Iconic models in science and psychology

Abstract

Scientists use a wide variety of models to formally represent their theories, including computational, quantitative, and mathematical models. A type of model often overlooked in discussions of modeling is an iconic model, which is a visual or physical representation of the structures and processes comprising some natural phenomenon. In the current paper, we present a brief history of iconic models as they have been employed in other sciences such as astronomy, physics, biology, and chemistry. We then present several iconic models that are well known to psychologists and show how they played important roles in theory construction and evaluation. Modern methods of iconic modeling are next presented and demonstrated with several examples. Finally, we close with a discussion of how iconic models can be used in tandem with computational, quantitative, or mathematical modeling procedures in the quest for rigorous theories of the human psyche.

Keywords

history of science iconic models models philosophy of science theoretical psychology

Recent special issues/sections from the journals Perspectives on Psychological Science (Proulx & Morey, 2021) and Child Development (Frankenhuis et al., 2023) were devoted to theory development and evaluation in the science of modern psychology. Twenty-two papers provided important insights into the roles that replication, measurement, validity, hypothesis testing, and effect sizes play in the construction and testing of psychological theories. The need for the increased utilization of formal modeling techniques in theory construction was also discussed (Borsboom et al., 2021; Robinaugh et al., 2021; van Rooij & Baggio, 2021), and examples of quantitative (Driver & Tomasik, 2023; Kellen et al., 2021), computational (Bhat et al., 2023; Grahek et al., 2021; Guest & Martin, 2021; Piantadosi, 2023), mathematical (Navarro, 2021), and neural network (Mihaela & Plunkett, 2023) models were demonstrated. Notably, iconic models, which have played historically important roles in the biological and physical sciences, were largely overlooked in these special issues/sections.

In this paper, we provide a review and analysis of iconic models and demonstrate how they are valuable tools for the generation, evaluation, and continuous development of scientific theories. We begin with a brief review of scholarly efforts to distinguish between theories and models and then move on to a more focused discussion of iconic models with examples from other sciences. We then present and discuss a number of historically relevant iconic models in psychology as well as examples from modern times. We close with a discussion of how iconic models can be used in tandem with other modeling techniques to advance psychological theories.

Theories and models

A scientific theory can generically—and admittedly insufficiently—be defined as a comprehensive set of propositions that explain how a system in nature (referred to as a “target system”) works. A scientific model can similarly be defined generically as a logical, mathematical, visual, or physical representation of the target system that is used as a tool for advancing a given theory.¹ In his recent book, Models and Theories, Frigg (2023) concludes that the dividing line between the two concepts is a matter of considerable philosophical debate. On the one hand, proponents of what Frigg calls the “received view” consider theories to be formal statements of target systems that do not necessitate the use of models. Newton’s (1725) theories of classical mechanics presented in his Principia, for instance, can be understood as sets of propositions, proofs, principles, and axioms that can be analyzed linguistically or logically. Any model derived from these propositions would serve only as an alternative interpretation of the theory and would therefore be superfluous. On the other hand, proponents of the “model-theoretical view” regard theories and models as intimately intertwined. Indeed, models are considered to be the building blocks of theories. In this view, a scientist seeks to establish an isomorphism between the structure of the posited model and the actual structure of the target system under investigation. James Maxwell, for example, imagined a system of hypothetical tubes filled with an incompressible fluid to develop his electromagnetic field equations. Theory development and evaluation depend upon building and testing models. To the extent a model–target system isomorphism is established and found convincing, the theory is supported.

Which of the two views regarding the relationship between theories and models is correct? It is difficult to say. While reviewing literature concerned with the study of scientists “in the wild” conducting their research, Frigg (2023, p. 366) presents examples in which models were constructed and evaluated without the aid of overarching theories. For instance, he lists the Schelling model of social segregation, the Fibonacci model of population growth, and the Akerlof model of the market for used cars as “theory-free” models. In contrast, Frigg (2023) presents examples in which theory and model building were intimately intertwined (p. 370). For instance, the j4 model is used to explore theoretical properties like symmetry breaking in quantum field theory, and causality and determinism in Newtonian mechanics are understood through use of the dome model. As noted above, the dividing line between theories and models cannot always be easily drawn because it is not etched into stone nor is it always clearly visible. Despite these philosophical complexities and uncertainties, however, one conclusion is inescapable: models play an important role in science. The sheer quantity of different types of models testifies to this fact. Frigg (2023, pp. 466–467) presents a list of 122 different kinds of scientific models, including the quantitative, mathematical, and computational models discussed in the special issues/sections of Perspectives on Psychological Science (Proulx & Morey, 2021) and Child Development (Frankenhuis et al., 2023). He also discusses iconic models, which are the focus of this paper.

Iconic models and examples from other sciences

As with definitions of “theory” and “model,” it is difficult to find a succinct and definitive definition of “iconic model.” Rothbart (1997) states, “An iconic model comprises an organized system of abstract variables and relations which presumably simulates some observable environment. This system comprises an idealized replica of the environment” (p. 4). Considering the chemical structure of hydrogen peroxide, he argues that “H – O – O – H” and “H₂O₂” can be regarded as simple iconic models. Hydrogen peroxide can also be represented materially (physically) by connecting balls and rods together or visually by drawing and connecting circles with lines in a 2-D image. These physical and visual representations are also regarded as iconic models. By comparison, Harré (2004) states, “The iconic model is a ‘picture’ of a possible mechanism for producing the phenomena” (p. 14). However, like Rothbart, he does not restrict iconic models to physical or visual representations, stating one page earlier in his book, “An iconic model stands in for the real mechanism of nature, of which we happen to be ignorant” (Harré, 2004, p. 14). With this description, an iconic model is not necessarily a visual image. Frigg (2023), too, recognizes the diverse meanings that have been attributed to the words “iconic model” and reviews much of the philosophical literature on the subject.

In this paper, we will focus on iconic models that are physical or visual representations of the structures and processes comprising the target systems they were created to represent. Moreover, we will borrow from Harré’s (1970, 2004) lexicon and discuss three types of iconic models: homeomorphic, paramorphic, and analogical. An exemplar of the first type of model is James Watson and Francis Crick’s double-helix representation of DNA shown in Figure 1. This model is homeomorphic because it is like an idealized version of the real target system it was constructed to represent. Such iconic models are primarily concerned with shape and form and may be presented as images (like Figure 1) or physically. One of the more famous photographs in all of science is of Watson and Crick posing with their large three-dimensional (physical) double-helix model in the Cavendish Laboratory at Cambridge University in 1953.

Figure 1.

Example Homeomorphic Model.

In the case of the structure of DNA, iconic modeling played a critical role in its discovery. As recounted and discussed by Evagorou et al. (2015), the only visual evidence Watson and Crick had to work with were the X-ray crystallography images produced by Rosalind Franklin. In his book The Double Helix, Watson (1968) recalled how he theorized that the black cross in one of the images could only be produced by a helical structure. He and Crick then created a physical, iconic model to represent what was theorized to have created the pattern in the X-ray images and that would also be consistent with evidence from other sources. Their first model was refuted by Franklin and her research group at Kings College, and Watson and Crick had to correct the model to make it more consistent with all of the available evidence (Evagorou et al., 2015). The final model was—as is well known—a success, resulting in the Nobel Prize in Physiology or Medicine being awarded in 1962. The theory of the structure of DNA was thus supported insofar as an isomorphism between the final model and the target system was established.

The creation of the DNA homeomorphic model can be traced to 1865 and the work of chemist Wilhelm Hofmann. He produced the ball and stick method of iconic modeling to render the molecular structure of chemical substances (Hofmann, 1865). Relying upon the ball and stick method and the van der Waals radius, chemists Robert Corey, Linus Pauling, and Walter Koltun later created accurately scaled space-filling homeomorphic models (see Figure 2), known as CPK or calotte models (Corey & Pauling, 1953; Koltun, 1965). By visually depicting the relations of the molecular structure of the chemical compounds via these models, chemists were better able to understand the nature of chemical reactions (see also Olmsted & Williams, 1997), thus advancing their scientific theories. It is also Linus Pauling who, using such iconic models, showed that helical structures are possible in nature, paving the way for Watson and Crick to imagine their double-helix model.

Figure 2.

Iconic Models of H₂O.

Whereas homeomorphic models are primarily concerned with shape and form, paramorphic iconic models are more dynamic in nature, depicting not only the structures that comprise a natural system but also mechanisms of movement, change, and generation (Harré, 2004). William Gilbert’s numerous sketches depicting the unseen forces within loadstones (magnets) and modern diagrams of biochemical processes (e.g., the Krebs cycle) are examples of paramorphic models (see Figure 3). The various arrows and lines in Gilbert’s iconic models would later play important roles in the construction of Michael Faraday’s theories of magnetism and electricity. Faraday was an outstanding experimentalist who developed his theories primarily through research in the laboratory rather than through mathematical formalisms written on paper. According to Gooding (2006), Faraday’s iconic models (sketches) served as “tools for reasoning with and about phenomena” (p. 59) and were therefore central to the development of his theories.

Figure 3.

Example Paramorphic Models.

Harré (2004) argues that some paramorphic models may also be considered as analogical. Specifically, such iconic models have their origins as analogues to (i.e., they are comparable to) existing models (see also Haig, 2014). Wallace (1996), for example, discusses the early diagram of the atom which was fashioned after the model of our solar system. This planetary model of the atom “is a pictorial or iconic model: it provides a picture or representation of the inner structure of the atom, showing how its nucleus and orbiting electrons are thought to be arranged in space” (Wallace, 1996, p. 41). Wallace goes on to describe how this picture led to the introduction of various circles or shells in which electrons can be distributed, as well as to a theoretical justification for the law of combining weights and to the methods of visualizing chemical bonding. In short, important strides in theories of atomic physics were made possible simply by imagining the structures and processes comprising an atom to be like those modeled for the solar system. History is replete with such examples in which iconic models of all sorts have played important roles in the generation and development of scientific theories.

Iconic models in psychology

In psychology, one of the most well-known, historic iconic models is Sigmund Freud’s so-called “iceberg model.” Most psychologists will be familiar with some version of the model shown in Figure 4, which is homeomorphic. As can be seen, the human psyche is divided into conscious, pre-conscious, and unconscious regions. Extending across the three regions are the ego and superego, two structural components of the psyche. The third component, the id, resides completely in the unconscious domain of the psyche. As one scans the image from top to bottom, one moves from the conscious realm of light to the twilight realm of the preconscious, to the dark realm of the unconscious, where one finds the bulk of the iceberg, the id. This representation makes clear the unconscious nature of the id and the significance it holds in Freud’s theory. If a psychologist wishes to explain human behavior using this theory, the id will play the dominant role.

Figure 4.

Iceberg Model of the Human Psyche.

The origins of the iceberg image demonstrate the importance of iconic modeling in the development of Freud’s theory of the human psyche. While there is no direct account of Freud sketching the structure of the human psyche as an iceberg (Green, 2019), he did draw the model shown in Panel A of Figure 5 in his New Introductory Lectures on Psycho-Analysis (Freud, 1933, p. 111). As can be seen, the model is a clear precursor to the iceberg image, particularly with Freud’s criticism of his own creation: “The space taken up by the unconscious id ought to be incomparably greater than that given to the ego or the preconscious. You must, if you please, correct that in your imagination” (Freud, 1933, p. 110). This particular image is itself a modification of several earlier models sketched in Freud’s (1900/1953) landmark book, The Interpretation of Dreams. These images, shown in Panels B and C of Figure 5, played important roles in Freud’s theorizing about the structure of the human psyche as well as the process of regression. Specifically, while attempting to explain how thoughts are converted to images in the psyche while dreaming, Freud (1900/1953) imagines “the instrument which carries out our mental functions as resembling a compound microscope or a photographic apparatus, or something of the kind” (p. 574). Furthermore, he posits components of this mental apparatus as arranged in spatial order to represent a temporal sequence. As can be seen in Panel B of Figure 5, the beginning of the sequence is labeled “Pcpt” to represent a component of the apparatus that receives perceptions. At the other end of the sequence is another component, labeled “M,” that serves as a gateway to motor activity. The model is analogous to a reflex arc, which Freud (1900/1953) makes clear: “Reflex processes remain the model of every psychical function” (p. 576). Freud then modifies the model by adding additional features, and the final version is shown in Panel C of Figure 5. As can be seen, a sequence of mnemic systems (each labeled “Mnem”) has been added, as well as the unconscious (“Uncs”) and preconscious (“Pcs”) systems. During waking life, the system represented by the model works from left to right, from the perceptual system to the preconscious endpoint. For hallucinatory dreams, however, the system works in reverse; that is, it is “regressive” in its operation. The unconscious is the source of dream material, but “instead of being transmitted towards the motor end of the apparatus it moves towards the sensory end and finally reaches the perceptual system” (Freud, 1900/1953, p. 581). In short, the unconscious material works its way backwards through the system and is transformed into an image by the perceptual system. The reverse operation of the system is referred to as regression. Finally, Freud lauds the power of the model for providing an explanation of the irrational nature of dreams:

And it is at this point that that picture begins to repay us for having constructed it. For an examination of it, without any further reflection, reveals a further characteristic of dream-formation. If we regard the process of dreaming as a regression occurring in our hypothetical mental apparatus, we at once arrive at the explanation of the empirically established fact that all the logical relations belonging to the dream-thoughts disappear during the dream-activity or can only find expression with difficulty. According to our schematic picture, these relations are contained not in the first Mnem systems but in later ones; and in case of regression they would necessarily lose any means of expression except in perceptual images. In regression the fabric of the dream-thoughts is resolved into its raw material. (Freud, 1900/1953, p. 582)

In summary, the history of the iceberg model reveals the intimate connection between modeling and theory development in the domain of psychoanalysis.

Figure 5.

Freud’s Early Models of the Human Psyche.

Another classic model in psychology which similarly represents inferred structures and processes within the human psyche is Atkinson and Shiffrin’s (1968) iconic model of memory, a version of which is presented in Figure 6. As can be seen, three domains of memory are represented: sensory, short term, and long term. Moreover, the arrows in the model represent changes, or processes, such as attention, rehearsal, and retrieval, within the memory system, thus making it a paramorphic model. In their 1968 paper, Atkinson and Shiffrin introduce and describe the model immediately after a brief overview of their manuscript in a section titled “Structural Features of the Memory System.” Like Freud’s (1933) model presented in his new introductory lectures, this model serves as a tool for reasoning and for guiding readers systematically through the theory of memory in the remainder of the manuscript. It is less clear, by comparison, if the image played a key role in the actual development of Atkinson and Shiffrin’s theory. It is likely they invoked such visual images when thinking about the distinct structural features of the memory system while developing and discussing their theory, but unlike Freud, they made no clear statement about the importance of such images. What cannot be argued, however, is that Atkinson and Shiffrin’s iconic model played an important role in subsequent memory research. At one point it became so influential that it was dubbed the “modal model [emphasis added] of memory” (Ellis & Hunt, 1983, p. 62), shaping the way researchers would think about and conduct research on human memory for many years after its publication. It is also quite telling that subsequent studies routinely used the term “model” rather than “theory” when describing Atkinson and Shiffrin’s (1968) contribution (e.g., see Raaijmakers, 1993). The iconic model itself served as the focal point and lynchpin for subsequent theoretical developments in memory research.

Figure 6.

Atkinson and Shiffrin’s Memory Model.

Considered generally, connected events like those in Atkinson and Shiffrin’s (1968) model are often represented in iconic models created from a common set of geometric figures, lines, and arrows. Figure 7 shows three such variable-association models: a structural equation model, a path model, and a directed acyclic graph. As can be seen, these models represent variables as squares, circles, or ellipses, and their associations are drawn as straight or curved lines or as directional or bidirectional arrows. The arrows may further be considered as representing hypothesized causal connections among variables—although this is not necessarily the case—and qualities that are not directly observable can be represented as ellipses (i.e., as latent variables). These types of iconic models are commonplace in modern psychology as they have the advantages of being both generic and flexible, making them useful for researchers working within a wide variety of theoretical domains. They are also easy to construct and, like all iconic models, they succinctly convey important features of a theory in a format that is easy to understand. In this regard, they represent theoretical commitments which may be compared, contrasted, critiqued, and developed. Is intelligence a single dimensive quality, or is it best theorized as a set of correlated dimensive qualities that can be arranged hierarchically (e.g., see Carroll, 1993)? Is it valid to posit a quality known as “emotional intelligence,” and if so, is it independent of what psychologists regard as traditional, general intelligence (e.g., see Lam & Kirby, 2002)? Inspired by the work of Spearman (1904) and Thurstone (1935) in the early 1900s and continuing to today, iconic models like those shown in Figure 7 have played an important role in seeking answers to such theoretical questions. It is also important to note that many of these contemporary models are tied to particular mathematical or statistical procedures, such as factor analysis and multiple regression, which facilitates the transition from theoretical exposition to empirical evaluation and validation.

Figure 7.

Variable-Association Models.

Integrated models in psychology

A type of iconic model that renders the structures and processes (that is, the causes and effects) entailed in a specific scientific study is an integrated model (Grice, 2011). Figure 8, for example, shows a closed-loop, integrated model for an experiment that is well known to students of cybernetics (Camhi, 1984; Yin, 2013). The iconic image shows a common housefly standing on a platform that is positioned in a striped cylinder. When the cylinder rotates, the fly turns in the same direction and at approximately the same speed as the cylinder. As can be seen in the iconic model, the experimenter sets and/or changes the direction and speed of the rotating cylinder which, according to Camhi (1984), causes changes in the fly’s rotational direction and speed. The experimenter can measure the difference between the cylinder’s and the fly’s rotational speeds, labeled “slip speed” in the model, and it is this difference that Camhi (1984) considered as the cause of the fly’s behavior. As the fly turns, it processes visual feedback in reference to the lines in the cylinder and then adjusts its behavior accordingly. This model has both homeomorphic and paramorphic features, but it also represents a particular experiment involving an organism (the fly) and variables controllable by the experimenter. In brief, the purpose of the integrated model in Figure 8 is to visually render the structures and processes involved in the experiment in order to elucidate a causal explanation of the fly’s behavior.

Figure 8.

Integrated Model for Rotational Behavior of a Fly.

Early cybernetic models (e.g., Rosenblueth et al., 1943; Wiener, 1948) served as the basis of and inspiration for Perceptual Control Theory (PCT), first developed by Powers (1973, 1978; see also, Marken, 2021; Runkel, 2005). Powers was an engineer trained to think about nature through images (iconic models) like that shown in Figure 8. However, while attempting to apply simple feedback models to a living organism, he realized they failed to include the internal state of the organism, which he regarded as important. From this insight, he developed PCT and unique ways to sketch iconic models of behavior. Figure 9 shows an example model based on PCT which provides a more complete explanation of the fly’s behavior in the rotating cylinder experiment. As can be seen, the visual slippage of the cylinder across the fly’s retina (called slip speed) is the perceptual input sensed by the organism. According to Yin (2013), the experimenter setting the rotational speed is not the input into the system but is better described as a disturbance in the system’s environment. The fly then behaves (e.g., rotates the same direction and same speed) because an error is computed between the amount of allowed slipping and the actual slipping which has occurred. Key to explaining the fly’s behavior, then, is the comparing process which occurs within the fly. Specifically, the fly compares the amount of “slippage” perceived across its retina with the amount of allowed slippage, also referred to as the referent signal in PCT.

Figure 9.

Perceptual Control Theory Model for Rotational Behavior of a Fly.

The iconic model in Figure 9 appears purely mechanical, but it takes into account the internal state of the organism in a way that is not entirely mechanistic. In other words, the perceptual control model indicates that the behavior of an organism cannot be sufficiently explained by time-ordered inputs and outputs. The changes in the models from Figures 8 and 9 therefore have important implications for our theoretical understanding of the very nature of behavior in living organisms. Arocha (2021) and Runkel (1990) provide excellent discussions of this dramatic shift in perspective and the impact it has on psychological research.

Model comparison and inference to best explanation

The integrated models in Figures 8 and 9 provide representations of competing explanations of the fly’s behavior in the rotating cylinder experiment. Which model possesses the greatest degree of verisimilitude (akin to the model/target system isomorphism discussed above); that is, which model represents the most plausible explanation of the fly’s behavior? Providing an answer to this question would be tantamount to drawing an inference to best explanation, which is a type of inference that entails the development, evaluation, and testing of competing explanations of some phenomenon (Lipton, 2004). For the fly’s behavior in the rotating drum experiment, the integrated models above would be compared according to their coherence, simplicity, and nonadhocery, among other criteria (see Haig, 2014), and the more plausible model judged as offering the best explanation. This type of inference is distinct from the inductive inferences drawn from null hypothesis significance tests and the deductive inferences drawn from the hypothetico-deductive method. It grew out of discussions of C. S. Peirce’s abductive inference (see Campos, 2011), which is an explanatory form of inference central to scientific reasoning.

Some competing integrated models can be evaluated empirically when attempting to draw this important form of inference. Grice et al. (2017), for instance, drew and compared the two models in Figure 10 for a study designed to test the plausibility of Nairne et al.’s (2008) theory of fitness-related processing. The methods of the study entailed four steps. First, participants imagined themselves in survival and nonsurvival scenarios. Second, while still imagining each scenario, participants rated supplied words on a 5-point scale regarding their relevance to the scenario. Third, a digital recall task was performed in order to clear the short-term memories of the participants. Fourth, a surprise recall task was completed in which the participants were asked to recall as many of the rated words as possible. These four steps are represented at the bottom of each integrated model in Figure 10 (note the sketched participant sitting at a desk at four different time points). As can also be seen in the models, and similar to the structural equation and path models above, different geometric shapes are included in the callouts above the sketched participant. These shapes represent distinct, unobserved psychological processes; namely, elongated hexagons represent visual imagery, circles represent simple predication, pentagons represent complex judgments, and hexagrams represent memory storage. In Figure 10, logical relations are presented as “if . . . then” statements or formal (Fo) causes. Final (Fi) and efficient (Ef) causes are also represented (for further details, see Grice et al., 2017). The models therefore differentiate Aristotle’s four species of cause; namely, material, formal, efficient, and final causes (Falcon, 2023; Rychlak, 1988).

Figure 10.

Competing Integrated Models.

Model #1 in Figure 10 renders the structures and processes entailed by the fitness-related processing theory in which words are memorized more efficiently when participants are asked to imagine themselves in a survival situation. As shown in the model, the word “truck” will be memorized as a result of the fitness-related processing mechanism. Model #2 is relatively atheoretical as it renders the structures and processes entailed by a well-known mnemonic device in which words that are relevant to some imagined scenario are more likely to be memorized. As shown, the word “truck” is memorized because it is rated as relevant (value of 4 or 5 on the scale) to the survival situation. Which of the two models offers the best explanation of the observations made from this study? Grice et al. (2017) collected data from 99 undergraduate students in order to answer this question. The data were analyzed at the level of the individual responses in such a way as to compare the two models. Results showed that 70.71% of the students recalled more relevant words than words in the survival condition of the study (randomization p-value < .001) thus supporting the verisimilitude of Model #2 in comparison to Model #1. Similar to goodness of fit indices that might be used for variable-association models (e.g., R², GFI, RMSEA) like those shown in Figure 7, this empirical information can be used when drawing conclusions about the validity of the fitness-related processing model of memory. Such information must of course be used judiciously with other important information when evaluating theories or models (e.g., see Roberts & Pashler, 2000; Rodgers & Rowe, 2002). As discussed above, criteria such as coherence, simplicity, and nonadhocery can be used as well as additional criteria such as dis-confirmability, novelty, and the capacity for generating surprising predictions. With particular regard to integrated models, issues such as overfitting, cross-validation, and generalizability are criteria that may also be relevant to judging their verisimilitude.

Finally, as an added benefit of drawing integrated models, Grice et al. (2017) discovered a potential confound in the experimental procedures which may have been responsible for the original findings in support of the fitness-related processing theory. This confound only became apparent in light of the integrated models and the necessity of analyzing the data in a manner consistent with the models; namely, analyzing the data at the level of the individual observations rather than in the aggregate. Consistent with the model–theoretic view, the integrated (iconic) models in this example study made clear the theoretical commitments of the researchers and the consequent predicted patterns of observations. If the confound is validated in subsequent studies, the iconic model will have played a critical role in paving the way for better tests of the fitness-related processing theory of memory.

Additional and concluding remarks

An additional point to be made about iconic models is that they can complement the mathematical, computational, and quantitative models discussed in the recent special issues/sections of Child Development (Frankenhuis et al., 2023) and Perspectives on Psychological Science (Proulx & Morey, 2021). In their paper on computational models, for instance, Guest and Martin (2021) demonstrated how psychologists can create and evaluate computational models using an example of an individual determining whether it is a better deal to buy two 12-inch pizzas or one 18-inch pizza for the same price. At first glance, one would expect to obtain a greater quantity of pizza by purchasing two medium pizzas, but in fact ordering the large pizza is the better deal. In their paper, Guest and Martin used scaled images of pizzas and presumed perfect circles (which are homeomorphic models) as well as geometry to show that, counter to one’s intuition, purchasing the single pizza is the better deal. If we were to sketch an iconic, integrated model of a student making the correct decision in this scenario, it would include the pizza images and geometric formulas (mathematical models), but it would also include an additional, important feature; namely, the goal of the student. The computational models are based on defining the best deal as ordering “the most pizza for our buck” (see Guest & Martin, 2021, p. 791). The “most pizza” is a final cause (i.e., a purpose; Falcon, 2023; Rychlak, 1988) wrapped around the computational formulas. Imagine a person, for instance, who correctly computes that the large pizza is the better deal but still purchases two medium pizzas. Why did the person make this choice? Imagine further that the pizza advertisement restricted the deal to one-topping pizzas but this person wished to have four toppings on one pizza and only one topping on another. The calculus of the computational formula for this person must change because the goal (final cause) has changed. The large pizza would be the better deal in terms of quantity, but the difference may not be judged by the person as sufficiently large to override the primary desire for ordering a pizza with diverse toppings. An integrated model for the pizza problem would make the subjective goal of the person clear and would therefore add important information to the pizza problem, akin to adding important auxiliary assumptions to a model (Saylors & Trafimow, 2021; Trafimow, 2012). If the goal is to purchase the largest quantity of pizza, then the computations shown by Guest and Martin (2021) must be used; but, if the goal is to diversify toppings, then another set of formal operations must instead be used to explain the individual’s behavior. By rendering the person’s goal, the integrated model would provide a more complete explanation of a given person’s behavior engaged in the pizza problem.

In modern psychology, the modeling of quantitative relationships is routinely wedded to the structural equation and path models shown above. Such models will continue to play important roles in research because of their generic and flexible nature. In light of the discussion of inference to best explanation above, another advantage of these types of models is that they can be constructed in ways that permit their direct comparison. It is common practice, for instance, to compare iconic factor models for overall fit to a sample of data. A specific example would entail comparing a second-order factor model to a first-order factor model in order to determine if adding the higher order factor would improve model fit. The better fitting model would be judged as providing the best explanation of the covariance structure of the observed variables. Inference to best explanation is also seen in the popular practice of comparing path models constructed with and without mediators in order to determine if the added mediation pathways improve model fit.

Insofar as model comparison in service to inference to best explanation is a fruitful way forward in psychology, and the life and social sciences as well, integrated models may hold unique promise. Like structural equation and path models, they are flexible and can represent relationships among variables, but they can also employ a richer set of symbols to represent the structures and processes underlying individual experiences. As the example memory study above regarding fitness-related processing showed, a more detailed understanding of the implied psychological processes was gained by representing acts of predication, judgment, and memory encoding with different geometric forms. The integrated models for the memory study also employed a richer vocabulary regarding causality, drawing upon Aristotle’s four species of cause (material, formal, efficient, and final), thus providing an explanation of the participants’ responses with greater nuance. Indeed, carefully sketching the structures and processes underlying the observations for a planned study requires the researcher to pay careful attention to every detail of the study design, including auxiliary assumptions. While difficult work, taking the time to sketch an integrated model may well be worth the effort. This statement holds true for iconic models in general, as history has shown they are valuable tools that should be included in every scientist’s toolbox.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

James W. Grice

Ian T. Jones

Notes

Author biographies

James W. Grice is professor of psychology at Oklahoma State University where he teaches courses on personality, statistics, and philosophy. His current research interests entail the development and dissemination of person-centered methods for data conceptualization and analysis. Recent representative publications include: “Drawing Inferences from Randomization Tests” in Personality and Individual Differences (2021), and (with E. Medellin, I. Jones, S. Horvath, H. McDaniel, C. O’lansen, & M. Baker) “Persons as Effect Sizes” in Advances in Methods and Practices in Psychological Science (2020). He is also author of the book Observation Oriented Modeling (Academic Press, 2011) and its accompanying OOM software as well as the Idiogrid computer program.

Ian T. Jones is currently a behavioral scientist and statistician at Applied Research and Analysis Company. His current research interests entail the development and dissemination of person-centered methods for data conceptualization and analysis, broadening the theoretical understanding of cause and effect in theoretical psychology, advancing visual modeling techniques to better depict and understand psychological phenomena, and more recently, tobacco harm reduction. Recent representative publications include: “Capturing the Heterogeneity of Word Learners by Analyzing Persons” in Behavioral Sciences (2024), and “Persons as Effect Sizes” (with J. Grice, E. Medellin, S. Horvath, H. McDaniel, C. O’Lansen, & M. Baker) in Advances in Methods and Practices in Psychological Science (2020).

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