Multidimensional Property Supplementation: A Method for Discovering and Describing Emergent Qualities of Concepts in Grounded Theory Research

Add properties until different codes are dimensionally distinguishable

According to the first heuristic, no two codes that differ in practically significant ways should be left to answer similarly the questions asked by their parent, for such a state of matters would indicate that some pertinent facts remain to be theorized. (In contrast, codes that do not differ significantly may safely co-exist; occasionally, they might even serve to inspire new thoughts and hypotheses.) To address untheorized differences between codes, the researcher adds properties until codes that are intuitively different are formally separated.

Consider an example from the study on the voice of the self. A recurrent theme was strain that threatened to overwhelm the GP’s cognitive capabilities. As I coded selectively, several codes came to suggest themselves as possible responses to such strain. One of them, tie up loose ends, was found to be too vague, subsuming incidents that were intuitively different. This led to the conception of a slightly different code, reduce the agenda, that could account for part of the incidents. To distinguish these two codes theoretically, a new property was then called for. Inspection revealed that reduce the agenda implied some degree of certainty without which a reductive move would entail a risk of missing something important, whereas tie up loose ends implied a degree of confusion by virtue of which the situation was already highly uncertain. The property that unified these contrasting dimensions was eventually named degree of certainty.

Properties, then, should never be added blindly; each must earn its place by making a difference elsewhere. This criterion saves us from the onus of adding properties forever, but also requires us to entertain already at this point some idea of the process that the concept will take part in.

Employ parsimony to keep the concept grounded and simple

A concept’s efficiency can be thought of as a function of its explanatory power and simplicity, both of which are arguably aspects of theory quality (Miller & Fredericks, 1999, p. 542; Soulliere et al., 2001, p. 256). Because we value efficiency, theorizing every practically significant difference is not always reasonable. There are two particular threats to watch out for.

One threat to efficiency is sparsity, which occurs whenever a combination of dimensions is unbacked by codes, and hence by data. It is often tempting to interpret sparsity as lack of theoretical saturation and to reflexively reach for the preferred remedy, theoretical sampling. Although theoretical sampling is indeed necessary to attain conceptual density (Becker, 1993, p. 256; Conlon et al., 2020, p. 948; Draucker et al., 2007, p. 1138), there are two caveats. First, if the properties that introduce sparsity interact, logically or empirically, in ways that preclude incidents manifesting certain combinations of dimensions, no amount of sampling will help. Second, sparsity limits the available material for making constant comparisons, without which one runs the risk of forcing data (Heath & Cowley, 2004, p. 144). While occasional gaps might be manageable by framing them as hypotheses and sampling theoretically, retracing one’s steps by removing some of the problematic properties is sometimes a safer course.

The other threat is rampant complexity. While high resolution—capacity for making fine distinctions—grants explanatory power to the model, it is also anathema to simplicity. Balancing these two concerns is a matter of identifying the point beyond which additional distinctions are conflatable with little loss in meaning, and then to apply Occam’s razor judiciously.

Address redundancy through reduction

Despite best efforts at parsimony, a potential problem may occur in the form of redundancy when one property includes another, or some aspect of it, in its definition. While theoretically possible with as few as two properties, redundancy becomes more of an issue as properties grow more numerous, particularly if insufficient attention is paid to how they interrelate. Like any dependency between properties, redundancy can entail sparsity: Given a redundant quality R such that dimension A implies R and dimension B implies not-R, sparsity occurs whenever one property declares A and another property declares B, for the case A and B is then contradictory. Barring coding idiosyncrasies, that particular combination of dimensions will then lack data to back it.

Redundancy differs from other kinds of dependency in that the gaps that it gives rise to will remain regardless of how much additional data are generated. This is because the cause is conceptual rather than empirical. To expose it, the researcher examines the properties and dimensions on a conceptual level, asking a series of questions such as the following:

Only when all such questions can be answered in the affirmative has redundancy been ruled out. In the case that two properties are dependent in a way that puzzles the researcher, reducing them into components that can be more easily checked for synonymy might be helpful.

In principle, redundancy can be prevented by ensuring that all properties are irreducible. In practice however, strictly irreducible properties may not always be efficient, particularly if some of their dimensions are rarely seen. We are thus impelled to look for the “sweet spot” between irreducibility and efficiency. One way of conceptualizing it is in terms of relative irreducibility, beyond which further reduction would cause “loss of an epistemologically indispensable level of description and explanation” (De Sousa, 1990, p. 91). What levels of explanation are indispensable depends, of course, on the purpose of the concept, but as a general rule, the loss to be avoided can be conceptualized as reduced practical utility from overcomplication. Given, for instance, two irreducible, rarely seen, partially overlapping qualities A and B that have almost the same effects with no significant interactions, it might be more reasonable (and certainly more economical) to construct from them a compound that distinguishes between the two cases A or B and neither A nor B. The compound property is, on this understanding, relatively irreducible.

As we saw earlier, the voice of the self came to declare a dichotomous property named cognition with the dimensions on top of it and strained. This was a simplification, for cognitive strain is arguably a function of at least two variables—the complexity of the problem and the amount of available time—and the former could probably be reduced even further. However, as the data indicated that the effects of rushing through a moderately complex encounter were comparable to those of slightly hastening a highly complex one, distinguishing between these two cases was superfluous in the current context. After the relatively irreducible property cognition had passed the redundancy test, I was therefore satisfied with it. Had I instead insisted on completely avoiding reducibility, the model’s resolution might have outgrown its explanatory power.

The spatial metaphor

A concept that has been defined in terms of its overall purpose and theorized through carefully selected properties has two important formal second-order qualities. First, given that redundancy has been avoided, its properties will be conceptually independent; they can thus be thought of as orthogonal axes. Second, the dimensions of any one property will be mutually exclusive; hence, they are coordinates. Taken together, these two qualities justify using the metaphor of space. Much like objects in physical space, codes that connect a concept to data can be thought of as occupying discrete positions in a space of conceptual possibilities. From this viewpoint, the spatial separability of codes is what marks them as theoretically distinct.

According to our spatial metaphor, the extension of a concept is an n-dimensional space defined by n axes (properties). Whenever a piece of data is assigned a coordinate (dimension), its vagueness is reduced because movement is no longer possible along the corresponding axis. Each coordinate thus corresponds to a hyperplane (an n–1-dimensional subspace) which is flat and perpendicular to the axis. Wherever two hyperplanes intersect, we find an n–2-dimensional subspace perpendicular to both. Compounding such intersections with additional hyperplanes yields subspaces of decreasing dimensionality, until we reach the 0-dimensional one, which is a point. As shown in Figure 3, flat hyperplanes in a three-dimensional space are planes.

OPEN IN VIEWER

Figure 3.

A 3-dimensional conceptual space is defined by three orthogonal axes.

A redundant yet convenient term is the codimension of a subspace, which is the number of coordinates that define it. For instance, within a 3-dimensional space, a 1-dimensional subspace—one that allows variation along a single axis—is defined by two coordinates; hence, it is 2-codimensional. Finally, representations of the conceptual space that take into account variation along less than the full number of axes are projections. A 2-dimensional projection, for instance, is constituted by two axes, regardless of the dimension of the ambient space. In the simplest possible 2-dimensional projection, both properties are dichotomous.

As an example, in the 4-dimensional conceptual space of the voice of the self, I sought the meaning of an intersection between three hyperplanes: unsuited, strained, and taken for granted. By considering its position in three 2-dimensional projections, I could also describe the resulting 3-codimensional subspace as a combination of being oppressed, cornered, and useless. This granted me an intuitively appealing explanation to why every incident in this subspace appeared to be, in some way or another, about regaining power. The subspace was 1-dimensional in that it still allowed variation along the fourth property, degree of certainty.

What this brief mathematical excursion grants us is the ability to define conceptual subspace as a subset of the concept’s extension, demarcated by coordinates (dimensions) along one or more axes (properties) that are declared or inherited by the concept.

Inferring emergent qualities of subspaces

The dimensions that define a subspace constitute it in the sense that dimensions cannot be added, redefined, or removed without altering its meaning. But the meaning of a subspace is not exhausted by the meanings of its dimensions. On the contrary, it should be possible to add, through repeated observation, practically significant facts to the definition of subspace A and B beyond what can be deduced from the definitions of A and B alone. In that case, the subspace in question has an emergent—unpredictable, or transcendent (Levers, 2013, p. 4)—quality.

For example, one of the 2-codimensional subspaces within the voice of the self came to represent the experience of being cornered (see Figure 4). This happened when the GP was both strained (being exposed to too much data, tasks and disturbances within a given time frame) and taken for granted (being expected to deliver predefined products rather than to exercise professional judgment). In these situations, setting one’s priorities would incur sanctions or disdain, whereas going along with expectations would aggravate the strain. Quite understandably, one emergent quality of cornered was that the GP, expecting their usual strategies for handling the component threats to do poorly, sought to escape the situation by persuasion or trickery. Another emergent quality, which applied to the complementary subspaces as well (that is to say, to the whole 2-dimensional projection), was that the properties of cognition and recognition both implied the possibility of someone or something intruding on the GP, thus implying a threat to integrity. Neither emergent quality could be deduced from the definitions of cognition and recognition alone; it was only after thinking about the subspaces and comparing their subsumed codes that such conclusions could be drawn.

OPEN IN VIEWER

Figure 4.

A 2-dimensional projection of the voice of the self in which properties cognition and recognition act as orthogonal axes reveals emergent qualities of the 2-codimensional subspaces.

To include emergent qualities in the definition of a subspace is to explicate it, improving upon its meaning by supplementing it (Quine, 1980, p. 25). Explication has the effect of making the subspace more convincing by itself and less dependent on purported examples, hence less vague in the sense suggested by Peirce (2011, p. 295). The other side of the coin is that the projection is rendered susceptible to falsification by examples that challenge its comprehensiveness, that is, cases that do not match any of its subspaces. As an example, I considered at one point trust as a property of the voice of the self, only to find that it introduced a false dichotomy between being trusted and distrusted that did not capture prevalent cases of indifference. This prompted a respecification of the concept that led up to the conception of another property, recognition.

Because emergent qualities can be explained by, but not analyzed in terms of, lower-level qualities (De Sousa, 1990, p. 32), explications of subspaces will always make reference to facts beyond those represented by the constituting dimensions. As an example, the experience of being cornered is emergent from the dimensions of being strained and taken for granted, just like the aesthetic features of a painting are emergent from how it reflects light; in neither case can the matter of that being experienced like this be deduced from the lower-order qualities of that. Once the connection has been pointed out however, it should be a simple matter to agree to it. Determining whether an observed pattern indicates a conceptual or causal relationship is a sometimes difficult task (Soulliere et al., 2001, p. 267) where explication can be helpful, for in the contrasting case that the emergent qualities make no sense in the light of their constituent dimensions, one may have stumbled upon incommensurability—the state of matters where one or several of the properties in question belong, if at all relevant, elsewhere in the theory.

Detecting skew as sparsity

Depending on how the researcher defines the dimensions of a property, it might come to suffer from skew (see Figure 5, middle). A skewed property is one that fails to partition the conceptual space into equally sized subspaces (where “size” is the unconditioned probability of containing a randomly sampled data point). Although based on the same raw data as the previous diagram, this one is markedly different. In particular, because both properties are skewed, subspace ab contains the majority of all data points, whereas subspace AB is sparsely populated.

Three aspects of skewness are worthy of note. First, a skewed property does not imply real-world skew; rather, skew (or its absence) reflects how we conceptualize the phenomenon of interest, including our choice of dimensional cutoffs. Second, skew incurs a loss of resolution that could impede modeling of cause–effect relationships. In the diagram, our inability to distinguish between the numerous cases within subspace ab is potentially problematic, as is the sparsity within subspace AB. Third, in a realistic setting, our only clue to the presence of skew is an unequal distribution of data points between subspaces. While skew can be detected by considering properties in isolation, a multidimensional approach provides a more sensitive test because the effects of individual skews are multiplied.

Because we value conceptual resolution, avoiding skew is—all else equal—a worthy goal. It is also a mostly attainable one, except in the case of categorical or multimodal distributions.

Detecting non-orthogonality as empirical dependence

The two diagrams discussed so far show no signs of significant non-orthogonality, which I shall define as an empirical correlation between properties to the effect that variation along one predicts variation along another, for in neither case does knowing that some hypothetical data point manifests dimension A help us predict whether it also manifests dimension B, or vice versa. The third diagram presents a contrasting case (see Figure 5, right). Notably, because equally many data points manifest dimensions A, a, B, and b, neither property is skewed. What sets this case apart is that some conditional probabilities differ from the corresponding unconditional ones. If we know, for instance, that a data point manifests A, we also know that it will be more likely to manifest B. That such predictions are possible indicates that the properties are non-orthogonal. Although I have drawn the Bb axis obliquely to illustrate the point, a real study will of course lack such giveaways. The researcher might therefore need to pay attention to relative frequencies, and perhaps even use some basic statistical tests to check for non-orthogonality.

There are several possible causes of non-orthogonality. As always, selection bias is one. A second, idiosyncratic coding, is particularly likely before codes have been clearly defined. A third is the presence of causal connections between properties, either directly or through confounders or mediators. While non-orthogonality may resemble redundancy, it is an empirical matter, hence our methods for revealing it are very much data-driven; a conceptual subspace might be logically consistent yet empty, indicating that the constituting combination of dimensions is perfectly possible yet unlikely to be seen for empirical (for instance, contextual) reasons. Non-orthogonality does not necessarily pose a serious problem, but is better thought of as a source of relative inefficiency, and sometimes one of lack of theoretical saturation.

The upshot of this venture into the realm of pseudo-quantification—which many qualitative researchers will no doubt find off-putting, but which I believe is in line with the original intent of grounded theory as a general methodology (Glaser, 1999, p. 842)—is this. Whenever the data are not naturally categorical, the researcher has good reason to explore signs of non-orthogonality or skew. To detect some sources of inefficiency, they need to pay at least passing attention to frequencies. While skew can in principle be identified by considering properties one by one, a multidimensional approach is clearly more sensitive. Non-orthogonality, being a multidimensional phenomenon, cannot be exposed without a multidimensional approach.

Expansion (code differentiation—property testing)

In the first step, the researcher expands the code base by making distinctions beyond those that the current model predicts. Expansion comes in two forms: splitting codes that appear vague, and overturning those that lack clear contraries.

A code is split by being replaced by more precise ones. Because splitting is intuitively rather than theoretically driven, a property that captures the distinction will often be lacking. As an example, the code reduce the agenda, representing the imperative to reduce the size of one’s to-do list to alleviate pressure, seemed heterogeneous in ways that could not yet be theorized. It was therefore split into two codes, turn a blind eye and set the agenda, both of which retained some of the meaning of the original code while being more precise.

To overturn a code is to use the “flip-flop” technique (Corbin & Strauss, 2008, p. 79) to generate a contrary one, using the concept as the frame that connects them and explains what it means for them to be “different.” For instance, the code return the initiative came over time to represent the imperative to reduce one’s burden by taking a step back—into the role of consultant, as it were—while leaving the other (the patient or co-worker) in charge of the problem. To complement it, I hypothesized the imperative stay in control which might also serve to protect the self in sufficiently different circumstances.

Besides adding codes that potentially increase the theoretical resolution of the concept, this step tests the hypothesis that the current set of properties suffice to make all necessary distinctions.

Abstraction (code reallocation—property differentiation)

To make the concept “applicable to many similar situations and contexts,” the researcher carries out “the analytic work of identifying attributes, moving beyond emic tag labels and developing careful definitions,” which Morse refers to as “decontextualization” (Morse, 2004, p. 1390). I prefer the term abstraction—extracting from “detached observations certain general characters in which the observed phenomena resemble one another” (Mill, 1884, p. 376)—because the purpose is to make the concept theoretically useful, whereas any loss of context is merely incidental. In MPS, the researcher abstracts by reallocating codes, framing their similarities and differences—so far only intuitively understood—in terms of properties and dimensions which are simultaneously differentiated. Depending on how similar or different the codes in question are, the procedure becomes predominantly one of separation or consolidation.

Separation is the theorizing of differences between mostly similar, but intuitively distinct, codes. One approach is to list dimensions that could be attributed to one code but not the other, select those that best bring out the difference, and create a property that pins down the disagreement. Recall the two codes set the agenda and turn a blind eye that emerged through splitting in the previous step. Closer inspection revealed that when GPs set the agenda, they felt that they were the right person to handle the problem—in effect, making a difference—whereas turn a blind eye indicated the feeling of being unsuited, with the implication that their skills could be put to better use elsewhere. To unify these dimensions, I devised the property purpose, which came to stand for variations along the range of being or not being in the right place.

Consolidation is about identifying the common denominator that unifies contrary codes. As there will be no shortage of candidate dimensions to mark them as different, the challenge is rather to identify the most crucial properties on which they “agree to disagree.” To this end, it can be effective to ask “In what regard do these codes stand for opposite things within the frame provided by the concept?” This allowed me to discover that the code return the initiative implied a greater degree of certainty, or capability of making predictions, than did stay in control.

Geometrization (code homogenization—property reallocation)

In this step, supplementation is driven by selection and juxtaposition of properties. Through these procedures, properties are reallocated to create subspaces of varying codimensionality.

Property selection involves grouping similar properties and selecting from each such group the property that best captures a practically significant difference. Good candidates are generally those that are relatively irreducible, lack skewness, and tie in with other concepts. As an example, the property purpose was chosen over responsibility because it was less reducible and showed no signs of redundancy with other properties. The property recognition turned out to be superior to social acceptance for similar reasons.

For a set of selected properties to be good all things considered, the implied conceptual space must be efficient. To this end, the researcher juxtaposes properties in a series of 2-dimensional projections and examines the codes subsumed in each of the resulting 2-codimensional subspaces. Each subspace is named and explicated, whereby emergent qualities are drawn out that add to—but are coherent with—the meanings of the constituting dimensions. If the properties are commensurable, the researcher should be able to name the projection as well. The same procedure is applied on each level of codimensionality until all practically significant properties (and all resulting n-codimensional subspaces) are considered simultaneously.

Property reallocation is accompanied by code homogenization, which sees codes subsumed under conceptual subspaces, thus reducing their code base footprint and theoretical relevance. In return, theoretical complexity increases with the proliferation of subspaces.

Unification (code testing—property homogenization)

Working with a narrow set of abstractions incurs a risk of losing contact with both data and other parts of the theory. In this final step of the supplementation cycle, the researcher therefore employs specification and contextualization to reduce the complexity of the concept and situate it firmly in data and context.

Specification involves revisiting codes and associated data and memos while paying attention to how codes reflect data in the light of the now slightly better characterized concept. Ensuring “fit” between codes and their containing subspaces is crucial to overcoming the concept-indicator problem (Layder, 1998, p. 79). As an example, in a projection juxtaposing the properties purpose and recognition (see Figure 7), the code refuse to clean up other people’s messes, signifying the GP’s exasperation at being implicitly assigned menial tasks, ended up in a subspace named oppressed. This subspace came to subsume instances where GPs were sidetracked from working toward valued goals and insufficiently respected to have their concerns heard unless they raised their voice. This drew my attention to the fact that autonomy can be infringed not only directly (through directives and threats of sanctions) but also indirectly (through negligence or indifference). Because its position in the other projections implied less severe component threats to the self, I inferred that refuse to clean up other people’s messes might be more about fighting for one’s autonomy than about, say, reducing one’s workload. By revisiting the backing data I could confirm this, and also found evidence supporting a number of corollary hypotheses, for instance, that being expected to clean up after others implied an infringement of integrity.

OPEN IN VIEWER

Figure 7.

Geometrizing dichotomous properties purpose and recognition reveals the autonomy projection.

In parallel, properties are homogenized by discarding those that fail to produce subspaces significant enough to warrant the increased complexity. Given that those that remain tie in strongly with other parts of the theory (as they should), this is a process of contextualization that can be applied to any concept regardless of its role in the emerging theory. For instance, conditions can be related to consequences that reinforce or challenge them using the same techniques as when interactions are contextualized through a theoretical account of causal and intervening conditions. Indeed, until a core category has been identified, it may not be at all obvious which categories represent the theory’s context (Scott, 2004, p. 121).

I defined the experience of being cornered as follows:

The GP works under the watchful eye of others, and the situation is too complex to be grasped within the allotted time. As the GP cannot simultaneously appease others and establish boundaries against the torrent of data and tasks, they are cornered, and their integrity seriously compromised. To escape, the GP must somehow convince others to cut them some slack.

This is clearly not a dictionary definition of the word “cornered,” but an explication that emerges from specific observations in a specific context. It goes well beyond the logical antecedents summarized in its first sentence and ties in—despite being abstract—with the context, embedding several factual claims that make reference to other parts of the theory: that the GP at least occasionally interacts with suspicious others; that the conditions surrounding such interactions may be cognitively complex; and that two potential consequences are gaining slack or losing integrity. Whether the definition contextualizes enough can be debated, but it clearly aspires to do so. Crucially, there is no trace of specification which, while procedurally important, would encumber the theory if allowed to linger in definitions.

In the unification step, any sparsity that indicates gaps in the concept’s empirical support is addressed. In the case of redundancy, skewness, or non-orthogonality, simplifying or modifying the model may be advisable. Barring theoretical causes, there are also procedural ones to consider, such as lack of theoretical saturation and coding idiosyncrasies. Before remedying such issues through theoretical sampling and recoding, respectively, the gaps should be reformulated as hypotheses.

Subspace-based nodes facilitate abductive inference

Causal hypotheses may appear spontaneously in the guise of emergent qualities of subspaces that echo other parts of the theory. Such overlaps being ripe targets for theorization, a reasonable next step is to investigate the extensions of the involved subspaces—the one currently under scrutiny, and the one that it resembles—while applying abductive inference (Conlon et al., 2020, p. 948) to spell out possible horizontal (temporal or causal) relationships between them. Testing such hypotheses against theoretically sampled data can provide grounds for either rejection or corroboration (Dewey, 2004, p. 89; Heath & Cowley, 2004, p. 144). In the case of the latter, the theory can now be enriched with another inductively derived piece of generalized knowledge.

For instance, it appeared that when GPs were pragmatic about their professional ethics, making use of their profession as a tool rather than tending to it as an end in itself, they were also likely to be, in the language of the voice of the self, certain yet struggling to regain power. I therefore hypothesized that such a message from the voice of the self would cause the pragmatic stance (given certain conditions that were not yet fully understood). Repeated observation confirmed that within the implied 4-codimensional subspace—which was eventually named being exploited—GPs almost always acted pragmatically. A generalized conclusion could now be formulated: When an exploited GP holds superior knowledge as their sole asset, they tend to use it for seizing back power rather than upholding professional ideals.

Property-based nodes facilitate parsimonious causal claims

If traditional conceptual hierarchies are blunt tools for understanding vertical conceptual relationships, they fare no better with horizontal ones. Whenever one subcategory differs from another in more than one respect, causal hypotheses that draw upon their differences will easily come to overstate the relevance of merely contingent qualities. One might reasonably worry that conceptual subspaces will lead back to this quandary. They have, however, the redeeming qualities of being explicit about their constituting dimensions and—purportedly—mutually exclusive and together comprehensive with regard to conceptual possibilities. This makes it possible to rework hypotheses that reference subspaces into ones that test the respective contributions of the constituting properties. This logico-deductive approach, which resembles how MacFarlane and O’Reilly-de Brún (2012) estimated likelihoods of an ideal outcome given certain conditions, makes for more parsimonious causal claims.

As an example, although it was obvious at an early stage that the voice of the self tended to drive GPs toward self-protection, an effect that was expected to be more pronounced the greater the perceived threat to the self (Johnsson & Nordgren, 2019), the precise nature of this relationship eluded me until I had elaborated the concept. An early insight was that GPs tended to act in their own interests when they perceived their labor to be useless rather than effective. Far from being content with this finding, I dug down into the relatively irreducible properties that defined these subspaces, namely purpose and cognition. I then found that the effect was mediated by the former through the experience of being unsuited, whereas the implied association with being strained was spurious. Without a multidimensional approach, I might have come to draw false conclusions about these relationships, especially because they were not deterministic.