Research

INTRODUCTION

In current sustainability research, evaluations of sustainability interventions in land use governance, natural resource management, and other related fields are hampered by the considerable incommensurability of many interventions and heterogeneity of the cases where they are attempted (Vatn 2005, Ostrom 2010, Meyfroidt et al. 2022). For the same reasons, it is also challenging to come up with a general explanation of crucial outcomes, e.g., successful collective action (Ostrom 2007). What works in certain ways in one case might work completely differently in other cases. However, sometimes seemingly different cases still produce similar outcomes.

Thus, different ways have been explored to address the challenge of dealing with heterogeneity among cases when producing generalized knowledge in sustainability research (e.g., Warren 2002, Ostrom 2007, Rudel 2008, Magliocca et al. 2018, Meyfroidt et al. 2018, Young 2019, Villamayor-Tomas et al. 2020, Castille et al. 2023, Eisenack 2024). If a single valid explanation is not feasible for the full set of cases one is interested in, we can group the cases into different classes (Young et al. 2006). If the cases are accurately and precisely classified (e.g., with a cluster analysis), each class might have its individual explanation. Class-wise explanations can then be tested by conventional methodological standards, but only applied to the separate classes (Ostrom and Cox 2010, Ratajczyk et al. 2016). From a general viewpoint, such approaches combine classification with explanation, as already discussed by Capecchi (1973). This is also implicit in qualitative comparative analysis (QCA; Schneider and Wagemann 2012), as it determines multiple sufficient conditions that are linked by logical disjunction (“OR”). It is also implicit in various studies in sustainability research that employ qualitative differential equations, fuzzy set indicators, or other methods (e.g., Lüdeke et al. 2004, Eisenack et al. 2006, Manuel-Navarrete et al. 2007, United Nations Environment Programme (UNEP) 2007, Eisenack and Roggero 2022).

Our methodological contribution aims at further improving approaches that combine explanation with classification. This task is also highly relevant for archetype analysis, an innovative approach that is increasingly used in case-based sustainability research (Neudert et al. 2019, Oberlack et al. 2023). Applications include research on land use (e.g., Gong and Tan 2021, Buchadas et al. 2022, Piemontese et al. 2022a), urban climate action (e.g., Roggero et al. 2025a, b), social-ecological systems and biodiversity (e.g., Nagel et al. 2024, Partelow et al. 2024, Harmáčková et al. 2025), or food systems (e.g., Sietz and Neudert 2022, Václavík et al. 2024). The basic idea of archetype analysis is to classify cases into archetypes (which do not need to be mutually exclusive). Each archetype is characterized by particular mechanisms that lead to an outcome (Eisenack et al. 2021). The approach can be operationalized with various methods like QCA (e.g., Sietz and van Dijk 2015, Kimmich and Villamayor Tomas 2019, Kurtmollaiev et al. 2023), cluster analysis (e.g., Levers et al. 2018, Orozco et al. 2024), or formal concept analysis (FCA; Oberlack et al. 2016, Oberlack and Eisenack 2018, Wang et al. 2019, Gotgelf et al. 2020). Yet, good criteria for the number of archetypes (i.e., the number of classes) are still acknowledged as a research gap (Eisenack et al. 2019).

Choosing the right number of classes is an overarching challenge when combining explanation with classification. A small number of classes has the advantage of a broader generality of the results and can be summarized in a shorter list. However, a small number of large classes might only lead to coarse-grained explanations or even to no explanations at all, e.g., in the extreme case of classifying a complete sample of cases into merely one single class (also called the “nomothetic trap” (Beltz et al. 2016, Lyon et al. 2017)). Small classes might be fine-grained enough to obtain rich explanations, but may lead to confusion in a long list of classes. In the extreme, every case would be its own class, so that there is no generalization at all, also called the “idiographic trap” (Beltz et al. 2016, Lyon et al. 2017). So, what is the best trade-off here? This trade-off is well known in cluster analysis, where various metrics are used to determine the appropriate number of clusters (e.g., Sietz et al. 2017), or in FCA which orders concepts by cardinality (Ganter and Wille 1999). For archetype analysis, a lack of good criteria for the number of archetypes is acknowledged as a crucial gap, as the approach is explicitly made to navigate between the idiographic and the nomothetic trap (Eisenack et al. 2019, Tan and Wang 2025).

To fill this methodological gap, this paper develops and tests operational and orderly procedures, along with rigorous and replicable metrics, to determine the (number of) classes. The procedures are also more efficient than previous methods that rely on extensive steps requiring potentially arbitrary human judgment. The new procedures can also engender more parsimonious findings. Specifically, we develop the procedures for archetype analyses using FCA (Ganter and Wille 1999), an increasingly used data-analytic approach (e.g., Oberlack and Eisenack 2018, Gotgelf et al. 2020, Harmáčková et al. 2025, Tan and Wang 2025). With FCA, the novel metrics are directly applicable to binary data, as frequently used in case-based sustainability studies. Some steps of the proposed procedure are computational and can be done with conventional software packages, whereas other steps are qualitative. We illustrate the metrics and procedure by replicating the data analysis part of previously published archetype analyses. We demonstrate the particularities of each step, show how to circumvent some possible pitfalls, and assess the procedures’ strengths in improving analytical efficiency and obtaining more parsimonious results. The conclusions further discuss extensions of the procedure, e.g., for other kinds of data or data-analytic methods.

DEVELOPING METHODOLOGY OF FORMAL CONCEPT-BASED ARCHETYPE ANALYSIS

As an innovative method for large-sample case study comparison, archetype analysis is increasingly used in sustainability research, and FCA serves as one major data-processing approach to identify archetypes. However, formal concept-based archetype analysis faces several methodological challenges—such as objectivity, replicability, and efficiency—in terms of combining explanation with classification.

Archetype analysis

In sustainability research, archetypes are recurrent patterns of interplay among elements of social-ecological systems (SES) that recur in multiple cases (Oberlack et al. 2019). An archetype is an analytical unit that features a combination of various attributes, their interactive mechanisms and outcomes, found in a set of cases. Archetype analysis aims to enable an analysis of various attributes that jointly affect the processes and outcomes of SES to reveal structural relations among these attributes (Tan and Wang 2025). To balance between the nomothetic and the idiographic trap, archetype analysis works on an intermediate level of abstraction: it neither intends to depict all details of cases (no abstraction), nor considers the details of cases as a black box (too abstract). In contrast, it intends to identify the recurrent configurations of attributes following coherent theories, realized in multiple but not necessarily all cases—which is a non-trivial challenge (Eisenack et al. 2021).

In this regard, archetype analysis combines classification with explanation by determining “recurrent patterns [...] that explain the phenomenon under particular conditions” (Oberlack et al. 2019). Each archetype in a set of identified archetypes (called “suite of archetypes”) is individually characterized by the set of cases that it covers, by a configuration of attributes that is common to the cases, and a mechanism that explains how the configuration leads to specified outcomes (Eisenack et al. 2021). For example, one archetype’s attribute configuration can express the conditions for government-led modes to consolidate rural construction land; this archetype cannot explain the cases where rural households self-organize. Otherwise, the configuration of attributes and outcomes reflected by this archetype may be too general; it might not even be identified as an archetype. Although one archetype should be present in a set of multiple (yet not all) cases, they can also function as building blocks: one case might be in the case set of multiple archetypes. Different aspects of the case might be explained by a combination of multiple archetypes (Eisenack et al. 2025). Archetype analysis thus admits a non-disjoined classification of cases. Importantly in this context, this makes FCA one highly relevant data-analytic method for archetype analysis.

Formal concept analysis (FCA)

Formal concept analysis, as developed by Ganter and Wille (1999), is a computational method that classifies data based on set theory and mathematical lattice theory. Here, we present some basics of FCA that are needed in the remainder of the paper.

In its simplest form, the input to an FCA is a table, with a set of cases (called “objects” in FCA jargon) in rows, and attributes in columns. For instance, cases can be a sample of municipalities in China, and the attributes express whether municipalities have a large area or whether there is a rising rural population (according to criteria defined prior to the assessment of the attributes). The table contains binary entries, which indicate which attributes are present (“hold”) in which cases.

With these data, the algorithm computes all possible classifications of the cases by their attributes. It does so by determining “concepts.” Each concept is described by its extent and its intent. Its extent is a set of cases that are covered by the concept, and its intent is a set of attributes. The challenge when determining the concepts is that extent and intent need to fit to each other: the extent of a concept contains exactly all cases that have all the intent’s attributes (but not more cases). For instance, a concept might contain all municipalities that are both large and have a rising rural population. In turn, the intent needs to contain all attributes that are common to all cases in the extent. So, if all large and rising-rural municipalities in the sample have a further shared feature, e.g., located in coastal areas, this feature should also be part of the concept’s intent. Thus, concepts are maximal classifications in the sense that they contain as many cases and attributes as possible, if they at least fit to each other in the above sense. Note that those concepts do not need to be mutually exclusive. Their extents can intersect if cases are contained in multiple concepts.

In addition to the concepts, FCA determines all sub-concept relations between them. A concept S is a sub-concept of concept C if S’s extent is a subset of C’s extent. This is equivalent to C’s intent being a subset of S’ intent. The concept of municipalities that are large, rising-rural, and coastal is a sub-concept of the large and rising-rural municipalities (as not all are coastal). One might say that the sub-concept C is less abstract/more concrete than S.

When the FCA is done, one can proceed with an archetype analysis by selecting several concepts at the same time to obtain a classification of the cases: each case is included in one of the selected concepts’ extents (or not in any extent). It is important to note that a case can also be in the extent of multiple concepts (the classification does not need to be disjoined). Every selection of multiple concepts is thus a candidate for a suite of archetypes. However, one can only speak about candidates here, because we still need to find a classification that can be combined with an explanation, so is consistent with theory. Explanation is not part of FCA and is addressed further below.

Methodological challenges

This paper addresses three main challenges, some of which are common to all research paradigms that combine explanation with classification, and some are particular to formal concept-based archetype analysis.

First, one needs to properly handle the level of abstraction to avoid nomothetic and idiographic traps. Increasing the number of archetypes depicts the characteristics of data sets and the actual phenomena more completely; however, each archetype may then contain too many attributes and draw too specific implications. Decreasing the number of archetypes (so that they, on average, cover more cases at the expense of comprising fewer attributes) can draw rather replicable and generalizable implications, but tends to neglect certain details deserving further investigation. For example, Wang and Tan (2020) identify eight archetypes of rural collective action, which capture the vital role of local leadership in fostering a self-organized mode to consolidate rural construction land. Although local leadership can be conceptualized in diverse aspects, such as village leaders, village elders, entrepreneurs, or local officials, it is not necessary to specify local leadership separately through more archetypes, and the current number of archetypes in this study identifies a recurrent and stable relation between local leadership and self-organized construction land consolidation. Yet, systematic ways for identifying the appropriate number of archetypes—and thus navigating the levels of abstraction in light of this trade-off—have been considered as one desideratum of archetype analysis (Eisenack et al. 2019).

Second, the previous literature has already proposed procedures and metrics to identify the appropriate number of archetypes in terms of attribute structures, reappearance of formal concepts, theoretical saliency and coherence, external and internal validity (e.g., Oberlack et al. 2016, Levers et al. 2018, Piemontese et al. 2022b, Partelow et al. 2024, Harmáčková et al. 2025). However, they still face controversial challenges. Specifically, the metrics are either too general or too study specific. On the general side, Eisenack et al. (2019) suggest general quality criteria to guide the design of archetype analyses, suggesting that an archetype analysis should specify the domain of validity for each archetype, ensure multiple archetypes to be combined in different ways to characterize single cases, explicitly navigate through different levels of abstraction, and obtain a fit between its configuration of attributes, theory, and empirical evidence. On the more specific side, Wang and Tan (2020) accomplish the archetype analysis based on a first-hand case set, presenting the process of case collection, coding, categorizing, and archetype identification with rather context-specific criteria. They require that an archetype appears in at least two cases, along with a configuration of at least one diagnostic attribute and one design attribute. Diagnostic attributes are considered contextual and reflect case-dependent conditions (e.g., biophysical conditions of rural construction land or characteristics of rural households), whereas design attributes are considered contextual and reflect case-dependent conditions (information or revenue distribution rules). Furthermore, an archetype must be illustrated by a coherent theory of collective action that provides a rationale for the configuration to occur. Also, sub-archetypes are identified to provide more detailed but not too specific or entirely repetitive information about the possibility of rural collective action. Except for the FCA, other steps for identifying the appropriate number of archetypes are rather qualitative and semi-qualitative, such as inspecting the extents and intents of formal concepts in detail.

Third, formal concept-based archetype analysis usually generates a large number of formal concepts in the first step, as the scale of a data set enlarges. For instance, Wang et al. (2019) obtained 131 formal concepts from 27 cases, whereas Oberlack and Eisenack (2018) got 216 formal concepts from 114 cases. In practice, researchers scrutinize hundreds of formal concepts by hand to identify the archetypes, impeding the process efficiency (the full data of Harmáčková et al. (2025) with 460 cases, e.g., obtains 982 concepts). We thus aim to propose procedures to speed up this approach.

In summary, the current procedures show the notable shortages in objectivity, replicability, and efficiency in identifying archetypes and the appropriate number of archetypes.

METRICS AND NEW PROCEDURES

To address the above methodological challenges, we introduce several quantitative metrics, and show how they can be used in the procedural steps we propose.

Quantitative metrics

The following metrics can be used in various ways to distill concepts and to later identify those that qualify as archetypes. Generally, we denote the number of cases in the data set by n, each described by a subset of the total number of attributes m (more precisely, m is the number of attributes except those that characterize outcomes). Recall that each concept is characterized by its intent (i.e., attributes) and extent (i.e., cases). To combine classification with explanation, we distinguish the outcome attribute(s) we are interested in (those we want to explain) from the remaining attributes. The number of cases that have a chosen outcome attribute “out” is denoted by n_out.

The size of a concept is the number of cases contained in the concept (i.e., the cardinality of its extent). If concepts contain more cases, it might be more reasonable to consider them as important archetypes. Yet, if they are too large, they might miss nuances to represent different kinds of cases. It can be convenient to normalize the size: by borrowing from qualitative comparative analysis (QCA; Ragin 1987, Schneider and Wagemann 2012), we define a concept’s coverage = size/n_out (which is below unity if size ≤ n_out).

The richness ρ of a concept is the number of its attributes (i.e., the cardinality of its intent). Concepts with low richness (e.g., ρ = 1 for a single attribute) express little nuance and tend to have a larger size. Usually, archetype analysis is interested in determining configurations of multiple attributes (so, richness is 2 or larger).

The lift of a concept (borrowed from association rule mining, e.g., McNicholas et al. 2008) combines data on attributes and cases. Let P(a_i) = #(a_i) / n denote the relative frequency of cases with attribute a_i appearing among all cases (possibly together with further attributes). For a concept with more attributes, say an intent set {a₁, a₂, ..., a_i, ..., a_k}, it is:

(1)

The numerator expresses the relative frequency of all the concept’s attributes appearing together in the cases. The denominator is the expected frequency that the concept’s attributes jointly co-occur exactly as frequently as expected under the assumption that the single attributes are stochastically independent. A lift = 1.0 thus means that the concept appears as frequently as expected. A greater value, above 1.0, means that the concept’s attributes co-occur more frequently than expected. Concepts with a high lift might be considered more surprising, so they warrant more attention when they explain an outcome.

To get a measure on whether a concept has potential to explain an outcome attribute, one can compute its consistency (again, borrowing from QCA; Schneider and Wagemann 2012, Roggero 2019). Consistency refers to the share of cases in a concept’s extent that also have the outcome attribute out. When considering a concept with intent set {a₁, a₂, ..., a_k}, some but possibly not all cases with these attributes also have the outcome out. We thus define the consistency = #(out AND a₁ AND a₂ AND ... AND a_k)/size. Here, “AND” denotes logical conjunction, so that the consistency is the number of all cases that have the outcome attribute together with all attributes of the concept’s intent, normalized to the concept’s size. So, consistency depends on the outcome attribute because the outcome can appear with or without the other attributes. A consistency of 100% means that all cases in a given concept have the outcome of interest. Such a concept is a good candidate for a strong explanation. With a lower consistency (more common in larger data sets), there are more false positives when we use the concept to explain the outcome.

The above metrics express quantitative features of single concepts. However, archetype analysis aims to determine a whole suite of multiple archetypes that explain a specific outcome. Some archetypes might cover cases that other archetypes do not speak about. Recall that concepts do not need to be disjoined (their extents can intersect): two concepts taken together (the union of their extents) might contain fewer cases than simply adding both concepts’ sizes. For any selection of concepts, we thus define their joint size, v, by the number of cases that are in the extent of at least one of those concepts (i.e., the cardinality of the union of their extents). When normalized, we have their joint coverage = v/n_out. The lower the joint coverage, the more cases are not covered by any concept in the selection. We have now introduced the main metrics. They can be calculated, e.g., by using the fcaR package in R (Lopez Rodriguez et al. 2020; see available scripts on https://github.com/ArchetypeAnalysis/Archetype-Identification).

Standard procedures

We propose the following standard procedures to determine the optimal number of archetypes, with core steps and optional steps (Table 1). Core steps should always be taken, whereas the optional steps can be adopted discretionally according to the research purposes, theoretical basis, and structure in the data. The novel metrics for archetype identification are summarized more technically in Fig. 1.

The first core step (CS1) conducts the FCA. It takes as input the data of all n cases. As attributes, it takes only those that are “not” considered as outcomes to be explained. The FCA results in a list of all concepts. Each selection from this list can be considered as a classification of the data. However, there are usually many ways to classify. The concepts yet differ a lot, for instance by size. More importantly, many will not have explanatory power. Thus, CS1 continues with computing, for every concept, its size, richness, lift, coverage, and consistency. Recall that the value of the last two metrics depends on the chosen outcome attribute.

These metrics are used in the concept filter in the second core step (CS2). The analyst sets (and reports) thresholds for all metrics (or, depending on the research, only some of these metrics). Only those concepts from CS1 that respect all thresholds are retained, whereas the remainder are discarded. This raises the question of how to set the thresholds. Although this generally depends on the research, some general considerations can be made. Most important for explanatory purposes is consistency, so ideally set to 100%. Then, all cases in the extent of the perfectly consistent concepts also have the outcome attribute we are interested in. Classifying a case with such a concept is sufficient for the outcome. For larger data sets, it might not be possible to find perfectly consistent concepts, possibly because there is so much variation in unobserved variables, or much noise. By referring to conventions from QCA, we propose a minimal consistency threshold of 80% (Ragin 2008).

For size and coverage, archetype analysis requires each archetype to appear repeatedly. The most relaxed quantification is to only accept concepts with size ≥ 2 or coverage ≥ 2 / n_out as archetypes. In larger data sets, however, we might set a higher threshold so that we require concepts to cover more cases, e.g., coverage ≥ 1%. A higher threshold implies that we ignore more unconventional cases.

For richness, one would require at least ρ ≥ 2 for a conjunctional causation approach (Schneider and Wagemann 2012), as is common in archetype analysis (e.g., Eisenack and Roggero 2022). If the explanatory power of single attributes is worth being checked, ρ ≥ 1 might also be reasonable. A higher threshold has the advantage that we retain only concepts with a richer characterization or which admit more complex theorizing. On the other hand, explanations are stronger if much follows from little. Richer concepts tend to become more difficult to comprehend qualitatively or to explain by theoretical considerations. Thus, an upper limit is recommended for richness, which yet depends on the research. We recommend ρ ≤ m / 2 to avoid certain attributes appearing in very many concepts.

For lift, a natural lower threshold is 100%. One might yet choose a higher threshold if the research aims for explanations that can be applied more frequently than expected from the single attribute frequency. Depending on the research, lift ≥ 80% can also be acceptable.

Next comes the theoretical analysis step (CS3), which is most difficult to formalize (partially because it strongly depends on disciplinary standards). The concepts that passed the previous steps only describe which attributes are present in the configurations and not how they are interlinked (although Eisenack and colleagues (2025) discuss ways to consider this in an FCA), and not even why they are interlinked (possibly even in a dynamic way). Thus, the aim of CS3 is to provide theoretical explanations. All the remaining concepts are inspected according to whether they illustrate established theories, corroborate theoretical inferences, are supported by case-based knowledge, and align with or extend the relevant literature. We obtain the final suite of archetypes by only keeping those concepts that pass this test. Such a test might be more or less easy. However, the advantage of our proposed procedure is that the previous steps have considerably reduced the list of concepts that warrant such a theorizing. So, one can concentrate this effort on those concepts that are repeatedly sufficient for the outcome. Theorizing can be further supported by findings from the previous steps. For instance, we can focus on less rich concepts first and later check whether sub-concepts of those qualify as sub-archetypes. Although we accept overlap between concepts, we might justify retaining those that are less repetitive in terms of other concepts. Sometimes, it can be useful to sort the m attributes into different kinds (e.g., referring to bio-physical or to socio-economic features) and only accept concepts as archetypes if they contain attributes of two kinds (e.g., in Gotgelf et al. 2020).

It may still be possible that too many concepts pass CS2 and CS3 or that too many concepts pass CS2 to accomplish CS3. We then propose two optional steps that can be done before or after some of the core steps.

Grouping (OS1) is a semi-qualitative optional step. It divides cases into groups defined by the analyst, according to their outcomes, critical attributes of interest, or research questions. The data are split into subsets by group. The other steps are then done for each group separately. The fewer cases in each group, the smaller the number of concepts determined with FCA. However, the number of groups should be small enough to be manageable in the subsequent steps. Grouping is possible before CS1 or between CS2 and CS3.

Optimal concept selection (OS2) optimizes the joint size v (ρ, N) for a given richness ρ and N. This can now build on tested algorithms (Harmáčková et al. 2025, Partelow et al. 2024). As input, it takes a set of concepts that have passed previous tests, and then determines optimal subsets (selections of a given number of N concepts). In other words, we want to cover as many cases as possible within the union of a small selection of concepts’ extents. If only N random concepts were selected, their extents might considerably intersect, so that their joint size is not much greater than the individual concepts’ sizes. The algorithm computes all selections that optimize the joint size for a given (ρ, N). Still, the analyst needs to justify which ranges for (ρ, N) to compute and then needs to choose among them. The following criteria can help.

For an archetype analysis, we should require N ≥ 2 (if N = 1, a nomothetic approach would be better). Furthermore, N < m is reasonable as otherwise each single attribute is admitted as a (not very interesting) concept. Pragmatically, there might be a maximum number of concepts one is able to cope with in terms of available resources, publication space, and so on (e.g., N ≤ 10 or N ≤ 40). The joint size is then optimized for this range for N, and the range for ρ is chosen as in previous steps. The more important task is to choose N among all computed optimal selections. This number is actually the size of the suite of archetypes. Generally, we want to cover many cases, so we could only accept an optimal selection of concepts with a high joint coverage. However, it is not reasonable to require 100% joint coverage because there might be (i) quite idiosyncratic cases that only appear in concepts of size 1, or (ii) cases that are only described by one attribute. Depending on the research, there might be a number of cases we can afford to leave uncovered. All this requires a lower threshold, e.g., joint coverage ≥ 80%. On the other hand, there might be many optimal selections of N concepts above the threshold. It is recommended to choose the lowest N that achieves the threshold of joint coverage (by parsimony).

Parsimony can also be operationalized in another way. Suppose we have, for a given (ρ, N), the optimal joint size v (ρ, N). Then, admitting one more concept can never decrease the optimal joint size: v(ρ, N+1) ≥ v(ρ, N) is always true. But what is the point of admitting one more concept if the joint size would be raised by just one more case? We can thus require that a further concept is only admitted to the suite of archetypes if it sufficiently improves coverage. With some preset threshold, e.g., ΔN = 2 or ΔN = 0.01 n_out, we can choose N so that it fulfills both v(ρ, N) ≥ v(ρ, N-1) + ΔN and v(ρ, N+1) <v(ρ, N) + ΔN. With these criteria, OS2 ends with (several) optimal selections of concepts. Only those are retained in subsequent steps. We recommend applying this step only after CS2 (although it also works after CS1).

RE-IDENTIFYING ARCHETYPES FROM TWO PREVIOUS STUDIES

To illustrate the proposed metrics and steps, we now attempt to replicate two previously published archetype analyses, one with a smaller and one with a larger data set. The comparison between the original and the novel procedure and findings also highlights the practical effectiveness of the metrics and steps. Further examples are available from the authors upon request.

Sustainable rural renewal in China

Wang and colleagues (2019) study archetypes of rural renewal in contemporary China involving activities that replan, consolidate, and redevelop the extant and idle rural construction land and then convert such land for alternative uses, including new rural settlement construction and rural industry development. The governance of rural renewal and its performance show great diversity in facilitating rural sustainability. The analysis is based on primary data from n = 27 cases from the eastern, central, and western parts of China. Each case is characterized by the presence/absence of m = 47 attributes, including three outcome attributes (unsustainable, semi-sustainable, or sustainable rural renewal). In total, eight archetypes and 17 further sub-archetypes were identified, the latter capturing more fine-grained but not repetitive information on the cases. These archetypes and sub-archetypes were originally obtained by FCA with four criteria concerning the configuration, recurrence, theoretical coherence, and generality/particularity. This procedure involved the manual inspection of 134 concepts, many of them with more than 10 attributes.

For the replication, we first used one optional step (OS1), and then the three core steps, with the quantitative metrics of consistency, coverage, and richness (Fig. 2).

The first step (OS1) followed the original study by grouping the cases by the three outcome attributes, including unstainable rural renewal (group 1), semi-sustainable rural renewal (group 2), and sustainable rural renewal (group 3) (Wang et al. 2019). In addition, the cases in group 3 were decomposed into three sub-groups according to further attributes for the scales of rural land and rural households involved in rural renewal (Wang et al. 2019).

Subsequently, CS1 was performed separately for each group (and sub-group), yielding between eight and 66 concepts per group (and 134 in total). The consistency, coverage, and richness of each concept was calculated. Considering the concepts in all the main groups and sub-groups, coverage varies from 22% to 100%, and richness from 2 to 21.

We then proceed with the concept filter (CS2) to filter out all concepts with consistency < 100%, i.e., those that do not always lead to the respective outcome (unsustainable, semi-sustainable, or sustainable rural renewal). The chosen thresholds for coverage differ between the groups, as they contain a quite different number of cases. Thus, in groups 1 and 2, and sub-groups 3.1-3.3, formal concepts were filtered out if their coverage was below certain thresholds, varying from 22% to 66%. The original study only accepted concepts as (sub-)archetype with size = 2 or more. In terms of richness, we only retained concepts with 2 ≤ ρ ≤ 21. The minimal richness ensures a configuration of attributes rather than an individual attribute. Concepts with 21 attributes are already quite complicated and specific, being nearly half of the total number of attributes (m = 47), and can contain multiple interactions in several analytic stages. In total, the step reduced the number of “surviving” concepts to 36 for the main groups, and to 67 for the sub-groups.

In CS3, all remaining concepts were inspected for theoretical saliency, i.e., whether these concepts can be explained by or develop the theories of transaction cost economics or/and rural collective action. The assessment of theoretical saliency was conducted by the authors of the original study. The remaining concepts were ultimately reduced to a suite of eight archetypes and 17 sub-archetypes. These are identical to those from the original study. The novel procedure (up to CS3) completely reproduced previous findings, yet with much less effort.

Barriers to water governance under climate change

The archetype analysis by Oberlack and Eisenack (2018) conducted a systematic literature review on barriers to climate change adaptation in water governance of river basins. Every “case” in this study is a causal statement made in an empirical publication (the study is a model-centered meta-analysis; Rudel 2008). Some publications made just one such claim, whereas others made several of them. The outcome is always the presence of a barrier (as other statements were not coded). In total, n = 114 statements about adaptation barriers provided by these peer-reviewed empirical publications were used in this study. The statements were coded with binary attributes based on the SES framework (m = 59) and performed with an FCA that led to 216 formal concepts. Only those 87 formal concepts that contained at least two different causal statements from two different publications were retained. In the original study, all of them were inspected by hand for theoretical saliency and led to a suite of 31 archetypes with richness 2, and a joint size of 93 cases. The study determined a further 10 archetypes with richness = 3, but those are all sub-archetypes of the previous.

The original data were re-analyzed through the three core steps and one optional step (OS3) with the quantitative metrics of coverage, richness, size, and lift (Fig. 3).

In CS1, the FCA again computed 216 formal concepts. The coverage, richness, size, and lift of each concept was calculated. We did not compute consistency as all concepts are consistent by definition in this study.

In CS2, in aligning with the original study, we only retained concepts with size ≥ 2 (which converts to a coverage of 1.75%) and that appear in at least two different papers. For richness, we imposed 2 ≤ ρ ≤ 3 to also align with the original study. We further required lift ≥ 120%. After the concept filter, 72 concepts were kept.

We then continued with optimal concept selection (OS2). We computed all selections of N concepts with the richness given in the previous step and that maximize the joint size. More specifically, we considered all optimal selections with N between 2 and 35 to have the original study’s N = 31 in the range, and to check whether there should be more archetypes than in the original study. We required the same joint coverage as in the original study (81.6%). The algorithm yields that this is already achieved for ρ = 2 and N = 25. So, the original coverage could have also been reached with considerably fewer archetypes if the proposed new procedures would have been available. If we additionally impose ΔN = 2, we cannot obtain a higher joint coverage than 79%. For ρ = 3, the threshold cannot be achieved (as in the original study). For re-identification, we thus required both joint coverage above 75% and ΔN = 2. The optimum (with a joint size of 90) is achieved with 10 different optimal selections, each with 22 concepts. As those selections are not so different, they contain 26 concepts in total.

All these are fed into the theoretical analysis (CS3). As our main aim is to explore the advantage of the new steps by comparing with the original study, we concentrated this step on scrutinizing the relevance of the concepts that disappeared in comparison to the original study, and that are new (2 for the most similar optimum, see Append. 1 for more details). They do not change the main insights from the original study.

Thus, 22 archetypes are obtained, fewer than the original study (i.e., 31 archetypes). The original study covers three more statements, but needs nine more archetypes to achieve this. In this regard, the novel procedures and metrics, while being more reproducible and faster, reproduce the essence of the original findings in a more parsimonious way.

CONCLUSION

Like other methodologies for combining explanation with classification, archetype analysis faces the trade-off between the number of classes and the strength of explanations. This requires purposefully navigating different levels of abstraction. If formal concept-based archetype analysis is used, this amounts to identifying the appropriate (number of) archetypes from the formal concepts. Our paper proposes standardized procedures with quantitative metrics to reduce unnecessary subjectivity and bias, to increase the process efficiency, and to obtain more parsimonious findings when selecting the archetypes. We propose using the metrics of consistency, coverage, richness, size, and lift. They can be computed and used in some of the three main steps: formal concept analysis, concept filtering, and theoretical analysis. Depending on the research question and data, two optional steps are suggested: grouping and optimal concept selection. The effectiveness of the novel procedures and metrics are assessed by reanalyzing two previously published studies. We demonstrate that the more quantitative steps lead to a manageable number of insightful formal concepts for further qualitative investigation—through a rather rigorous, objective, and much less resource-intensive procedure than previous studies. The novel procedure completely replicates the original archetypes or the original findings in a more parsimonious way. Our paper thus contributes to the further methodological development of archetype analysis in case-based sustainability research by suggesting a more objective and efficient way toward identifying archetypes with the appropriate granularity. In other words, an intermediate level of abstraction is achieved that avoids the “nomothetic trap” and the “idiographic trap” when combining classification with explanation.

Obviously, the quantitative metrics and standard procedures for formal concept-based archetype analysis remain to be further tested and improved in the broader field of sustainability research. In particular, reasonable ranges for the thresholds for the metrics can be further tested in a heuristic manner that accumulates experience from individual studies. The lift metric has limits in exploiting more information on how attributes are configured by assuming stochastic independence, and when comparing among concepts with different richness. Technical advances are still possible to refine the standard procedures and improve their efficiency, especially on the stages of grouping and optimal concept selection.

Our proposal is tailored to binary data, analyzed with FCA. Natural extensions would address non-binary or fuzzy data. Extending the procedures and metrics for methods other than FCA might be promising, e.g., for cluster analysis (one might compute correlation coefficients with an outcome variable for each level in a hierarchical cluster analysis to identify the best level). In addition, machine learning algorithms such as artificial neural networks or self-organizing maps offer entry points (Frey and Rusch 2013, Levers et al. 2018). Unsupervised classification realized by machine learning algorithms can further mitigate the subjective and biased identification of the appropriate number of archetypes, while likely involving trade-offs with theoretical saliency or consistency. In this regard, we argue that the methodological advances suggested in this paper show promising avenues for improved archetype analysis in case-based sustainability research, particularly by increasing analytical efficiency and obtaining more parsimonious findings.

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