Acta Psychologica 74 (1990) 297-318 North-Holland 297 DISCRETENESS AND CONTINUITY IN MODELS OF HUMAN INFORMATION PROCESSING * Jeff MILLER University of California, San Diego, L.a Jolla, USA Accepted April 1990 For over 100 years, researchers have sought to learn about the operations of the mind by studying the time needed to perform cognitive tasks, an endeavor which has come to be called ‘mental chronometry’ (cf. Meyer et al. 1988b). Mental processes are not directly observable, however, so their durations must be inferred using models. Following Donders’ ([1869] 1969), many researchers have based their inferences on so-called ‘discrete’ models, in which total reaction time (RT) is the sum of the times needed by a series of distinct mental processes intervening between stimulus and response. For example, the Additive Factor Method (AFM; Sternberg 1969) is one particularly powerful inferential technique that can be used if certain assumptions of discreteness are met. In the past ten years there have been a number of suggestions that RT may not be decomposable into a sum of stage times, and some alternative ‘continuous’ models have been formulated (e.g., Eriksen and Schultz 1979; McClelland 1979; Taylor 1976). Some of these models are theoretically attractive, because they are constructed from neuron- like processing elements. Methodologically, however, they are all very * This research was supported by National Institute of Mental Health grant PHSMH40733. Requests for reprints should be addressed to the author at the Department of Psychology, C-009, USCD, La Jolla, CA, 92093. I would like to thank Patricia Haden and Maurits van der Molen for helpful comments on earlier drafts of the paper. The manuscript was prepared while the author was a visiting researcher in the Laboratoire de Neuroscience Fonctionelles at the Centre National de la Recherche Scientifique in Marseille. Author’s address: J. Miller, Dept. of Psychology, University of California, San Diego, La Jolla, CA 92093, U.S.A. OOOl-6918/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Transcript
Acta Psychologica 74 (1990) 297-318
North-Holland
297
DISCRETENESS AND CONTINUITY IN MODELS OF HUMAN INFORMATION PROCESSING *
Jeff MILLER
University of California, San Diego, L.a Jolla, USA
Accepted April 1990
For over 100 years, researchers have sought to learn about the operations of the mind by studying the time needed to perform cognitive tasks, an endeavor which has come to be called ‘mental chronometry’ (cf. Meyer et al. 1988b). Mental processes are not directly observable, however, so their durations must be inferred using models. Following Donders’ ([1869] 1969), many researchers have based their inferences on so-called ‘discrete’ models, in which total reaction time (RT) is the sum of the times needed by a series of distinct mental processes intervening between stimulus and response. For example, the Additive Factor Method (AFM; Sternberg 1969) is one particularly powerful inferential technique that can be used if certain assumptions of discreteness are met.
In the past ten years there have been a number of suggestions that RT may not be decomposable into a sum of stage times, and some alternative ‘continuous’ models have been formulated (e.g., Eriksen and Schultz 1979; McClelland 1979; Taylor 1976). Some of these models are theoretically attractive, because they are constructed from neuron- like processing elements. Methodologically, however, they are all very
* This research was supported by National Institute of Mental Health grant PHSMH40733.
Requests for reprints should be addressed to the author at the Department of Psychology, C-009,
USCD, La Jolla, CA, 92093. I would like to thank Patricia Haden and Maurits van der Molen for helpful comments on earlier drafts of the paper. The manuscript was prepared while the author
was a visiting researcher in the Laboratoire de Neuroscience Fonctionelles at the Centre National de la Recherche Scientifique in Marseille.
Author’s address: J. Miller, Dept. of Psychology, University of California, San Diego, La Jolla, CA 92093, U.S.A.
298 J. Miller / Commentate: discreteness and coniinuir~
unattractive, because they are cumbersome to use in making inferences about mental processes.
At present, the relative validity of discrete and continuous models is a matter of considerable debate, with some arguing for discrete models (e.g., Sanders 1980, this issue; Van Galen, this issue), others arguing for continuous ones (e.g., Eriksen and Schultz 1979; Smid et al., this issue), and some trying to construct intermediate positions (e.g., Miller 1982, 1988). The other papers in this issue were presented to a conference held to facilitate this debate. 1 Since I had recently reviewed a large number of arguments on each side of the debate (Miller 1988) the editors of this volume kindly invited me to comment on these articles, indicating how each moves the debate forward, and I am happy to do so.
Before embarking on this commentary, I would first like to try to clarify the boundaries of the debate. Because they are general terms, the adjectives ‘discrete’ and ‘continuous’ have been applied to chrono- metric models in several very different senses. If the different senses are not carefully distinguished, it is easy to draw overly general conclusions from data, because empirical results may support discreteness or con- tinuity in one sense without implying anything at all about discreteness or continuity in the other senses. I will first briefly recapitulate three senses distinguished previously (Miller 1988), and then present a fourth sense not explicitly identified earlier. This new sense is especially relevant to the problem of applying chronometric models to energetics and psychopathology, and it figures prominently in the subsequent discussion of the papers by Kok (this issue) and Sergeant and Van der Meere (this issue).
Following the discussion of the fourth sense of discreteness and continuity and its relevance to the paper of Kok (this issue), to the paper of Sergeant and Van der Meere (this issue), and to models of energetics and psychopathology in general, I will evaluate the new evidence offered in the other articles in this volume and also comment briefly on the theoretical perspective of Sanders (this issue). In sum, the new arguments seem to strengthen slightly the case for discrete models.
1 C.W. Eriksen, one of the conference participants representing the continuous viewpoint, was unfortunately not able to prepare a paper for inclusion in this volume.
J. Miller / Commentary: discrefeness and continuity 299
Discrete versus continuous representation, transformation, and transmission
Almost all chronometric models are based on the premise that information processing tasks are performed by a series of distinct mental processes that accomplish logically different functions (e.g., perception, decision, response execution). These processes are typically called stages when they are embedded in discrete models and ‘processing levels’ in continuous ones; here, for want of a more general term, we will call them all ‘stages’ without intending to prejudge the debate.
For a model to process information, each stage must receive some input information, transform that input in some way to compute its output, and then transmit its output to the next stage in the processing sequence (e.g., Bower 1975). Previously, I have argued that stages can be discrete or continuous in each of these three activities (Miller 1988, in press). Thus, a stage can be discrete or continuous in the sense that it (1) receives categorical versus quantitative representations of its input information, (2) carries out its transformation in an abrupt versus gradual fashion, or (3) transmits its output to the next stage in a single complete message versus a long series of partial messages.
The debate between discrete and continuous models is complicated for three reasons. First, an individual stage can be discrete or continu- ous in each of the above three senses almost independently (cf. Miller 1988). Thus, evidence can support discreteness or continuity in one sense but not in the others. Second, models generally have multiple stages, and different stages need not all be discrete or continuous in the same senses. Thus, evidence may support discreteness or continuity (in one sense) within a given stage, but may be neutral with respect to the properties of other stages. Third, discreteness and continuity are them- selves quantitatively rather than qualitatively different possibilities. Consider, for example, the transmission of information. If a stage transmits all of its output information in a single message, transmission is fully discrete. If a stage transmits its output in an infinite number of arbitrarily small messages, transmission is fully continuous. But sup- pose there are 2, 10, or 50 transmissions, each containing l/2, l/10, or l/50 of the total output. Then transmission is neither fully discrete nor fully continuous, but somewhere in between. As this example il- lustrates, discreteness and continuity should be thought of as the two
300 J. MiNer / Commenmy: discreteness and continuity
ends of a ‘grain size’ dimension rather than as qualitatively different possibilities, so a stage may be rel&iueIy discrete or continuous in any of the three different senses (Miller 1988, in press). 2
Discrete versus continuous variation in a priori state
A fourth sense in which a stage may be relatively discrete or continuous arises from consideration of how the stage varies from trial to trial. Although it is very convenient to think of processing stages as invariant across trials, there is considerable evidence that they are not. For example, trial-to-trial fluctuations at one or more stages of processing are indicated by the well-documented effects of practice (e.g., Schneider and Shiffrin 1977), stimulus sequence (e.g., Kornblum 1973), attentional fluctuations (e.g., Ward 1982), time on a vigilance task (e.g., Beatty 1982) and the type of task performed on a previous trial (e.g., Long 1977) as well as autocorrelation of both successive responses and successive RTs (e.g., John 1973; Laming 1989; Lute and Green 1978).
Clearly, many factors might cause a stage to vary in its a priori state at the moment it begins to process information. The state is a priori in the sense that it is influenced only by previous informational and biological factors, not by the actual content of the upcoming informa- tion. The state, in turn, determines how the stage operates on its impending input. The state may have a general influence independent of the stimulus input; for example, it may control an overall processing rate, determine the algorithm (strategy) used to carry out the transfor- mation, or select the form of output transmission to the next stage. Alternatively, the state may have input-specific influences; for example, it may involve particular readiness for a given stimulus or response.
Variation in a priori state provides a fourth sense in which a stage can be discrete or continuous. 3 According to a model at the discrete
* As should be clear by now, I have abandoned the formal, mathematical definitions of ‘discrete’ and ‘continuous’. As far as I am concerned. for example, a variable which takes on integer values
from 1 to 100 is relatively continuous, because it spans a range of values and includes very many
intermediates between the end points. The motivations for this change in terminology, discussed
in Miller (1988). are largely practical. 3 Miller (in press) came close to identifying this sense, but tried to categorize it as a special case of
transformations. I now believe it is better to consider it to be a separate sense of its own.
J. Miller / Commentary: discreteness and continuity 301
extreme, a stage would have to be in one of two distinct states at the start of each trial, with no intermediate possibilities. The fast guess model (e.g., Ollman 1966; Yellott 1971) is a perfect example. According to this model the subject decides, before each trial, either to process the forthcoming information and make a slow, accurate response, or to ignore the forthcoming information and make a rapid guess. At some stage in the information processing chain, then, there are exactly two distinct a priori states.
According to a model at the continuous extreme, a stage might be in any one of a large number of a priori states. For example, almost all models involving the concepts of ‘resources’ and ‘capacities’ allow continuous variation in a priori states (e.g., Kahneman 1973; Navon and Gopher 1979, 1980). Depending on task demands and perhaps subject strategy, a given stage is allocated a certain amount of some fuel-like quantity, and the stage operates at a speed which increases with the size of its fuel allottment. The amount of fuel is a continuous variable, so there are an infinite number of a priori states (i.e., amounts of fuel) in which the stage might exist at the start of processing.
The distinction between discrete and continuous variation in a priori state is quantitative rather than qualitative, just as it was for the three previous distinctions, because the a priori states may span a given range with a relatively large or small grain size. The self-terminating memory scanning model of Theios et al. (1973) provides one illustra- tion. In this model the memory scanning stage keeps an ordered list of stimulus-response (S-R) pairs, and the order of pairs changes from trial to trial depending on stimulus presentations. A probe item is sequentially compared against the stimulus components in the list until a match is found, at which time the paired response is initiated. The a priori state of the stage corresponds to the ordering of the S-R pairs. Note that a given impending stimulus may be located in the first position of the list, the second, the third, and so on. But of course it cannot occupy position 1.5. Thus, at least for medium-length lists of S-R pairs (say, four to ten), the a priori states vary neither extremely discretely nor extremely continuously.
A serious theoretical complication is that there are potentially many different dimensions of variation within a priori state. Over a given set of trials, for example, there may be variation in both processing rate and strategy, and these two sorts of variation may be independent of one another. In principle, variation may be continuous along one
302 J. Miller / Commentary: discreteness and continuity
dimension (e.g., processing rate) and discrete along another (e.g., strategy). Thus, variation in a priori state refers not to a single theoretical dimension, but rather to a whole family of dimensions, along each of which variation may be relatively discrete or continuous.
An additional complication is that there are as yet no clear empirical criteria for identifying a priori states and determining whether they vary discretely or continuously. It is all too difficult to determine from behavioral data whether the subject’s state changes gradually or abruptly, as evidenced by the very difficult debate over whether learn- ing proceeds in an incremental or all-or-none fashion (e.g., Restle and Green0 1970). While psychophysiological measures seem potentially very useful in this connection, they contain too much noise to allow direct readout of a subject’s state, and the possibility of averaging artifacts prevents the identification of subject states using average measures (cf., Meyer et al. 1988b).
Two points about discreteness or continuity of variation in a priori
state are particularly important for the present purposes. First, this sense is largely independent of discreteness or continuity in the other three senses. The others involve the operation of a stage within a trial, and the new one concerns changes in the operation of the stage between trials. Thus, the operation of a stage on a given trial could be relatively discrete, in all the various senses, even if the a priori state of the stage varied continuously from trial to trial. In Sternberg’s (1969) model of memory search, for instance, the discrete nature of the model’s operation on each trial would not be lost if, for example, fluctuations in the subject’s level of effort caused the rate of memory search to vary continuously from trial to trial. 4
Second, discreteness of variation in a priori state is not a prerequisite for use of the AFM. As discussed by Sanders (this issue) in connection with presetting processes, the AFM is quite compatible with trial-to-trial variations in the durations of component stages (see also Sternberg 1969). These variations could result from variations in a priori state, as well as other, random fluctuations (e.g., momentary internal noise).
4 I have used Sternberg’s model to exemplify this point because it is widely known. As
emphasized by Miller (1988, in press), this model does not really deserve its reputation as a fully
discrete model, because it requires neither discrete transformations nor discrete information
representations.
J. Miller / Commentary: discreteness and continuity 303
Discussion of papers
Kok, and Sergeant & Van der Meere
It is natural to assume that chronometric models of normal adult cognition should be very useful in describing performance changes, both between and within individuals, due to biological factors like psychopathology, energetics, intelligence, stress, drugs, maturation, and aging. Yet Kok (this issue) and Sergeant and Van der Meere (this issue), like others concerned with such biological factors (e.g., Rabbitt 1981, 1986), express deep dissatisfaction with current chronometric models, especially discrete ones. I argue that discrete models have been criticized too severely, based partly on the concept of discrete or continuous variation in a priori state.
The dissatisfaction with discrete models seems to stem from two main sources. 5 The first is that performance differences are gradual, not abrupt (cf. Norman and Bobrow 1975). Sergeant and Van der Meere (this issue), for example, note that ‘search and decision are [gradually] influenced by practice and adjustment factors’, which sug- gests to them ‘a more continuous state of affairs in the processing of information rather than the discrete processing suggested by the AFM’ (p. 285). But gradual changes in average performance can result from gradual changes in the relative proportions of two distinct (i.e., dis- crete) performance levels (cf. Meyer et al. 1988a; Miller 1988; Wickelgren 1977), so it is not appropriate to regard continuous changes in average performance as evidence of continuous changes in instanta- neous performance.
Even if there were continuous changes in instantaneous perfor- mance, such changes would only indicate continuity in variation of a priori state. As noted above, gradual changes in a priori state are not incompatible with discreteness in the other three senses, including those required by the AFM. Thus, this particular source of dissatisfaction with discrete models in general and the AFM in particular is far too severe because it completely ignores the distinctions among different senses of discreteness versus continuity.
s Some authors (e.g., Rabbitt 1986) give much longer lists of what they regard as specific
shortcomings of the discrete approach. The motivations behind these lists, however, seem to
reduce to two main dissatisfactions discussed here.
304 J. Miller / Commentary: discreteness and contim+
The other source of dissatisfaction with discrete models is, to put it bluntly, that they have not always led to very nice interpretations of experimental results. Many investigators, for example, have tried to use the AFM to determine what stage or stages of processing are altered when overall performance changes due to psychopathology, energetic manipulations, maturation, aging, stress, and so on. The results have often been difficult to interpret, and this has led many to wonder whether the problem is that the underlying class of models is wrong.
Another possibility, however, is that discrete models are inap- propriate for the questions being asked, rather than wrong. Chronomet- ric models have generally been designed to account for the micro-struc- ture of cognitive activity underlying relatively steady-state perfor- mance, and in this domain discrete models have been quite fruitful. Psychopathology, stress, practice, energetic manipulations, and other such variables, however, all produce major changes in a priori state. Thus, in studying these variables, chronometric models have been stretched far beyond the domains for which they were developed.
Of course, it would be nice if discrete micro-models would explain the macro-effects of the more biological variables. In an ideal world, perhaps the effect of each of these variables would be localized on a single parameter of information processing. That is exactly the outcome that would make these manipulations easiest to understand in terms of current chronometric models. But given the global, biological nature of these manipulations, it may never have been realistic to expect that they would have localized effects on information processing. Instead, many of them could quite plausibly have effects along the entire information processing sequence, possibly producing changes in a priori state as drastic as a complete restructuring of the sequence itself. Can anyone who has ever talked to an acute schizophrenic, for exam- ple, really imagine that the deficit would turn out to be something as informationally local as, say, the rate of memory scanning? Indeed, the global nature of these manipulations makes it all the more impressive when one does seem to have effect on a specific subsystem (e.g., Callaway 1984).
In short, the difficulty of applying discrete models to biological manipulations may stem more from the complexity of the resulting effects than the limitations of the models. After all, the Newtonian model of gravity is not falsified by its failure to offer a simple account of the trajectory of a falling leaf. Though it would be legitimate to seek
J. Miller / Commentary: discreteness and continuity 305
another model to understand such motion, it would not be legitimate to conclude that the Newtonian model was a poor starting point, let alone wrong.
Actually, there is reason to believe that stage models can be ex- tended to the point where they may be useful in understanding global changes in a priori state, in spite of the complexity of such changes. Variations in a priori state have always been considered to some extent in discrete models. Mostly, such consideration has been at a micro level, corresponding directly to individual parameters of the hypothe- sized information processing (e.g., speed-accuracy criterion, stimulus- or response-specific preparation, amount of attention). Perhaps the best illustration is in the study of sequential effects on performance (e.g., Kornblum 1973), where the goal of the modeling endeavor is to describe the subject’s state as a function of the sequence of previous stimuli.
In the past ten years, furthermore, there have been significant extensions of this approach. Attempts to combine stages and resources into a single model (e.g., Gopher and Sanders 1984; Sanders 1983, 1986; Wickens 1984) have spawned the new field of cognitive energet- its, and researchers are now starting to ask how many different types of energy there are, how the total amount of each type of energy is determined, how energy is allocated between stages, and how particular stages are sensitive to their energy allocations. Thus, a good start has already been made toward understanding effort-related and drug-re- lated energetic effects, and further extensions may be quite useful in understanding psychopathology and strategy changes. In particular, both of these areas seem amenable to analysis in terms of mixture models (e.g., Meyer et al. 1988a), because they appear to involve qualitative changes in performance.
A further virtue of discrete models is that, through the AFM, they encourage researchers to perform factorial experiments. As discussed by various authors (Sergeant and Van der Meere, this issue), factorial experiments are more informative than simple between-group compari- sons, because they allow the separation of group main effects from treatment X group interactions. It is important to realize that factorial experiments have this advantage whether or not the underlying system conforms to the assumptions of the AFM. Furthermore, the AFM suggests a rather specific strategy for interpreting additivities and interactions in factorial experiments, which continuous models do not,
306 J. Mtller / Commentu~~: drscreteness und continuit~~
and this strategy may lead to incorrect conclusions if the assumptions are not met. But the AFM should be regarded as useful to some extent simply because it promotes the use of relatively informative experimen- tal designs. It is true that one can do factorial experiments without thinking in terms of the AFM, but, in practice, it is moot.
Even if one disagrees with this optimistic assessment of the value of and prospects for discrete models in the study of biological factors, it is not clear that a more useful model is available. Continuous models have generally been applied to the same tasks and variables as discrete ones, and it is by no means clear that they are more easily extended to variables influencing a priori states. It is natural to have some en- thusiasm for the new, especially when the old does not always produce the desired results, but this enthusiasm should not replace objective criteria. Before abandoning discrete models in favor of continuous ones, it is necessary to see (1) substantial data for which continuous interpretations are much more compelling with discrete ones, and (2) new methods of data collection and interpretation prescribed by con- tinuous models.
Smid, Mulder, and Mulder
Smid, Mulder, and Mulder (this issue) claim to have strong evidence in favor of continuous transmission between stages. They present evidence that response preparation can begin before stimulus analysis has finished, and claim that this demonstrates continuous transmission from perceptual to response processes. The latter claim does not seem justified, however. With respect to this issue, the new data do not fundamentally extend previous results (e.g., Coles 1989; Coles and Gratton 1986; Coles et al. 1985; Coles et al. 1988; Eriksen et al. 1985; Gratton et al. 1988) that have already been reconciled with discrete transmission (Miller 1988; Molenaar, this issue). In essence, though the data of Smid et al. (this issue) contribute additional evidence about how unattended flanker letters activate responses, they do not show that such activation is based on continuous output from the perceptual system.
Smid et al. studied performance in the response competition para- digm of Eriksen and Eriksen (1974). Subjects responded with the left or right hand to indicate the identity of a relevant target letter in the center of a display. Response-compatible, neutral, or response-incom-
J. Miller / Commentary: discreteness and continuity 307
patible flanker letters were presented on both sides of the target, and subjects were instructed to ignore these. In spite of their task-irrele- vance, flankers had substantial effects on RT; for example, responses were slowest with response-incompatible flankers.
In addition to RT and percentage correct, Smid et al. took several psychophysiological measures, including the lateralized readiness potential (LRP) derived from the EEG, and EMG activity. As in previous studies, these measures provided evidence that flankers activate responses, and that this activation is at least partly responsible for the effects of flankers on RT (e.g., Coles 1989; Coles and Gratton 1986; Coles et al. 1985; Coles et al. 1988; Eriksen et al. 1985; Gratton et al. 1988). For example, incompatible flankers caused an increase in both LRP and EMG activity associated with the incorrect response. Further- more, the timing of these effects suggested that flankers activate responses before perceptual analysis of the stimulus display is com- plete. 6
According to Smid et al. (this issue), the early response activation produced by flankers contradicts models with discrete transmission, because it shows that response processes have information about the flankers before they have complete information about the target. Clearly, the perceptual process must have transmitted some stimulus information (i.e., flanker identity) earlier than other stimulus informa- tion (i.e., target identity), rather than transmitting stimulus information all at once.
Contrary to Smid et al.‘s (this issue) assertion, however, a relatively discrete model can allow flankers to produce response activation before perceptual analysis is complete (Miller 1988; Molenaar, this issue). Suppose the perceptual system saves stimulus information until a letter is fully identified, and then transmits this identity in a single message. This would be a relatively discrete model, because there would be no transmission of partial information about a stimulus letter (e.g., its features). Yet, flanker identity could still be transmitted out of the perceptual system - and thus cause early response activation - on trials when a flanker was recognized before the target.
6 Some might question whether the data of Smid et al. (this issue) even demonstrate these
empirical points, because only six subjects were used, and two of these showed anomolous results.
We will not dispute these points, however, because they have been demonstrated before (cf., Coles 1989) and because the more important conclusion of continuous transmission does not follow
even if these points are granted.
308 J. Mdler / Commentmy: dwcreteness and continuity
The flaw in Smid et al’s argument is that it ignores the issue of grain size. 7 To demonstrate continuous transmission, one needs to show that messages are passed in very small information units (i.e., a small ‘grain size’). Flanker identity, however, is a very large grain of information, because it corresponds to full information about a distinct display element. In defense of discrete models, one could argue that this early transmission of flanker identity shows overlap of perceptual processing of one stimulus (i.e., the target) with response processing of another stimulus (i.e., the flanker), not overlap of perceptual and response processing for a single stimulus, as assumed by continuous models. Such early transmission falsifies onb models in which an entire multi- element or multi-attribute stimulus display is evaluated as a single indivisible unit. It is therefore consistent not only with continuous flow models (e.g., Eriksen and Schultz 1979) but also with relatively discrete models like the ADC model (e.g., Miller 1982), as has recently been acknowledged even by proponents of continuous transmission (Coles et al. 1989).
Finally, it should be noted that Smid et al. do consider and reject a discrete model that might be proposed for this paradigm. In this model, the subject responds based on the identity of either the target or a flanker, whichever is recognized first. As they point out, this model is contradicted by the high accuracy on response-incompatible trials, not only in their experiment but also in all the previous ones using this paradigm (e.g., Eriksen and Eriksen 1974). The model also fails to explain the basic RT effects and the psychophysiological evidence of response competition, since on every trial the response is determined entirely by the first stimulus letter recognized.
The discrete model considered by Smid et al. (this issue), though attributed to me (Miller 1988), is a straw man. I suggested a model in which flankers could generate response activation, not responses, when they were recognized before the target. In fact, Molenaar (this issue) has implemented a model of the form I had in mind and has shown that it can generate the basic mean RT results. Thus, it is clear that Smid et al. (this issue) have not given discrete models fair consideration before rejecting them.
’ Sanders (this issue) describes the flaw by saying that the argument ignores the problem of what constitutes a signal. I believe that our objections to Smid et al. are equivalent, in spite of our
different terminology.
J. Miller / Commentary: discreteness and continuity 309
Molenaar
The new evidence provided by Molenaar (this issue) concerns the capabilities of a certain class of mathematical models. Most previous comparisons of discrete and continuous transmission have avoided formal modeling (but see McClelland 1979; and Molenaar and Van der Molen 1986) and emphasized verbal descriptions of models that seem to account for the results (e.g., Eriksen and Schultz 1979; Miller 1988). Molenaar (this issue) addressed the questions of whether discrete-stage models might really exist, given the neuron-like units from which they must be constructed, and, if so, whether they really account for the flanker-compatibility effect as suggested earlier (Miller 1988).
With respect to the first question, the answer is clearly in the affirmative. Extending previous work (Molenaar and Van der Molen 1986) Molenaar shows how to build a stage model with a neural network and shows that it does produce additive effects on RT when parameters affecting different stages are varied orthogonally. To at least this extent, then, the neural network may be regarded as an instantiation of the sequential stage model underlying the additive factor method (AFM). This contribution is extremely important be- cause it answers the existence question, demonstrating that stage mod- els (and the AFM) actually are biologically feasible. With a parameter change, of course, the same neural network can operate continuously, so it is clear that the neural network approach supplies a common framework within which both discrete and continuous models can be studied.
With respect to the second question - whether discrete models can account for flanker effects in the focused attention task - the answer is also affirmative, though less strongly so. Molenaar showed that a discrete model of the general form suggested by Miller (1988) can account for the effects of flankers on mean correct RT. This strengthens the verbal arguments given previously (Miller 1988) and refutes the claim that flanker effects are incompatible with discrete transmission (e.g., Smid et al., this issue).
The affirmative answer to the second question must be qualified, however, because Molenaar uncovered a problem with the discrete model he simulated: errors were slower than correct responses, rather than faster, as in the data. Rather pessimistically, Molenaar concluded that ‘the quantitative behavior of Miller’s model is opposite to em- pirical results pertaining to the latency of incorrect responses’ (p. 255).
310 J. Miller / Commentury: dmretenesr and rontinult.)
Molenaar’s conclusion is overly pessimistic, because the model that doesn’t fit is simply one of many possible implementations of a discrete model for this task. Even if this implementation does not account for the fast errors, there could well be another implementation that does. Even within his rather general modeling framework, Molenaar was forced to choose certain assumptions, and the results might have been different had he made different choices. For example, random variabil- ity was added to the model using one of four separate and apparently not mutually exclusive techniques. Indeed, Molenaar himself considers two possible modifications of his model that would probably account for the fast errors, though he considers one implausible on theoretical grounds. Thus, though Molenaar’s work shows that one discrete model can account for the effects of flankers on mean RT, it does not show that all discrete models are falsified by the fast errors. With respect to discrete models for this task, it is not a case of the glass being half empty or half full, but rather of the glass being at least half full.
Before leaving Molenaar’s paper, I must disagree strongly with his assertion that ‘the distinction between discrete and continuous models of information processing may not be a fundamental one’ (p. 256). Of course, what seems fundamental to one researcher may seem incidental to another, so this dispute could be just a matter of semantics. On the other hand, scientific strategy is heavily oriented toward settling the most fundamental distinctions, so this assertion is not without conse- quences for research programs.
Molenaar’s main justification for this assertion seems to have been the formal equivalence of discrete and continuous transmission within his overall modeling framework. Specifically, both types of models can be represented as instances of the same class of neural networks, varying only in the size of a single parameter (analogous to the ‘grain size of information transmission’ parameter discussed by Miller (1988)). Any time two models are formulated so that they differ with respect to a single parameter, they are not - by Molenaar’s definition - funda- mentally different.
There are various problems with this argument. A practical criticism is that it ignores the very important (‘fundamental?‘) methodological consequences of the value of this parameter, such as those concerning the AFM. Surely a fundamental difference can result from a very important change in a single parameter (e.g., day vs. night), unless one makes multiparametricity a defining characteristic of a ‘fundamental’
J. Miller / Commentary: discreteness and continuity 311
difference. A logical criticism is that it presumes that all discrete and continuous models can be represented within the particular formal structure Molenaar has adopted. Although his classes of discrete and continuous models differ only in the value of a parameter, other classes may differ more substantially, and be fundamentally different even by his definition. A philosophical criticism is that the argument hinges critically on the unspecified definition of a ‘parameter’. Given ade- quate leeway in what can be called a parameter, one might find that any two models can always be recast into versions that differ in only one parameter, in which case no two models would ever by fundamen- tally different.
A secondary justification for the assertion seems to have been that both types of models can produce graded (i.e., continuous) effects, but I cannot discern the logical force of this justification. The fact that two types of models produce similar effects does not mean that the models are fundamentally similar. If it did, it is hard to see how two plausible models could ever be fundamentally different.
Van Galen
Van Galen (this issue) argued for a model of handwriting with a number of hierarchically arranged processors (e.g., graphemic analysis, allographic analysis, force parametrization and muscular initiation). Each processor in the hierarchy deals with successively smaller and more specific chunks of information; for example, the processor at the top level may encode a word or phrase, whereas the processor at the bottom level may generate the muscle commands to write a single stroke of a letter.
The different processors in the model operate in parallel yet dis- cretely, much like workers on an assembly line. All processors are active at same time, but each processor must finish its processing of a particular part of the message before it can transmit output to the next processor. For example, the processes preparing for the writing of one letter operate concurrently with the movements accomplishing the writing of a previously prepared letter. In effect, the discrete transmis- sions between simultaneous processes serve to insulate the ongoing action of one process from disturbing effects of inputs it will soon receive from a previous process in the hierarchy.
312 J. Miller / Commentary: discreteness and continuity
The evidence for the discrete character of Van Galen’s model is circumstantial rather than decisive, because the experiments were desig- ned to identify the processes involved in handwriting rather than to test explicitly between discrete and continuous transmission. One point in favor of discreteness is simply that the model is easy to reconcile with results from multiple dependent variables in this relatively complex task. Of course, it is possible that a continuous model could be equally successful, but until such a model is constructed the research tends to support discreteness.
A second source of evidence for the discrete model comes from additive effects of letter form, letter size, and musculature on RT (Van Galen and Teulings 1983). Additivity is generally regarded as support for discrete models, simply because it is easy for them to produce (Sanders 1980, this issue; Sternberg 1969). The support is weak, how- ever, because continuous models may also produce additive effects (e.g., McClelland 1979).
A third source of evidence for the discrete model is that specific task demands influence specific aspects of task performance, often with divergent time courses (see also Van Galen et al. 1986; Van Galen et al. 1989). In the experiment of Van Galen (this issue), phonological and motoric task demands were varied. In general, phonological demands influenced dependent variables related to early stages in handwriting (e.g., handwriting initiation time), whereas motoric demands influenced dependent variables related to later stages (e.g., writing time). This is evidence for discreteness, because it suggests that the early processes affected by one factor (i.e., phonology) were completely finished before the start of later processes reflected in a subsequent dependent variable (i.e., writing time). In others words, this pattern is consistent with models in which factor effects are localized to a given stage of processing because of discrete transmission. With continuous flow, factor effects tend to be propagated rather than localized (e.g., Eriksen and Schultz 1979). 8
Again, however, the argument is only circumstantial, because there is no direct evidence that a continuous model is incapable of explaining
* There was also a small, reversed effect of phonological demands on writing time, and this effect weakens the above case for discreteness. Because this effect might be an artifact of inhibition
associated with repetition of identical stroke sequences (Van Galen, this issue), it does not disconfirm the entire agrument for discreteness.
J. Miller / Commentary: discreteness and continuity 313
such results. An interesting theoretical consideration, which might be called the ‘distance criterion’, arises in this context. When there is evidence that one process finishes before a subsequent one begins, this evidence is more problematic for continuous models if the two processes are relatively close together than if they are relatively far apart. Two distant processes may be sequential even in a relatively continuous model, because the delays introduced by the intervening processes may be sufficient to allow the earlier process to finish before the later process begins. If two nearby processes are sequential, however, it is much stronger evidence that the earlier one transmits discretely, pre- venting the later one from starting until it is finished. 9
In the experiment of Van Galen (this issue), the distance criterion weakens the evidence for discreteness provided by the localized effect of syllable repetition. The effect of syllable repetition should be on a planning process which is relatively distant from the motor execution process involved in producing strokes. Because of this distance, it is not surprising that the evidence suggests that the process influenced by syllable repetition is finished before the start of the process reflected in stroke production. Thus, these two processes might be sequential even in a continuous model.
Sanders
Sanders’ (this issue) interesting paper is difficult to discuss briefly, because it is itself a review and discussion covering a great many topics. Nonetheless, a few comments on his article are particularly relevant to the major themes of this article.
Firstly, Sanders reviewed two types of empirical results that add to the evidence in support of discrete stage models. He showed that discrete stage models give a good account of a large number of experiments on motor preparation and execution, and that the inferred
9 The distance criterion is the opposite of the adjacency criterion discussed by Miller (1988).
Specifically, it was argued that evidence of overlapping processes is less problematic for discrete
models when the overlapping processes are closer together in the processing chain. For example,
overlap of perception of a stimulus and motor preparation in response to that stimulus would be highly problematic, because it would show continuous flow from the beginning to the end of the
processing sequence. Overlap of line segment perception and letter perception would be less
problematic, however, because it would show continuous flow only between subprocesses involved in word perception.
314 J. Miller / Commenrary: discreteness und mntinuit~
pattern of processing stages is fairly consistent or robust across para- digms. Both types of evidence are of the same logical form as Van Galen’s (this issue); that is, they constitute cases where the discrete explanation is clear and the continuous explanation is not, but they do not show that continuous models are directly contradicted. Some of the effects discussed by Sanders are especially impressive with respect to the distance criterion, however. For example, additivity of average movement velocity and response specificity (e.g., Spijkers 1987) sug- gests that sequential processes are influenced by these two variables, in spite of the fact that they would seem to affect very similar aspects of motor programming.
Secondly, much of Sanders’ theoretical analysis focused on the relevance of various types of discreteness and continuity to the AFM, a subject which deserves considerable attention because of the power of this method. I disagree, however, with Sanders’ conclusions about the types of discreteness necessary for the AFM to work. Sanders suggested that discrete coding is most important, whereas I think discrete trans- mission is most important.
As to the great importance of discrete coding, I believe that Sanders is wrong in theory, although he may be right in practice. The AFM requires the output of a stage of be uncorrelated with its duration (Sternberg 1969). In principle, this means that the AFM could work in a system with continuous information codes, as long as the value of the output code was not correlated with the duration of the stage. In practice, however, Sanders may be right, because such models do seem unlikely. In most models, factors that influence stage durations seem very likely to influence the strength of a continuous output code, too.
Arguing against the importance of discrete transmission, Sanders noted that there is at least one known case in which the AFM is valid in spite of continuous transmission: specifically, the Cascade model (McClelland 1979) with factor effects on processing rates. Given the clear connection between sequential stages and additive effects, how- ever, I would argue that this is an unusual case and that the AFM will generally lead to incorrect conclusions in models with overlapping stages (cf. Taylor 1976). Clearly, more theoretical work is needed to determine the exact conditions under which the AFM breaks down. Until this work is done, it may be pointless to speculate about whether the AFM will fail more often because of continuous coding or continu- ous transmission.
J. Miller / Commentary: discreteness and continuity 315
Another issue considered by Sanders is the effect of presetting processes, especially those involved in controlling speed-accuracy tradeoff. As Sanders noted, the AFM is not inconsistent with the idea of presetting processes, although it does require a strictly feed-forward system from the moment of stimulus onset. Clearly, this aspect of Sanders’ discussion is one forerunner of the present distinction between discrete and continuous variation in a priori state.
Finally, Sanders offers a pessimistic assessment of the usefulness of psychophysiological measures in mental chronometry. While I agree with Sanders about many of the difficulties inherent in this approach (see also Meyer et al. 1988b), I believe that the methods hold more promise than one would think from reading his analysis. In particular, lateralized readiness potential and lateralized EMG have been reasona- bly well validated as measures of hand-specific response preparation (e.g., Coles 1989; Smid et al., this issue), and they seem to be useful supplements to RT in chronometric analysis.
Conclusions
The papers in this issue contribute to the ongoing debate between proponents of discrete and continuous models. The new evidence provided by these articles tends to favor discrete rather than continu- ous transmission, though the debate will certainly not end here. Molenaar (this issue) has shown that discrete models are biologically feasible -and can account for some effects on mean RT that have previously been thought to require continuous models. Van Galen (this issue) has shown that a discrete model gives a good account of factor effects on multiple dependent variables in a complex task, providing circumstantial support for discrete transmission. Sanders (this issue) reviews recent work showing that discrete models given a good account of motor preparation and execution and that stage structures are robust a good deal of the time. On the continuous side, Smid et al. (this issue) have shown that the perceptual system transmits partial output, but they have not shown that the grain size of this transmission is small, as required by continuous models. Kok (this issue) and Sergeant and Van der Meere (this issue) discuss problems of extending discrete models to new domains, but this may have more to do with the complexity of the new domains than any shortcomings of the models.
316 J. Miller / Commentary: discreteness and continuir);
The debate appears to be productive in two ways. Theoretically, it forces us to examine the various meanings of discreteness and continu- ity, the interrelationships among these meanings, and the relationship of each meaning to RT methods like the AFM. Empirically, it directs us toward an important class of experiments to perform - experiments directed at the overall architecture of information processing rather than the existence and capabilities of specific mental processes. Given this progress, it seems reasonable to hope that the detabe will reach a satisfactory resolution. This is not to hope that everything will turn out to be discrete or continuous, but rather that it will become possible to determine the discreteness or continuity - in each of the various senses of those terms - of particular mental processes involved in the perfor- mance of specific tasks.
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