Mathematical models are formal systems designed to describe behavioral processes. The goal is to increase understanding of behavior by working through an abstract model. This approach has rarely been used in the compliance field, so this article may be regarded more as a preview than as a review.

To illustrate the way in which a modeler might approach a behavior, let us consider a new model for pill-taking. I have not seen such a model proposed in the literature. While I have no supporting data, the proposed model seems to have some intuitive appeal.

It is plausible to envision pill-taking by an outpatient as a probabilistic process. The probability of taking an assigned pill may be determined to some extent by controllable aspects of the regimen, such as the required frequency of ingestion and the help provided in the patient-physician relationship, including technological support. Also likely to be of importance are individual characteristics of the patient, such as memory capability and severity of physical condition. One might also expect that belief in the utility of the medication, which depends in part on the patient-physician interaction, will play a role in compliance. While this list of inputs is not exhaustive, it

© Springer Publishing Company 95

96 Weiss

should be sufficient to convey the flavor of the enterprise. One might speculate that the following simple model gives a first order description of the process:

p(i,j) = m_i,j (s_j + (b₀ - (s₀-s_j)))

where p(i,j) = the probability of ingesting the ith pill on the jth day

m_i,j = the strength of the memory trace at the ith opportunity on the jth day

s₀ = the subjective prominence of the initial symptoms

s_j = the subjective prominence of the symptoms on the jth day

b₀ = the initial belief in the efficiency of the pills

The model is multiplicative in character, with the strength of the memory trace being crucial. If the patient forgets the schedule, the pill will not be taken as assigned. Although the current formulation does not attempt a detailed specification of the forgetting curve, it is presumed that memory strength decreases over time, and also that it may be manipulable by the physician; for example, by using mechanical reminders or by careful scheduling of the pill-taking. How apparent the symptoms are to the patient is also important. This model posits a direct relationship between current symptom severity and the likelihood of pill taking. However, symptom reduction also promotes compliance, as expressed by the quantity (s₀- s_j); the justification for this aspect of the formulation is that the patient s belief in the efficacy of treatment should be increased if the physical condition seems to be improving. But even after symptoms disappear (that is when s_j = 0), pill taking can still occur if there was sufficient initial faith in treatment effectiveness.

The beauty of the model is certainly in the eye of the beholder. What sort of practical value might it have? The model tells us that improving the patient's ability to remember the regimen will increase compliance. This idea seems obvious and scarcely justifies the labor of construction. But within the algebra is an insight that may not be so obvious. The parameter s_j, current symptom prominence, appears twice in the definition, so that the right side of the equation may be rewritten as m_i,j (2s_j – s₀ + b₀). Thus, current symptom severity

counts more heavily than original symptom severity. The implication of this derivation is that a patient will adhere more closely to a regimen while suffering continues. Dare one suggest that a medication that cures the underlying disease without alleviating the discomfort (e.g., pain) may be more effective than a medication that patients would generally prefer? I do not wish to advocate sadistic treatment, but if it is crucial that a patient follow a regimen, then pain relievers, for example, might in some circumstances be counter-productive because pill-taking would be reduced since the pain has been relieved. From the model builder's perspective, such a provocative recommendation is desirable because it helps to publicize the model.

Is the proposed model true? Surprisingly, that question is not of central interest to the mathematical psychologist. Not because the theorist's head is in the clouds, I hope, but because no model is ever complete enough to account for a given complex behavior in detail. It is not particularly troubling, for example, that the model does not account for the palatability of the medication. If an experiment were to demonstrate that palatability affects pill-taking, then the model would be false, but not necessarily in a fundamental way. Nor does the model consider excessive numbers of pills taken on schedule, or any taken outside the schedule. If it were deemed to be of interest to make the model more complex, additional parameters could be included. However, if it were shown that an experimental manipulation that increased belief in the effectiveness of the treatment did not at the same time increase the pill-taking rate, then the model would require severe alteration. The empirical evidence of which I am currently aware is consistent with the formulation, but that is hardly surprising since the model was constructed with the evidence in mind. The model's value lies in its descriptive simplicity and in new ideas that may be derived from it, rather than in its ability to accommodate all present and future data.

The testing of any model requires that a number of assumptions be made. Some parameters of a model must be identified with experimental operations, which is usually straightforward. More troubling are the assumptions that connect parameters to data. The pill-taking model is concerned with probabilities that change during the period of observation. In this sense the model is similar to other time-dependent probabilistic models employed in experimental psychology, principally for judgmental tasks (Luce, 1986). However, the laboratory studies have in common two features that distinguish

98 Weiss

them from medical settings. First, the time frame for each judgment is very brief, usually less than one second. This brevity makes it easy to collect many responses from an individual. Second, the stimuli are well-defined and may be presented repeatedly, so that the probability associated with each stimulus value may be identified with an obtained proportion. These features make it a straightforward matter to estimate the values specified by the model under consideration.

In the pill-taking case, on the other hand, each response is governed by parameter values held to be unique to the individual and to the moment. The relevant behaviors take place at most a few times a day. They are carded out over periods of days, weeks, months, or even years. Surmounting the consequent difficulties in estimating probabilities will probably require the pragmatic assumption that responses from different individuals be combined, so that the model describes a hypothetical typical patient rather than any actual participant. This kind of grouping assumption is familiar in research.

Because the relevant behavior is carried out privately, technology is needed that will dependably record pill-taking or its absence. An electronic pill dispenser (medication monitor) that records time and date in a form suitable for computer input will be indispensable. Using such instrumentation (Eisen, Hanpeter, Kreuger, and Gard (1987) have developed a prototype), it becomes feasible to take the model seriously.

One could perhaps devise an experimental program to manipulate the variables that control the unobservable parameters and thus evaluate the model. Assessment of the model's ability to give a quantitatively accurate description of the observed pill-taking behavior (the goodness-of-fit problem) is crucial as the model goes through successive stages of improvement, but a perfect account may remain an ideal.

VERBAL VS. MATHEMATICAL MODELS

The above illustration may have struck the reader as somewhat elaborate. Certainly the compliance literature, in this journal and elsewhere, contains many models. While these proposals provide a framework for integrating empirical results, they do not fulfill my criterion for the term model. What they have in common is that they offer qualitative descriptions of the effects of varying the input (usually some kind of information given to a patient) on the observed

output (patient behavior). A key missing element is a specification of whether the inputs combine additively or interactively. As they stand, these efforts seem precursors to models, rather than working models.

A model should specify exactly what is implied about a class of behaviors, which is an advantage of the mathematical model that has been stressed by Bjork (1973). A similar but more caustically phrased advocacy is that of Harris (1976), who cites published examples demonstrating the difficulty researchers have had in deriving testable predictions from verbally stated theories.

There have been few mathematical models proposed in the compliance field. Stout (1988) has been studying temporal aspects of relapse using life-table approaches. These methods enable researchers to capture the enormous complexity of a class of behaviors that extend over long time periods and that seem to have multiple causes. Stout's theoretical work may be considered a logical successor to the empirical summarization of relapse rotes in addiction programs for three different drugs made by Hunt, Barnett, and Branch (1971). The concordance these authors extracted from scattered reports inspired the seminal work of Marlatt (1985) on designing programs for the prevention of relapse. The key insight is that similar behavioral processes ought to underlie similar survival curves.

An alternative to modeling actual behavior is modeling the mental processes presumed to precede the behavior. A practical example has been furnished by Carter, Beach, and Inui (1986), who were able to increase the likelihood that recalcitrant patients would get flu shots. They used a weighted averaging model (multiattribute utility theory) to distinguish patients who had received flu vaccination with noncompliant ones who had not. These authors designed an informational brochure based on the obtained weightings, and used this brochure on new patients to increase their compliance.

A recent mathematical development that I believe may be of value is catastrophe theory (Fararo, 1978). Despite the frightening name, it is merely a geometrical framework concerned with systems that fluctuate between different states. Recent applications in the compliance domain are those of Guastello (1984), who has modeled drug addiction, and Callahan (1982), who has modeled anorexia nervosa. I look forward to the union of these catastrophe models with clinical data. This union may provide a powerful means of studying factors influencing relapse.

100. Weiss

Acknowledgements: I wish to thank Raymond Ulmer for suggesting the potential utility of this essay, and D. L. Walker for comments on an earlier draft.

Reprints: Requests for reprints should be addressed to David J. Weiss, Department of Psychology, California State University, Los Angeles, CA 90032.

Received: January 1988

Revised: February 1989

Accepted: February 1989