypothesis that the correct checks resulted from subjects’ being aware of the values of Ft. Now, if we find that the number of values of which subjects were actually aware is less than four, then we may justifiably conclude that the efficiency of the Catch model is greater than the efficiency of the awareness hypothesis, even in the present simple case where m = 4 and n = 2. Clearly, the awareness hypothesis has no hope in a real-life situation where, for all practical purposes, the number of dimensions and values reaches infinity. The correct number of checks made by chance was estimated in the following way. Given a facial dimension in a chosen face a subject may check either t or nt as reason for selecting that face, or he may check neither. Hence, given a facial dimension and assuming a random process of making checks, the probability of randomly checking a value belonging to F,say a,- is v3 (i.e., checking t out of the following three possibilities: checking t or nt or checking neither). Each subject chose the 21 faces most similar to Ft. In some of these faces, t (e.g., u2) appears. We counted the number of times t appears in the 21 faces chosen for each subject, and we multiplied these numbers by the probability of correct checking, namely VS.This yielded the number of correct checkings made by chance. For example, if a2 appeared 18 times in the 21 faces chosen by a particular subject, then the number of correct checkings made by chance by that subject is 18 X % = 6. Given these estimations of correct random checkings, we used a correlated t-test to confirm the hypothesis that subjects were aware of the

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value of F,, namely that the number of observed correct checks is significantly greater than the number of correct checks made by chance. The results showed that checking was nonrandom for all facial dimensions except D:t(,,) = 11.05 p < .Ol for dimension AJ~,,~ = 7.27 p < .Ol for dimension B, to,) = 12.68 p < .Ol for dimension C, and to,) = 1.33 p > .05 for dimension D. These results suggest that subjects were probably aware of the following facial components of F,: u2, b,, and c, but not of d2. Given these results, it is clear that Ft cannot be reconstructed in full on the basis of the subject’s awareness of the values of the target. On the other hand Ft can be completely reconstructed on the basis of the Catch model. Hence, we can conclude that in order to account for all the data we have to assume that unconscious processes were involved in addition to those processes of which the subjects are aware. It seems that retrospective reports provide us with little help in recognition of faces. This conclusion accords with the proposals of other researchers that both semantic and visual analyses are involved in the recognition of faces (e.g., Baddeley, 1979; Bruce, 1979; Ellis, 1981). And that verbal information is of limited use in face recognition (e.g., Ellis, 1984; Goldstein & Chance, 1981; Shepherd, Davies, & Ellis, 1981; Yarmey, 1979). This conclusion does not stand in contradiction to several successful attempts to identify Ft automatically through the use of the eyewitness verbal description and certain computer search programs (e.g., Harmon, 1976; Harmon, Kuo, Ramig, & Raudkiv, 1978; Harmon, Khan, Lash, & Ramig, 1981; Lenorovitz & Laughery, 1984). The major idea in these studies is to develop a computer program for searching Ft from a given set of faces including Ft. These faces, such as human face profiles and “mug files,” are stored in the computer’s memory as certain vectors of facial dimensions. One of the faces-Ft-is shown to the witness and he describes it to the computer, dimension by dimension, from his memory (or from a picture). Utilizing this information the computer eliminates parts of the faces as nontarget. Each additional description eliminates more faces stored in the computer as nontargets. The process continues until Ft is identified. There are two fundamental differences between the computer search program and our procedure which supports the previously mentioned conclusion that verbal reports have little to do with face recognition. First, while the computer program’s goal is to search a particular member from a given set of faces, our procedure attempts to use a given set of faces in order to reconstruct a completely new face-that of Ft. And finally, while the computer program is first of all a technique for searching for Ft in a given set of faces, our procedure attempts, first, to theorize how the eyewitness recognition memory works, and then to utilize this theory for reconstructing Ft.

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Saliency of facial dimensions and values. Tables 4 and 6 provide us with information concerning the values’ saliency. The percent t(t + nt*) can be interpreted as reflecting the ease with which F,‘s values are identified. The higher the percentage of a particular value the easier is the detection of that particular value as belonging to Ft. The order of the detecting-easiness of the male values (see Experiment 1) is as follows: Mouth smiling (75%); Head with hair (63%); Chin with beard (61%); and Eyeglasses, without (42%). And the order of the female values (see Experiment 2) is: Hair length (79%); Mouth closed (71%); Earrings and necklace, with (58%); and Eyeglasses, without (56%). Without going into the differences between the differential saliency of female vs male values, it is clear that hair and mouth (and beard) are detected much more easily than earrings and necklace, and eyeglasses. This finding, on the one hand, supports our model since it corresponds with what is known from the literature (e.g., Ellis, 1984; Shepherd, Davies, & Ellis, 1981). On the other hand, it proposes that attributing different weights to different dimensions and values could improve the predictability capacity of the Catch model. Experiment 3 The purpose of this experiment is to test the model’s major implications when shifting from a low level to a high level of approximation to reality. First, in comparison to the previous two experiments in which line drawings of faces were used, the present experiment used photographs of male faces. Second, whereas these two experiments used four facial dimensions, consisting of two values per dimension, the present experiment used five dimensions with two values per dimension. And, finally, while the first two experiments used a sample of 21 test pairs composed of seven profiles which were selected nonrandomly from Z, the present experiment presented a sample of 100 test pairs randomly selected from a population of 465 test pairs. Method Procedure and subjects. Essentially the procedure of the present experiment was the same as for the first two experiments. Briefly, subjects were shown black and white photographs of male faces (F, and test pairs) on a television screen. They had to decide wltich photograph of a test pair best resembled F,. The target face and each of the test pairs were presented for 15 s. The experimental session lasted for about 1% h. Subjects were students in the Department of Psychology at Haifa University in Israel. There were 30 subjects whose average age was 25.8 years, of whom 11 were male and 19 were female. Stimuli and design. The Ft and the test pairs were taken from Penry’s Photo-Fit kit (Penry, 1971, a, b). In this technique faces are reconstructed from five facial dimensions, each consisting of many different values. For the present experiment the male faces were made up of five dimensions, each of which consisted of two values as listed below:

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449

(A) Foreheads (at = F19, a2 = F17)

(B) Eyes and Eyebrows (b, = E16, b, = El) (C) Noses (c, = N20, c2 = N26)

(0) Mouths (d, = Ml5, d, = M17) (E) Chins (e, = C45, e2 = C60). The capital letters and numbers in the parentheses designate the specific value taken from Penry’s Photo-Fit kit. Figure 3 presents F,: a,, b,, c2, d2, e, and two test pairs FL: a,, b,, cz, d,, elr and FR: a2, b,, c,, d2, e,, where the p(W) of FL is 4, and the p,(W) of FR is 1. All faces were reconstructed with the same distances between the forehead and the eyes (at the D-level in Penry’s Photo-Fit kit), and between the mouth and the chin (at the Y-level in Penry’s Photo-Fit kit). A set of 32 photographs of faces was made up. All those pictures were retouched in order to eliminate the contours created among the five values constituting a face. These pictures (except F,) were used to generate the 100 test pairs, which were photographed by a television video camera. There were 32 male faces (i.e., 25 = 32) and 465 test pairs (l/2.31.30 = 465). Out of these test pairs a sample of 100test pairs was randomly drawn with the following restriction: each profile of the 31 test faces appeared six to eight times among the test pairs. There were 19 equal test pairs in the sample of 100 test pairs. Locations of faces within test pairs were determined randomly. The order of the presentation of the 100 test pairs was also randomly determined. As can be seen from the percentage expected of (t&t + nt*)) in Table 7, F, can be detected with or without the equal pairs, even though the particular sample used in the present experiment consists of only 100 test pairs. The probability of detecting F, with a sample of 100 test pairs is .8868. A sample of 358 test pairs is needed in order to detect F, with the probability of unity.

Results Ft detection and the effect of number of subjects. Table 7 presents the main results of Experiment 3. As can be seen from the table, Ft was detected both when equal pairs were present and when they were absent. While group data allow the complete identification of Ft (i.e., GI = lOO%), individual data do not permit the reconstruction of the whole face of F,: II = 78.0% with equal pairs, and II = 78.7% without equal pairs. The data support the present model’s main predictions that Ft would be identified with a high probability and that an increase in the number of witnesses would enhance Ft detection. As in the other two experiments, here, too, the data illustrates the difficulties created by equal pairs for identification of Ft (i.e., in Table 7 compare percentage t/(t + nt*), in the case of equal pairs, to the same percentage in the absence of equal pairs). Fast method. Among the 100 test pairs, five test pairs yield one differentiating value a,, three yield b,, two c2, four d2, and five e2. Data based on these nineteen test pairs show that four out of five values of Ft were identified: a, = 124, a2 = 26; 6, = 41, b, = 49; cl = 13, c2 = 47; d, = 49, d2 = 71; and e, = 35, e2 = 115.

450

RAKOVERANDCAHLON The Tagat‘s

Face

Ft : a, b, c2d2 e2

The Test - Facss

Fl:alb2c2d2e2

Fr:a2

b2c,d2e,

FIG. 3. The target face and a test pair used in Experiment 3.

Additional analysis also supports Corollary 2, according to which increasing the number of subjects enhances Ft detection; whereas GI = 80.0%, II = 65.33%. Correct choices and errors. As in the previous two experiments, the results tend to support our two predictions concerning correct choices (see Table 5, Experiment 3). First, percentage correct choices is signifi-

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TABLE I Frequencies of t and nt* Summated over 100 Test Pairs and 30 Subjects in Experiment 3 Values: Conditions: With equal pairs Frequencies Percentage expected 100 . tl(t + rlt*) Percentage observed 100 . t/(t + nt*) Without equal pairs Frequencies Percentage expected 100 . tl(t + rlt*) Percentage observed 100 . t(t + rlt*)

B b,

1109481

-833 697

572 1018

551 -739

636 954 -

83.02

84.31

61.92

76.14

69.81

69.15

54.44

64.03

57.29

60.00

-962 388

-693 531

426 -804

372 -618

464 -766

80.0

82.93

80.49

87.88

85.37

71.26

56.34

65.37

62.42

62.28

(t)

b

C Cl G

D

E e, e2

A aI a2

4

d,

(t)

(t)

(t)

Note. See text for further explanations.

cantly above the chance level except for test pairs 2/O. Second, percentage correct choices tends to increase as a function of the increase of t.~(w~ in the following test pairs: 2/O, 3/Oand 4/O(x’(2) = 10.65 p < .Ol); 2/l, 3/l and 4/l (x2(2) = 8.03 p < .05); 3/2 and 412(x2(1) = 16.70~ < .Ol). Percentage of correct choices tends to decrease as a function of increase in t~,(W”)nin the following test pairs: 410, 4/l 4/2 and 4/3 (x2(3) = 24.79 p < .Ol); 3/O, 3/l and 3/2 (x2(2) = 16.77 p < .Ol), 2/O and 2/l (x2(1) = 1.50 p > .05). Percentage correct choices tends to increase as a function of the increase in (&W), - t~,(W)a)difference giving 1, 2, 3, or 4 (x2(3) = 41.81 p < .Ol). As in the other two experiments, the results of the present experiment show that when the number of errors increases, the number of detected facial components of Ft is reduced: in the case of equal pairs R = - .704 p < .Ol; and without equal pairs R = - .887 p < .Ol. Saliency of facial dimensions and values. As in the previous two experiments, the order of saliency of the facial dimensions in the present experiment corresponds-except for the eyes-to what is reported in the literature (see percentage tl(t + nt*)) in Table 7). No tenable explanation for this deviation has been found. Discussion Correspondence

between the Catch Model and the Results

The major predictions of the Catch model have been confirmed. Ft was reconstructed in full in Experiments 2 and 3, and three out of the four values were detected in Experiment 1. Detection of Ft both when equal

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pairs were present and absent was more or less equal. Ft was reconstructed in full by using the fast method in Experiments 1 and 2, and four out of five values were detected in Experiment 3. Reconstruction of Ft was better when group data were considered than when individual data were used. The number of correct choices was above chance level in all three experiments. The greater the number of errors in making the choices, the smaller the number of detected values of Ft. And, finally, analysis of the postdecision data in Experiment 2 suggests that it would be hard to explain our results solely in terms of subjects’ awareness of the values of Ft. Given the above, and taking into account that only 21 out of a possible 105 test pairs were presented in Experiments 1 and 2, and 100 out of 465 test pairs in Experiment 3, we propose that the Catch model stood up well to our initial empirical tests. Awareness Obviously if subjects were aware of all the correct values of F,, then reconstruction of the target would be a trivial matter. For example, there would be no need for choice data, since it would be sufficient to have the subject describe the Ft. However, our experimental results show that awareness is not the whole story. First of all, awareness is corrigible, and second there are processes involved in memory which are unconscious. Memory comprises a number of interacting processes of which awareness is merely one. The Catch model does not specify which of the assumed processes of memory are part of the subject’s consciousness and which are not. Nevertheless, we would suggest that in real-life situations, where the number of dimensions of Ft is large (if not infinite to all intents and purposes) most of the processes that we have assumed-such as coding, storing, retrieval, and comparison-probably take place without awareness coming into play. These processes simply seem to be out of our conscious attentional mechanisms (e.g., Shiffrin & Schneider, 1977). Shifting from the Ideal Situation to Reality The strategy of constructing the Catch model began with an ideal situation in which we assumed a single subject with perfect memory, independence of facial dimensions and values from one another, the absence of equal pairs, and a test situation that included all possible test faces. In the process of developing the model we increasingly approximated the real-life situation by taking into account imperfect memory, summating the results over a group of subjects, considering the effect of unequal pairs, and undertaking preliminary efforts to estimate the effect of reducing the number of test faces. The latter has been handled by proposition

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3 and the fast method. Proposition 3 describes the probability of identifying Ft as a function of the size of a random sample of all possible test pairs. The fast method introduces the method of hypothesis testing for detecting F,, i.e., the method tests whether or not a particular value belongs to Ft. We believe that it is possible to combine both these techniques and develop a computer program for identifying F,, which will reduce the time of probing the subject’s memory considerably. We retained the assumption of the independence of facial dimensions and values. Further experiments involving a large number of facial dimensions and values will be required in order to determine whether the assumption of the independence of dimensions and values has to be replaced by a much more complicated theory based on interactive assumptions (see alternative models below). Similarly, we tested the model by increasingly approximating a real-life situation. We used photographs of real faces instead of line drawings, and we increased the number of facial dimensions. Finally, it should be noted that the assumption that Ft can be reconstructed as a finite set of organized values cannot be eliminated. We have assumed a finite set of facial dimensions and values, although for all practical purposes a real face is composed of an infinite number of elements. In effect, we used a finite set of elements in trying to deal with a gestalt consisting of an infinite number of elements. However, this case falls within the purview of the mathematics of approximations, as when functions are approximated from infinite dimensional space by finite dimensional space (e.g., Lorentz, 1966). Unit of Measurement

From the Catch model it is obvious that the basic unit of measurement of Ft is the value that is presented to the subject. (Note that the assumption made in Postulate 2 cannot be dispensed with.) For each facial dimension one determines its relevant value. In the experiments reported here, Ft consisted of the same values that accounted for the configuration of the test faces. Hence, there was a complete correspondence between the theoretical and the observable basic units of the model. This perfect correspondence breaks down when we shift to a real-life situation. Even when the highest attainable correspondence exists between a picture of a real face and its reconstruction by a given set of values (e.g., the set of values contained in the identikit, or in the photokit), it would still be impossible to achieve a complete resemblance between the reconstruction and the original (e.g., Davies, 1981). Moreover, the perfect correspondence also breaks down when Ft is not detected in full, even if the same values go to make up Ft and all of the test faces. For example, let us consider the case in which three out of a total

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RAKOVERANDCAHLON

of four correct values have been detected. Which of the four possible faces (each of which is a 75% reconstruction of the FJ most resembles the target? The model does not provide us with an answer. Even worse, although according to the model a 75% reconstruction of Ft is better than, say, a 50% reconstruction, subjects may nevertheless judge the 50% reconstruction to be more similar to Ft than the 75% reconstruction. This apparent contradiction derives from the fact that people do not attribute the same weight to all values (i.e., weight = 1) as we did in our model (see Ellis, 1984; Shepherd and Ellis, 1981; Yarmey, 1979). Hence, a 50% reconstruction might resemble Ft better than a 75% reconstruction because important values were detected in the former case and only negligible values were detected in the latter (see also alternative models below). Alternative Models We have already discarded one possible alternative explanation for our data-the hypothesis of subject awareness. But this is not the only possible alternative explanation. We shall discuss briefly three ways in which our model can be changed. The first involves assigning different weights to each value; the second involves the interaction among values; and the third involves various assumptions concerning the similarity between Ft and the test faces. Weights and interactions. Many experiments as well as common experience and our own results indicate that people tend to ascribe various degrees of significance, or different weights, to different facial dimensions and values (for a review of the subject, see Ellis, 1984; Shepherd, Davies, & Ellis, 1981). Furthermore, it has been shown that recognition of values is context dependent, that values interact with one another, and that faces are probably processed in terms of preexisting schema (e.g., Ellis, 1981; Goldstein & Chance, 1981). Our assumption concerning the independence of facial dimensions and values might therefore be criticized for not being in accord with these findings. But it is obvious that our model would have been complicated enormously had these findings been taken into consideration. As mentioned before, we therefore decided to start with a simple model in order to see how far it would take us. This only means that further experiments involving a greater number of dimensions and values will have to be undertaken. Nevertheless, the present experiments provide us with some empirical support for the assumptions about equal weights and noninteractive values. First, subjects have chosen the test face predicted by the Catch model above the chance level. Second, correct choice tends to increase as a function of (F(W)~ - t#V)n). These results show that what determines one’s choice is not a particular configuration of facial dimensions and

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values but the relationship between k(W), and k(W), of a test pair. However, this relationship is not simple. The result that correct choices tend to decrease as a function of the increase in u(W),, suggests a discrimination process according to which the greater the similarity among the test faces and F,, the greater the difficulty in selecting the predicted face. These results are in line with the formula for prediction of event frequency which shows that the probability of choosing the more frequent of two events is a negative accelerated function of the difference between the frequencies of occurrence of two events, where the weight of the difference decreases with increasing frequencies of occurrence of these events (e.g., Rakover, in press). The similarity between this formula and the present findings is particularly salient if one compares frequency of occurrence with p(w). Similarity or psychological distance. Postulate 4 is a major assumption in the present model. This postulate describes a mental operation that determines which of two possible test faces (FL and Fa) most resembles Ft in terms of number of matches. The greater the number of matches between a test face and F,, the smaller the psychological distance between them-in other words, the greater their similarity. Now it is clear that with different definitions of similarity we obtain different models for catching a thief. A definition of psychological distance similar to ours has been proposed by Tversky (1977) and Tversky & Gati (1978). According to Tversky’s definition, which he calls the “contrast model,” the similarity between two objects is described as a linear function of the measures of the common and distinctive features. As can be seen, Postulate 4 is partially the definition of Tversky’s: in the present study similarity or psychological distance equals the number of common values shared by a given test face and Ft. Although distinctive features as such-or what we call the “differentiating values”-are not counted in Postulate 4, they nevertheless constitute the basic operation in the policeman’s procedure. These differences between Tversky’s model and our own arise from the different aims of the two research programs. While Tversky is generally concerned with predicting how people judge similarity among objects, our main purpose was to reconstruct an unknown face by employing a very simple assumption concerning psychological similarity and psychological distinctiveness. Some Theoretical Implications There is a great amount of empirical work on face memory discovering many interesting facts about the identification of Ft (for review see Baddeley, 1979; Davies, 1981; Ellis, 1981, 1984; Goldstein & Chance, 1981; Yarmey, 1979). Some of these findings discuss the effects of cue salience,

456

RAKOVERANDCAHLON

race, duration of viewing, context, prior training, delay, and retroactive interference on the detection of Ft. How does our model handle these findings? We think that the attempt to incorporate these results within the framework of the present model is premature. Nevertheless, a general idea of how this can come about could be outlined. Since our model is developed within the general framework of the information-processing approach (i.e., feature analysis), and since the widely accepted assumption is that, in general, recognition memory is much better than recall, the results cited in the literature can be explained by the customary cognitive approach. For example, the model would predict that the longer the duration of viewing Ft the deeper the level of its processing and therefore the better the identification of Ft. And as another example, since our model presupposes that “recalling a face is usually more difficult than recognizing it” (Ellis, 1984, p. 35), we are tempted to predict that retroactive interference would reduce the accuracy of identification of Ft by the present method to a lesser degree than other techniques based on recall memory. Practical

Implications

Although our present model serves to illuminate a number of theoretical issues discussed in this study, it is also proposed with a practical end in mind. We believe that our technique for detecting a criminal suspect’s face is better than the method in current use by the police. (For one thing, while the cutomary identikit technique uses one witness without having a procedure for summation of several witness’ information, the present model can handle information of one witness or more.) Ultimately we envisage a general computer program that would be linked to various police stations and would produce as its end product a series of faces approximating Ft. Witnesses at any of the stations would be able to respond to such a program displayed on a TV screen by choosing among different test faces. Their responses would be analyzed by the computer, and the result, consisting of a series of possible faces of the suspect, would then be sent back to the station. This procedure could well contribute to police efforts to identify and apprehend criminal suspects. APPENDIX

I: PROOF OF PROPOSITION

1

Given a pair of test faces, we shall define X as the set of the differentiating values in FL, and Y as the set of differentiating values in FR, X = FL - (FL f~ FR) Y = FR - (FL fl FR),

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From these we get FL = x u (FL r-l FR)

(3)

FR = YU (F,n

(4)

FR).

The number of matches for FL, p(W,), is given by PAW,) = Wt n FL)

(5)

and the number of matches for FR, p,(WR), is given by PO’,) = I@, n FRI.

(6)

Since the subject has chosen FL, it follows that k(WL) > p(W,) or that kw,) - p(W,) > 0. Given the above, in order to demonstrate Proposition 1 we must show that p(F, n X) > p(F, n Y) or that the number of the differentiating values in Ft and FL is greater than the number of differentiating values in Ft and FR, JILL)

= PUS n FL) = d&

IJWL)

= dU5

n x) u (6

= IN?

n x)

n (X u FL n FR)I

according to (3)

n FL n FR)I

+ PUS n FL n FR)

- PUS n x)

n (Ft n FL n FRII.

(7) However, since the last term equals zero because the intersection of X with (FL n FR) is an empty set, then k(W,) = k(Ft rl X) + p,(F, fl FL n W. And in a manner similar to that of (7), k(wR) = t@t n Y) + IJ@, n FL fl Fd.

(8)

Therefore, according to (7) and (8) P(~L)

- P(WR) = id& -

n x) 1.45

= 145

n

n m

+ F.(& n FL f-7 FR) FL

n

- let

I.@? n Y)

FR)

(9)

f-l n

Now, since p(W,) - k(WR) > 0, then also p,(F, n X) - p(F, fl I’) > 0. APPENDIX

II: PROOF OF PROPOSITION

2

Let (AJiEl be the set of facial dimensions, where i = 1, . . . , m and u$ E Ai is a value of a facial dimension where j = 1, . . . , n. Let fl be the facial space-i.e., R = A, X A, X . . . X A,. A point in fi is a face (F) given by F = (a’, a*, . . . , am).

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RAKOVERANDCAHLON

We denote by Ft the target face as a certain face from a; for example, Ft = (a:, ai, . . . , $J and ai E A,, . . . , ajm, E A,,,. We shall use the following definitions: Distance. The distance between F (a face of a) and Ft is a nonnegative integer d, where d is the number of the nonequal values differentiating between F and Ft. For example, if d = 2, then F and Ft have the same values except two. We shall denote the distance between F and Ft by d = P.(F) - I@ n Ft) = 1~07 - P.(W).

We shall use the following sets: A = all faces which contain t as their first value except Ft. B = all faces which contain nt* as their first value.

C = all faces which contain nt as their first value. D, = all faces in A such that their distance from Ft is d, where d =

1,2, . . . ) m - 1. R, = all faces in B such that their distance from Ft is d, where d = 1,2, . . . , m. Sd = all the faces in C such that their distance from Ft is d, where d = 1,2, . . . , m. The numbers of the faces in each of the sets Dd, Rd, and Sd and which are denoted respectively by k(DJ, p(RJ, and p&S,) are d=l,...,m-1 d=l,...,m

d = 1, . . . , m.

These formulas for the number of faces are obtained by simple calculation of the number of possible faces for each case. Take for example t&3,) = (“; ‘)(n - l)(n - 2). The first value of each face in Sz is selected from (n - 2) possibilities. The other values which are different from t, are selected from (“i ‘) possibilities. Given that this is the case, there are (n - 1) different values and hence &S,) = (“T ‘)(n - l)(n - 2). The basic idea of the proof of the proposition is as follows: Given the choices of a subject over Z, we shall show that the number of times that t is selected is greater than the number of times nt* is selected. To do this we take the following steps: Step I: We design a table like Table 6 for all possible test pairs. Step 2: We find the number of times that t occurs in the chosen test faces, and we denote this number as p(t).

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Step 3: We find the number of times that nt* occurs in the chosen test faces, and we denote this number as p(nt*). Step 4: We find the number CI = p(t) - p,(nt*) and show that CY> 0. Step 1: Given Table 8 we shall consider all those test pairs which contain different faces or profiles. These test faces are obtained from triangles (l), (5), and (9), and from squares (2), (3), and (6). Step 2: In order to find p(t), we shall consider squares (2) and (3) in Table 8. Accordingly, p(t) is calculated as

+ Pan-1)

k4w.

Step 3: Similarly, p,(nt*) is calculated from squares (3) and (6), and triangle (9) in Table 8,

b-1

7

m-l

ht*)=/@I)2 @d) 1 1

+ kR2)

+

[ 1 [ 1 c

p(Dd)

+

* . ’

d=3

d=2

dR1)

5

i@d)

d=2

m

+ i4R2)

c

hsd)

L d=3

1

+ * * ’ + IMm-1)

P,(Sm)*

Note that in Steps 2 and 3 we did not take into account equal pairs. Step 4: To find the value of ct = p(t) - F(nt*), we use the equations in Steps 2 and 3. Since u(R,) = 1, p&) = l@,), . . . , k(R,) = t.@,- i), we obtain m-l

m cx =

c d=2

!@d)

h@d-1)

-

11 + c d=l

m-2 h@d)12

+

c

t@d)

b(Dd-I)

-

11.

d=2

It is clear that 01> 0. The expression [l.~(D~-i) - l] is greater than or equal to 1, and u(Dd) is not zero.

460

RAKOVERANDCAHLON

As mentioned before, p(t), l&t*), and cxwere calculated without taking into account equal pairs. We shall now prove that the Ft can be detected even if equal pairs are considered. In order to do this, we shall examine squares (3) and (6) in Table 8. Given squares (3) and (6) in Table 8, let @a) = 1 and l3 denote the number of points contributed to nt* by all possible contributory equal pairs; then l3 is calculated as m-l

P = c d=l

t@d)

i@d)

+ 5

@d)

i-@d)

d=l

m-1

m

= c

t-@d)

@d-l)

+

d=l

t@d)

c

i@d).

d=l

We shall calculate y = OL- p, m-l

b( D d-l ) -

Y = $ i@d) d=2

m-l

11 + 2

h@d)12

+ c @d) d=2

d=l

b@d-1)

-

11

m-l

-

c

i@d)

t@d-1)

-

d=l

5

@d)

p@d)

d=l

or m-l

m

Y = c k@d) d=2

b(Dd-1)

-

11 +

b@d)

c

h@d)

-

11

d=I

m-l

-

c

i@d)

i-@d)*

d=l

Utilizing the definitions of l.@,), l.&),

and uJDd) we obtain

m-l

‘Y = 2 d=l

@d)

h@d)

-

21 -

(n -

2).

For m - 1 2 d 2 2, y is positive. Hence, Ft will be detected even if the number of contributory equal pairs is subtracted from (Y. That is p,(t) > p&t*), even if all the points contributed by the equal pairs are calculated to t’s disadvantage.

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TEST

III: PROOF OF PROPOSITION

3

In this proof we shall consider the case in which only a random sample of the test pairs is presented to the subjects. Our goal is to find the relation between the number of pairs presented to the subjects and the probability of detecting Ft. For computational purposes we shall use the following definitions: p(k) is the number of random test pairs which are not presented to the subject. 2 is the set of all test pairs which do not include Ft. The number of all possible pairs which do not include Ft is Pm =

(nrn - 2)(nrn - 1) 2 *

The number of test pairs presented to the witness is p(S) = p,(Z) dk). We shall define Xi as a random variable which denotes a random elimination of a test pair i from Z, with the values

I 1

if t loses a point if nt* loses a point otherwise.

Given that the subject is presented with l&S), we shall define p(t) as the probability [(p(f) - p(nt*) - Zriki Xi)] > 0 where there are no equal pairs, and the probability [(p(t) - l&t*) - p - EyLkiXi)] > 0 where equal pairs are present. The ErLkl Xi is a random variable which counts the number of points k.(t) earns or loses by eliminating the k-pairs. For example, if all the k-pairs were to contribute points to t, then p(t) would be reduced by p(k). It is clear that p(t) = P@(t) - &zt*) - C;iki Xi > 0] = P[Z;iki Xi < l.~(t) - p(nt*)l = P lE$kj Xi < al (and P [ZrLki Xi < r] in the case with equal pairs). The random variable varies between the limits -F(k)dCXib

+k(k)

and

ifp(k)
i=l

that is, p(t) = 1. When &Z) > k(k) > (Y,then p(t) < 1, or P (ZrLk! Xi > 0~)is less than one. In what follows we shall develop a formula for calculating the values of p(t), when p(k) > a.

462

RAKOVER

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We shall define P (Ej!i’j) Xi = J) as the probability that eliminating k-pairs from Z results in p,(t) - t&t*) = J. Hence,

=

i

PEXi=J)+zP($Xi=J).

J= -&)

Given this result, we shall consider two cases where J 6 0 and where J > 0. J > 0: The P (Zr~‘j Xi = J) is the probability of having J times more nt* than t. Now, in order to obtain I@1 Xi = J, test pairs must be eliminated from Z in the following way: J times t and no nt*, or (J + 1) t and one Ott*, or (J + 2) t and two nt*, etc., until all k-pairs (i.e., p,(k)) are eliminated. If we denote L as the number of nt* eliminated, then L = 0, 1,2,3, . . . , [(p(k) - J)/2]. Similarly, (J + L) denotes the number of t eliminated. Hence, 2L + J = k(k), and since L is an integer, 0 s L < (p(k) - 1)/2). For fixed J and k(k), we obtain

f’

dk) [ 1 xxi=

J

=

i=l

J 6 0:

We shalt now consider the limits of L and J. As for J, since J s 0 we can eliminate a large number of test pairs which contribute to nt*. The value of J is J = -Min[p(k),

&zt*)].

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For J > 0, the lowest value from which J can start is

a = Max Mk) - tdz) - I.@*) + p(t), I]. The largest value J can be is J < a. - 1. As for L, when .I s 0

L + IJI s p(m*), P(k) -

2L

- VI s lJ.(z)- p(1) - p(rzt*).

(1) (2)

Therefore ; . (k&v - CL0 - (JI + p(t) + &zt*)) s L s (p(nt”) - pj) L 2 0. L starts with Maxl(b + Vz), 01, b = 1/2(I - lo@) - IJJ + I ~(nt*)). When J > 0, L c [(CL(~)- 4121 and L < &zt*). Hence

+

where 0 v,

=

c J=-Min[~(k),&w*)]

L=Max(O,[b+

l/2])

a = max(p(k) - p(Z) + ct, 1) b = ‘/T&,(k) - p(Z) + CY- IJI) c = p(nt*) - IJI dk) - J 2 . > Note that if equal pairs are taken into consideration, then P (,TyL’j Xi < y) < P (Zrz’j Xi < CY),since y < CLIn order to obtain the formula of p(t) when equal pairs are present the a’s must be replaced with the y’s,

RAKOVERANDCAHLON

where 0 v;

5

=

(;w)(;y;1+

“)

c

.T=Min[p(k),&tr*)+p] PQ

-

Y

( dk)

-

~25 -

a’ = MaxW)

L=Max[O,b’+1/2]

IJI )

- 14.3 + Y, 1)

b’ = ~2 h-44 - 1.43 + Y - 14) c’ = p(nt*) + p - IJJ E’ = min(F(nt*) + p, M@(k) - J)). APPENDIX IV: PROOF OF COROLLARY 1 Given that an error (e) takes place under the most error-prone condition, then for Ft to be identified we require the following: (1) Under the condition in which equal pairs are excluded p(t) - e > p(nt*) + e, or p(t) - p(nt*) > 2e. Since p(t) - k(nt*) = (Y, we require e< eS

when a is odd and when OLis even.

The percentage of errors (pe) is given by e pe = p(t) + p(nt*) *

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(2) Under the condition in which equal pairs are considered, we require similarly

e-c

when y is odd and

e < ($2 - 1) when y is even, e pe = p(t) + &It*) APPENDIX

+ p *

V: PROOF OF COROLLARY

2

Let k(s) be the number of test pairs presented, and let J be the number of subjects. Further, let Xi be a random variable representing the correct answers of any one witness. Finally, let P be the probability of obtaining a correct answer for a given test pair. Accordingly, Corollary 2 proposes that

J c

xi

l

P(Xi 3 k(S) . P) < P +

i

2 p(s) . P .

That is, the probability of obtaining better results in identifying Ft from J number of subjects when J is a large number is greater than the probability of identifying Ft where there is only one subject. If J + 03,then it follows from the central limit theorem that J

xi

c

i=l -+

yj’

J

Y,

when Y is a random variable with a normal distribution, and with mean = b(s) * P and variance = p,(s) * P(1 - P)/J (see Feller, 1968). Since J + ~0 it follows that the variance goes to zero and therefore Y becomes a constant &)P, and P(Y 3 k.(s) * P) = 1. And since P(X, > p,(s)P) is less than unity, while P( Yi 2 p(s)P) approaches 1 as J + m, then it follows that

J c

xi

P(Xi 2 p(S) * P) < P

i

+

3 CL(s)* p

1

for large J.

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APPENDIX

Vi: NOMENCLATURE

Contributory pairs-test faces which contribute one point to either t or nt* given a particular facial dimension. Diffeerentiuting values-values which are not common to FL and FR. Equal pair-a pair of test faces for which p(w) is the same. Facial dimensions-general categories of the face such as the nose, eyes, and mouth. FL-a test face presented on the left side of the subjects. Fa-a test face presented on the right side of the subjects. Frthe target face. K-all test pairs in 2 which will not be presented to the subjects. m-the number of independent facial dimensions. Most error-prone condition-the condition in which the subject’s errors generate the greatest interference with the attempt to identify Ft (i.e., t is reduced by one point and nt* is increased by one point). n-the number of independent values in a facial dimension. nt-a value not belonging to Ft. nt*-an nt arbitrarily chosen to be compared with t in terms of frequency of occurrence. Profile-a face defined as a set of a number of organized values such as dvd2. p(t)-the probability of identifying t (i.e., that k(t) > lo (nt*)) as a function of Pm. S-a random sample of 2 to be presented to a subject. t-a value belonging to Ft. Values-specific elements of a given facial dimension, such as a long, short, wide, or round nose. Z-all possible pairs of test faces excluding Ft. o-the target’s advantage oft over nt*, given a facial dimension: (Y = p(t) - k(nt)*. p-the number of points contributed to nt* by all possible contributory equal pairs. ‘y--Y = (a - P). CL-symbol denoting the number of elements in a set. l&v--P(S) = P(z) - clva. p,(U)-the number of all profiles minus Ft. @V)--the number of values common to Ft and FR or to Ft and FL (note in Appendix. I we use the symbol d which is equal to k(F,) - p(W)). I@)-the number of pairs in Z: k(Z) = F(U) * (p(v) - l)v2. l&+-the number of profiles in a facial space: p(O) = n”. La facial space, or the set of all possible profiles.

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TEST

TABLE 8 Appendix I: All Possible Test Pairs which include t, nr*, and nt

D,D, . . D,., D,

s,s, . . . s,

R,R, . . . R,

(1)

c-4

(3)

(4)

(5)

(6)

(7)

03)

(9)

D2

Dm.1 s, s2

s, RI R2

Note. See text.

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Harmon, L. D., Kuo, S. C., Ramig, P. F., & Raudkiv, U. (1978). Identification of human face profiles by computer. Pattern Recognition, 10, 301-312. Lenorovitz, D. R., & Laughery, K. R. (1984). A witness-computer interactive system for searching mug files. In G. L. Wells & E. F. Loftus (Eds.), Eyewitness testimony: Psychological perspectives. New York: Cambridge Univ. Press. Levin, S. M. (1961). The effects of awareness on verbal conditioning. Journal of Experimental Psychology, 61, 67-75. New York: Holt, Rinehart 62 Winston. Lorentz, G. G. (1%6). Approximation offunctions. Patterson, K. E., & Baddeley, A. D. (1977). When face recognition fails. Journal of Experimental Psychology: Human Learning and Memory, 1, 246-252. Penry, J. (197la) Photo-fit kit. England: John Waddington of Kirkstall Ltd. Penry, J. (197lb). Looking at faces and remembering them: A guide to facial identification. London: Blek Books. Rakover, S. S. (1983). In defense of memory viewed as stored mental representation. Behaviorism, 11, 53-62. Rakover, S. S. The prediction of event frequency: The frequency-learning, conlidencematching and subjective-probability distribution hypotheses. Acta Psychologica, in press. Shepherd, J., Davies, G., & Ellis, H. (1981). Studies of cue saliency. In G. Davies, H. Ellis, & J. Shepherd (Eds.), Perceiving and remembering faces. New York: Academic Press. ShitTrin, R. M., & Schneider, W. (1977). Controlled and automatic human information processing: II Perceptual learning automatic attending and general theory. Psychological Review A, 84, 127-190. Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327-352. Tversky, A., & Gati, I. (1978). Studies of similarity. In E. Rosch and B. B. Lloyd (Eds.), Cognition and categorization. Hillsdale, NY: Erlbaum. Yarmey, A. D. (1979). The psychology of eyewitness testimony. New York: Free Press. Accepted February 22, 1989