3 78 MOBILITY

MODELS, MATHEMATICAL

Although mathematical models are applied in many areas of the social sciences, this article is limited to mathematical models of individual be- havior. For applications of mathematical models in econometrics, see ECONOMETRICMODELS, AGGRE- GATE. Other articles discussing modeling in general include CYBERNETICS,PROBABILITY, SCALING, SIM- ULATION, and SIMULTANEOUSEQUATION ESTIMA- TION. Specific models are discussed in various articles dealing with substantive topics.

Theories of behavior that have been developed and presented verbally, such as those of Hull or Tolman or Freud, have attempted to describe and predict behavior under any and all circumstances. Mathematical models of individual behavior, by contrast, have been much less ambitious: their goal has been a precise description of the data obtained from restricted classes of behavioral experiments concerned with simple and discrimination learn- ing; with detection, recognition, and discrimination of simple physical stimuli; with the patterns of preference exhibited among outcomes; and so on. Models that embody very specific mathematical assumptions, which are at best approximations applicable to highly limited situations, have been analyzed exhaustively and applied to every con- ceivable aspect of available data. From this work broader classes of models, baskd on weaker as- sumptions and thus providing more general pre- dictions, have evolved in the past few years. The successes of the special models have stimulated, and their failures have demanded, these general- izations. The number and variety of experiments to which these mathematical models have been ap- plied have also grown, but not as rapidly as the catalogue of models. Most of the models so far developed are re- stricted to experiments having discrete trials. Each trial is composed of three types of events: the MODELS, MATHEMATICAL 3 7 9 presentation of a stimulus configuration selected times be recorded. Each of these is unique to cer- by the experimenter from a limited set of possible tain experimental realizations, and so they have presentations; the subject's selection of a response not been much studied by theorists. from a specified set of possible responses; and the Probability measures. The stimulus presenta- experimenter's feedback of information, rewards, tions, the responses, and the outcomes can each and punishments to the subject. Primarily because be thought of as a sequence of selections of ele- the response set is fixed and feedback is used, these ments from known sets of elements, i.e., as a are called choice experiments (Bush et al. 1963). schedule over trials. It is not usual to work with Most psychophysical and preference experiments, the specific schedules that have occurred but, as well as many learning experiments, are of this rather, with the probability rules that were used to type. Among the exceptions are the experiments generate them. For the stimulus presentations and without trials-e.g., vigilance experiments and the the outcomes, the rules are selected by the experi- operant conditioning methods of Skinner. Cur- menter, and so there is no question about what rently, models for these experiments are beginning they are. Not only are the rules not known for the to be developed. responses, but even their general form is not cer- Measures. With attention confined to choice tain. Each response theory is, in fact, a hypothesis experiments, three broad classes of variables nec- about the form of these rules, and certain relative essarily arise-those concerned with stimuli, with frequencies of responses are used to estimate the responses, and with outcomes. The response vari- postulated conditional response probabilities. ables are, of course, assumed to depend upon the Often the schedules for stimulus presentations (experimentally) independent stimuli and upon are simple random ones in the sense that the prob- the outcome variables, and each model is nothing ability of a stimulus' being presented is independ- more or less than an explicit conjecture about the ent of the trial number and of the previous history nature of this dependency. Usually such conjec- of the experiment; but sometimes more complex tures are stated in terms of some measures, often contingent schedules are used in which various numerical ones, that are associated with the vari- conditional probabilities must be specified. Most ables. Three quite different types of measures are outcome schedules are to some degree contingent, used: physical, probabilistic, and psychological. usually on the immediately preceding presentation The first two are objective and descriptive; they and response, but sometimes the dependencies can be introduced and used without reference to reach further back into the past. Again, conditional any psychological theory, and so they are especially probabilities are the measures used to summarize popular with atheoretical experimentalists, even the schedule. [See PROBABILITY.I though the choice of a measure usually reflects a Psychological measures. Most psychological theoretical attitude about what is and is not psy- models attempt to state how either a physical chologically relevant. Although we often use phys- measure or a probability measure of the response ical measures to characterize the events for which depends upon measures of the experimental inde- probabilities are defined, this is only a labeling pendent variables, but in addition they usually in- function which makes little or no use of the power- clude unknown free parameters-that is, numerical ful mathematical structure embodied in many phys- constants whose values are specified neither by the ical measures. The psychological measures are con- experimental conditions nor by independent meas- structs within some specifiable psychological theory, urements on the subject. Such parameters must, and their calculation in terms of observables is therefore, be estimated from the data that have possible only within the terms of that theory. Ex- been collected to test the adequacy of the theory, amples of each type of measure should clarify the which thereby reduces to some degree the strin- meaning. gency of the test. It is quite common for current Physical measures. In experimental reports, the psychological models to involve only probability stimuli and outcomes are usually described in measures and unknown numerical parameters, but terms of standard physical measures : intensity, fre- not any physical measures. When the numerical quency, size, weight, time, chemical composition, parameters are estimated from different sets of amount, etc. Certain standard response measures data obtained by varying some independent vari- are physical. The most ubiquitous is response ables under the experimenter's control, it is often latency (or reaction time), and it has received the found that the parameters vary with some variables attention of some mathematical theorists (McGill and not with others. In other words, the parameters 1963). In addition, force of response, magnitude are actually functions of some of the experimental of displacement, speed of running, etc., can some- variables, and so they can be, and often are, viewed 3 8 0 MODELS, MATHEMATICAL

as psychological measures (relative to the model second level of decision : the specific assumptions within which they appear) of the variables that made. affect them. Theories are sometimes then provided Probability us. determinism. One of the most for this dependence, although so far this has been basic decisions is whether to treat the behavior as the exception rather than the rule. if it arises from some sort of probabilistic mech- The theory of signal detectability, for example, anism, in which case detailed, exact predictions involves two parameters: the magnitude, d', of the are not possible, or whether to treat it as determi- psychological difference between two stimuli; and nistic, in which case each specific response is sus- a response criterion, c, which depends upon the ceptible to exact prediction. If the latter decision is outcomes and the presentation schedule. Theories made, one is forced to provide some account of the for the dependence of d' and c upon physical meas- observed inconsistencies of responses before it is ures have been suggested (Luce 1963; Swets 1964). possible to test the adequacy of the model. Usually Most learning theories for experiments with only one falls back on either the idea of errors of meas- one presentation simply involve the conditional out- urement or on the idea of systematic changes with come probabilities and one or more free param- time (or experience), but in practice it has not eters. Little is known about the dependence of been easy to make effective use of either idea, and these parameters upon experimentally manipulable most workers have been content to develop prob- variables. In certain scaling theories, numerical ability models. It should be pointed out that, as far parameters are assigned to the response alterna- as the model is concerned, it is immaterial whether tives and are interpreted as measures of response the model builder believes the behavior to be in- strength (Luce & Galanter 1963). In some models herently probabilistic, or its determinants to be too these parameters are factored into two terms, one complex to give a detailed analysis, or that there of which is assumed to measure the contribution are uncontrolled factors which lead to experimental of the stimulus to response strength and the other errors. of which is the contribution due to the outcome Static us. dynamic models. A second decision structure. is whether the model shall be dynamic or static. The phrasing of psychological models in terms (We use these terms in the way they are used in only of probability measures and parameters (psy- physics; static models characterize systems which chological measures) has proved to be an effective do not change with time or systems which have research strategy. Nonetheless, it appears impor- reached equilibrium in time, whereas dynamic tant to devise theories that relate psychological models are concerned with time changes.) Some measures to the physical and probability measures dynamic models, especially those for learning, state that describe the experiments. The most extensive how conditional response probabilities change with mathematical models of this type can be found in experience. Usually these models are not very help- audition and vision (Hurvich et al. 1965; Zwislocki ful in telling us what would happen if, for exam- 1965). The various theories of utility are, in part, ple, we substituted a different but closely related attempts to relate the psychological measure called set of response alternatives or outcomes. In static utility to physical measures of outcomes, such as models the constraints embodied in the model con- amounts of money, and probability measures of cern the relations among response probabilities in their schedules, such as probabilities governing several different, but related, choice situations. The gambles (Luce & Suppes 1965). In spite of the utility models for the study of preference are typi- fact that it is clear that the utilities of outcomes cal of this class. must be related to learning parameters, little is The main characteristic of the existing dynamic known about this relation. [See GAMBLING;GAME models is that the probabilities are functions of a THEORY; UTILITY.] discrete time parameter. Such processes are called The nature of the models. The construction of stochastic, and they can be thought of as generat- a mathematical model involves decisions on at ing branching processes through the fanning out least two levels. There is, first, the over-all perspec- of new possibilities on each trial (Snell 1965). tive about what is and is not important and about Each individual in an experiment traces out one the best way to secure the relevant facts. Usually path of the over-all tree, and we attempt to infer this is little discussed in the presentation of a from a small but, it is hoped, typical sample of model, mainly because it is so difficult to make the these paths something about the probabilities that discussion coherent and convincing. Nonetheless, supposedly underlie the process. Usually, if enough this is what we shall attempt to deal with in this time is allowed to pass, such a process settles down section. In the following section we turn to the --becomes asymptotic-in a statistical sense. This MODELS, MATHEMATICAL 3 8 1 is one way to arrive at a static model; and when sequential dependencies among responses, and the we state a static model, we implicitly assume that like. it describes (approximately) the asymptotic be- Recurrent theoretical themes. Beyond a doubt, havior of the (unknown) dynamic process govern- the most recurrent theme in models is independ- ing the organisms. ence. Indeed, one can fairly doubt whether a seri- Psychological us. mathematical assumptions. ous theory exists if it does not include statements Another distinction is that between psychological to the effect that certain measures which contrib- and formal mathematical assumptions. This is by ute to the response are in some way independent no means a sharp one, if for no other reason than of other measures which contribute to the same that the psychological assumptions of a mathe- response. Of course, independence assumes differ- matical model are ultimately cast in formal terms ent mathematical forms and therefore has different and that psychological rationales can always be names, depending upon the problem, but one evolved for formal axioms. Roughly, however, the should not lose sight of the common underlying distinction is between a structure built up from intuition which, in a sense, may be simply equiv- elementary principles and a postulated constraint alent to what we mean when we say that a model concerning observable behavior. Perhaps the sim- helps to simplify and to provide understanding of plest example of the latter is the axiom of transi- some behavior. tivity of preferences; if a is preferred to b and b is Statistical independence. In quite a few models preferred to c, then a will be preferred to c. This is simple statistical independence is invoked. For ex- not usually derived from more basic psychological ample, two chance events, A and B, are said to be postulates but, rather, is simply asserted on the independent when the conditional probability of A, grounds that it is (approximately) true in fact. A given B, is equal to the unconditional probability somewhat more complex, but essentially similar, of A; equivalently, the probability of the joint event example is the so-called choice axiom which postu- AB is the product of the separate probabilities of lates how choice probabilities change when the set A and B. of possible choices is either reduced or augmented A very simple substantive use of this notion is (Luce 1959). Again, no rationale was originally contained in the choice axiom which says, in effect, given except plausibility; later, psychological mech- that altering the membership of a choice set does anisms were proposed from which it derives as a not affect the relative probabilities of choice of two consequence. alternatives (Luce 1959). More complex notions of The most familiar example of a mathematical independence are invoked whenever the behavior is model which is generally viewed as more psycho- assumed to be described by a stochastic process. logical and less formal is stimulus sampling theory. Each such process states that some, but not all, of In this theory it is supposed that an organism is the past is relevant in understanding the future: exposed to a set of stimulus "elements" from which some probabilities are independent of some earlier one or more are sampled on a trial and that these events. For example, in the "operator models" of elements may become "conditioned to the per- learning, it is assumed that the process is "path formed response, depending upon the outcome that independent" in the sense that it is sufficient to follows the response (Atkinson & Estes 1963). The know the existing choice probability and what has concepts of sampling and conditioning are inter- happened on that trial in order to calculate the preted as elementary psychological processes from choice probability on the next trial (Bush & Mostel- which the observed properties of the choice behav- ler 1955). In the "Markovian" learning models, the ior are to be derived. Lying somewhere between the organism is always in one of a finite number of two extremes just cited are, for example, the linear states which control the choice probabilities, and operator learning models (Bush & Mosteller 1955; the probabilities of transition from one state to Sternberg 1963). The trial-by-trial changes in re- another are independent of time, i.e., trials (Atkin- sponse probabilities are assumed to be linear, son & Estes 1963). Again, the major assumption mainly because of certain formal considerations; of the model is a rather strong one about independ- the choice of the limit points of the operators in ence of past history. [See MARKOVCHAINS.] specific applications is, however, usually based Additivity and linearity. Still another form of upon psychological considerations; and the result- independence is known as additivity. If r is a re- ing mathematical structure is not evaluated di- sponse measure that depends upon two different rectly but, rather, in terms of its ability to account variables assuming values in sets A, and A,, then for the observed choice behavior as summarized in we say that the measure is additive (over the inde- such observables as the mean learning curve, the pendent variables) if there exists a numerical 3 8 2 MODELS, MATHEMATICAL measure r, on A, and r, on A, such that for x, in A, responses into classes, then only the linear learning and x, in A,, r(x,,x2)= r, (x,) + r,(x,). This as- models are appropriate. sumption for particular experimental measures r At a somewhat more detailed level, but still en- is frequently postulated in the models of analysis compassing several different models, are predic- of variance as well as derived from certain theories tions such as the mean learning curve, response of fundamental measurement. A special case of operating characteristics, and stochastic transitivity additivity known as linearity is very important. of successive choices among pairs of alternatives. Here there is but one variable ( that is, A, = A, = A) ; Sometimes it is not realized that conceptually quite any two values of that variable, x and x' in A, com- different models, which make some radically dif- bine through some physical operation to form a ferent predictions, may nonetheless agree com- third value of that variable, denoted x * x'; and pletely on other features of the data, often on ones there is a single measure r on A (that is, r, = r, = r) that are ordinarily reported in experimental studies. such that r(x * x') = r(x) + r(x'). Such a require- Perhaps the best example of this phenomenon ment captures the superposition principle and leads arises in the analysis of experiments in which sub- to models of a very simple sort. These linear mod- jects learn arbitrary associations between verbal els have played an especially important role in the stimuli and responses. A linear incremental model, study of learning, where it is postulated that the of the sort described above, predicts exactly the choice probability on one trial, p,, can be expressed same mean learning curve as does a model that linearly in terms of the probability, p,-, , on the pre- postulates that the arbitrary association is acquired ceding trial. Other models also postulate linear on an all-or-none basis. On the face of it, this result transformations, but not necessarily on the response seems paradoxical. It is not, because in the latter probability itself. In the "beta" model, the quantity model, different subjects acquire the association p,/(l - p,) is assumed to be transformed linearly; on different trials, and averaging over subjects this quantity is interpreted as a measure of re- thereby leads to a smooth mean curve that happens sponse strength (Luce 1959). to be identical with the one predicted by the linear Commutativity. The "beta" model exhibits an- model. Actually, a wide variety of models predict other property that is of considerable importance, the same mean learning curve for many probabilis- namely, commutativity. The essence of commuta- tic schedules of reinforcement, and so one must tivity is that the order in which the operators are turn to finer-grained features of the data to distin- applied does not matter; that is, if A and B are guish among the models. Among these differential operators, then the composite operator AB (apply predictions are the distribution of runs of the same B first and then A) is the same as the operator BA. response, the expected number of such runs, the Again, there is a notion of independence-inde- variance of the number of successes in a fixed pendence of the order of application. It is an ex- block of trials, the mean number of total errors, tremely powerful property that permits one to the mean trial of last error, etc. [See STATISTICAL derive a considerable number of properties of the IDENTIFIABILITY.] resulting process; however, it is generally viewed The classical topic of individual differences raises with suspicion, since it requires the distant past to issues of a different sort. For the kinds of predic- have exactly the same effect as the recent past. A tions discussed above it is customary to pool indi- commutative model fails to forget gradually. vidual data and to analyze them as if they were Nature of the predictions. As would be ex- entirely homogeneous. Often, in treating learning pected, models are used to make a variety of pre- data this way, it is argued that the structural con- dictions. Perhaps the most general sorts of predic- ditions of the experiment are sufficiently more im- tions involve broad classes of models. For example, portant determinants of behavior than are individ- probabilistic reinforcement schedules for a certain ual differences so that the latter may be ignored class of distance-diminishing models, i.e., ones that without serious distortion. For many experiments require the behavior of two subjects to become to which models have been applied with consider- increasingly similar when they are identically re- able success, simple tests of this hypothesis of inforced, can be shown to be ergodic, which means homogeneity are not easily made. For example, that these models exhibit the asymptotic properties when a group of 30 or 40 subjects is run on 12 to that are commonly taken for granted. A second 15 paired-associate items, it is not useful to analyze example is the combining-of-classes theorem, which each subject item because of the large relative asserts that if the theoretical descriptions of be- variability which accompanies a small number of havior are to be independent of the grouping of observations. On the other hand, in some psycho- MODELS, MATHEMATICAL 3 8 3

physical experiments in which each subject is run ones that use only a limited portion of the immedi- for thousands of trials under constant conditions ate past. For processes that are approximately sta- of presentation and reinforcement, it is possible to tionary, a small part of the past sometimes provides treat in detail the data of individuals. The final a very good approximation to the full chain of justification for using group data, on the assump- infinite order, and then pseudo-maximum-likeli- tion of identical subjects, is the fact that for ergodic hood estimates can be good approximations to the processes, which most models are, the predictions exact ones. Because of mathematical complexities for data averaged over subjects are the same as in applying even these simplified techniques, Monte those for the data of an individual averaged over Carlo and other numerical methods are frequently trials. used. [Set? ESTIMATION.] Another issue, which relates to group versus in- Once the parameters have been estimated, the dividual data, is parameter invariance. One way of number of predictions that can be derived is, in asking if a group of individuals is homogeneous is principle, enormous: the values of the parameters to ask whether, within sampling error, the param- of the model, together with the initial conditions eters for individuals are identical. Thus far, how- and the outcome schedule, uniquely determine the ever, more experimental attention has been devoted probability of all possible combinations of events. to the question of parameter invariance for sets of In a sense, the investigator is faced with a plethora group data collected under different experimental of riches, and his problem is to decide what pre- conditions. For instance, the parameters of most dictions are the most significant from the stand- learning models should be independent of the par- point of providing telling tests of a model. In more ticular reinforcement schedule adopted by the ex- classical statistical terms, what can be said about perimenter. Although in many cases a reasonable the goodness of fit of the model? degree of parameter invariance has been obtained Just as with estimation, it might be desirable to for different schedules, it is fair to say that the evaluate goodness of fit by a likelihood ratio test. results have not been wholly satisfactory. But, a fortiori, this is not practical when maximum- For a detailed discussion of the topics of this likelihood estimators themselves are not feasible. section, see Sternberg (1963) and Atkinson and Rather, a combination of minimum chi-square Estes ( 1963). techniques for both estimation and testing good- Model testing. Most of the mathematical mod- ness of fit have come to be widely used in recent els used to analyze psychological data require that years. No single statistic, however, serves as a sat- at least one parameter, and often more, be esti- isfactory over-all evaluation of a model, and so the mated from the data before the adequacy of the report usually summarizes its successes and fail- model can be evaluated. In principle, it might be , ures on a rather extensive list of measures of fit. desirable to use maximum-likelihood methods for A model is never rejected outright because it estimation. Perhaps the central difficulty which pre- does not fit a particular set of data, but it may dis- vents our using such estimators is that the observ- appear from the scene or be rejected in favor of able random variables, such as the presentation, another model that fits the data more adequately. response, and outcome random variables, form Thus, the classical statistical procedure of accept- chains of infinite order. This means that their ing or rejecting a hypothesis-or model-is in fact probabilities on any trial depend on what actually seldom directly invoked in research on mathe- happened in all preceding trials. When that is so, matical models; rather, the strong and weak points it is almost always impractical to obtain a useful of the model are brought out, and new models are maximum-likelihood estimator of a parameter. In sought that do not have the discovered weaknesses. the face of such difficulties, less desirable methods [See GOODNESSOF FIT; more detail on these topics of estimation have perforce been used. Theoretical can be found in Bush 19631. expressions showing the dependency on the un- Impact on psychology. Although the study of known parameter of, for example, the mean num- mathematical models has come to be a subject in ber of total errors, the mean trial of first success, its own right within psychology, it is also pertinent and the mean number of runs, have been equated to ask in what ways their development has had an to data statistics to estimate the parameters. The impact on general experimental psychology. classical methods of moments and of least squares For one, it has almost certainly raised the stand- have sometimes been applied successfully. And, in ards of systematic experimentation : the applica- certain cases, maximum-likelihood estimators can tion of a model to data prompts a number of de- be approximated by pseudo-maximum-likelihood tailed questions frequently ignored in the past. A 3 8 4 MODELS, MATHEMATICAL model permits one to squeeze more information occur. In contrast, the simple all-or-none model out of the data than is done by the classical tech- postulates that the subject is either completely nique of comparing experimental and control conditioned to make the correct response, or he is groups and rejecting the null hypothesis whenever not so conditioned. No intermediate states exist, the difference between the two groups is suf- and until the correct conditioning association is ficiently large. A successful test of a mathematical established on an all-or-none basis, his responses model often requires much larger experiments are determined by a constant guessing probability. than has been customary. It is no longer unusual This means that learning curves for individual sub- for a quantitative experiqent to consist of 100,000 jects are flat until conditioning occurs, at which responses and an equal number of outcomes. In point they exhibit a strong discontinuity. The prob- addition to these methodological effects on experi- lem of discriminating the two models must be ap- mentation and on data analysis, there have been proached with some care since, for instance, the substantive ones. Of these we mention a few of mean learning curve obtained by averaging data the more salient ones. over subjects, or over subjects and a list of items Probability matching. A well-known finding, as well, is much the same for the two models. On which dates back to Humphreys ( 1939), is that of the other hand, analyses of such statistics as the probability matching. If either one of two responses variance of total errors, the probability of an is rewarded on each trial, then in many situations error before the last error, and the distribution of organisms tend to respond with probabilities equal last errors exhibit sharp differences between the to the reward probabilities rather than to choose models. For paired-associates learning, the all-or- the more often rewarded response almost all of the none model is definitely more adequate than the time. Since Humphreys' original experiment, many linear incremental model (Atkinson & Estes 1963). similar ones have been performed on both human Of course, the issue of all-or-none versus incre- and animal subjects to discover the extent and mental learning is not special to mathematical nature of the phenomenon, and a great deal of psychology; however, the application of formal effort has been expended on theoretical analyses models has raised detailed questions of data anal- of the results. Estes (1964) has given an extensive ysis and posed additional theoretical problems not review of both the experimental and the theoretical raised, let alone answered, by previous approaches literature. Perhaps the most important contribu- to the problem. tion of mathematical models to this problem was Reward and punishment. The classic psycho- to provide sets of simple general assumptions about logical question of the relative effects of reward behavior which, coupled with the specification of and punishment (or nonreward) has also arisen the experimenter's schedule of outcomes, predict in work on models, and it has been partially an- probability matching. As noted above, investigators swered. In some models, such as the linear one, have not been content with just predicting the there are two rate parameters, one of which repre- mean asymptotic values but have dealt in detail sents the effect of reward on a single trial and the with the relation between predicted and observed other of which represents the effect of nonreward. conditional expectations, run distributions, vari- Their estimated values provide comparable meas- ances, etc. Although this experimental paradigm ures of the effects of these two events for those for probability learning did not originate in data from which they are estimated. For example, mathematical psychology, its thorough exploration Bush and Mosteller (1955) found that a trial on and the resulting interpretations of the learning which a dog avoided shock (reward) in an avoid- process have been strongly promoted by the ance training experiment produced about the same many predictions made possible by models for this change in response probabilities as three trials of paradigm. nonavoidance (punishment). No general law has The all-or-none model. A second substantive emerged, however. The relative effects of reward issue to which a number of investigators have ad- and nonreward seem to vary from one experiment dressed mathematical models is whether or not to another and to depend on a number of experi- simple learning is of an all-or-none character. As mental variables. noted earlier, the linear model assumes learning When using a model to estimate the relative to be incremental in the sense that whenever a effects of different events, the results must be in- stimulus is presented, a response made, and an terpreted with some care. The measures are mean- outcome given, the association reinforced by the ingful only in terms of the model in which they are outcome is thereby made somewhat more likely to defined. A different model with corresponding re- MODELS, MATHEMATICAL 3 8 5 ward and nonreward parameters may lead to the from 0 to 1. The data points appear to fall on a opposite conclusion. Thus, one must decide which smooth, convex curve, which shows the relation, model best accounts for the data and use it for for the subject, between correct responses to measuring the relative effects of the two events. stimuli and incorrect responses to no-stimulus Very delicate issues of parameter estimation arise, trials (false alarms). Its curvature, in effect, char- and examples exist where opposite conclusions acterizes the subject's sensitivity, and the location have been drawn, depending on the estimators of the data point along the curve represents the used. The alternative is to devise more nonpara- amount of bias, i.e., his over-all tendency to say metric methods of inference which make weaker "Yes," which varies with rr, with the payoffs used, assumptions about the learning process. A detailed and with instructions. Several conceptually differ- discussion of these problems is given by Sternberg ent theories, which are currently being tested, ac- (1963, pp. 109-116). [See LEARNING,article on count for such curves; it is clear that any new REINFORCEMENT.] theory will be seriously entertained only if it admits Homogenizing a group. If one wishes to obtain to some such partition of the response behavior a homogeneous group of subjects after a particular into sensory and bias components. This point of experimental treatment, should all subjects be run view is, of course, applicable to any two-stimulus- for a fixed number of trials, or should each subject two-response experiment, and often it alters signifi- be run until he meets a specific performance cri- cantly the qualitative interpretation of data. [See terion? Typically it is assumed by those who use ATTENTION;PSYCHOPHY SICS.] such a criterion that individual subjects differ; that, for example, some are fast learners and some Although one cannot be certain about what will are slow. It is further assumed that all subjects happen next in the application of mathematical will achieve the same performance level if each is models to problems of individual behavior, certain run to a criterion such as ten successive successes. trends seem clear. (1) The ties that have been Now it is clear that for identical subjects, it is established between mathematical theorists and simpler to run them all for the same number of experimentalists appear firm and productive; they trials and perhaps use a group performance cri- probably will be strengthened. (2) The general terion. It is, however, less obvious whether it would level of mathematical sophistication in psychology be better to do this than to run each to a criterion. can be expected to increase in response to the in- An analysis of stochastic learning models has creasing numbers of experimental studies that shown that running each of identical subjects to a stem from mathematical theories. (3) The major criterion introduces appreciable variance in the applications will continue to center around well- terminal performance levels. One can study indi- defined psychological issues for which there are vidual differences only in terms of a model and accepted experimental paradigms and a consider- assumptions about the distributions of the model able body of data. One relatively untapped area is parameters. When this is done, it becomes evident operant (instrumental) conditioning. (4) Along that very large individual differences must exist to with models for explicit paradigms, abstract prin- justify using the criterion method of homogenizing ciples (axioms) of behavior that have wide poten- a group of subjects. tial applicability are being isolated and refined, Psychophysics. The final example is selected and attempts are being made to explore general from psychophysics. With the advent of signal de- qualitative properties of whole classes of models. tection theory it became increasingly apparent that (5) Even though the most successful models to the classical methods for measuring sensory thresh- date are probabilistic, the analysis of symbolic and olds are inherently ambiguous, that they depend conceptual processes seems better handled by other not only, as they are supposed to, on sensitivity mathematical techniques, and so more nonproba- but also on response biases (Luce 1963; Swets bilistic models can be anticipated. 1964). Consider a detection experiment in which ROBERTR. BUSH,R. DUNCANLUCE, the stimulus is presented only on a proportion rr of AND PATRICKSUPPES the trials. Let p(Y1s) and p(Y(n) be the proba- bilities of a 'Yes" response to the stimulus and to [See also DEC~S~ONMAKING, article on PSYCHOLOG~CAL no stimulus respectively. If the experiment is run ASPECTS; MULA LA TI ON, article On INDlVlDUAL BE- several times with different values of rr between 0 HAVIOR. Other relevant material may be found in and 1, then p(YIn), as well as p(Yls), which is a ATTENTION; LEARNING;MATHEMATIC s ; PROBABIL- classical threshold measure, varies systematically ITY; PSYCHOMETRICS;PSYCHOPHYSICS; SCALING.] 3 8 6 MODERNIZATION : Social Aspects

BIBLIOGRAPHY Luce, Robert R. Bush, and Eugene Galanter (editors), ATKINSON,RICHARD C.; and ESTES, WILLIAM K. 1963 Handbook of Mathematical Psychology. New York: Stimulus Sampling Theory. Volume 2, pages 121-268 Wiley. in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psy- chology. New York: Wiley. BUSH, ROBERTR. 1963 Estimation and Evaluation. Vol- ume 1, pages 429-469 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psychology. New York: Wiley. BUSH, ROBERTR.; GALANTER,EUGENE; and LUCE,R. DUN- CAN 1963 Characterization and Classification of Choice Experiments. Volume 1, pages 77-102 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psychology. New York: Wiley. BUSH, ROBERT R.; and MOSTELLER, FREDERICK 1955 Stochastic Models for Learning. New York: Wiley. ESTES, WILLIAM K. 1964 Probability Learning. Pages 89-128 in Symposium on the Psychology of Human Learning, University of Michigan, 1962, Categories of Human Learning. Edited by Arthur W. Melton. New York: Academic Press. HUMPHREYS,LLOYD G. 1939 Acquisition and Extinction of Verbal Expectations in a Situation Analogous to Conditioning. journal of Experimental Psychology 25: 294-301. HURVICH,LEO M.; JAMESON,DOROTHEA; and KRANTZ, DAVIDH. 1965 Theoretical Treatments of Selected Visual Problems. Volume 3, pages 99-160 in R. Dun- can Luce, Robert R. Bush, and Eugene Galanter (edi- tors), Handbook of Mathematical Psychology. New York: Wiley. LUCE, R. DUNCAN 1959 Individtcal Choice Behavior. New York: Wiley. LUCE, R. DUNCAN 1963 Detection and Recognition. Volume 1, pages 103-190 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psychology. New York: Wiley. LUCE, R. DUNCAN;and GA'ANTER, EUGENE 1963 Psy- chophysical Scaling. Volume 1, pages 245-308 in R. Duncan Luce, Robert R. Bush, and Eugene Galan- ter (editors), Handbook of Mathematical Psychology. New York: Wiley. LUCE, R. DUNCAN;and SUPPES, PATRICK 1965 Prefer- ence, Utility, and Subjective Probability. Volume 3, pages 249-410 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathe- matical Psychology. New York: Wiley. MCGILL, WILLIAM J. 1963 Stochastic Latency Mecha- nisms. Volume 1, pages 309-360 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psychology. New York: Wiley. SNELL, J. LAURIE 1965 Stochastic Recesses. Volume 3, pages 411-486 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathe- matical Psychology. New York: Wiley. STERNBERG,SAUL 1963 Stochastic Learning Theory. Volume 2, pages 1-120 in R. Duncan Luce, Robert R. Bush, and Eugene Galanter (editors), Handbook of Mathematical Psychology. New York: Wiley. SWETS,JOHN A. (editor) 1964 Signal Detection and Rec- ognition by Human Observers: Contemporary Readings. New York: Wiley. ZWISLOCKI, JOZEF 1965 Analysis of Some Auditory Characteristics. Volume 3, pages 1-98 in R. Duncan